Finite Group Databasebeta

← #15682678

Subgroups — #15682678

Order 2944.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625158C211yesmaximal, characteristic, complemented
2C2×#10625085C211yesmaximal, characteristic, complemented
3C2×SD16×D23C211yesmaximal, characteristic, complemented
4#13092983C211yesmaximal, characteristic, complemented
5#13092565C211yesmaximal, characteristic, complemented
6#13092564C211yesmaximal, characteristic, complemented
7#13092561C211yesmaximal, characteristic, complemented
8#13092560C211yesmaximal, characteristic, complemented
9#13092557C211yesmaximal, characteristic, complemented
10#13092556C211yesmaximal, characteristic, complemented
11#13092554C211yesmaximal, characteristic, complemented
12#13092424C211yesmaximal, characteristic, complemented
13#13092412C211yesmaximal, characteristic, complemented
14#13092382C211yesmaximal, characteristic, complemented
15C23×#448C211yesmaximal, characteristic, complemented
16C22×Q8×C23C2211yescharacteristic, complemented
17#10625158—24no—
18#10625158—24no—
19C2×#67292—21no—
20C2×#67292C2211yescharacteristic
21C2×#67292C2211yescharacteristic, complemented
22C2×Q8×D23—21no—
23C2×Q8×D23C2211yescharacteristic
24C2×Q8×D23C2211yescharacteristic, complemented
25C2×D4×D23C2211yescharacteristic
26C2×#67288—21no—
27C2×#67288—21no—
28C2×#67288C2211yescharacteristic
29C2×#67288C2211yescharacteristic, complemented
30C2×SD16×C23C2211yescharacteristic
31C2×SD16×C23C2211yescharacteristic
32C23×#121C2211yescharacteristic
33C2×#67274C2211yescharacteristic
34C2×#67274C2211yescharacteristic
35#10625111—21no—
36#10625109C2211yescharacteristic
37#10625108—21no—
38C2×#67273C2211yescharacteristic
39C2×#67272C2211yescharacteristic
40#10625101C2211yescharacteristic
41#10625085—24no—
42SD16×D23—24no—
43C2×D184C2211yescharacteristic
44C2×#67263C2211yescharacteristic
45C2×#67262C2211yescharacteristic
46C2×C8×D23C2211yescharacteristic
47#10625066C2211yescharacteristic
48#10625063C2211yescharacteristic
49#10625056—21no—
50#10625053—21no—
51C23×#103C2211yescharacteristic, complemented
52C23×#103C2211yescharacteristic, complemented
53C23×C22⋊C8C2211yescharacteristic, complemented
54#10625017C2211yescharacteristic, complemented
55#10625017C2211yescharacteristic, complemented
56C22⋊(C23⋊C8)C2211yescharacteristic, complemented
57#10625003C2211yescharacteristic, complemented
58(C2×D23)⋊C8C2211yescharacteristic, complemented
59(C2×D23)⋊C8C2211yescharacteristic, complemented
60#10624998C2211yescharacteristic, complemented
61#10624992C2211yescharacteristic, complemented
62#10624991C2211yescharacteristic, complemented
63C23×D23—21no—
64C2×Q8×C23—22no—
65C2×Q8×C23—22no—
66C2×Q8×C23—21no—
67C2×Q8×C23C2311yescharacteristic
68C2×Q8×C23C2311yescharacteristic
69C2×D4×C23—21no—
70C2×D4×C23C2311yescharacteristic
71C22×C4×C23—21no—
72C22×C4×C23C2311yescharacteristic
73C2×C22⋊D23—21no—
74C2×C22⋊D23—21no—
75C2×C22⋊D23—21no—
76C2×C22⋊D23—21no—
77C2×C22⋊D23D411yescharacteristic
78C2×C22⋊D23D411yescharacteristic
79#67292—28no—
80#67292—22no—
81#67292—22no—
82#67292—22no—
83#67292D414yescomplemented
84Q8×D23—28no—
85Q8×D23—22no—
86Q8×D23—22no—
87Q8×D23—22no—
88Q8×D23D414yescomplemented
89D4×D23—22no—
90D4×D23—22no—
91#67288—28no—
92#67288—28no—
93#67288—22no—
94#67288—22no—
95#67288—22no—
96#67288—22no—
97C2×D92—21no—
98C2×D92—21no—
99C2×D92—21no—
100C2×D92D411yescharacteristic
101C2×D92C2311yescharacteristic
102C2×D92C2311yescharacteristic
103C2×C4×D23—21no—
104C2×C4×D23—21no—
105C2×C4×D23—21no—
106C2×C4×D23—21no—
107C2×C4×D23—21no—
108C2×C4×D23C2311yescharacteristic
109C2×C4×D23C2311yescharacteristic
110C2×Dic46—21no—
111C2×Dic46—21no—
112C2×Dic46D411yescharacteristic
113C2×Dic46C2311yescharacteristic
114SD16×C23—22no—
115SD16×C23—22no—
116C2×C8×C23C2311yescharacteristic
117C2×C8×C23C2311yescharacteristic
118C4⋊C4×C23C2311yescharacteristic
119C22⋊C4×C23—21no—
120C22⋊Dic23—21no—
121#67274—22no—
122#67274—22no—
123#67273—22no—
124#67272—22no—
125(C2×D23)⋊C4—21no—
126(C2×D23)⋊C4—21no—
127(C2×D23)⋊C4—21no—
128C4⋊Dic23C2311yescharacteristic
129C4×Dic23—21no—
130C2×C23⋊C8C2311yescharacteristic
131C2×C23⋊C8C2311yescharacteristic
132D184—22no—
133#67263—22no—
134#67262—22no—
135C8×D23—22no—
136C23×C23—21no—
137C23×C23C2×D411yescharacteristic
138C22×D23—21no—
139C22×D23—21no—
140C22×D23—42no—
141C22×D23—81no—
142C22×D23—81no—
143C22×D23C2×D411yescharacteristic
144C22×D23C2×D411yescharacteristic
145Q8×C23—24no—
146Q8×C23—21no—
147Q8×C23—21no—
148Q8×C23C2×D412yes—
149Q8×C23C2×D412yes—
150D4×C23—21no—
151D4×C23—21no—
152D4×C23—81no—
153C2×C4×C23—24no—
154C2×C4×C23—21no—
155C2×C4×C23—21no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23C2×D411yescharacteristic
160C2×C4×C23C2411yescharacteristic
161C22⋊D23—24no—
162C22⋊D23—22no—
163C22⋊D23—22no—
164C22⋊D23—42no—
165C22⋊D23—81no—
166C22⋊D23—81no—
167C2×Dic23—21no—
168C2×Dic23—21no—
169C2×Dic23C2×D411yescharacteristic
170C2×Dic23C2×D411yescharacteristic
171D92—24no—
172D92—24no—
173D92—22no—
174D92—21no—
175D92—21no—
176D92—21no—
177D92—81no—
178D92C2×D412yes—
179C4×D23—24no—
180C4×D23—24no—
181C4×D23—24no—
182C4×D23—24no—
183C4×D23—21no—
184C4×D23—21no—
185C4×D23—24no—
186C4×D23C2×D412yes—
187C4×D23C2×D412yes—
188Dic46—24no—
189Dic46—21no—
190Dic46—24no—
191Dic46—22no—
192Dic46C2×D412yes—
193C8×C23—21no—
194C8×C23—21no—
195C23⋊C8—21no—
196C23⋊C8—21no—
197#3010—231nomaximal, Sylow
198C22×C23—21no—
199C22×C23—21no—
200C22×C23—21no—
201C22×C23—41no—
202C22×C23—41no—
203C22×C23—41no—
204C22×C23C22×D411yescharacteristic
205C2×D23—21no—
206C2×D23—21no—
207C2×D23—21no—
208C2×D23—22no—
209C2×D23—21no—
210C2×D23—21no—
211C2×D23—41no—
212C2×D23—41no—
213C2×D23—41no—
214C2×D23—41no—
215C2×D23—41no—
216C2×D23—84no—
217C2×D23#12012yes—
218C4×C23—22no—
219C4×C23—22no—
220C4×C23—21no—
221C4×C23—41no—
222C4×C23C22×D411yescharacteristic
223C4×C23C22×D411yescharacteristic
224Dic23—21no—
225Dic23—22no—
226Dic23—41no—
227Dic23#12012yes—
228C2×2−1+4—231no—
229C2×AGL(1,ℤ/8)—231no—
230C22×SD16—231no—
231#535—231no—
232#449—231no—
233#449—231no—
234#449—231no—
235#449—231no—
236#448—231no—
237#448—231no—
238#448—231no—
239#448D2311yescharacteristic, complemented
240#419—231no—
241#416—231no—
242#408—231no—
243C2×C23—41no—
244C2×C23—81no—
245C2×C23C2×#12011yescharacteristic
246C2×C23#57711yescharacteristic
247C2×C23#57711yescharacteristic
248D23—22no—
249D23—41no—
250D23—42no—
251D23—81no—
2522−1+4—464no—
2532−1+4—464no—
254C2×#41—231no—
255C2×#41—231no—
256C2×#41—231no—
257C2×#41—231no—
258C2×#41—461no—
259C2×#41—461no—
260C2×#41—461no—
261C22×Q8—231no—
262C22×Q8—231no—
263C22×Q8—461no—
264C22×Q8C2×D2311yescharacteristic, complemented
265C22×D4—231no—
266AGL(1,ℤ/8)—464no—
267C2×SD16—231no—
268C2×SD16—231no—
269C2×SD16—231no—
270C2×SD16—231no—
271C2×SD16—464no—
272C2×SD16C2×D2311yescharacteristic
273C2×SD16C2×D2311yescharacteristic
274C2×D8—231no—
275C2×D8—231no—
276C2×M16—231no—
277C22×C8—231no—
278#127—461no—
279#124—461no—
280#121—231no—
281#121—231no—
282#121—231no—
283#121C2×D2311yescharacteristic
284#120—461no—
285#120—461no—
286#117—231no—
287#103—231no—
288#103—231no—
289#103—231no—
290#103—231no—
291#103C2×D2311yescharacteristic, complemented
292#103C2×D2311yescharacteristic, complemented
293#102—231no—
294#102—231no—
295C22⋊C8—231no—
296C22⋊C8—231no—
297C22⋊C8—231no—
298C22⋊C8C2×D2311yescharacteristic, complemented
299C23#301011yescharacteristic, complemented, Sylow
300C24—461no—
301#41—234no—
302#41—462no—
303#41—462no—
304#41—462no—
305#41—462no—
306#41—468no—
307#41—468no—
308#41—462no—
309#41—462no—
310#41—468no—
311#41—462no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—21no—
315C2×Q8—231no—
316C2×Q8—234no—
317C2×Q8—231no—
318C2×Q8—461no—
319C2×Q8—461no—
320C2×Q8—462no—
321C2×Q8—462no—
322C2×Q8—468no—
323C2×Q8—462no—
324C2×Q8C22×D2311yescharacteristic
325C2×Q8C22×D2311yescharacteristic
326C2×D4—21no—
327C2×D4—231no—
328C2×D4—231no—
329C2×D4—231no—
330C2×D4—231no—
331C2×D4—231no—
332C2×D4—461no—
333C2×D4—461no—
334C2×D4—461no—
335C2×D4—462no—
336C2×D4—462no—
337C2×D4—461no—
338C2×D4—461no—
339C2×D4—461no—
340C2×D4—461no—
341C2×D4C22×D2311yescharacteristic
342C22×C4—21no—
343C22×C4—231no—
344C22×C4—231no—
345C22×C4—461no—
346C22×C4—461no—
347C22×C4—461no—
348C22×C4—461no—
349C22×C4—461no—
350C22×C4C22×D2311yescharacteristic
351SD16—22no—
352SD16—22no—
353SD16—462no—
354SD16—462no—
355SD16—462no—
356SD16—462no—
357D8—462no—
358D8—462no—
359M16—462no—
360C2×C8—231no—
361C2×C8—231no—
362C2×C8—462no—
363C2×C8C22×D2311yescharacteristic
364C2×C8C22×D2311yescharacteristic
365C4⋊C4—231no—
366C4⋊C4C22×D2311yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—461no—
369C22⋊C4—461no—
370C22⋊C4—461no—
371C22⋊C4—461no—
372C42—461no—
373C23—21no—
374C23—231no—
375C23—231no—
376C23—461no—
377C23—461no—
378C23—922no—
379C23—1841no—
380C23—1841no—
381C23D4×D2311yescharacteristic
382Q8—24no—
383Q8—21no—
384Q8—21no—
385Q8—232no—
386Q8—464no—
387Q8—462no—
388Q8—464no—
389Q8—461no—
390Q8D4×D2312yes—
391Q8D4×D2312yes—
392D4—21no—
393D4—21no—
394D4—81no—
395D4—232no—
396D4—462no—
397D4—464no—
398D4—464no—
399D4—462no—
400D4—462no—
401D4—464no—
402D4—461no—
403D4—461no—
404D4—461no—
405D4—922no—
406D4—1841no—
407D4—1841no—
408D4—1841no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—231no—
416C2×C4—232no—
417C2×C4—232no—
418C2×C4—231no—
419C2×C4—464no—
420C2×C4—464no—
421C2×C4—464no—
422C2×C4—464no—
423C2×C4—464no—
424C2×C4—461no—
425C2×C4—461no—
426C2×C4—461no—
427C2×C4—461no—
428C2×C4D4×D2311yescharacteristic
429C2×C4C23×D2311yescharacteristic
430C8—21no—
431C8—21no—
432C8—461no—
433C8—461no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—232no—
441C22—461no—
442C22—461no—
443C22—462no—
444C22—461no—
445C22—461no—
446C22—461no—
447C22—921no—
448C22—921no—
449C22—921no—
450C22—921no—
451C22—921no—
452C22—1844no—
453C22C2×D4×D2311yescharacteristic
454C4—22no—
455C4—22no—
456C4—21no—
457C4—41no—
458C4—232no—
459C4—461no—
460C4—462no—
461C4—921no—
462C4C2×D4×D2311yescharacteristic
463C4C2×D4×D2311yescharacteristic
464C2—41no—
465C2—81no—
466C2—462no—
467C2—921no—
468C2—922no—
469C2—1841no—
470C2#120×D2311yescharacteristic
471C2#1309322311yescharacteristic
472C2#1309322611yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#448C2
socleC22×C23C22×D4
hypercenter#448D23
hypocenterC23#3010
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#10625158C2semidirect
2. C2×#10625085C2semidirect
3. C2×SD16×D23C2semidirect
4. #13092983C2semidirect
5. #13092565C2semidirect
6. #13092564C2semidirect
7. #13092561C2semidirect
8. #13092560C2semidirect
9. #13092557C2semidirect
10. #13092556C2semidirect
11. #13092554C2semidirect
12. #13092424C2semidirect
13. #13092412C2semidirect
14. #13092382C2semidirect
15. C23×#448C2semidirect
16. C22×Q8×C23C22semidirect
21. C2×#67292C22semidirect
24. C2×Q8×D23C22semidirect
29. C2×#67288C22semidirect
51. C23×#103C22semidirect
52. C23×#103C22semidirect
53. C23×C22⋊C8C22semidirect
54. #10625017C22semidirect
55. #10625017C22semidirect
56. C22⋊(C23⋊C8)C22semidirect
57. #10625003C22semidirect
58. (C2×D23)⋊C8C22semidirect
59. (C2×D23)⋊C8C22semidirect
60. #10624998C22semidirect
61. #10624992C22semidirect
62. #10624991C22semidirect
83. #67292D4semidirect
88. Q8×D23D4semidirect
239. #448D23semidirect
264. C22×Q8C2×D23semidirect
291. #103C2×D23semidirect
292. #103C2×D23semidirect
298. C22⋊C8C2×D23semidirect
299. C23#3010semidirect

Maximal subgroups

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

subgroupindex
C2×#106251582
C2×#106250852
C2×SD16×D232
#130929832
#130925652
#130925642
#130925612
#130925602
#130925572
#130925562
#130925542
#130924242
#130924122
#130923822
C23×#4482
#301023