Finite Group Databasebeta

← #28404251

Subgroups — #28404251

Order 14464.

1522 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813670C211yesmaximal, characteristic, complemented
2C22×#14268648C211yesmaximal, characteristic, complemented
3Q8×C22⋊D113C211yesmaximal, characteristic, complemented
4#20299260C211yesmaximal, characteristic, complemented
5#20299259C211yesmaximal, characteristic, complemented
6#20299251C213yesmaximal, complemented
7#20299250C213yesmaximal, complemented
8#20299244C213yesmaximal, complemented
9#20299239C213yesmaximal, complemented
10#20299178C213yesmaximal, complemented
11#20299173C213yesmaximal, complemented
12#20299107C213yesmaximal, complemented
13C22⋊Q8×D113C213yesmaximal, complemented
14#20299080C211yesmaximal, characteristic, complemented
15D4×Q8×C113C211yesmaximal, characteristic, complemented
16#16813670—28no—
17C2×#14268648—28no—
18C2×#14268648C2213yes—
19C2×#14268648C2213yes—
20C2×#14268648C2211yescharacteristic
21C2×#14268648C2211yescharacteristic
22C2×#14268648C2213yescomplemented
23C2×#14268647C2213yescomplemented
24C2×#14268647C2213yescomplemented
25C2×Q8×D113C2213yescomplemented
26C2×Q8×D113C2211yescharacteristic, complemented
27C22×D452C2213yes—
28C22×C4×D113C2213yes—
29#16813620C2213yescomplemented
30#16813611C2213yescomplemented
31#16813610C2213yescomplemented
32#16813607C2213yescomplemented
33#16813607C2216yescomplemented
34#16813607C2216yescomplemented
35#16813604C2213yescomplemented
36#16813602C2213yescomplemented
37#16813601C2213yes—
38#16813600C2216yescomplemented
39#16813600C2213yescomplemented
40#16813599C2213yes—
41#16813598C2213yescomplemented
42#16813598C2213yescomplemented
43#16813597C2216yescomplemented
44#16813597C2213yescomplemented
45#16813596C2213yescomplemented
46#16813595C2213yescomplemented
47#16813595C2216yescomplemented
48#16813594C2211yescharacteristic
49#16813592C2213yes—
50#16813590C2213yes—
51Q8×Dic113C2211yescharacteristic, complemented
52C4×D452C2213yescomplemented
53C4×D452C2213yescomplemented
54#16813584C2213yescomplemented
55C4×C22⋊D113C2213yes—
56C4×C22⋊D113C2213yescomplemented
57#16813579C2213yescomplemented
58C4⋊C4×D113C2213yescomplemented
59C4⋊C4×D113C2216yescomplemented
60C22⋊C4×D113C2216yes—
61C22×Q8×C113C2211yescharacteristic
62C22×Q8×C113C2211yescharacteristic, complemented
63C4⋊Q8×C113C2213yescomplemented
64C22⋊Q8×C113C2213yes—
65C22⋊Q8×C113C2213yescomplemented
66C4×Q8×C113C2211yescharacteristic, complemented
67C4×D4×C113C2213yes—
68C23×D113C2313yes—
69#14268648—24no—
70#14268648—212no—
71#14268648—212no—
72#14268648—212no—
73#14268648D418yescomplemented
74#14268647—212no—
75#14268647—212no—
76C2×Dic226C2313yes—
77C2×Dic226C2313yes—
78Q8×D113—24no—
79Q8×D113—212no—
80C2×D452—212no—
81C2×D452C2313yes—
82C2×D452C2316yes—
83C2×D452C2313yes—
84C2×D452C2313yes—
85C2×D452C2313yes—
86C2×C22⋊D113C2311yescharacteristic
87C2×C22⋊D113C2313yes—
88C2×C22⋊D113C2311yescharacteristic
89C2×C22⋊D113C2313yes—
90C22×Dic113C2311yescharacteristic
91C2×C4×D113—212no—
92C2×C4×D113C2316yes—
93C2×C4×D113C2313yes—
94C2×C4×D113C2313yes—
95C2×C4×D113C2313yes—
96C2×C4×D113C2313yes—
97C2×C4×D113C2316yes—
98C2×C4×D113C2313yes—
99C4⋊Dic113C2313yes—
100C4⋊Dic113C2313yes—
101#14268626C2313yes—
102#14268626C2313yes—
103#14268626C2316yes—
104#14268626C2311yescharacteristic
105#14268626C2313yes—
106C22⋊Dic113C2313yes—
107C22⋊Dic113C2311yescharacteristic
108#14268624C2316yes—
109#14268624C2316yes—
110#14268624C2313yes—
111#14268624C2313yes—
112#14268624C2313yes—
113#14268624C2313yes—
114#14268624C2311yescharacteristic
115#14268624C2313yes—
116C4×Dic113C2313yes—
117C2×Q8×C113—24no—
118C2×Q8×C113—24no—
119C2×Q8×C113C2313yes—
120C2×Q8×C113C2313yes—
121C2×Q8×C113C2311yescharacteristic
122C2×D4×C113C2311yescharacteristic
123C22×C4×C113C2313yes—
124C22×C4×C113C2313yes—
125C4⋊C4×C113C2313yes—
126C4⋊C4×C113C2316yes—
127C4⋊C4×C113C2313yes—
128C22⋊C4×C113C2313yes—
129C22⋊C4×C113C2313yes—
130C42×C113C2313yes—
131C23×C113C2411yescharacteristic
132C23×C113C2411yescharacteristic
133C22×D113—26no—
134C22×D113—26no—
135C22×D113—43no—
136C22×D113C2413yes—
137C22×D113C2411yescharacteristic
138C22×D113C2413yes—
139Q8×C113—26no—
140Q8×C113—26no—
141Q8×C113C2×D414yes—
142D4×C113—22no—
143C2×C4×C113—26no—
144C2×C4×C113—26no—
145C2×C4×C113C2411yescharacteristic
146C2×C4×C113C2413yes—
147C2×C4×C113C2413yes—
148C2×C4×C113C2413yes—
149C2×C4×C113C2413yes—
150C22⋊D113—22no—
151C22⋊D113—26no—
152C22⋊D113—22no—
153C22⋊D113—43no—
154C2×Dic113—22no—
155C2×Dic113C2411yescharacteristic
156C2×Dic113C2413yes—
157C2×Dic113C2413yes—
158C2×Dic113C2411yescharacteristic
159C2×Dic113C2411yescharacteristic
160D452—212no—
161D452—26no—
162D452—26no—
163D452—43no—
164D452C2×D4112yes—
165C4×D113—26no—
166C4×D113—212no—
167C4×D113—26no—
168C4×D113—26no—
169C4×D113—26no—
170C4×D113—46no—
171C4×D113C2×D4112yes—
172Dic226—26no—
173Dic226—26no—
174C22×C113—21no—
175C22×C113—21no—
176C22×C113—21no—
177C22×C113—21no—
178C22×C113—21no—
179C22×C1132−1+412yescomplemented
180C22×C113C2511yescharacteristic
181C2×D113—23no—
182C2×D113—23no—
183C2×D113—21no—
184C2×D113—23no—
185C2×D113—21no—
186C2×D113—23no—
187C2×D113—21no—
188C2×D113—23no—
189C2×D113—412no—
190C2×D113C22×D416yes—
191C4×C113—23no—
192C4×C113—23no—
193C4×C113—21no—
194C4×C113—23no—
195C4×C113C22×D416yes—
196Dic113—23no—
197Dic113—21no—
198Dic113—21no—
199Dic113—23no—
200Dic113C22×D412yes—
201C2×C113—22no—
202C2×C113—41no—
203C2×C113C23×D411yescharacteristic
204C2×C113C2×2−1+411yescharacteristic
205C2×C113#58511yescharacteristic
206D113—26no—
207D113—41no—
208D113—43no—
209#3489—1131nomaximal, Sylow
210C113#348911yescharacteristic, complemented, Sylow
211C2×2−1+4—1131no—
212C22×#41—1131no—
213D4×Q8—1131no—
214D4×Q8D11311yescharacteristic, complemented
215#548—1131no—
216#548—1131no—
217#547—1133no—
218#547—1133no—
219#546—1133no—
220#546—1133no—
221#537—1133no—
222#536—1133no—
223#525—1133no—
224C2×C22⋊Q8—1133no—
225#520—1131no—
2262−1+4—2268no—
227C2×#41—1131no—
228C2×#41—1133no—
229C2×#41—1133no—
230C2×#41—1133no—
231C2×#41—1131no—
232C2×#41—1133no—
233C2×#41—1133no—
234C2×#41—2268no—
235C22×Q8—1133no—
236C22×Q8—1131no—
237C22×Q8C2×D11311yescharacteristic
238C22×Q8C2×D11311yescharacteristic, complemented
239C22×D4—1133no—
240C23×C4—1133no—
241C4⋊Q8—1133no—
242C4⋊Q8C2×D11313yescomplemented
243#124—1133no—
244#124—1133no—
245#123—1133no—
246#123—1136no—
247#123—1136no—
248#123—1133no—
249C22⋊Q8—1133no—
250C22⋊Q8—1133no—
251C22⋊Q8—1133no—
252C22⋊Q8—1136no—
253C22⋊Q8—1133no—
254C22⋊Q8—1133no—
255C22⋊Q8—1133no—
256C22⋊Q8C2×D11313yes—
257C22⋊Q8C2×D11313yescomplemented
258#121—1131no—
259#121—1133no—
260#121—1136no—
261#121—1133no—
262#121—1133no—
263#121—1136no—
264#120—1133no—
265#120—1133no—
266C4×Q8—1131no—
267C4×Q8C2×D11311yescharacteristic, complemented
268C4×D4—1133no—
269C4×D4—1133no—
270C4×D4—1133no—
271C4×D4—1133no—
272C4×D4—1133no—
273C4×D4C2×D11313yes—
274#117—1133no—
275C2×C4⋊C4—1133no—
276C2×C4⋊C4—1136no—
277C2×C22⋊C4—1136no—
278C24—1133no—
279#41—1138no—
280#41—22612no—
281#41—22612no—
282#41—22612no—
283#41—22612no—
284#41—22612no—
285#41—2264no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—1133no—
289C2×Q8—1133no—
290C2×Q8—22612no—
291C2×Q8—2264no—
292C2×Q8C22×D11313yes—
293C2×Q8C22×D11313yes—
294C2×Q8C22×D11311yescharacteristic
295C2×D4—1133no—
296C2×D4—1131no—
297C2×D4—1133no—
298C2×D4—1131no—
299C2×D4—1133no—
300C2×D4—1133no—
301C2×D4—1133no—
302C2×D4—1136no—
303C2×D4—1133no—
304C2×D4—22612no—
305C2×D4C22×D11311yescharacteristic
306C22×C4—1133no—
307C22×C4—1131no—
308C22×C4—1136no—
309C22×C4—1136no—
310C22×C4—1133no—
311C22×C4—1133no—
312C22×C4—1133no—
313C22×C4—1133no—
314C22×C4—22612no—
315C22×C4C22×D11313yes—
316C22×C4C22×D11313yes—
317C4⋊C4—1133no—
318C4⋊C4—1133no—
319C4⋊C4—1133no—
320C4⋊C4—1133no—
321C4⋊C4—1136no—
322C4⋊C4—1131no—
323C4⋊C4—1133no—
324C4⋊C4C22×D11313yes—
325C4⋊C4C22×D11316yes—
326C4⋊C4C22×D11313yes—
327C22⋊C4—1133no—
328C22⋊C4—1131no—
329C22⋊C4—1136no—
330C22⋊C4—1136no—
331C22⋊C4—1133no—
332C22⋊C4—1133no—
333C22⋊C4—1131no—
334C22⋊C4—1133no—
335C22⋊C4—1133no—
336C22⋊C4—1133no—
337C22⋊C4C22×D11313yes—
338C22⋊C4C22×D11313yes—
339C42—1133no—
340C42C22×D11313yes—
341C23—1133no—
342C23—1133no—
343C23—1131no—
344C23—4523no—
345C23—2266no—
346C23—2266no—
347C23C23×D11311yescharacteristic
348C23C23×D11311yescharacteristic
349Q8—26no—
350Q8—26no—
351Q8—2266no—
352Q8—2266no—
353Q8D4×D11314yes—
354D4—22no—
355D4—11312no—
356D4—4523no—
357D4—4523no—
358D4—22612no—
359D4—2266no—
360D4—2262no—
361D4—2266no—
362D4—2262no—
363D4—2266no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—1131no—
367C2×C4—11312no—
368C2×C4—1131no—
369C2×C4—1133no—
370C2×C4—1133no—
371C2×C4—1131no—
372C2×C4—4526no—
373C2×C4—2266no—
374C2×C4—2266no—
375C2×C4—22612no—
376C2×C4—2262no—
377C2×C4—2266no—
378C2×C4—2266no—
379C2×C4C23×D11311yescharacteristic
380C2×C4C23×D11313yes—
381C2×C4C23×D11313yes—
382C2×C4C23×D11313yes—
383C2×C4C23×D11313yes—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—1136no—
390C22—45212no—
391C22—2261no—
392C22—2263no—
393C22—2261no—
394C22—2263no—
395C22—2261no—
396C22—2263no—
397C22—2263no—
398C22—2263no—
399C22#1681367012yescomplemented
400C22C24×D11311yescharacteristic
401C4—23no—
402C4—23no—
403C4—21no—
404C4—23no—
405C4—1132no—
406C4—2261no—
407C4—2263no—
408C4—2261no—
409C4—2263no—
410C4C2×D4×D11316yes—
411C2—22no—
412C2—41no—
413C2—4523no—
414C2—4521no—
415C2—2266no—
416C2C22×D4×D11311yescharacteristic
417C2C2×#1681367011yescharacteristic
418C2#2029943511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D113
commutator, socle, cosocleC22×C113C25
FittingD4×Q8×C113C2
hypercenterD4×Q8D113
hypocenterC113#3489
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16813670C2semidirect
2. C22×#14268648C2semidirect
3. Q8×C22⋊D113C2semidirect
4. #20299260C2semidirect
5. #20299259C2semidirect
6. #20299251C2semidirect
7. #20299250C2semidirect
8. #20299244C2semidirect
9. #20299239C2semidirect
10. #20299178C2semidirect
11. #20299173C2semidirect
12. #20299107C2semidirect
13. C22⋊Q8×D113C2semidirect
14. #20299080C2semidirect
15. D4×Q8×C113C2semidirect
22. C2×#14268648C22semidirect
23. C2×#14268647C22semidirect
24. C2×#14268647C22semidirect
25. C2×Q8×D113C22semidirect
26. C2×Q8×D113C22semidirect
29. #16813620C22semidirect
30. #16813611C22semidirect
31. #16813610C22semidirect
32. #16813607C22semidirect
33. #16813607C22semidirect
34. #16813607C22semidirect
35. #16813604C22semidirect
36. #16813602C22semidirect
38. #16813600C22semidirect
39. #16813600C22semidirect
41. #16813598C22semidirect
42. #16813598C22semidirect
43. #16813597C22semidirect
44. #16813597C22semidirect
45. #16813596C22semidirect
46. #16813595C22semidirect
47. #16813595C22semidirect
51. Q8×Dic113C22semidirect
52. C4×D452C22semidirect
53. C4×D452C22semidirect
54. #16813584C22semidirect
56. C4×C22⋊D113C22semidirect
57. #16813579C22semidirect
58. C4⋊C4×D113C22semidirect
59. C4⋊C4×D113C22semidirect
62. C22×Q8×C113C22semidirect
63. C4⋊Q8×C113C22semidirect
65. C22⋊Q8×C113C22semidirect
66. C4×Q8×C113C22semidirect
73. #14268648D4semidirect
179. C22×C1132−1+4semidirect
210. C113#3489semidirect
214. D4×Q8D113semidirect
238. C22×Q8C2×D113semidirect
242. C4⋊Q8C2×D113semidirect
257. C22⋊Q8C2×D113semidirect
267. C4×Q8C2×D113semidirect
399. C22#16813670semidirect

Maximal subgroups

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

subgroupindex
C2×#168136702
C22×#142686482
Q8×C22⋊D1132
#202992602
#202992592
#202992512
#202992502
#202992442
#202992392
#202991782
#202991732
#202991072
C22⋊Q8×D1132
#202990802
D4×Q8×C1132
#3489113