Finite Group Databasebeta

← #11857546

Subgroups — #11857546

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×#64361C211yesmaximal, characteristic
2#10596642C212yesmaximal, complemented
3C2×#63786C214yesmaximal, complemented
4#10593818C212yesmaximal, complemented
5#10593817C214yesmaximal, complemented
6#10593792C212yesmaximal, complemented
7C22×C32×D4C411yescharacteristic, complemented
8C22×#638C412yescomplemented
9C22×#638C2212yes—
10#64361—28no—
11C2×#3830C412yescomplemented
12C2×#3830C411yescharacteristic, complemented
13C2×#3830C2211yescharacteristic
14C2×D4×C3⋊D3C411yescharacteristic, complemented
15C2×D4×C3⋊D3C2212yes—
16C2×D4×C3⋊D3C2211yescharacteristic
17C2×#3828C411yescharacteristic, complemented
18C2×#3828C2211yescharacteristic
19C2×C22⋊#157C2218yescomplemented
20D4×#157—28no—
21#64286C2214yescomplemented
22#63786C2214yescomplemented
23#63786C2218yescomplemented
24#63774C2214yescomplemented
25C2×#1004—62no—
26C24×C32C2×C412yescomplemented
27C23×C3⋊D3C2×C412yescomplemented
28C23×C3⋊D3C2312yes—
29C22×#157—28no—
30C2×C32×D4—24no—
31C2×C32×D4C2×C411yescharacteristic
32C2×C32×D4C2×C411yescharacteristic, complemented
33C2×C32×D4C2×C412yescomplemented
34C22×C32×C4C2×C411yescharacteristic, complemented
35C2×#638—28no—
36C2×#638—28no—
37C2×#638C2×C412yes—
38C2×#638C2×C412yescomplemented
39C2×#638C2×C411yescharacteristic, complemented
40C2×#638C2×C412yescomplemented
41C2×#638C2×C412yescomplemented
42C2×#638C2311yescharacteristic
43C2×#638C2312yes—
44C22×C3⋊Dic3C2×C412yescomplemented
45C22×C3⋊Dic3C2×C412yescomplemented
46#3830—24no—
47#3830—28no—
48#3830—24no—
49D4×C3⋊D3—28no—
50D4×C3⋊D3—24no—
51D4×C3⋊D3D418yes—
52#3828—24no—
53#3828—24no—
54C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
55C2×C4⋊(C3⋊D3)C2311yescharacteristic
56C2×C4×C3⋊D3C2×C412yescomplemented
57C2×C4×C3⋊D3C2×C411yescharacteristic, complemented
58C2×C4×C3⋊D3C2311yescharacteristic
59C2×#634C2×C411yescharacteristic, complemented
60C2×#634C2×C411yescharacteristic, complemented
61C22⋊#157—28no—
62C22⋊#157C2318yes—
63C4⋊#157—24no—
64C4×#157—24no—
65#2883—91nomaximal, Sylow
66C22×C3×D4—22no—
67C22×#67—64no—
68C22×#67—64no—
69#1004—632no—
70C2×#237—62no—
71C2×#237—64no—
72C2×#237—62no—
73C2×D3×D4—62no—
74C2×D3×D4—62no—
75C2×D3×D4—64no—
76C2×#235—62no—
77C2×#235—62no—
78C23×C32—24no—
79C23×C32—82no—
80C23×C32C22×C412yes—
81C23×C32C22×C411yescharacteristic
82C23×C32C22×C412yes—
83C22×C3⋊D3—24no—
84C22×C3⋊D3—82no—
85C22×C3⋊D3—82no—
86C22×C3⋊D3C22×C411yescharacteristic
87C22×C3⋊D3C22×C412yes—
88C22×C3⋊D3C2×D418yes—
89C22×C3⋊D3C2411yescharacteristic
90C2×#157—28no—
91C2×#157—88no—
92C32×D4—24no—
93C32×D4—22no—
94C32×D4C22⋊C414yes—
95C2×C32×C4—22no—
96C2×C32×C4C22×C411yescharacteristic
97C2×C32×C4C22×C411yescharacteristic
98#638—24no—
99#638—24no—
100#638—24no—
101#638—22no—
102#638—24no—
103#638—22no—
104#638C22⋊C418yes—
105C2×C3⋊Dic3—24no—
106C2×C3⋊Dic3—24no—
107C2×C3⋊Dic3C22×C412yes—
108C2×C3⋊Dic3C22×C411yescharacteristic
109C2×C3⋊Dic3C22×C411yescharacteristic
110C4⋊(C3⋊D3)—22no—
111C4⋊(C3⋊D3)C22⋊C414yes—
112C4×C3⋊D3—24no—
113C4×C3⋊D3—22no—
114C4×C3⋊D3C2×D414yes—
115#634—22no—
116#634—22no—
117C2×H(5,𝔽2)—91no—
118#518—92no—
119#410—92no—
120#410—92no—
121C2×#99—94no—
122C2×#99—94no—
123C24×C3—24no—
124C23×D3—64no—
125C23×D3—64no—
126C2×C3×D4—216no—
127C2×C3×D4—22no—
128C2×C3×D4—22no—
129C2×C3×D4—24no—
130C22×C3×C4—22no—
131C2×#67—64no—
132C2×#67—62no—
133C2×#67—62no—
134C2×#67—64no—
135C2×#67—64no—
136C2×#67—64no—
137C2×#67—64no—
138C2×#67—632no—
139C2×#67—632no—
140C22×Dic3—64no—
141C22×Dic3—64no—
142#237—616no—
143#237—632no—
144#237—616no—
145D3×D4—616no—
146D3×D4—616no—
147D3×D4—632no—
148#235—616no—
149#235—616no—
150C2×D12—62no—
151C2×D12—62no—
152C2×C4×D3—64no—
153C2×C4×D3—62no—
154C2×C4×D3—62no—
155C2×Dic6—62no—
156C2×Dic6—62no—
157C22×C32—22no—
158C22×C32—21no—
159C22×C32—42no—
160C22×C32—42no—
161C22×C32—41no—
162C22×C32—84no—
163C22×C32C2×C22⋊C414yes—
164C22×C32C23×C411yescharacteristic
165C2×C3⋊D3—21no—
166C2×C3⋊D3—42no—
167C2×C3⋊D3—41no—
168C2×C3⋊D3—41no—
169C2×C3⋊D3—48no—
170C2×C3⋊D3—84no—
171C2×C3⋊D3C2×C22⋊C414yes—
172C2×C3⋊D3C22×D412yes—
173#157—48no—
174C32×C4—21no—
175C32×C4C2×C22⋊C412yes—
176C3⋊Dic3—22no—
177C3⋊Dic3—21no—
178C3⋊Dic3C2×C22⋊C412yes—
179H(5,𝔽2)—188no—
180C2×#41—91no—
181C2×#41—91no—
182C2×#41—92no—
183C2×#41—91no—
184C2×#41—91no—
185C22×D4—91no—
186C22×D4—92no—
187C22×D4—92no—
188C22×D4—92no—
189C22×D4—91no—
190C22×D4#15711yescharacteristic, complemented
191C4×D4—188no—
192#117—94no—
193C2×C22⋊C4—98no—
194#99—94no—
195#99—94no—
196#99—98no—
197C23×C3—216no—
198C23×C3—24no—
199C23×C3—24no—
200C23×C3—22no—
201C23×C3—88no—
202C22×D3—64no—
203C22×D3—62no—
204C22×D3—62no—
205C22×D3—616no—
206C22×D3—616no—
207C22×D3—248no—
208C22×D3—248no—
209C3×D4—216no—
210C3×D4—28no—
211C3×D4—28no—
212C2×C3×C4—28no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215#67—616no—
216#67—68no—
217#67—616no—
218#67—68no—
219#67—616no—
220#67—616no—
221#67—616no—
222C2×Dic3—62no—
223C2×Dic3—64no—
224C2×Dic3—62no—
225C2×Dic3—616no—
226C2×Dic3—616no—
227D12—68no—
228D12—68no—
229C4×D3—68no—
230C4×D3—616no—
231C4×D3—68no—
232Dic6—68no—
233Dic6—68no—
234C2×C32—21no—
235C2×C32—24no—
236C2×C32—42no—
237C2×C32—41no—
238C2×C32C22×C22⋊C411yescharacteristic
239C3⋊D3—22no—
240C3⋊D3—24no—
241C3⋊D3—41no—
242C24—92no—
243C24—92no—
244C24C2×#15712yescomplemented
245#41—184no—
246#41—184no—
247#41—184no—
248#41—188no—
249#41—184no—
250C2×Q8—91no—
251C2×Q8—91no—
252C2×D4—24no—
253C2×D4—92no—
254C2×D4—98no—
255C2×D4—92no—
256C2×D4—91no—
257C2×D4—91no—
258C2×D4—92no—
259C2×D4—92no—
260C2×D4—92no—
261C2×D4—91no—
262C2×D4—91no—
263C2×D4—184no—
264C2×D4—188no—
265C2×D4—188no—
266C2×D4—188no—
267C2×D4C2×#15711yescharacteristic
268C2×D4C2×#15712yescomplemented
269C2×D4C2×#15711yescharacteristic, complemented
270C22×C4—91no—
271C22×C4—92no—
272C22×C4—92no—
273C22×C4—92no—
274C22×C4—91no—
275C22×C4—188no—
276C22×C4C2×#15711yescharacteristic, complemented
277C4⋊C4—184no—
278C22⋊C4—98no—
279C22⋊C4—188no—
280C42—184no—
281C22×C3—28no—
282C22×C3—24no—
283C22×C3—28no—
284C22×C3—22no—
285C22×C3—48no—
286C22×C3—48no—
287C22×C3—44no—
288C22×C3—816no—
289C2×D3—68no—
290C2×D3—64no—
291C2×D3—64no—
292C2×D3—128no—
293C2×D3—124no—
294C2×D3—124no—
295C2×D3—2416no—
296C2×D3—2416no—
297C3×C4—24no—
298C3×C4—24no—
299Dic3—64no—
300Dic3—68no—
301Dic3—64no—
302C32#288311yescharacteristic, complemented, Sylow
303C23—24no—
304C23—82no—
305C23—98no—
306C23—91no—
307C23—92no—
308C23—91no—
309C23—184no—
310C23—722no—
311C23—722no—
312C23C22×#15712yes—
313C23C22×#15711yescharacteristic
314C23C22×#15712yes—
315Q8—182no—
316Q8—182no—
317D4—24no—
318D4—22no—
319D4—98no—
320D4—94no—
321D4—182no—
322D4—182no—
323D4—182no—
324D4—184no—
325D4—184no—
326D4—184no—
327D4—184no—
328D4C22⋊#15714yes—
329C2×C4—22no—
330C2×C4—91no—
331C2×C4—94no—
332C2×C4—92no—
333C2×C4—91no—
334C2×C4—184no—
335C2×C4—184no—
336C2×C4—184no—
337C2×C4—182no—
338C2×C4—188no—
339C2×C4—728no—
340C2×C4C22×#15711yescharacteristic
341C2×C4C22×#15711yescharacteristic
342C2×C3—24no—
343C2×C3—22no—
344C2×C3—48no—
345C2×C3—48no—
346C2×C3—44no—
347D3—124no—
348D3—128no—
349D3—124no—
350C22—21no—
351C22—22no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—84no—
356C22—92no—
357C22—94no—
358C22—181no—
359C22—368no—
360C22—361no—
361C22—362no—
362C22—361no—
363C22—724no—
364C22C2×C22⋊#15714yes—
365C22C23×#15711yescharacteristic
366C4—21no—
367C4—92no—
368C4—182no—
369C4—181no—
370C4—368no—
371C4C2×C22⋊#15712yes—
372C3—22no—
373C2—21no—
374C2—24no—
375C2—41no—
376C2—42no—
377C2—182no—
378C2—184no—
379C2—361no—
380C2C22×C22⋊#15711yescharacteristic
subgroupG/H
centerC2C22×C22⋊#157
commutatorC22×C32C23×C4
FrattiniC22C23×#157
FittingC22×C32×D4C4
socleC2×C32C22×C22⋊C4
prime radicalC2×#64361C2
cosocleC22×C3⋊D3C24
hypercenterC22×D4#157
hypocenterC32#2883
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×D4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #10596642C2semidirect
3. C2×#63786C2semidirect
4. #10593818C2semidirect
5. #10593817C2semidirect
6. #10593792C2semidirect
7. C22×C32×D4C4semidirect
8. C22×#638C4semidirect
11. C2×#3830C4semidirect
12. C2×#3830C4semidirect
14. C2×D4×C3⋊D3C4semidirect
17. C2×#3828C4semidirect
19. C2×C22⋊#157C22semidirect
21. #64286C22semidirect
22. #63786C22semidirect
23. #63786C22semidirect
24. #63774C22semidirect
26. C24×C32C2×C4semidirect
27. C23×C3⋊D3C2×C4semidirect
32. C2×C32×D4C2×C4semidirect
33. C2×C32×D4C2×C4semidirect
34. C22×C32×C4C2×C4semidirect
38. C2×#638C2×C4semidirect
39. C2×#638C2×C4semidirect
40. C2×#638C2×C4semidirect
41. C2×#638C2×C4semidirect
44. C22×C3⋊Dic3C2×C4semidirect
45. C22×C3⋊Dic3C2×C4semidirect
56. C2×C4×C3⋊D3C2×C4semidirect
57. C2×C4×C3⋊D3C2×C4semidirect
59. C2×#634C2×C4semidirect
60. C2×#634C2×C4semidirect
190. C22×D4#157semidirect
244. C24C2×#157semidirect
268. C2×D4C2×#157semidirect
269. C2×D4C2×#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×#643612
#105966422
C2×#637862
#105938182
#105938172
#105937922
#28839