Finite Group Databasebeta

← #11857547

Subgroups — #11857547

Order 1152.

1698 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1698 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64366C211yesmaximal, characteristic
2#10596642C212yesmaximal, complemented
3#10593832C212yesmaximal, complemented
4#10593817C214yesmaximal, complemented
5#10593792C212yesmaximal, complemented
6C2×#63774C214yesmaximal, complemented
7C2×C32×#41C411yescharacteristic, complemented
8#64366—28no—
9C2×#3832C411yescharacteristic, complemented
10C2×#3832C2211yescharacteristic
11C2×D4×C3⋊D3C412yescomplemented
12C2×D4×C3⋊D3C411yescharacteristic, complemented
13C2×D4×C3⋊D3C2212yes—
14C2×D4×C3⋊D3C2211yescharacteristic
15C2×#3828C412yescomplemented
16C2×#3828C2211yescharacteristic
17C22×C4⋊(C3⋊D3)C411yescharacteristic, complemented
18C22×C4⋊(C3⋊D3)C2212yes—
19C2×C22⋊#157C2218yescomplemented
20D4×#157—28no—
21#64286C2214yescomplemented
22#63786C2214yescomplemented
23#63774C2214yescomplemented
24#63774C2218yescomplemented
25C2×#1009—62no—
26C23×C3⋊D3C2×C412yescomplemented
27C23×C3⋊D3C2×C412yescomplemented
28C23×C3⋊D3C2312yes—
29C22×#157—28no—
30C32×#41—24no—
31C2×C32×Q8C2×C411yescharacteristic, complemented
32C2×C32×D4C2×C411yescharacteristic
33C2×C32×D4C2×C412yescomplemented
34C22×C32×C4C2×C411yescharacteristic, complemented
35C22×C32×C4C2×C412yescomplemented
36C2×#638C2×C412yes—
37C2×#638C2×C412yescomplemented
38C2×#638C2×C411yescharacteristic, complemented
39C2×#638C2311yescharacteristic
40#3832—24no—
41#3832—24no—
42D4×C3⋊D3—28no—
43D4×C3⋊D3—24no—
44D4×C3⋊D3—28no—
45D4×C3⋊D3D418yes—
46#3828—28no—
47#3828—24no—
48C2×C4⋊(C3⋊D3)—28no—
49C2×C4⋊(C3⋊D3)—24no—
50C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
51C2×C4⋊(C3⋊D3)C2×C412yescomplemented
52C2×C4⋊(C3⋊D3)C2×C411yescharacteristic, complemented
53C2×C4⋊(C3⋊D3)C2×C412yescomplemented
54C2×C4⋊(C3⋊D3)C2311yescharacteristic
55C2×C4⋊(C3⋊D3)C2312yes—
56C2×C4×C3⋊D3C2×C412yescomplemented
57C2×C4×C3⋊D3C2×C412yescomplemented
58C2×C4×C3⋊D3C2×C411yescharacteristic, complemented
59C2×C4×C3⋊D3C2311yescharacteristic
60C2×#634C2×C411yescharacteristic, complemented
61C22⋊#157—28no—
62C22⋊#157C2318yes—
63C4⋊#157—24no—
64C4×#157—24no—
65#2883—91nomaximal, Sylow
66C2×C3×#41—22no—
67#1009—632no—
68C2×#239—62no—
69C2×#239—62no—
70C2×D3×D4—62no—
71C2×D3×D4—64no—
72C2×D3×D4—62no—
73C2×D3×D4—64no—
74C2×#235—64no—
75C2×#235—62no—
76C22×D12—62no—
77C22×D12—64no—
78C23×C32C22×C411yescharacteristic
79C23×C32C22×C412yes—
80C22×C3⋊D3—24no—
81C22×C3⋊D3—24no—
82C22×C3⋊D3—82no—
83C22×C3⋊D3—82no—
84C22×C3⋊D3—82no—
85C22×C3⋊D3C22×C412yes—
86C22×C3⋊D3C22×C411yescharacteristic
87C22×C3⋊D3C22×C412yes—
88C22×C3⋊D3C2×D418yes—
89C22×C3⋊D3C2411yescharacteristic
90C2×#157—28no—
91C2×#157—88no—
92C32×Q8—22no—
93C32×D4—24no—
94C32×D4C22⋊C414yes—
95C2×C32×C4—24no—
96C2×C32×C4—22no—
97C2×C32×C4C22×C411yescharacteristic
98C2×C32×C4C22×C412yes—
99C2×C32×C4C22×C411yescharacteristic
100#638—24no—
101#638—22no—
102#638—22no—
103#638C22⋊C418yes—
104C2×C3⋊Dic3C22×C411yescharacteristic
105C2×C3⋊Dic3C22×C411yescharacteristic
106C4⋊(C3⋊D3)—24no—
107C4⋊(C3⋊D3)—24no—
108C4⋊(C3⋊D3)—22no—
109C4⋊(C3⋊D3)—24no—
110C4⋊(C3⋊D3)—22no—
111C4⋊(C3⋊D3)C22⋊C414yes—
112C4×C3⋊D3—24no—
113C4×C3⋊D3—22no—
114C4×C3⋊D3—24no—
115C4×C3⋊D3C2×D414yes—
116#634—22no—
117C2×H(5,𝔽2)—91no—
118#518—92no—
119#410—92no—
120#410—92no—
121C2×#99—94no—
122C2×#99—94no—
123C23×D3—64no—
124C23×D3—64no—
125C23×D3—64no—
126C3×#41—216no—
127C2×C3×Q8—22no—
128C2×C3×D4—22no—
129C2×C3×D4—24no—
130C22×C3×C4—22no—
131C22×C3×C4—24no—
132C2×#67—64no—
133C2×#67—64no—
134C2×#67—62no—
135C2×#67—62no—
136#239—616no—
137#239—616no—
138D3×D4—616no—
139D3×D4—632no—
140D3×D4—632no—
141D3×D4—616no—
142#235—616no—
143#235—632no—
144C2×D12—64no—
145C2×D12—62no—
146C2×D12—62no—
147C2×D12—62no—
148C2×D12—64no—
149C2×D12—64no—
150C2×D12—616no—
151C2×D12—632no—
152C2×C4×D3—64no—
153C2×C4×D3—64no—
154C2×C4×D3—62no—
155C2×C4×D3—62no—
156C2×Dic6—62no—
157C22×C32—21no—
158C22×C32—42no—
159C22×C32—41no—
160C22×C32C2×C22⋊C414yes—
161C22×C32C23×C411yescharacteristic
162C2×C3⋊D3—21no—
163C2×C3⋊D3—22no—
164C2×C3⋊D3—42no—
165C2×C3⋊D3—41no—
166C2×C3⋊D3—42no—
167C2×C3⋊D3—41no—
168C2×C3⋊D3—48no—
169C2×C3⋊D3—84no—
170C2×C3⋊D3—84no—
171C2×C3⋊D3C2×C22⋊C414yes—
172C2×C3⋊D3C22×D412yes—
173#157—48no—
174C32×C4—21no—
175C32×C4—22no—
176C32×C4C2×C22⋊C412yes—
177C3⋊Dic3—21no—
178C3⋊Dic3C2×C22⋊C412yes—
179H(5,𝔽2)—188no—
180C2×#41—91no—
181C2×#41—91no—
182C2×#41—91no—
183C2×#41—92no—
184C2×#41#15711yescharacteristic, complemented
185C22×D4—91no—
186C22×D4—91no—
187C22×D4—92no—
188C22×D4—92no—
189C22×D4—92no—
190C22×D4—91no—
191C4×D4—188no—
192#117—94no—
193C2×C22⋊C4—98no—
194#99—94no—
195#99—98no—
196#99—94no—
197C23×C3—22no—
198C23×C3—24no—
199C22×D3—64no—
200C22×D3—64no—
201C22×D3—62no—
202C22×D3—62no—
203C22×D3—616no—
204C22×D3—616no—
205C22×D3—616no—
206C22×D3—248no—
207C22×D3—248no—
208C22×D3—248no—
209C3×Q8—28no—
210C3×D4—216no—
211C3×D4—28no—
212C2×C3×C4—216no—
213C2×C3×C4—28no—
214C2×C3×C4—22no—
215C2×C3×C4—24no—
216C2×C3×C4—22no—
217#67—616no—
218#67—68no—
219#67—68no—
220#67—616no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223D12—68no—
224D12—616no—
225D12—68no—
226D12—616no—
227D12—616no—
228D12—68no—
229C4×D3—68no—
230C4×D3—616no—
231C4×D3—616no—
232C4×D3—68no—
233Dic6—68no—
234C2×C32—21no—
235C2×C32—24no—
236C2×C32—41no—
237C2×C32C22×C22⋊C411yescharacteristic
238C3⋊D3—22no—
239C3⋊D3—24no—
240C3⋊D3—42no—
241C3⋊D3—41no—
242C24—92no—
243C24—92no—
244C24—92no—
245#41—24no—
246#41—184no—
247#41—188no—
248#41—184no—
249#41—184no—
250C2×Q8—91no—
251C2×Q8C2×#15711yescharacteristic, complemented
252C2×D4—92no—
253C2×D4—98no—
254C2×D4—91no—
255C2×D4—91no—
256C2×D4—92no—
257C2×D4—91no—
258C2×D4—92no—
259C2×D4—91no—
260C2×D4—92no—
261C2×D4—92no—
262C2×D4—91no—
263C2×D4—184no—
264C2×D4—188no—
265C2×D4—188no—
266C2×D4—188no—
267C2×D4—184no—
268C2×D4C2×#15711yescharacteristic
269C2×D4C2×#15712yescomplemented
270C22×C4—91no—
271C22×C4—92no—
272C22×C4—92no—
273C22×C4—91no—
274C22×C4—188no—
275C22×C4C2×#15712yescomplemented
276C22×C4C2×#15711yescharacteristic, complemented
277C4⋊C4—184no—
278C22⋊C4—98no—
279C22⋊C4—188no—
280C42—184no—
281C22×C3—24no—
282C22×C3—28no—
283C22×C3—22no—
284C22×C3—48no—
285C22×C3—44no—
286C2×D3—68no—
287C2×D3—64no—
288C2×D3—68no—
289C2×D3—64no—
290C2×D3—128no—
291C2×D3—124no—
292C2×D3—128no—
293C2×D3—124no—
294C2×D3—2416no—
295C2×D3—2416no—
296C2×D3—2416no—
297C3×C4—24no—
298C3×C4—28no—
299C3×C4—24no—
300Dic3—64no—
301Dic3—64no—
302C32#288311yescharacteristic, complemented, Sylow
303C23—98no—
304C23—91no—
305C23—92no—
306C23—92no—
307C23—91no—
308C23—184no—
309C23—184no—
310C23—722no—
311C23—722no—
312C23—722no—
313C23C22×#15711yescharacteristic
314C23C22×#15712yes—
315Q8—22no—
316Q8—182no—
317D4—24no—
318D4—98no—
319D4—94no—
320D4—182no—
321D4—184no—
322D4—184no—
323D4—184no—
324D4—182no—
325D4—182no—
326D4—184no—
327D4—182no—
328D4C22⋊#15714yes—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—91no—
332C2×C4—94no—
333C2×C4—91no—
334C2×C4—184no—
335C2×C4—184no—
336C2×C4—182no—
337C2×C4—188no—
338C2×C4—728no—
339C2×C4C22×#15712yes—
340C2×C4C22×#15711yescharacteristic
341C2×C4C22×#15711yescharacteristic
342C2×C3—24no—
343C2×C3—22no—
344C2×C3—48no—
345C2×C3—44no—
346D3—124no—
347D3—128no—
348D3—124no—
349D3—128no—
350C22—21no—
351C22—42no—
352C22—41no—
353C22—92no—
354C22—94no—
355C22—182no—
356C22—181no—
357C22—368no—
358C22—361no—
359C22—362no—
360C22—361no—
361C22—362no—
362C22—724no—
363C22—724no—
364C22C2×C22⋊#15714yes—
365C22C23×#15711yescharacteristic
366C4—21no—
367C4—22no—
368C4—92no—
369C4—181no—
370C4—368no—
371C4C2×C22⋊#15712yes—
372C3—22no—
373C2—21no—
374C2—24no—
375C2—41no—
376C2—182no—
377C2—184no—
378C2—361no—
379C2—362no—
380C2C22×C22⋊#15711yescharacteristic
subgroupG/H
centerC2C22×C22⋊#157
commutatorC22×C32C23×C4
FrattiniC22C23×#157
FittingC2×C32×#41C4
socleC2×C32C22×C22⋊C4
prime radicalC2×#64366C2
cosocleC22×C3⋊D3C24
hypercenterC2×#41#157
hypocenterC32#2883
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #10596642C2semidirect
3. #10593832C2semidirect
4. #10593817C2semidirect
5. #10593792C2semidirect
6. C2×#63774C2semidirect
7. C2×C32×#41C4semidirect
9. C2×#3832C4semidirect
11. C2×D4×C3⋊D3C4semidirect
12. C2×D4×C3⋊D3C4semidirect
15. C2×#3828C4semidirect
17. C22×C4⋊(C3⋊D3)C4semidirect
19. C2×C22⋊#157C22semidirect
21. #64286C22semidirect
22. #63786C22semidirect
23. #63774C22semidirect
24. #63774C22semidirect
26. C23×C3⋊D3C2×C4semidirect
27. C23×C3⋊D3C2×C4semidirect
31. C2×C32×Q8C2×C4semidirect
33. C2×C32×D4C2×C4semidirect
34. C22×C32×C4C2×C4semidirect
35. C22×C32×C4C2×C4semidirect
37. C2×#638C2×C4semidirect
38. C2×#638C2×C4semidirect
51. C2×C4⋊(C3⋊D3)C2×C4semidirect
52. C2×C4⋊(C3⋊D3)C2×C4semidirect
53. C2×C4⋊(C3⋊D3)C2×C4semidirect
56. C2×C4×C3⋊D3C2×C4semidirect
57. C2×C4×C3⋊D3C2×C4semidirect
58. C2×C4×C3⋊D3C2×C4semidirect
60. C2×#634C2×C4semidirect
184. C2×#41#157semidirect
251. C2×Q8C2×#157semidirect
269. C2×D4C2×#157semidirect
275. C22×C4C2×#157semidirect
276. C22×C4C2×#157semidirect
302. C32#2883semidirect

Maximal subgroups

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

subgroupindex
C2×#643662
#105966422
#105938322
#105938172
#105937922
C2×#637742
#28839