Finite Group Databasebeta

← #36363358

Subgroups — #36363358

Order 22144.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012047C211yesmaximal, characteristic, complemented
2C4×D4×D173C211yesmaximal, characteristic
3#26011792C211yesmaximal, characteristic, complemented
4#26011791C211yesmaximal, characteristic, complemented
5#26011788C211yesmaximal, characteristic, complemented
6#26011780C211yesmaximal, characteristic, complemented
7#26011779C211yesmaximal, characteristic, complemented
8#26011770C211yesmaximal, characteristic, complemented
9#26011769C211yesmaximal, characteristic, complemented
10#26011532C211yesmaximal, characteristic, complemented
11#26011530C211yesmaximal, characteristic, complemented
12#26011513C211yesmaximal, characteristic, complemented
13#26011494C211yesmaximal, characteristic
14#26011494C211yesmaximal, characteristic
15#478×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C22×C4×D173—21no—
18C2×#14583809C2211yescharacteristic
19C2×#14583809C2211yescharacteristic
20#19149678C2211yescharacteristic, complemented
21#19149678C2211yescharacteristic, complemented
22#19149671C2211yescharacteristic, complemented
23#19149657—21no—
24#19149657—21no—
25#19149644—21no—
26#19149644—21no—
27C4×D692C2211yescharacteristic
28D4×Dic173C2211yescharacteristic
29#19149641C2211yescharacteristic
30C4×C22⋊D173—21no—
31#19149639—21no—
32#19149637C2211yescharacteristic
33#19149637C2211yescharacteristic
34C4⋊C4×D173C2211yescharacteristic
35C22⋊C4×D173—21no—
36C42×D173C2211yescharacteristic
37#19149613C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149611C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43C4⋊#13063436C2211yescharacteristic, complemented
44#19149608C2211yescharacteristic, complemented
45#19149608C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149599C2211yescharacteristic
51#19149599C2211yescharacteristic
52#125×C173C2211yescharacteristic, complemented
53C4×D4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56C4⋊C8×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64D4×D173—22no—
65D4×D173Q814yescomplemented
66C22×Dic173—21no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—44no—
71C2×C4×D173D411yescharacteristic
72C2×C4×D173D411yescharacteristic
73C2×C4×D173C2311yescharacteristic
74#14583809—22no—
75#14583809—22no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173D411yescharacteristic
94C4×Dic173D411yescharacteristic
95C4×Dic173C2311yescharacteristic
96C2×D4×C173C2311yescharacteristic
97C22×C4×C173—21no—
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C42×C173C2311yescharacteristic
107C22×D173—21no—
108C22×D173—24no—
109C22×D173—42no—
110C22×D173—42no—
111C22×D173C2×D411yescharacteristic
112Dic346—42no—
113Dic346—42no—
114D692—21no—
115D692C2×Q812yes—
116C22⋊D173—24no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—42no—
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C4×D173—22no—
125C4×D173—41no—
126C4×D173—42no—
127C4×D173—42no—
128C4×D173—42no—
129C4×D173C2×Q812yes—
130C4×D173#4112yes—
131#13063436—21no—
132#13063436—21no—
133C23×C173—21no—
134D4×C173—21no—
135D4×C173C2×Q812yes—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—42no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2411yescharacteristic
143C8×C173—21no—
144C8×C173—21no—
145C22×C173—22no—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173C22×D411yescharacteristic
149C2×D173—21no—
150C2×D173—21no—
151C2×D173—22no—
152C2×D173—48no—
153C2×D173—41no—
154C2×D173—41no—
155C2×D173C22⋊Q812yes—
156C4×C173—21no—
157C4×C173—21no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173C22×Q811yescharacteristic
162C4×C173C2×#4111yescharacteristic
163Dic173—21no—
164Dic173—41no—
165Dic173—41no—
166Dic173—41no—
167Dic173C22⋊Q812yes—
168Dic173AGL(1,ℤ/8)12yes—
169C2×C173—42no—
170C2×C173C2×C22⋊Q811yescharacteristic
171C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C173#57511yescharacteristic
173D173—22no—
174D173—42no—
175C173#307711yescharacteristic, complemented, Sylow
176#3077—1731nomaximal, Sylow
177#533—1731no—
178C2×C4×D4—1731no—
179#478—1731no—
180#478—1731no—
181#478—1731no—
182#478D17311yescharacteristic, complemented
183#476—1731no—
184#476—1731no—
185#474—1731no—
186#474—1731no—
187#428—1731no—
188#428—1731no—
189#423—1731no—
190#418—1731no—
191#418—1731no—
192C22×D4—1731no—
193C23×C4—3461no—
194C2×M16—1731no—
195C2×M16—1731no—
196C4⋊Q8—1731no—
197C4⋊Q8—1731no—
198#125—1731no—
199#125C2×D17311yescharacteristic, complemented
200C22⋊Q8—3461no—
201C22⋊Q8—3461no—
202C4×Q8—3461no—
203C4×Q8—3461no—
204C4×D4—3461no—
205C4×D4—3461no—
206C4×D4—1731no—
207C4×D4—1731no—
208C4×D4—1731no—
209C4×D4C2×D17311yescharacteristic
210#117—1731no—
211#117—1731no—
212C2×C4⋊C4—1731no—
213C2×C22⋊C4—3461no—
214C2×C42—1731no—
215#107—1731no—
216#107—1731no—
217#107—1731no—
218#107C2×D17311yescharacteristic, complemented
219#106—1731no—
220#106—1731no—
221#106—1731no—
222#106C2×D17311yescharacteristic, complemented
223C4⋊C8—1731no—
224C4⋊C8—1731no—
225C4⋊C8—1731no—
226C4⋊C8C2×D17311yescharacteristic, complemented
227#102—1731no—
228#102—1731no—
229#102—1731no—
230#102—1731no—
231#102—1731no—
232#102—1731no—
233#102C2×D17311yescharacteristic
234#102C2×D17311yescharacteristic
235C24—3461no—
236C2×Q8—3461no—
237C2×Q8—3461no—
238C2×D4—3462no—
239C2×D4—3461no—
240C2×D4—1734no—
241C2×D4—1731no—
242C2×D4C22×D17311yescharacteristic
243C22×C4—21no—
244C22×C4—3461no—
245C22×C4—3461no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4—1731no—
249C22×C4—1731no—
250C22×C4—1731no—
251C22×C4—6924no—
252M16—3462no—
253M16—3462no—
254C2×C8—1731no—
255C2×C8—1731no—
256C2×C8C22×D17311yescharacteristic
257C2×C8C22×D17311yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—3461no—
264C4⋊C4—3461no—
265C4⋊C4—1731no—
266C4⋊C4—1731no—
267C4⋊C4—1731no—
268C4⋊C4C22×D17311yescharacteristic
269C4⋊C4C22×D17311yescharacteristic
270C4⋊C4C22×D17311yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C42—3461no—
278C42—3461no—
279C42—1731no—
280C42—1731no—
281C42—1731no—
282C42C22×D17311yescharacteristic
283C23—21no—
284C23—3464no—
285C23—3461no—
286C23—1731no—
287C23—6922no—
288C23—6922no—
289Q8—6922no—
290Q8—6922no—
291D4—21no—
292D4—3464no—
293D4—3461no—
294D4—1732no—
295D4Q8×D17312yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—3462no—
301C2×C4—3461no—
302C2×C4—3461no—
303C2×C4—3461no—
304C2×C4—1731no—
305C2×C4—1731no—
306C2×C4—1731no—
307C2×C4—1732no—
308C2×C4—1732no—
309C2×C4—6921no—
310C2×C4—6922no—
311C2×C4—6922no—
312C2×C4—6922no—
313C2×C4—6922no—
314C2×C4D4×D17311yescharacteristic
315C2×C4D4×D17311yescharacteristic
316C2×C4C23×D17311yescharacteristic
317C8—21no—
318C8—21no—
319C8—3461no—
320C8—3461no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—3461no—
325C22—3461no—
326C22—3462no—
327C22—1732no—
328C22—6928no—
329C22—6921no—
330C22—6921no—
331C22C2×D4×D17311yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3461no—
338C4—1732no—
339C4—1732no—
340C4—6921no—
341C4—6921no—
342C4—6921no—
343C4C2×Q8×D17311yescharacteristic
344C4#41×D17311yescharacteristic
345C2—42no—
346C2—3462no—
347C2—6922no—
348C2C22⋊Q8×D17311yescharacteristic
349C2AGL(1,ℤ/8)×D17311yescharacteristic
350C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#478×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×D173C22
hypercenter#478D173
hypocenterC173#3077
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #26012047C2semidirect
3. #26011792C2semidirect
4. #26011791C2semidirect
5. #26011788C2semidirect
6. #26011780C2semidirect
7. #26011779C2semidirect
8. #26011770C2semidirect
9. #26011769C2semidirect
10. #26011532C2semidirect
11. #26011530C2semidirect
12. #26011513C2semidirect
15. #478×C173C2semidirect
20. #19149678C22semidirect
21. #19149678C22semidirect
22. #19149671C22semidirect
37. #19149613C22semidirect
38. #19149612C22semidirect
39. #19149612C22semidirect
40. #19149611C22semidirect
41. #19149610C22semidirect
42. #19149610C22semidirect
43. C4⋊#13063436C22semidirect
44. #19149608C22semidirect
45. #19149608C22semidirect
52. #125×C173C22semidirect
54. #107×C173C22semidirect
55. #106×C173C22semidirect
56. C4⋊C8×C173C22semidirect
65. D4×D173Q8semidirect
175. C173#3077semidirect
182. #478D173semidirect
199. #125C2×D173semidirect
218. #107C2×D173semidirect
222. #106C2×D173semidirect
226. C4⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120472
C4×D4×D1732
#260117922
#260117912
#260117882
#260117802
#260117792
#260117702
#260117692
#260115322
#260115302
#260115132
#260114942
#478×C1732
#3077173