Finite Group Databasebeta

← C2×#72238016 #72311218

Subgroups — C2×#72238016 #72311218

Order 2048.

At least 6346 conjugacy classes of subgroups; only the 1576 normal subgroups are shown.

The subgroup lattice has 491 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#139736C214yesmaximal, complemented
2C2×#139812C214yesmaximal, complemented
3C2×#6437452C211yesmaximal, characteristic
4C2×#128079C212yesmaximal, complemented
5C2×#493430C211yesmaximal, characteristic, complemented
6C2×#6385980C211yesmaximal, characteristic, complemented
7C2×#493397C212yesmaximal, complemented
8#72238016C2116yesmaximal, direct product factor
9C2×#62852C2211yescharacteristic
10C22×#3498C2211yescharacteristic
11C22×#3018C411yescharacteristic, complemented
12C22×#3021C411yescharacteristic, complemented
13#6437452C418yescomplemented
14#6437452C2218yes—
15C2×#23915C412yescomplemented
16C2×#23915C2212yes—
17C2×#23909C412yescomplemented
18C2×#23909C2212yes—
19#6385980C2218yescomplemented
20C2×#21794C2212yescomplemented
21C2×#21750C2212yescomplemented
22C2×#20956C411yescharacteristic, complemented
23C2×#20952C411yescharacteristic, complemented
24C2×#20592C2211yescharacteristic
25C2×#20585C2212yes—
26C2×#20585C2212yes—
27#493430C2218yescomplemented
28#493397C22116yescomplemented
29C2×#12708C2218yescomplemented
30C2×#12697C2212yescomplemented
31C2×#12695C2212yescomplemented
32#139812C22132yescomplemented
33#139736C22132yescomplemented
34#128079C22116yescomplemented
35C2×#7313C2218yescomplemented
36C22×#585C2×C411yescharacteristic, complemented
37C23×H(5,𝔽2)D411yescharacteristic
38C23×H(5,𝔽2)C2311yescharacteristic
39#62852C2×C414yescomplemented
40#62852C2314yes—
41C2×#3498C2×C414yes—
42C2×#3498D414yes—
43C2×#3498D414yes—
44C2×#3498C2314yes—
45C2×#3487C2×C412yescomplemented
46C2×#3487C2312yes—
47C22×#535C2×C411yescharacteristic
48C22×#535C2×C411yescharacteristic
49C22×#535D411yescharacteristic
50C22×#535D411yescharacteristic
51C22×#535D411yescharacteristic
52C22×#535C2311yescharacteristic
53C22×#532C2×C411yescharacteristic
54C22×#532C2×C411yescharacteristic
55C22×#532C2311yescharacteristic
56C23×#127D411yescharacteristic
57C23×#127D411yescharacteristic
58C23×#127D411yescharacteristic
59C2×#3428C2×C411yescharacteristic, complemented
60C22×C4×#41D411yescharacteristic
61C2×#3021C2×C418yescomplemented
62C2×#3018C2×C418yescomplemented
63C22×#456C2×C412yescomplemented
64C22×#454C2×C412yescomplemented
65C22×#454C2×C412yescomplemented
66C22×#454C2×C412yescomplemented
67C22×#413C2×C411yescharacteristic, complemented
68C22×#413C2×C411yescharacteristic, complemented
69C23×#99D414yes—
70#23915C2×C418yescomplemented
71#23915C2×C4116yescomplemented
72#23915C2318yes—
73#23909C2×C418yescomplemented
74#23909C2×C4116yescomplemented
75#23909C2318yes—
76#21794C2318yescomplemented
77#21750C2318yescomplemented
78#20956C2×C418yescomplemented
79#20952C2×C418yescomplemented
80C2×#1888C2×C412yescomplemented
81C2×#1888C2×C412yescomplemented
82C2×#1888C2×C412yescomplemented
83C2×#1886C2×C412yescomplemented
84#20592C2×C414yescomplemented
85#20592C2314yes—
86C2×#1785C2×C412yescomplemented
87C2×#1785C2312yes—
88#20585D418yes—
89#20585C2318yes—
90#20585C2318yes—
91C2×#1782C2314yes—
92C2×#1782C2314yes—
93C2×#1781D414yes—
94#12708C23132yescomplemented
95#12697C2318yescomplemented
96#12695C2318yescomplemented
97#7313C23132yescomplemented
98C2×#585C22×C414yescomplemented
99C22×2−1+4C22×C412yes—
100C22×H(5,𝔽2)C22⋊C414yes—
101C22×H(5,𝔽2)C22×C412yes—
102C22×H(5,𝔽2)C2×D414yes—
103C22×H(5,𝔽2)C2×D412yes—
104C22×H(5,𝔽2)C2×D412yes—
105C22×H(5,𝔽2)C2×D414yes—
106C22×H(5,𝔽2)#4114yes—
107C22×H(5,𝔽2)C2412yes—
108C23×#41C22⋊C411yescharacteristic
109C23×#41C22⋊C411yescharacteristic
110C23×#41C22×C411yescharacteristic
111C23×#41C22×C411yescharacteristic
112C23×#41C2×D411yescharacteristic
113C23×#41C2×D411yescharacteristic
114C23×#41#4112yes—
115C23×#41C2411yescharacteristic
116C24×D4C22⋊C411yescharacteristic
117C24×D4C22⋊C411yescharacteristic
118C24×D4C2×D411yescharacteristic
119C24×D4C2×D411yescharacteristic
120C24×D4C2×D412yes—
121C24×D4C2×D412yes—
122C24×D4C2×D411yescharacteristic
123C24×D4C2×D411yescharacteristic
124C24×D4#4112yes—
125#3487C22×C414yes—
126#3487C22×C418yescomplemented
127#3487C2414yes—
128C2×#535C22⋊C414yes—
129C2×#535C22⋊C414yes—
130C2×#535C22⋊C414yes—
131C2×#535C22⋊C414yes—
132C2×#535C22⋊C414yes—
133C2×#535C22×C414yes—
134C2×#535C22×C412yes—
135C2×#535C22×C414yes—
136C2×#535C2×D412yes—
137C2×#535C2×D414yes—
138C2×#535C2×D414yes—
139C2×#535C2×D412yes—
140C2×#535C2×D414yes—
141C2×#535C2412yes—
142C2×#532C22⋊C414yes—
143C2×#532C22⋊C414yes—
144C2×#532C22×C414yes—
145C2×#532C22×C412yes—
146C2×#532C22×C414yes—
147C2×#532C2×D412yes—
148C2×#532C2×D412yes—
149C2×#532C2412yes—
150C22×C4⋊Q8C22×C411yescharacteristic
151C22×#127C22⋊C414yes—
152C22×#127C22⋊C411yescharacteristic
153C22×#127C22⋊C411yescharacteristic
154C22×#127C22⋊C411yescharacteristic
155C22×#127C22⋊C414yes—
156C22×#127C22⋊C411yescharacteristic
157C22×#127C22⋊C411yescharacteristic
158C22×#127C22⋊C414yes—
159C22×#127C22⋊C411yescharacteristic
160C22×#127C2×D414yes—
161C22×#127C2×D414yes—
162C22×#127C2×D411yescharacteristic
163C22×#127C2×D411yescharacteristic
164C22×#127C2×D411yescharacteristic
165C22×#127C2×D414yes—
166C22×#124C22⋊C411yescharacteristic
167C22×#124C22⋊C411yescharacteristic
168C22×#124C22×C411yescharacteristic
169C22×#124C22×C411yescharacteristic
170C22×#124C22×C411yescharacteristic
171C22×#124C2×D411yescharacteristic
172#3428C22×C414yescomplemented
173C2×#519C22×C412yes—
174C2×C4×#41C22⋊C414yes—
175C2×C4×#41C2×D414yes—
176C22×#117C22⋊C411yescharacteristic
177C22×#117C22⋊C411yescharacteristic
178C22×#117C2×D411yescharacteristic
179C23×C42C22⋊C411yescharacteristic
180C23×C42C22⋊C411yescharacteristic
181C23×C42C2×D411yescharacteristic
182C2×#456C22×C418yescomplemented
183C2×#454C22×C418yescomplemented
184C2×#454C22×C418yescomplemented
185C2×#454C22×C418yescomplemented
186C2×#413C22×C414yescomplemented
187C2×#413C22×C414yescomplemented
188C22×#101C22×C412yes—
189C22×#101C22×C412yes—
190C22×#99C2×D418yes—
191C22×#99C2×D4116yes—
192#1888C22×C418yescomplemented
193#1888C22×C418yescomplemented
194#1888C22×C418yescomplemented
195#1886C22×C418yescomplemented
196#1785C22×C414yes—
197#1785C22×C418yescomplemented
198#1785C2414yes—
199#1782C2×D418yes—
200#1782C2418yes—
201#1782C2418yes—
202#1781C2×D4116yes—
203C26#12012yes—
204C26#12112yes—
205C26#12112yes—
206C2×2−1+4C23×C414yes—
207C22×#41C2×C22⋊C414yes—
208C22×#41C2×C22⋊C414yes—
209C22×#41C2×C22⋊C412yes—
210C22×#41C2×C22⋊C412yes—
211C22×#41C2×C22⋊C412yes—
212C22×#41C2×C22⋊C412yes—
213C22×#41C4×D414yes—
214C22×#41C4×D414yes—
215C22×#41C4×D414yes—
216C22×#41#12012yes—
217C22×#41#12114yes—
218C22×#41#12312yes—
219C22×#41#12314yes—
220C22×#41C23×C412yes—
221C22×#41C23×C412yes—
222C22×#41C23×C411yescharacteristic
223C22×#41C22×D412yes—
224C22×#41C22×D412yes—
225C22×#41C22×D411yescharacteristic
226C22×#41C22×D412yes—
227C22×#41C22×D411yescharacteristic
228C22×#41C2×#4114yes—
229C22×#41C2×#4112yes—
230C22×#41C2511yescharacteristic
231C23×Q8C2×C22⋊C411yescharacteristic
232C23×Q8#12311yescharacteristic
233C23×Q8C23×C411yescharacteristic
234C23×D4C2×C22⋊C411yescharacteristic
235C23×D4C2×C22⋊C412yes—
236C23×D4C2×C22⋊C411yescharacteristic
237C23×D4C2×C22⋊C412yes—
238C23×D4C2×C22⋊C412yes—
239C23×D4C2×C22⋊C414yes—
240C23×D4C2×C22⋊C411yescharacteristic
241C23×D4C2×C22⋊C411yescharacteristic
242C23×D4C2×C22⋊C414yes—
243C23×D4C2×C22⋊C412yes—
244C23×D4C2×C22⋊C411yescharacteristic
245C23×D4C4×D412yes—
246C23×D4C4×D412yes—
247C23×D4C4×D414yes—
248C23×D4C4×D414yes—
249C23×D4C4×D414yes—
250C23×D4#12012yes—
251C23×D4#12011yescharacteristic
252C23×D4#12014yes—
253C23×D4#12014yes—
254C23×D4#12011yescharacteristic
255C23×D4#12114yes—
256C23×D4#12112yes—
257C23×D4#12112yes—
258C23×D4#12114yes—
259C23×D4#12114yes—
260C23×D4#12114yes—
261C23×D4#12114yes—
262C23×D4#12311yescharacteristic
263C23×D4#12312yes—
264C23×D4#12312yes—
265C23×D4#12314yes—
266C23×D4C22×D411yescharacteristic
267C23×D4C22×D412yes—
268C23×D4C22×D412yes—
269C23×D4C22×D412yes—
270C23×D4C22×D414yes—
271C23×D4C22×D414yes—
272C23×D4C22×D412yes—
273C23×D4C22×D412yes—
274C23×D4C2×#4114yes—
275C24×C4C2×C22⋊C411yescharacteristic
276C24×C4C2×C22⋊C411yescharacteristic
277C24×C4C4×D412yes—
278C24×C4C4×D412yes—
279C24×C4C22×D411yescharacteristic
280C24×C4C2×#4112yes—
281C2×C4⋊Q8C2×C22⋊C412yes—
282C2×C4⋊Q8C23×C412yes—
283C2×#127#9912yes—
284C2×#127#9912yes—
285C2×#127#9914yes—
286C2×#127#9914yes—
287C2×#127#9914yes—
288C2×#127C2×C22⋊C414yes—
289C2×#127C2×C22⋊C414yes—
290C2×#127C2×C22⋊C414yes—
291C2×#127C2×C22⋊C412yes—
292C2×#127C2×C22⋊C412yes—
293C2×#127C2×C22⋊C412yes—
294C2×#127C2×C22⋊C414yes—
295C2×#127C2×C22⋊C414yes—
296C2×#127C2×C22⋊C414yes—
297C2×#127C22×D412yes—
298C2×#127C22×D412yes—
299C2×#127C22×D412yes—
300C2×#124C2×C22⋊C414yes—
301C2×#124C2×C22⋊C412yes—
302C2×#124C2×C22⋊C412yes—
303C2×#124C2×C22⋊C412yes—
304C2×#124C2×C22⋊C412yes—
305C2×#124C2×C22⋊C414yes—
306C2×#124C23×C412yes—
307C2×#124C23×C412yes—
308C2×#124C23×C412yes—
309C2×#124C22×D412yes—
310#519C23×C414yes—
311C2×#117C2×C22⋊C414yes—
312C2×#117C2×C22⋊C414yes—
313C2×#117C2×C22⋊C412yes—
314C2×#117C22×D412yes—
315C22×C42C2×C22⋊C411yescharacteristic
316C22×C42C2×C22⋊C411yescharacteristic
317C22×C42C2×C22⋊C412yes—
318C22×C42C2×C22⋊C414yes—
319C22×C42C2×C22⋊C414yes—
320C22×C42C2×C22⋊C411yescharacteristic
321C22×C42C2×C22⋊C411yescharacteristic
322C22×C42C22×D412yes—
323C2×#101C23×C414yes—
324C2×#101C23×C414yes—
325C2×#99C22×D4116yes—
326C25#38612yes—
327C25#38614yes—
328C25#38614yes—
329C25#38611yescharacteristic
330C25#38612yes—
331C25#38611yescharacteristic
332C25#38614yes—
333C25C2×#12014yes—
334C25C2×#12011yescharacteristic
335C25C2×#12112yes—
336C25C2×#12114yes—
337C25C2×#12114yes—
338C22×Q8#38612yes—
339C22×Q8C22×C22⋊C412yes—
340C22×Q8C22×C22⋊C411yescharacteristic
341C22×Q8C2×C4×D412yes—
342C22×Q8C2×#12312yes—
343C22×Q8C24×C411yescharacteristic
344C22×D4#38612yes—
345C22×D4#38612yes—
346C22×D4#38614yes—
347C22×D4#38614yes—
348C22×D4#38614yes—
349C22×D4#38614yes—
350C22×D4#38614yes—
351C22×D4#38612yes—
352C22×D4#38612yes—
353C22×D4#38612yes—
354C22×D4C2×#9912yes—
355C22×D4C2×#9912yes—
356C22×D4C2×#9912yes—
357C22×D4C2×#9912yes—
358C22×D4C2×#9912yes—
359C22×D4C2×#9912yes—
360C22×D4C2×#9912yes—
361C22×D4C2×#9912yes—
362C22×D4B(4,𝔽2)12yes—
363C22×D4B(4,𝔽2)12yes—
364C22×D4B(4,𝔽2)14yes—
365C22×D4C22×C22⋊C412yes—
366C22×D4C22×C22⋊C411yescharacteristic
367C22×D4C22×C22⋊C412yes—
368C22×D4C22×C22⋊C412yes—
369C22×D4C22×C22⋊C412yes—
370C22×D4C22×C22⋊C412yes—
371C22×D4C2×C4×D414yes—
372C22×D4C2×C4×D412yes—
373C22×D4C2×C4×D414yes—
374C22×D4C2×#12012yes—
375C22×D4C2×#12012yes—
376C22×D4C2×#12012yes—
377C22×D4C2×#12112yes—
378C22×D4C2×#12114yes—
379C22×D4C2×#12114yes—
380C22×D4C2×#12112yes—
381C22×D4C2×#12312yes—
382C22×D4C2×#12314yes—
383C22×D4C23×D411yescharacteristic
384C22×D4C23×D412yes—
385C23×C4#38612yes—
386C23×C4#38611yescharacteristic
387C23×C4#38612yes—
388C23×C4#38614yes—
389C23×C4#38611yescharacteristic
390C23×C4#38614yes—
391C23×C4C22×C22⋊C412yes—
392C23×C4C22×C22⋊C412yes—
393C23×C4C22×C22⋊C411yescharacteristic
394C23×C4C22×C22⋊C411yescharacteristic
395C23×C4C2×C4×D412yes—
396C23×C4C2×C4×D414yes—
397C23×C4C2×C4×D414yes—
398C23×C4C2×C4×D412yes—
399C23×C4C2×#12011yescharacteristic
400C23×C4C2×#12112yes—
401C23×C4C2×#12311yescharacteristic
402C23×C4C2×#12311yescharacteristic
403C23×C4C2×#12312yes—
404C23×C4C23×D411yescharacteristic
405C23×C4C22×#4112yes—
406#127C2×#9918yes—
407#127C2×#9918yes—
408#127C2×#99116yes—
409C2×C42C2×#9912yes—
410C2×C42C2×#9912yes—
411C2×C42C2×#9911yescharacteristic
412C2×C42C2×#9911yescharacteristic
413C2×C42C2×#9912yes—
414C2×C42C22×C22⋊C412yes—
415C2×C42C22×C22⋊C412yes—
416C2×C42C22×C22⋊C412yes—
417C2×C42C22×C22⋊C412yes—
418C24#188912yes—
419C24#188912yes—
420C24#189814yes—
421C24#189814yes—
422C24#191312yes—
423C24#191312yes—
424C24C2×#38612yes—
425C24C2×#38614yes—
426C24C2×#38612yes—
427C24C2×#38611yescharacteristic
428C24C2×#38612yes—
429C24C2×#38614yes—
430C24C2×#38611yescharacteristic
431C24C2×B(4,𝔽2)12yes—
432C24C2×B(4,𝔽2)12yes—
433C24C22×#12011yescharacteristic
434C24C22×#12112yes—
435C2×D4#188914yes—
436C2×D4#188914yes—
437C2×D4#189814yes—
438C2×D4#189814yes—
439C2×D4C22×#9914yes—
440C2×D4C22×#9914yes—
441C2×D4C22×#9914yes—
442C2×D4C22×#9914yes—
443C2×D4C2×B(4,𝔽2)14yes—
444C2×D4C2×B(4,𝔽2)14yes—
445C22×C4#188911yescharacteristic
446C22×C4#188911yescharacteristic
447C22×C4#188912yes—
448C22×C4C2×#38611yescharacteristic
449C22×C4C2×#38612yes—
450C22×C4C2×#38612yes—
451C22×C4C2×#38612yes—
452C22×C4C2×#38611yescharacteristic
453C22×C4C22×#9911yescharacteristic
454C22×C4C22×#9911yescharacteristic
455C22×C4C22×#9911yescharacteristic
456C22×C4C22×#9911yescharacteristic
457C22×C4C23×C22⋊C411yescharacteristic
458C22×C4C22×C4×D412yes—
459C22×C4C22×#12311yescharacteristic
460C42C22×#9912yes—
461C42C22×#9912yes—
462C42C22×#9914yes—
463C23C2×#188911yescharacteristic
464C23C2×#188912yes—
465C23C2×#188914yes—
466C23C2×#188911yescharacteristic
467C23C2×#189811yescharacteristic
468C23C2×#189818yes—
469C23C2×#189811yescharacteristic
470C23C2×#191311yescharacteristic
471C23C2×#191314yes—
472C23C2×#191311yescharacteristic
473C23C22×#38611yescharacteristic
474C23C22×B(4,𝔽2)12yes—
475C2×C4C2×#188912yes—
476C2×C4C2×#188912yes—
477C2×C4C2×#189811yescharacteristic
478C2×C4C2×#189811yescharacteristic
479C2×C4C23×#9911yescharacteristic
480C2×C4C23×#9911yescharacteristic
481C22#20166411yescharacteristic
482C22#20166411yescharacteristic
483C22#20166411yescharacteristic
484C22#20166411yescharacteristic
485C22C22×#188911yescharacteristic
486C22C22×#189811yescharacteristic
487C22C22×#191311yescharacteristic
488C2C2×#20166411yescharacteristic
489C2#7223801612yesdirect product factor
subgroupG/H
center, socleC22#201664
commutatorC22×Q8C24×C4
Frattini, cosocleC22×#41C25
prime radicalC2×#6437452C2
LCSG ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×H(5,𝔽2) ◁ G
LECSG ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#139736C2semidirect
2. C2×#139812C2semidirect
4. C2×#128079C2semidirect
5. C2×#493430C2semidirect
6. C2×#6385980C2semidirect
7. C2×#493397C2semidirect
8. #72238016C2direct (N × H)
11. C22×#3018C4semidirect
12. C22×#3021C4semidirect
13. #6437452C4semidirect
15. C2×#23915C4semidirect
17. C2×#23909C4semidirect
19. #6385980C22semidirect
20. C2×#21794C22semidirect
21. C2×#21750C22semidirect
22. C2×#20956C4semidirect
23. C2×#20952C4semidirect
27. #493430C22semidirect
28. #493397C22semidirect
29. C2×#12708C22semidirect
30. C2×#12697C22semidirect
31. C2×#12695C22semidirect
32. #139812C22semidirect
33. #139736C22semidirect
34. #128079C22semidirect
35. C2×#7313C22semidirect
36. C22×#585C2×C4semidirect
39. #62852C2×C4semidirect
45. C2×#3487C2×C4semidirect
59. C2×#3428C2×C4semidirect
61. C2×#3021C2×C4semidirect
62. C2×#3018C2×C4semidirect
63. C22×#456C2×C4semidirect
64. C22×#454C2×C4semidirect
65. C22×#454C2×C4semidirect
66. C22×#454C2×C4semidirect
67. C22×#413C2×C4semidirect
68. C22×#413C2×C4semidirect
70. #23915C2×C4semidirect
71. #23915C2×C4semidirect
73. #23909C2×C4semidirect
74. #23909C2×C4semidirect
76. #21794C23semidirect
77. #21750C23semidirect
78. #20956C2×C4semidirect
79. #20952C2×C4semidirect
80. C2×#1888C2×C4semidirect
81. C2×#1888C2×C4semidirect
82. C2×#1888C2×C4semidirect
83. C2×#1886C2×C4semidirect
84. #20592C2×C4semidirect
86. C2×#1785C2×C4semidirect
94. #12708C23semidirect
95. #12697C23semidirect
96. #12695C23semidirect
97. #7313C23semidirect
98. C2×#585C22×C4semidirect
126. #3487C22×C4semidirect
172. #3428C22×C4semidirect
182. C2×#456C22×C4semidirect
183. C2×#454C22×C4semidirect
184. C2×#454C22×C4semidirect
185. C2×#454C22×C4semidirect
186. C2×#413C22×C4semidirect
187. C2×#413C22×C4semidirect
192. #1888C22×C4semidirect
193. #1888C22×C4semidirect
194. #1888C22×C4semidirect
195. #1886C22×C4semidirect
197. #1785C22×C4semidirect
489. C2#72238016direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#1397362
C2×#1398122
C2×#64374522
C2×#1280792
C2×#4934302
C2×#63859802
C2×#4933972
#722380162