Finite Group Databasebeta

← #15682671

Subgroups — #15682671

Order 2944.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625160C211yesmaximal, characteristic
2C2×#10625089C211yesmaximal, characteristic, complemented
3C2×SD16×D23C211yesmaximal, characteristic, complemented
4#13092990C211yesmaximal, characteristic, complemented
5#13092582C211yesmaximal, characteristic, complemented
6#13092579C211yesmaximal, characteristic, complemented
7#13092578C211yesmaximal, characteristic, complemented
8#13092570C211yesmaximal, characteristic, complemented
9#13092569C211yesmaximal, characteristic, complemented
10#13092567C211yesmaximal, characteristic, complemented
11#13092566C211yesmaximal, characteristic, complemented
12#13092420C211yesmaximal, characteristic, complemented
13#13092412C211yesmaximal, characteristic, complemented
14#13092382C211yesmaximal, characteristic, complemented
15C23×#452C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17#10625160—24no—
18#10625160—24no—
19C2×#67292C2211yescharacteristic
20C2×#67292C2211yescharacteristic
21C2×Q8×D23C2211yescharacteristic, complemented
22C2×D4×D23—21no—
23C2×D4×D23—21no—
24C2×D4×D23C2211yescharacteristic
25C2×D4×D23C2211yescharacteristic
26C2×#67288—21no—
27C2×#67288C2211yescharacteristic
28C22×D92—21no—
29C22×D92C2211yescharacteristic
30C2×Q16×C23C2211yescharacteristic, complemented
31C2×SD16×C23C2211yescharacteristic, complemented
32C23×C22⋊Q8C2211yescharacteristic, complemented
33#10625117C2211yescharacteristic, complemented
34C2×#67275C2211yescharacteristic, complemented
35C2×#67274C2211yescharacteristic, complemented
36C2×#67274C2211yescharacteristic, complemented
37C2×#67273C2211yescharacteristic, complemented
38#10625095C2211yescharacteristic, complemented
39#10625089—24no—
40SD16×D23—24no—
41C2×#67263C2211yescharacteristic, complemented
42C2×#67263C2211yescharacteristic, complemented
43C2×#67262C2211yescharacteristic, complemented
44C2×C8×D23C2211yescharacteristic, complemented
45#10625068C2211yescharacteristic, complemented
46#10625065—21no—
47#10625063C2211yescharacteristic, complemented
48#10625059—21no—
49#10625056—21no—
50#10625054—21no—
51C23×#103C2211yescharacteristic, complemented
52C23×#102C2211yescharacteristic, complemented
53C23×C22⋊C8C2211yescharacteristic, complemented
54#10625017C2211yescharacteristic, complemented
55#10625014C2211yescharacteristic, complemented
56C22⋊(C23⋊C8)C2211yescharacteristic, complemented
57#10625003C2211yescharacteristic, complemented
58#10625003C2211yescharacteristic, complemented
59(C2×D23)⋊C8C2211yescharacteristic, complemented
60(C2×D23)⋊C8C2211yescharacteristic, complemented
61#10624991C2211yescharacteristic, complemented
62#10624991C2211yescharacteristic, complemented
63C23×D23—21no—
64C23×D23—21no—
65C23×D23—21no—
66#41×C23—22no—
67#41×C23—22no—
68C2×Q8×C23C2311yescharacteristic
69C2×Q8×C23C2311yescharacteristic
70C2×D4×C23—21no—
71C2×D4×C23C2311yescharacteristic
72C22×C4×C23—21no—
73C22×C4×C23C2311yescharacteristic
74C2×C22⋊D23—21no—
75C2×C22⋊D23—21no—
76C2×C22⋊D23D411yescharacteristic
77C2×C22⋊D23D411yescharacteristic
78#67292—22no—
79#67292—22no—
80#67292—22no—
81#67292D414yes—
82Q8×D23—22no—
83Q8×D23—22no—
84D4×D23—28no—
85D4×D23—28no—
86D4×D23—22no—
87D4×D23—22no—
88D4×D23—22no—
89D4×D23D414yes—
90#67288—28no—
91#67288—22no—
92#67288—22no—
93C2×D92—28no—
94C2×D92—21no—
95C2×D92—21no—
96C2×D92—21no—
97C2×D92—22no—
98C2×D92—22no—
99C2×D92D411yescharacteristic
100C2×D92C2311yescharacteristic
101C2×D92C2311yescharacteristic
102C2×C4×D23—21no—
103C2×C4×D23—21no—
104C2×C4×D23—21no—
105C2×C4×D23C2311yescharacteristic
106C2×C4×D23C2311yescharacteristic
107C2×Dic46—21no—
108C2×Dic46D411yescharacteristic
109C2×Dic46C2311yescharacteristic
110Q16×C23—22no—
111SD16×C23—22no—
112C2×C8×C23C2311yescharacteristic
113C2×C8×C23C2311yescharacteristic
114C4⋊C4×C23—21no—
115C4⋊C4×C23C2311yescharacteristic
116C22⋊C4×C23—21no—
117C22⋊Dic23—21no—
118#67275—22no—
119#67274—22no—
120#67274—22no—
121#67273—22no—
122(C2×D23)⋊C4—21no—
123(C2×D23)⋊C4—21no—
124(C2×D23)⋊C4—21no—
125C4⋊Dic23—21no—
126C4⋊Dic23C2311yescharacteristic
127Dic23⋊C4—21no—
128Dic23⋊C4—21no—
129C4×Dic23—21no—
130C2×C23⋊C8C2311yescharacteristic
131C2×C23⋊C8C2311yescharacteristic
132#67263—22no—
133#67263—22no—
134#67262—22no—
135C8×D23—22no—
136C23×C23—21no—
137C23×C23C2×D411yescharacteristic
138C22×D23—21no—
139C22×D23—21no—
140C22×D23—24no—
141C22×D23—24no—
142C22×D23—24no—
143C22×D23—81no—
144C22×D23—81no—
145C22×D23—81no—
146C22×D23—81no—
147C22×D23—81no—
148C22×D23—81no—
149C22×D23C2×D411yescharacteristic
150C22×D23C2×D411yescharacteristic
151Q8×C23—21no—
152Q8×C23—21no—
153Q8×C23—21no—
154Q8×C23C2×D412yes—
155D4×C23—24no—
156D4×C23—21no—
157D4×C23C2×D412yes—
158C2×C4×C23—24no—
159C2×C4×C23—21no—
160C2×C4×C23—21no—
161C2×C4×C23—21no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23C2×D411yescharacteristic
165C2×C4×C23C2411yescharacteristic
166C22⋊D23—24no—
167C22⋊D23—22no—
168C22⋊D23—22no—
169C22⋊D23—24no—
170C2×Dic23—21no—
171C2×Dic23—21no—
172C2×Dic23C2×D411yescharacteristic
173C2×Dic23C2×D411yescharacteristic
174D92—24no—
175D92—24no—
176D92—22no—
177D92—24no—
178D92—21no—
179D92—21no—
180D92C2×D412yes—
181D92C2×D412yes—
182C4×D23—24no—
183C4×D23—21no—
184C4×D23—21no—
185C4×D23—24no—
186C4×D23—42no—
187C4×D23C2×D412yes—
188C4×D23C2×D412yes—
189Dic46—22no—
190Dic46—21no—
191Dic46—21no—
192Dic46—42no—
193C8×C23—21no—
194C8×C23—21no—
195C23⋊C8—21no—
196C23⋊C8—21no—
197#3009—231nomaximal, Sylow
198C22×C23—21no—
199C22×C23—22no—
200C22×C23—21no—
201C22×C23—21no—
202C22×C23—41no—
203C22×C23—41no—
204C22×C23C22×D411yescharacteristic
205C2×D23—21no—
206C2×D23—21no—
207C2×D23—21no—
208C2×D23—22no—
209C2×D23—22no—
210C2×D23—21no—
211C2×D23—21no—
212C2×D23—41no—
213C2×D23—41no—
214C2×D23—41no—
215C2×D23—41no—
216C2×D23—84no—
217C2×D23—84no—
218C2×D23—84no—
219C2×D23#12012yes—
220C4×C23—21no—
221C4×C23—22no—
222C4×C23—41no—
223C4×C23—41no—
224C4×C23C22×D411yescharacteristic
225C4×C23C22×D411yescharacteristic
226Dic23—21no—
227Dic23—41no—
228Dic23—41no—
229Dic23#12012yes—
230C2×H(5,𝔽2)—231no—
231C2×#137—231no—
232C22×SD16—231no—
233#536—231no—
234#452—231no—
235#452—231no—
236#452—231no—
237#452D2311yescharacteristic, complemented
238#450—231no—
239#450—231no—
240#450—231no—
241#450—231no—
242#418—231no—
243#416—231no—
244#408—231no—
245C2×C23—41no—
246C2×C23—42no—
247C2×C23C2×#12011yescharacteristic
248C2×C23#57711yescharacteristic
249C2×C23#57711yescharacteristic
250D23—22no—
251D23—41no—
252D23—42no—
253D23—42no—
254H(5,𝔽2)—464no—
255H(5,𝔽2)—464no—
256C2×#41—231no—
257C2×#41—231no—
258C2×#41—231no—
259C2×#41—461no—
260C2×#41C2×D2311yescharacteristic
261C22×Q8—231no—
262C22×D4—231no—
263C22×D4—231no—
264C22×D4—231no—
265C22×D4—461no—
266C22×D4—461no—
267C22×D4—461no—
268#137—464no—
269C2×Q16—231no—
270C2×Q16C2×D2311yescharacteristic, complemented
271C2×SD16—231no—
272C2×SD16—231no—
273C2×SD16—231no—
274C2×SD16—231no—
275C2×SD16—231no—
276C2×SD16—464no—
277C2×SD16C2×D2311yescharacteristic, complemented
278C2×M16—231no—
279C22×C8—231no—
280C4⋊Q8—461no—
281#124—461no—
282#123—461no—
283#123—461no—
284C22⋊Q8—231no—
285C22⋊Q8—231no—
286C22⋊Q8—231no—
287C22⋊Q8C2×D2311yescharacteristic, complemented
288#117—231no—
289#103—231no—
290#103C2×D2311yescharacteristic, complemented
291#102—231no—
292#102—231no—
293#102—231no—
294#102—231no—
295#102—231no—
296#102C2×D2311yescharacteristic, complemented
297C22⋊C8—231no—
298C22⋊C8—231no—
299C22⋊C8—231no—
300C22⋊C8C2×D2311yescharacteristic, complemented
301C23#300911yescharacteristic, complemented, Sylow
302C24—461no—
303C24—461no—
304C24—461no—
305#41—22no—
306#41—22no—
307#41—234no—
308#41—462no—
309#41—462no—
310#41—468no—
311#41—462no—
312#41—462no—
313#41—462no—
314C2×Q8—231no—
315C2×Q8—231no—
316C2×Q8—461no—
317C2×Q8—462no—
318C2×Q8—462no—
319C2×Q8C22×D2311yescharacteristic
320C2×Q8C22×D2311yescharacteristic
321C2×D4—21no—
322C2×D4—234no—
323C2×D4—231no—
324C2×D4—231no—
325C2×D4—231no—
326C2×D4—231no—
327C2×D4—231no—
328C2×D4—462no—
329C2×D4—462no—
330C2×D4—461no—
331C2×D4—461no—
332C2×D4—468no—
333C2×D4—462no—
334C2×D4—461no—
335C2×D4—461no—
336C2×D4—462no—
337C2×D4—468no—
338C2×D4—468no—
339C2×D4—462no—
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×C4C22×D2311yescharacteristic
349Q16—22no—
350Q16—462no—
351SD16—22no—
352SD16—462no—
353SD16—462no—
354SD16—462no—
355SD16—462no—
356SD16—462no—
357M16—462no—
358C2×C8—231no—
359C2×C8—231no—
360C2×C8—462no—
361C2×C8C22×D2311yescharacteristic
362C2×C8C22×D2311yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—231no—
365C4⋊C4—461no—
366C4⋊C4—461no—
367C4⋊C4—461no—
368C4⋊C4C22×D2311yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—461no—
371C22⋊C4—461no—
372C22⋊C4—461no—
373C22⋊C4—461no—
374C42—461no—
375C23—21no—
376C23—231no—
377C23—231no—
378C23—464no—
379C23—464no—
380C23—464no—
381C23—461no—
382C23—461no—
383C23—1841no—
384C23—1841no—
385C23—1841no—
386C23—1841no—
387C23—1841no—
388C23—1841no—
389C23D4×D2311yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—21no—
393Q8—462no—
394Q8—461no—
395Q8—461no—
396Q8—922no—
397Q8D4×D2312yes—
398D4—24no—
399D4—21no—
400D4—232no—
401D4—232no—
402D4—462no—
403D4—464no—
404D4—464no—
405D4—464no—
406D4—462no—
407D4—464no—
408D4—462no—
409D4—464no—
410D4—461no—
411D4—461no—
412D4D4×D2312yes—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—231no—
420C2×C4—232no—
421C2×C4—232no—
422C2×C4—231no—
423C2×C4—464no—
424C2×C4—464no—
425C2×C4—461no—
426C2×C4—461no—
427C2×C4—461no—
428C2×C4—461no—
429C2×C4—922no—
430C2×C4D4×D2311yescharacteristic
431C2×C4C23×D2311yescharacteristic
432C8—21no—
433C8—21no—
434C8—461no—
435C8—461no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22—232no—
443C22—461no—
444C22—461no—
445C22—462no—
446C22—462no—
447C22—461no—
448C22—461no—
449C22—461no—
450C22—921no—
451C22—921no—
452C22—921no—
453C22—921no—
454C22—1844no—
455C22—1844no—
456C22—1844no—
457C22C2×D4×D2311yescharacteristic
458C4—21no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—232no—
463C4—461no—
464C4—921no—
465C4—921no—
466C4C2×D4×D2311yescharacteristic
467C4C2×D4×D2311yescharacteristic
468C2—41no—
469C2—42no—
470C2—462no—
471C2—921no—
472C2—922no—
473C2—922no—
474C2#120×D2311yescharacteristic
475C2#1309322411yescharacteristic
476C2#1309322611yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#452C2
socleC22×C23C22×D4
prime radicalC2×#10625160C2
hypercenter#452D23
hypocenterC23#3009
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#10625089C2semidirect
3. C2×SD16×D23C2semidirect
4. #13092990C2semidirect
5. #13092582C2semidirect
6. #13092579C2semidirect
7. #13092578C2semidirect
8. #13092570C2semidirect
9. #13092569C2semidirect
10. #13092567C2semidirect
11. #13092566C2semidirect
12. #13092420C2semidirect
13. #13092412C2semidirect
14. #13092382C2semidirect
15. C23×#452C2semidirect
21. C2×Q8×D23C22semidirect
30. C2×Q16×C23C22semidirect
31. C2×SD16×C23C22semidirect
32. C23×C22⋊Q8C22semidirect
33. #10625117C22semidirect
34. C2×#67275C22semidirect
35. C2×#67274C22semidirect
36. C2×#67274C22semidirect
37. C2×#67273C22semidirect
38. #10625095C22semidirect
41. C2×#67263C22semidirect
42. C2×#67263C22semidirect
43. C2×#67262C22semidirect
44. C2×C8×D23C22semidirect
45. #10625068C22semidirect
47. #10625063C22semidirect
51. C23×#103C22semidirect
52. C23×#102C22semidirect
53. C23×C22⋊C8C22semidirect
54. #10625017C22semidirect
55. #10625014C22semidirect
56. C22⋊(C23⋊C8)C22semidirect
57. #10625003C22semidirect
58. #10625003C22semidirect
59. (C2×D23)⋊C8C22semidirect
60. (C2×D23)⋊C8C22semidirect
61. #10624991C22semidirect
62. #10624991C22semidirect
237. #452D23semidirect
270. C2×Q16C2×D23semidirect
277. C2×SD16C2×D23semidirect
287. C22⋊Q8C2×D23semidirect
290. #103C2×D23semidirect
296. #102C2×D23semidirect
300. C22⋊C8C2×D23semidirect
301. C23#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#106251602
C2×#106250892
C2×SD16×D232
#130929902
#130925822
#130925792
#130925782
#130925702
#130925692
#130925672
#130925662
#130924202
#130924122
#130923822
C23×#4522
#300923