Finite Group Databasebeta

← C23×#19007 #19166633

Subgroups — C23×#19007 #19166633

Order 5888.

862 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#563C211yesmaximal, characteristic
2C23×#3236C211yesmaximal, characteristic, complemented
3C23×#3215C211yesmaximal, characteristic
4C23×#3166C211yesmaximal, characteristic, complemented
5C23×#1664C214yesmaximal, complemented
6C23×#1660C214yesmaximal, complemented
7C23×#1534C211yesmaximal, characteristic, complemented
8C23×#1528C211yesmaximal, characteristic, complemented
9C23×#1527C211yesmaximal, characteristic
10C23×#563—22no—
11C23×#563—22no—
12C23×#563—22no—
13C23×#563C2214yes—
14C23×#533C2211yescharacteristic
15C2×C23×C4⋊Q8—21no—
16C2×C23×#126—21no—
17C2×C23×#126—21no—
18C2×C23×#125C2211yescharacteristic
19C2×C23×#125C2211yescharacteristic
20C2×C23×#124C2211yescharacteristic
21C2×C23×#123—21no—
22C2×C23×C22⋊Q8—21no—
23C2×C23×C22⋊Q8—21no—
24C23×#501C2212yescomplemented
25C23×#501C2212yescomplemented
26C23×#501C2212yescomplemented
27C23×#499—22no—
28C23×#497C2212yes—
29C23×#491C2212yescomplemented
30C23×#489C2212yes—
31C23×#484—21no—
32C23×#483—21no—
33C23×#479—22no—
34C23×#464—22no—
35C23×#451—21no—
36C23×#450—21no—
37C23×#432C2211yescharacteristic
38C23×#432C2211yescharacteristic
39C23×#428—21no—
40C23×#428—21no—
41C4×M16×C23C2211yescharacteristic
42C23×#330C2214yescomplemented
43C23×#330C2214yescomplemented
44C23×#329C2214yes—
45C22×Q8×C23—21no—
46C22×Q8×C23D411yescharacteristic
47C22×D4×C23D411yescharacteristic
48C23×C4×C23—21no—
49C2×Q16×C23—22no—
50C2×SD16×C23—22no—
51C2×M16×C23—21no—
52C23×C4⋊Q8—24no—
53C23×C4⋊Q8—24no—
54C23×C4⋊Q8C2311yescharacteristic
55C23×C4⋊Q8C2311yescharacteristic
56C23×#126—24no—
57C23×#126—24no—
58C23×#126—24no—
59C23×#126—24no—
60C23×#125—21no—
61C23×#125—21no—
62C23×#125—21no—
63C23×#125—21no—
64C23×#125—21no—
65C23×#125D412yes—
66C23×#125D412yes—
67C23×#125C2312yes—
68C23×#125C2312yes—
69C23×#124—21no—
70C23×#124—21no—
71C23×#124D412yes—
72C23×#124C2312yes—
73C23×#123—24no—
74C23×#123—24no—
75C23×C22⋊Q8—24no—
76C23×C22⋊Q8—24no—
77C23×C22⋊Q8—24no—
78C23×C22⋊Q8—24no—
79C23×C22⋊Q8—21no—
80C23×C22⋊Q8—21no—
81C4×Q8×C23—21no—
82C4×Q8×C23—21no—
83C23×#117—21no—
84C2×C4⋊C4×C23—21no—
85C2×C4⋊C4×C23—21no—
86C2×C4⋊C4×C23—21no—
87C2×C4⋊C4×C23—21no—
88C2×C4⋊C4×C23—21no—
89C2×C4⋊C4×C23—21no—
90C2×C4⋊C4×C23D411yescharacteristic
91C2×C4⋊C4×C23D411yescharacteristic
92C2×C4⋊C4×C23D411yescharacteristic
93C2×C4⋊C4×C23D411yescharacteristic
94C2×C22⋊C4×C23—21no—
95C2×C22⋊C4×C23—21no—
96C2×C22⋊C4×C23—21no—
97C2×C22⋊C4×C23—21no—
98C2×C22⋊C4×C23—21no—
99C2×C42×C23—21no—
100C2×C42×C23C2311yescharacteristic
101C23×#107—22no—
102C23×#107—22no—
103C23×#107—22no—
104C23×#106—22no—
105C23×#106—22no—
106C23×#106—22no—
107C23×C4⋊C8—21no—
108C23×C4⋊C8—21no—
109C23×#103—22no—
110C23×#103—22no—
111C23×#103—22no—
112C23×#102—22no—
113C23×C22⋊C8—21no—
114C23×C22⋊C8—21no—
115C23×#97C2312yes—
116C23×#97C2312yes—
117C23×#97C2312yes—
118C4×C8×C23—21no—
119C24×C23—21no—
120C2×Q8×C23—24no—
121C2×Q8×C23—21no—
122C2×Q8×C23—21no—
123C2×Q8×C23—21no—
124C2×Q8×C23—81no—
125C2×Q8×C23—82no—
126C2×Q8×C23C2×D412yes—
127C2×D4×C23—21no—
128C2×D4×C23—81no—
129C2×D4×C23C2×D412yes—
130C22×C4×C23—21no—
131C22×C4×C23—21no—
132C22×C4×C23—21no—
133C22×C4×C23—21no—
134C22×C4×C23—21no—
135C22×C4×C23—24no—
136C22×C4×C23—82no—
137C22×C4×C23C2×D411yescharacteristic
138C22×C4×C23C2×D411yescharacteristic
139C22×C4×C23C2×D411yescharacteristic
140Q16×C23—82no—
141SD16×C23—82no—
142M16×C23—24no—
143C2×C8×C23—22no—
144C2×C8×C23—22no—
145C2×C8×C23—22no—
146C4⋊C4×C23—24no—
147C4⋊C4×C23—24no—
148C4⋊C4×C23—22no—
149C4⋊C4×C23—22no—
150C4⋊C4×C23—22no—
151C4⋊C4×C23—22no—
152C4⋊C4×C23—21no—
153C4⋊C4×C23—21no—
154C4⋊C4×C23—21no—
155C4⋊C4×C23—21no—
156C4⋊C4×C23—22no—
157C4⋊C4×C23—22no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—21no—
161C4⋊C4×C23—21no—
162C4⋊C4×C23—21no—
163C4⋊C4×C23—21no—
164C4⋊C4×C23—41no—
165C4⋊C4×C23—41no—
166C4⋊C4×C23C2×D412yes—
167C4⋊C4×C23C2×D412yes—
168C4⋊C4×C23C2×D412yes—
169C4⋊C4×C23C2×D412yes—
170C22⋊C4×C23—24no—
171C22⋊C4×C23—24no—
172C22⋊C4×C23—24no—
173C22⋊C4×C23—22no—
174C22⋊C4×C23—22no—
175C22⋊C4×C23—22no—
176C22⋊C4×C23—22no—
177C22⋊C4×C23—41no—
178C42×C23—24no—
179C42×C23—41no—
180C42×C23C2×D411yescharacteristic
181C42×C23C2×D411yescharacteristic
182C42×C23C2×D411yescharacteristic
183C42×C23C2411yescharacteristic
184#19007C2311yesmaximal, characteristic, direct product factor, Sylow
185C23×C23—22no—
186C23×C23—42no—
187C23×C23—42no—
188C23×C23—42no—
189C23×C23#12011yescharacteristic
190Q8×C23—42no—
191Q8×C23—42no—
192Q8×C23—42no—
193Q8×C23—81no—
194Q8×C23—84no—
195D4×C23—42no—
196D4×C23—81no—
197C2×C4×C23—22no—
198C2×C4×C23—22no—
199C2×C4×C23—22no—
200C2×C4×C23—22no—
201C2×C4×C23—22no—
202C2×C4×C23—22no—
203C2×C4×C23—41no—
204C2×C4×C23—41no—
205C2×C4×C23—42no—
206C2×C4×C23—42no—
207C2×C4×C23—42no—
208C2×C4×C23—42no—
209C2×C4×C23—42no—
210C2×C4×C23—41no—
211C2×C4×C23—41no—
212C2×C4×C23—84no—
213C2×C4×C23#12011yescharacteristic
214C2×C4×C23#12011yescharacteristic
215C2×C4×C23#12011yescharacteristic
216C2×C4×C23C22×D411yescharacteristic
217C2×C4×C23C22×D411yescharacteristic
218C2×C4×C23C22×D411yescharacteristic
219C8×C23—24no—
220C8×C23—42no—
221C8×C23—42no—
222C2×#563C2×C2311yescharacteristic
223#3236C2×C2311yescharacteristic, complemented
224#3215C2×C2311yescharacteristic
225#3166C2×C2311yescharacteristic, complemented
226#1664C2×C2314yescomplemented
227#1660C2×C2314yescomplemented
228#1534C2×C2311yescharacteristic, complemented
229#1528C2×C2311yescharacteristic, complemented
230#1527C2×C2311yescharacteristic
231C22×C23—21no—
232C22×C23—21no—
233C22×C23—42no—
234C22×C23—42no—
235C22×C23—42no—
236C22×C23—84no—
237C22×C23#45612yes—
238C22×C23C2×#12011yescharacteristic
239C4×C23—21no—
240C4×C23—21no—
241C4×C23—21no—
242C4×C23—21no—
243C4×C23—21no—
244C4×C23—21no—
245C4×C23—42no—
246C4×C23—42no—
247C4×C23—42no—
248C4×C23—42no—
249C4×C23—42no—
250C4×C23—81no—
251C4×C23—81no—
252#563—22no—
253#563—22no—
254#563—22no—
255#563C22×C2314yes—
256#533C22×C2311yescharacteristic
257C2×C4⋊Q8—21no—
258C2×#126—21no—
259C2×#126—21no—
260C2×#125C22×C2311yescharacteristic
261C2×#125C22×C2311yescharacteristic
262C2×#124C22×C2311yescharacteristic
263C2×#123—21no—
264C2×C22⋊Q8—21no—
265C2×C22⋊Q8—21no—
266#501C22×C2312yescomplemented
267#501C22×C2312yescomplemented
268#501C22×C2312yescomplemented
269#499—22no—
270#497C22×C2312yes—
271#491C22×C2312yescomplemented
272#489C22×C2312yes—
273#484—21no—
274#483—21no—
275#479—22no—
276#464—22no—
277#451—21no—
278#450—21no—
279#432C22×C2311yescharacteristic
280#432C22×C2311yescharacteristic
281#428—21no—
282#428—21no—
283C4×M16C22×C2311yescharacteristic
284#330C22×C2314yescomplemented
285#330C22×C2314yescomplemented
286#329C22×C2314yes—
287C2×C23—22no—
288C2×C23—82no—
289C2×C23C2×#45611yescharacteristic
290C2×C23#301911yescharacteristic
291C2×C23#302011yescharacteristic
292C22×Q8—21no—
293C22×Q8D4×C2311yescharacteristic
294C22×D4D4×C2311yescharacteristic
295C23×C4—21no—
296C2×Q16—22no—
297C2×SD16—22no—
298C2×M16—21no—
299C4⋊Q8—24no—
300C4⋊Q8—24no—
301C4⋊Q8C23×C2311yescharacteristic
302C4⋊Q8C23×C2311yescharacteristic
303#126—24no—
304#126—24no—
305#126—24no—
306#126—24no—
307#125—21no—
308#125—21no—
309#125—21no—
310#125—21no—
311#125—21no—
312#125D4×C2312yes—
313#125D4×C2312yes—
314#125C23×C2312yes—
315#125C23×C2312yes—
316#124—21no—
317#124—21no—
318#124D4×C2312yes—
319#124C23×C2312yes—
320#123—24no—
321#123—24no—
322C22⋊Q8—24no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325C22⋊Q8—24no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330#117—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4D4×C2311yescharacteristic
338C2×C4⋊C4D4×C2311yescharacteristic
339C2×C4⋊C4D4×C2311yescharacteristic
340C2×C4⋊C4D4×C2311yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C42—21no—
347C2×C42C23×C2311yescharacteristic
348#107—22no—
349#107—22no—
350#107—22no—
351#106—22no—
352#106—22no—
353#106—22no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356#103—22no—
357#103—22no—
358#103—22no—
359#102—22no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362#97C23×C2312yes—
363#97C23×C2312yes—
364#97C23×C2312yes—
365C4×C8—21no—
366C23#1900711yescharacteristic, direct product factor, Sylow
367C24—21no—
368C2×Q8—24no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—81no—
373C2×Q8—82no—
374C2×Q8C2×D4×C2312yes—
375C2×D4—21no—
376C2×D4—81no—
377C2×D4C2×D4×C2312yes—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—82no—
385C22×C4C2×D4×C2311yescharacteristic
386C22×C4C2×D4×C2311yescharacteristic
387C22×C4C2×D4×C2311yescharacteristic
388Q16—82no—
389SD16—82no—
390M16—24no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4C2×D4×C2312yes—
415C4⋊C4C2×D4×C2312yes—
416C4⋊C4C2×D4×C2312yes—
417C4⋊C4C2×D4×C2312yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—41no—
426C42—24no—
427C42—41no—
428C42C2×D4×C2311yescharacteristic
429C42C2×D4×C2311yescharacteristic
430C42C2×D4×C2311yescharacteristic
431C42C24×C2311yescharacteristic
432C23—22no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23C23×#12011yescharacteristic
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—81no—
441Q8—84no—
442D4—42no—
443D4—81no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—84no—
460C2×C4C23×#12011yescharacteristic
461C2×C4C23×#12011yescharacteristic
462C2×C4C23×#12011yescharacteristic
463C2×C4C22×D4×C2311yescharacteristic
464C2×C4C22×D4×C2311yescharacteristic
465C2×C4C22×D4×C2311yescharacteristic
466C8—24no—
467C8—42no—
468C8—42no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—84no—
475C22C23×#45612yes—
476C22C2×C23×#12011yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—81no—
489C4—81no—
490C2—22no—
491C2—82no—
492C2C2×C23×#45611yescharacteristic
493C2C23×#301911yescharacteristic
494C2C23×#302011yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC22×D4×C23D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3236C2semidirect
4. C23×#3166C2semidirect
5. C23×#1664C2semidirect
6. C23×#1660C2semidirect
7. C23×#1534C2semidirect
8. C23×#1528C2semidirect
24. C23×#501C22semidirect
25. C23×#501C22semidirect
26. C23×#501C22semidirect
29. C23×#491C22semidirect
42. C23×#330C22semidirect
43. C23×#330C22semidirect
184. #19007C23direct (N × H)
223. #3236C2×C23semidirect
225. #3166C2×C23semidirect
226. #1664C2×C23semidirect
227. #1660C2×C23semidirect
228. #1534C2×C23semidirect
229. #1528C2×C23semidirect
266. #501C22×C23semidirect
267. #501C22×C23semidirect
268. #501C22×C23semidirect
271. #491C22×C23semidirect
284. #330C22×C23semidirect
285. #330C22×C23semidirect
366. C23#19007direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5632
C23×#32362
C23×#32152
C23×#31662
C23×#16642
C23×#16602
C23×#15342
C23×#15282
C23×#15272
#1900723