Finite Group Databasebeta

← #28404365

Subgroups — #28404365

Order 14464.

1540 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813670C211yesmaximal, characteristic, complemented
2C2×#16813668C211yesmaximal, characteristic, complemented
3#20299260C211yesmaximal, characteristic, complemented
4#20299259C211yesmaximal, characteristic, complemented
5#20299255C211yesmaximal, characteristic, complemented
6#20299245C213yesmaximal, complemented
7#20299244C213yesmaximal, complemented
8#20299241C213yesmaximal, complemented
9#20299239C213yesmaximal, complemented
10#20299171C213yesmaximal, complemented
11#20299164C213yesmaximal, complemented
12#20299145C213yesmaximal, complemented
13#124×D113C213yesmaximal, complemented
14C4×#14268648C211yesmaximal, characteristic, complemented
15#548×C113C211yesmaximal, characteristic, complemented
16#16813670—28no—
17#16813668—28no—
18C2×#14268648C2211yescharacteristic
19C2×#14268648C2211yescharacteristic
20C2×#14268648C2213yescomplemented
21C2×#14268647C2213yes—
22C2×#14268647C2213yescomplemented
23C2×#14268647C2213yescomplemented
24C2×Q8×D113C2213yescomplemented
25C2×Q8×D113C2211yescharacteristic, complemented
26C22×D452C2213yes—
27C2×D4×D113C2213yes—
28C2×D4×D113C2213yes—
29#16813621C2213yescomplemented
30#16813618C2213yescomplemented
31#16813611C2213yescomplemented
32#16813610C2213yescomplemented
33#16813609C2213yescomplemented
34#16813608C2213yescomplemented
35#16813608C2216yescomplemented
36#16813608C2211yescharacteristic, complemented
37#16813607C2216yescomplemented
38#16813604C2216yescomplemented
39#16813602C2213yescomplemented
40#16813601C2213yescomplemented
41#16813598C2213yescomplemented
42#16813597C2213yescomplemented
43#16813597C2213yescomplemented
44#16813596C2213yescomplemented
45#16813595C2213yescomplemented
46#16813595C2213yescomplemented
47#16813593C2213yescomplemented
48#16813593C2213yescomplemented
49#16813592C2216yescomplemented
50#16813590C2216yescomplemented
51Q8×Dic113C2211yescharacteristic, complemented
52C4×D452C2213yescomplemented
53C4×D452C2213yescomplemented
54#16813584C2213yescomplemented
55C4×C22⋊D113C2213yescomplemented
56C4×C22⋊D113C2213yescomplemented
57#16813580C2213yescomplemented
58C22⋊C4×D113C2216yescomplemented
59C22⋊C4×D113C2216yescomplemented
60C42×D113C2213yescomplemented
61C2×#41×C113C2211yescharacteristic
62C22×Q8×C113C2211yescharacteristic, complemented
63#124×C113C2213yescomplemented
64C22⋊Q8×C113C2213yescomplemented
65#121×C113C2213yescomplemented
66C4×Q8×C113C2211yescharacteristic, complemented
67C4×D4×C113C2213yescomplemented
68C23×D113C2316yes—
69#14268648—24no—
70#14268648—212no—
71#14268648D418yescomplemented
72#14268647—212no—
73#14268647—212no—
74#14268647—212no—
75C2×Dic226C2313yes—
76C2×Dic226C2313yes—
77C2×Dic226C2311yescharacteristic
78Q8×D113—24no—
79Q8×D113—212no—
80C2×D452—212no—
81C2×D452C2313yes—
82C2×D452C2313yes—
83C2×D452C2313yes—
84C2×D452C2313yes—
85C2×D452C2313yes—
86C2×C22⋊D113C2313yes—
87C2×C22⋊D113C2313yes—
88C2×C22⋊D113C2313yes—
89C2×C22⋊D113C2311yescharacteristic
90C2×C22⋊D113C2313yes—
91C2×C22⋊D113C2313yes—
92D4×D113—212no—
93D4×D113—212no—
94C2×C4×D113C2316yes—
95C2×C4×D113C2313yes—
96C2×C4×D113C2313yes—
97C2×C4×D113C2313yes—
98C2×C4×D113C2313yes—
99C2×C4×D113C2313yes—
100C4⋊Dic113C2313yes—
101C4⋊Dic113C2313yes—
102#14268626C2313yes—
103#14268626C2313yes—
104C22⋊Dic113C2313yes—
105C22⋊Dic113C2316yes—
106C22⋊Dic113C2311yescharacteristic
107#14268624C2313yes—
108#14268624C2313yes—
109#14268624C2316yes—
110#14268624C2316yes—
111#14268624C2316yes—
112#14268624C2311yescharacteristic
113#14268624C2311yescharacteristic
114C4×Dic113C2311yescharacteristic
115C4×Dic113C2313yes—
116C4×Dic113C2313yes—
117#41×C113—24no—
118C2×Q8×C113—24no—
119C2×Q8×C113C2313yes—
120C2×Q8×C113C2311yescharacteristic
121C2×D4×C113C2313yes—
122C2×D4×C113C2313yes—
123C22×C4×C113C2313yes—
124C22×C4×C113C2313yes—
125C4⋊C4×C113C2313yes—
126C4⋊C4×C113C2313yes—
127C22⋊C4×C113C2313yes—
128C22⋊C4×C113C2316yes—
129C22⋊C4×C113C2311yescharacteristic
130C42×C113C2313yes—
131C23×C113C2413yes—
132C23×C113C2411yescharacteristic
133C22×D113—212no—
134C22×D113—46no—
135C22×D113—46no—
136C22×D113C2413yes—
137C22×D113C2411yescharacteristic
138C22×D113C2413yes—
139Q8×C113—26no—
140Q8×C113C2×D414yes—
141D4×C113—26no—
142D4×C113—43no—
143C2×C4×C113—26no—
144C2×C4×C113—26no—
145C2×C4×C113C2411yescharacteristic
146C2×C4×C113C2413yes—
147C2×C4×C113C2413yes—
148C2×C4×C113C2411yescharacteristic
149C2×C4×C113C2413yes—
150C22⋊D113—26no—
151C22⋊D113—26no—
152C22⋊D113—26no—
153C22⋊D113—22no—
154C22⋊D113—43no—
155C22⋊D113—43no—
156C2×Dic113C2411yescharacteristic
157C2×Dic113C2413yes—
158C2×Dic113C2413yes—
159C2×Dic113C2411yescharacteristic
160C2×Dic113C2411yescharacteristic
161D452—26no—
162D452—26no—
163D452—26no—
164D452—43no—
165D452C2×D4112yes—
166C4×D113—212no—
167C4×D113—26no—
168C4×D113—26no—
169C4×D113—43no—
170C4×D113—43no—
171C4×D113C2×D4112yes—
172Dic226—26no—
173Dic226—26no—
174Dic226—22no—
175C22×C113—21no—
176C22×C113—23no—
177C22×C113—21no—
178C22×C113—23no—
179C22×C113—21no—
180C22×C113—23no—
181C22×C113C2511yescharacteristic
182C2×D113—23no—
183C2×D113—23no—
184C2×D113—21no—
185C2×D113—23no—
186C2×D113—21no—
187C2×D113—23no—
188C2×D113—21no—
189C2×D113—23no—
190C2×D113—812no—
191C2×D113C22×D416yes—
192C4×C113—23no—
193C4×C113—21no—
194C4×C113—21no—
195C4×C113—23no—
196C4×C113C22×D416yes—
197Dic113—23no—
198Dic113—21no—
199Dic113—21no—
200Dic113—23no—
201Dic113C22×D412yes—
202C2×C113—43no—
203C2×C113—41no—
204C2×C113C23×D411yescharacteristic
205C2×C113#58511yescharacteristic
206C2×C113#58511yescharacteristic
207D113—26no—
208D113—41no—
209D113—43no—
210#3500—1131nomaximal, Sylow
211C113#350011yescharacteristic, complemented, Sylow
212C2×2−1+4—1131no—
213C2×H(5,𝔽2)—1131no—
214#548—1131no—
215#548—1131no—
216#548—1131no—
217#548D11311yescharacteristic, complemented
218#546—1133no—
219#546—1133no—
220#546—1133no—
221#546—1133no—
222#536—1133no—
223#535—1133no—
224#532—1133no—
225C2×#124—1133no—
226C4×#41—1131no—
2272−1+4—2268no—
228H(5,𝔽2)—2268no—
229C2×#41—1131no—
230C2×#41—1133no—
231C2×#41—1131no—
232C2×#41—1133no—
233C2×#41—1133no—
234C2×#41—1133no—
235C2×#41C2×D11311yescharacteristic
236C22×Q8—1133no—
237C22×Q8—1131no—
238C22×Q8C2×D11311yescharacteristic, complemented
239C22×D4—1133no—
240C22×D4—1133no—
241C22×D4—1133no—
242C4⋊Q8—1133no—
243#127—1133no—
244#124—1133no—
245#124—1133no—
246#124—1133no—
247#124—1136no—
248#124—1131no—
249#124—1133no—
250#124C2×D11313yescomplemented
251#123—1136no—
252#123—1136no—
253C22⋊Q8—1133no—
254C22⋊Q8—1133no—
255C22⋊Q8—1133no—
256C22⋊Q8C2×D11313yescomplemented
257#121—1133no—
258#121—1133no—
259#121—1133no—
260#121—1133no—
261#121—1133no—
262#121—1133no—
263#121—1133no—
264#121C2×D11313yescomplemented
265#120—1136no—
266#120—1136no—
267C4×Q8—1131no—
268C4×Q8C2×D11311yescharacteristic, complemented
269C4×D4—1133no—
270C4×D4—1133no—
271C4×D4—1133no—
272C4×D4—1133no—
273C4×D4—1133no—
274C4×D4C2×D11313yescomplemented
275#117—1133no—
276C2×C22⋊C4—1136no—
277C2×C22⋊C4—1136no—
278C2×C42—1133no—
279C24—1136no—
280#41—24no—
281#41—1138no—
282#41—22612no—
283#41—22612no—
284#41—22612no—
285#41—22612no—
286#41—2264no—
287C2×Q8—24no—
288C2×Q8—1133no—
289C2×Q8—1133no—
290C2×Q8—1131no—
291C2×Q8—22612no—
292C2×Q8—2264no—
293C2×Q8C22×D11313yes—
294C2×Q8C22×D11311yescharacteristic
295C2×D4—1133no—
296C2×D4—1133no—
297C2×D4—1133no—
298C2×D4—1133no—
299C2×D4—1133no—
300C2×D4—1131no—
301C2×D4—1133no—
302C2×D4—1133no—
303C2×D4—1133no—
304C2×D4—1133no—
305C2×D4—1133no—
306C2×D4—22612no—
307C2×D4—22612no—
308C2×D4—22612no—
309C2×D4C22×D11313yes—
310C2×D4C22×D11313yes—
311C22×C4—1133no—
312C22×C4—1133no—
313C22×C4—1133no—
314C22×C4—1136no—
315C22×C4—1133no—
316C22×C4—1133no—
317C22×C4C22×D11313yes—
318C22×C4C22×D11313yes—
319C4⋊C4—1133no—
320C4⋊C4—1133no—
321C4⋊C4—1133no—
322C4⋊C4—1133no—
323C4⋊C4C22×D11313yes—
324C4⋊C4C22×D11313yes—
325C22⋊C4—1133no—
326C22⋊C4—1136no—
327C22⋊C4—1131no—
328C22⋊C4—1133no—
329C22⋊C4—1133no—
330C22⋊C4—1136no—
331C22⋊C4—1136no—
332C22⋊C4—1136no—
333C22⋊C4—1131no—
334C22⋊C4—1131no—
335C22⋊C4C22×D11313yes—
336C22⋊C4C22×D11316yes—
337C22⋊C4C22×D11311yescharacteristic
338C42—1131no—
339C42—1133no—
340C42—1133no—
341C42C22×D11313yes—
342C23—1133no—
343C23—1133no—
344C23—1131no—
345C23—4526no—
346C23—4526no—
347C23—22612no—
348C23C23×D11313yes—
349C23C23×D11311yescharacteristic
350Q8—26no—
351Q8—2266no—
352Q8—2262no—
353Q8—2266no—
354Q8D4×D11314yes—
355D4—26no—
356D4—43no—
357D4—11312no—
358D4—4523no—
359D4—4523no—
360D4—4523no—
361D4—2266no—
362D4—2266no—
363D4—2266no—
364D4—2262no—
365D4—2266no—
366D4—2266no—
367D4—2266no—
368C2×C4—26no—
369C2×C4—26no—
370C2×C4—1131no—
371C2×C4—11312no—
372C2×C4—1131no—
373C2×C4—1133no—
374C2×C4—1133no—
375C2×C4—1131no—
376C2×C4—4523no—
377C2×C4—4523no—
378C2×C4—22612no—
379C2×C4—2266no—
380C2×C4—2266no—
381C2×C4C23×D11311yescharacteristic
382C2×C4C23×D11313yes—
383C2×C4C23×D11313yes—
384C2×C4C23×D11311yescharacteristic
385C2×C4C23×D11313yes—
386C22—21no—
387C22—23no—
388C22—21no—
389C22—23no—
390C22—21no—
391C22—23no—
392C22—1136no—
393C22—90412no—
394C22—2261no—
395C22—2263no—
396C22—2261no—
397C22—2263no—
398C22—2261no—
399C22—2263no—
400C22—2263no—
401C22—2263no—
402C22C24×D11311yescharacteristic
403C4—21no—
404C4—23no—
405C4—23no—
406C4—21no—
407C4—1132no—
408C4—2261no—
409C4—2263no—
410C4—2261no—
411C4—2263no—
412C4C2×D4×D11316yes—
413C2—43no—
414C2—41no—
415C2—4523no—
416C2—4521no—
417C2—2266no—
418C2C22×D4×D11311yescharacteristic
419C2#2029943411yescharacteristic
420C2#2029943511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D113
commutator, socle, cosocleC22×C113C25
Fitting#548×C113C2
hypercenter#548D113
hypocenterC113#3500
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#16813670C2semidirect
2. C2×#16813668C2semidirect
3. #20299260C2semidirect
4. #20299259C2semidirect
5. #20299255C2semidirect
6. #20299245C2semidirect
7. #20299244C2semidirect
8. #20299241C2semidirect
9. #20299239C2semidirect
10. #20299171C2semidirect
11. #20299164C2semidirect
12. #20299145C2semidirect
13. #124×D113C2semidirect
14. C4×#14268648C2semidirect
15. #548×C113C2semidirect
20. C2×#14268648C22semidirect
22. C2×#14268647C22semidirect
23. C2×#14268647C22semidirect
24. C2×Q8×D113C22semidirect
25. C2×Q8×D113C22semidirect
29. #16813621C22semidirect
30. #16813618C22semidirect
31. #16813611C22semidirect
32. #16813610C22semidirect
33. #16813609C22semidirect
34. #16813608C22semidirect
35. #16813608C22semidirect
36. #16813608C22semidirect
37. #16813607C22semidirect
38. #16813604C22semidirect
39. #16813602C22semidirect
40. #16813601C22semidirect
41. #16813598C22semidirect
42. #16813597C22semidirect
43. #16813597C22semidirect
44. #16813596C22semidirect
45. #16813595C22semidirect
46. #16813595C22semidirect
47. #16813593C22semidirect
48. #16813593C22semidirect
49. #16813592C22semidirect
50. #16813590C22semidirect
51. Q8×Dic113C22semidirect
52. C4×D452C22semidirect
53. C4×D452C22semidirect
54. #16813584C22semidirect
55. C4×C22⋊D113C22semidirect
56. C4×C22⋊D113C22semidirect
57. #16813580C22semidirect
58. C22⋊C4×D113C22semidirect
59. C22⋊C4×D113C22semidirect
60. C42×D113C22semidirect
62. C22×Q8×C113C22semidirect
63. #124×C113C22semidirect
64. C22⋊Q8×C113C22semidirect
65. #121×C113C22semidirect
66. C4×Q8×C113C22semidirect
67. C4×D4×C113C22semidirect
71. #14268648D4semidirect
211. C113#3500semidirect
217. #548D113semidirect
238. C22×Q8C2×D113semidirect
250. #124C2×D113semidirect
256. C22⋊Q8C2×D113semidirect
264. #121C2×D113semidirect
268. C4×Q8C2×D113semidirect
274. C4×D4C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136702
C2×#168136682
#202992602
#202992592
#202992552
#202992452
#202992442
#202992412
#202992392
#202991712
#202991642
#202991452
#124×D1132
C4×#142686482
#548×C1132
#3500113