Finite Group Databasebeta

← #381692

Subgroups — #381692

Order 512.

847 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23627C211yesmaximal, characteristic
2#21046C211yesmaximal, characteristic
3#20437C211yesmaximal, characteristic, complemented
4#11359C211yesmaximal, characteristic, complemented
5#11326C211yesmaximal, characteristic, complemented
6#11291C212yesmaximal, complemented
7#11125C212yesmaximal
8#11053C211yesmaximal, characteristic
9#10934C212yesmaximal
10#10910C212yesmaximal, complemented
11#10797C211yesmaximal, characteristic, complemented
12C22×#108C2212yes—
13C2×#424C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×#104C2212yes—
16C2×#416C2212yes—
17C2×#408C2211yescharacteristic
18C2×#408C2211yescharacteristic
19C2×#405C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2297C2211yescharacteristic
22#2285C2211yescharacteristic
23#2074C412yescomplemented
24#2073C412yescomplemented
25#2072C411yescharacteristic, complemented
26#2072C411yescharacteristic, complemented
27#2071C411yescharacteristic, complemented
28#2071C411yescharacteristic, complemented
29#1923—21no—
30#1923—21no—
31#1923—21no—
32#1923—21no—
33#1770—21no—
34#1770—21no—
35C8×C4⋊C4—21no—
36C8×C4⋊C4—21no—
37C2×#341C2212yes—
38#1735C2212yes—
39C2×#339C2212yes—
40C2×#339C2212yes—
41C2×#339C2212yes—
42#1729C2212yes—
43#1729C2211yescharacteristic
44#1728C2211yescharacteristic
45C2×#336C2212yes—
46C2×#336C2212yes—
47C2×#336C2211yescharacteristic
48C4×#95C2211yescharacteristic
49C22×#41C2311yescharacteristic
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2312yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#117C2312yes—
57C2×#117C2311yescharacteristic
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×#108—24no—
61C2×#108—42no—
62C2×#108—42no—
63#424—42no—
64#423—24no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×#104—24no—
70C2×#104—42no—
71C2×#104—42no—
72#416—42no—
73#416—42no—
74C2×#103C2×C412yes—
75C2×#103C2×C412yes—
76C2×#102C2×C412yes—
77C2×#102C2×C412yes—
78#408—24no—
79#408—42no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84#405—42no—
85#405—42no—
86C2×#97—22no—
87C2×C4×C8—24no—
88C2×C4×C8—22no—
89C2×C4×C8D411yescharacteristic
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93#391C2×C411yescharacteristic
94#391C2×C411yescharacteristic
95#390C2×C411yescharacteristic
96#390C2×C411yescharacteristic
97#386—21no—
98#386—21no—
99#384C2×C411yescharacteristic
100#384C2×C411yescharacteristic
101#382C2×C412yes—
102#380—21no—
103#380—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C2×#95—21no—
109C2×#95—21no—
110#341—42no—
111#340C2×C412yes—
112#340C2×C412yes—
113#340C2×C412yes—
114#340C2×C412yes—
115#339—42no—
116#339—42no—
117#339—42no—
118#336—21no—
119#336—21no—
120#336—22no—
121#336—22no—
122#336—24no—
123#336—21no—
124#336—21no—
125#336—42no—
126#336C2×C411yescharacteristic
127#336C2×C411yescharacteristic
128#336C2×C412yes—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—22no—
132C2×#41—21no—
133C2×#41—21no—
134C2×#41—21no—
135C2×#41—22no—
136C2×#41—21no—
137C22×Q8C22×C411yescharacteristic
138C22×D4—21no—
139C22×D4C22×C411yescharacteristic
140C23×C4—22no—
141C23×C4—21no—
142C23×C4#4111yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4C2411yescharacteristic
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16#4114yes—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8C22×C412yes—
176C22×C8C22×C412yes—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8#4114yes—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216#108—48no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221#104—48no—
222#103—42no—
223#103—42no—
224#102—42no—
225#102—42no—
226C22⋊C8—42no—
227C22⋊C8—42no—
228C22⋊C8—42no—
229C22⋊C8—42no—
230#97—44no—
231C4×C8—22no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—44no—
236#95—24no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—42no—
244C24—21no—
245C24C2×#4111yescharacteristic
246#41—42no—
247#41—42no—
248#41—42no—
249#41—42no—
250#41—42no—
251#41—42no—
252#41—42no—
253#41—42no—
254C2×Q8—22no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—22no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4112yes—
297C22×C4C2×#4112yes—
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C4H(5,𝔽2)12yes—
302C22×C42−1+412yes—
303C22×C42−1+411yescharacteristic
304M16—24no—
305M16—24no—
306M16—42no—
307M16—42no—
308M16—42no—
309M16—42no—
310M16—42no—
311M16—42no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—24no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8#12312yes—
346C2×C8#12312yes—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C42—22no—
365C42—22no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—42no—
371C42—41no—
372C42—41no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23C2×C4×D411yescharacteristic
390C23#52912yes—
391C23#54011yescharacteristic
392C23#54211yescharacteristic
393C23#54212yes—
394Q8—42no—
395Q8—42no—
396D4—42no—
397D4—42no—
398D4—82no—
399D4—82no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4#44312yes—
439C2×C4#44412yes—
440C2×C4C4×#4112yes—
441C2×C4#51811yescharacteristic
442C2×C4#52011yescharacteristic
443C2×C4#52012yes—
444C2×C4C2×#12311yescharacteristic
445C2×C4#52511yescharacteristic
446C2×C4#53511yescharacteristic
447C2×C4#53511yescharacteristic
448C2×C4#53611yescharacteristic
449C2×C4#53711yescharacteristic
450C2×C4#53812yes—
451C2×C4#53812yes—
452C2×C4#53912yes—
453C2×C4#53912yes—
454C2×C4#54012yes—
455C2×C4#54112yes—
456C2×C4#54212yes—
457C2×C4#54212yes—
458C8—22no—
459C8—22no—
460C8—42no—
461C8—42no—
462C8—41no—
463C8—41no—
464C8—44no—
465C8—81no—
466C8—81no—
467C22—22no—
468C22—22no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—22no—
474C22—22no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22C4×#12311yescharacteristic
483C22#232411yescharacteristic
484C22#232611yescharacteristic
485C22#265012yes—
486C22#265112yes—
487C22C2×#44311yescharacteristic
488C22C2×#44411yescharacteristic
489C22#295511yescharacteristic
490C22#295511yescharacteristic
491C4—22no—
492C4—22no—
493C4—41no—
494C4—41no—
495C4—84no—
496C4—82no—
497C4—82no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4#233412yes—
502C4#233512yes—
503C4#262311yescharacteristic
504C4#262411yescharacteristic
505C4#263111yescharacteristic
506C4#263711yescharacteristic
507C2—22no—
508C2—22no—
509C2—161no—
510C2#1430611yescharacteristic
511C2#2159211yescharacteristic
512C2#2159311yescharacteristic
513C2#2216511yescharacteristic
514C2#2216611yescharacteristic
515C2#2223411yescharacteristic
516C2#2223511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20437C2semidirect
4. #11359C2semidirect
5. #11326C2semidirect
6. #11291C2semidirect
10. #10910C2semidirect
11. #10797C2semidirect
23. #2074C4semidirect
24. #2073C4semidirect
25. #2072C4semidirect
26. #2072C4semidirect
27. #2071C4semidirect
28. #2071C4semidirect

Maximal subgroups

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

subgroupindex
#236272
#210462
#204372
#113592
#113262
#112912
#111252
#110532
#109342
#109102
#107972