Finite Group Databasebeta

← #36367052

Subgroups — #36367052

Order 22144.

1648 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14583827C211yesmaximal, characteristic, complemented
2C2×#41×D173C211yesmaximal, characteristic, complemented
3#26012160C211yesmaximal, characteristic, complemented
4Q8×C22⋊D173C211yesmaximal, characteristic, complemented
5#26012153C211yesmaximal, characteristic, complemented
6#26012143C213yesmaximal, complemented
7#26012137C213yesmaximal, complemented
8#26012134C213yesmaximal, complemented
9#26012125C213yesmaximal, complemented
10#26012040C213yesmaximal, complemented
11#26012004C213yesmaximal, complemented
12#26012001C213yesmaximal, complemented
13C4×#14583827C211yesmaximal, characteristic, complemented
14C4×D4×D173C213yesmaximal, complemented
15#548×C173C211yesmaximal, characteristic, complemented
16C2×#14583827—28no—
17C2×#14583827C2211yescharacteristic
18C2×#14583827C2211yescharacteristic, complemented
19C2×#14583827C2213yescomplemented
20C2×#14583827C2213yescomplemented
21C2×#14583826C2213yescomplemented
22C2×#14583825C2213yescomplemented
23#41×D173—28no—
24C2×Q8×D173C2211yescharacteristic, complemented
25C22×D692C2213yescomplemented
26C2×D4×D173C2213yes—
27C2×D4×D173C2213yescomplemented
28C22×C4×D173C2213yescomplemented
29C22×C4×D173C2213yescomplemented
30#19149677C2213yescomplemented
31#19149675C2213yescomplemented
32#19149667C2213yescomplemented
33#19149664C2216yescomplemented
34#19149661C2216yescomplemented
35#19149659C2213yescomplemented
36#19149658C2213yescomplemented
37#19149655C2213yescomplemented
38#19149654C2213yescomplemented
39#19149653C2213yescomplemented
40#19149652C2216yescomplemented
41#19149652C2216yescomplemented
42#19149650C2213yescomplemented
43#19149649C2216yescomplemented
44#19149647C2216yescomplemented
45Q8×Dic173C2211yescharacteristic, complemented
46C4×D692C2213yescomplemented
47C4×D692C2213yescomplemented
48D4×Dic173C2213yescomplemented
49#19149641C2213yescomplemented
50#19149641C2213yescomplemented
51C4×C22⋊D173C2213yes—
52C4×C22⋊D173C2213yescomplemented
53#19149639C2211yescharacteristic
54#19149639C2213yescomplemented
55#19149639C2216yescomplemented
56#19149637C2213yescomplemented
57C4⋊C4×D173C2213yescomplemented
58C22⋊C4×D173C2213yescomplemented
59C22⋊C4×D173C2213yescomplemented
60C42×D173C2213yescomplemented
61C2×#41×C173C2211yescharacteristic, complemented
62C22×Q8×C173C2211yescharacteristic, complemented
63#124×C173C2213yescomplemented
64C22⋊Q8×C173C2213yescomplemented
65#121×C173C2213yescomplemented
66C4×Q8×C173C2211yescharacteristic, complemented
67C4×D4×C173C2213yescomplemented
68C23×D173C2313yes—
69C23×D173C2313yes—
70#14583827—212no—
71#14583827—212no—
72#14583827—24no—
73#14583827D418yescomplemented
74#14583826—212no—
75#14583825—212no—
76C2×Dic346C2313yes—
77Q8×D173—24no—
78C2×D692—212no—
79C2×D692C2313yes—
80C2×D692C2316yes—
81C2×D692C2313yes—
82C2×D692C2313yes—
83C2×C22⋊D173C2313yes—
84C2×C22⋊D173C2316yes—
85C2×C22⋊D173C2316yes—
86C2×C22⋊D173C2311yescharacteristic
87D4×D173—212no—
88D4×D173—212no—
89C22×Dic173C2311yescharacteristic
90C22×Dic173C2313yes—
91C2×C4×D173—212no—
92C2×C4×D173—212no—
93C2×C4×D173C2313yes—
94C2×C4×D173C2313yes—
95C2×C4×D173C2313yes—
96C2×C4×D173C2311yescharacteristic
97C2×C4×D173C2313yes—
98C2×C4×D173C2313yes—
99C2×C4×D173C2313yes—
100C2×C4×D173C2313yes—
101C2×C4×D173C2313yes—
102C4⋊Dic173C2313yes—
103C4⋊Dic173C2313yes—
104#14583805C2311yescharacteristic
105#14583805C2313yes—
106#14583805C2316yes—
107C22⋊Dic173C2313yes—
108C22⋊Dic173C2313yes—
109#14583803C2311yescharacteristic
110#14583803C2313yes—
111#14583803C2313yes—
112#14583803C2313yes—
113#14583803C2316yes—
114#14583803C2316yes—
115C4×Dic173C2311yescharacteristic
116C4×Dic173C2313yes—
117C4×Dic173C2313yes—
118#41×C173—24no—
119C2×Q8×C173—24no—
120C2×Q8×C173C2313yes—
121C2×Q8×C173C2311yescharacteristic
122C2×D4×C173C2313yes—
123C2×D4×C173C2313yes—
124C22×C4×C173C2313yes—
125C22×C4×C173C2313yes—
126C4⋊C4×C173C2313yes—
127C4⋊C4×C173C2313yes—
128C22⋊C4×C173C2311yescharacteristic
129C22⋊C4×C173C2313yes—
130C22⋊C4×C173C2316yes—
131C42×C173C2313yes—
132C22×D173—26no—
133C22×D173—26no—
134C22×D173—26no—
135C22×D173—26no—
136C22×D173—43no—
137C22×D173—43no—
138C22×D173C2413yes—
139C22×D173C2413yes—
140C22×D173C2411yescharacteristic
141Dic346—26no—
142D692—212no—
143D692—26no—
144D692—26no—
145D692C2×D4112yescomplemented
146C22⋊D173—212no—
147C22⋊D173—26no—
148C22⋊D173—46no—
149C22⋊D173#4114yescomplemented
150C2×Dic173—22no—
151C2×Dic173—26no—
152C2×Dic173C2411yescharacteristic
153C2×Dic173C2413yes—
154C2×Dic173C2413yes—
155C2×Dic173C2411yescharacteristic
156C2×Dic173C2411yescharacteristic
157C4×D173—26no—
158C4×D173—26no—
159C4×D173—26no—
160C4×D173—26no—
161C4×D173—22no—
162C4×D173—26no—
163C4×D173—26no—
164C4×D173—43no—
165C4×D173C2×D4112yescomplemented
166C23×C173C2413yes—
167C23×C173C2411yescharacteristic
168Q8×C173—26no—
169Q8×C173C2×D414yescomplemented
170D4×C173—26no—
171D4×C173—26no—
172C2×C4×C173—26no—
173C2×C4×C173—26no—
174C2×C4×C173C2411yescharacteristic
175C2×C4×C173C2411yescharacteristic
176C2×C4×C173C2413yes—
177C2×C4×C173C2413yes—
178C2×C4×C173C2413yes—
179C22×C173—21no—
180C22×C173—21no—
181C22×C173—23no—
182C22×C173—23no—
183C22×C173—23no—
184C22×C173C2×#4112yescomplemented
185C22×C173C2511yescharacteristic
186C2×D173—21no—
187C2×D173—23no—
188C2×D173—23no—
189C2×D173—21no—
190C2×D173—23no—
191C2×D173—23no—
192C2×D173—23no—
193C2×D173—412no—
194C2×D173—412no—
195C2×D173C22×D416yes—
196C2×D173C2×#4112yescomplemented
197C4×C173—21no—
198C4×C173—23no—
199C4×C173—23no—
200C4×C173—21no—
201C4×C173C22×D416yes—
202Dic173—21no—
203Dic173—23no—
204Dic173—23no—
205Dic173C22×D412yes—
206Dic173C2×#4112yescomplemented
207C2×C173—22no—
208C2×C173—43no—
209C2×C173C23×D411yescharacteristic
210C2×C173C22×#4111yescharacteristic
211C2×C173#58511yescharacteristic
212D173—22no—
213D173—26no—
214D173—43no—
215C173D4×#4111yescharacteristic, complemented, Sylow
216D4×#41—1731nomaximal, Sylow
217C22×#41—1731no—
218C22×#41—1731no—
219#550—1731no—
220D4×Q8—1731no—
221#548—1731no—
222#548D17311yescharacteristic, complemented
223#547—1733no—
224#546—1733no—
225#546—1733no—
226D42—1733no—
227#532—1733no—
228#525—1733no—
229#525—1733no—
230C4×#41—1731no—
231C2×C4×D4—1733no—
232C2×#41—3468no—
233C2×#41—3468no—
234C2×#41—1733no—
235C2×#41—1733no—
236C2×#41—1731no—
237C2×#41—1731no—
238C2×#41—1733no—
239C2×#41—1733no—
240C2×#41C2×D17311yescharacteristic, complemented
241C22×Q8—1731no—
242C22×Q8C2×D17311yescharacteristic, complemented
243C22×D4—1733no—
244C22×D4—1733no—
245C22×D4—1733no—
246C23×C4—1733no—
247C23×C4—1733no—
248C4⋊Q8—1733no—
249#127—1733no—
250#124—1733no—
251#124C2×D17313yescomplemented
252#123—1736no—
253#123—1736no—
254C22⋊Q8—1733no—
255C22⋊Q8—1733no—
256C22⋊Q8—1733no—
257C22⋊Q8C2×D17313yescomplemented
258#121—1733no—
259#121—1733no—
260#121—1733no—
261#121—1736no—
262#121—1736no—
263#121C2×D17313yescomplemented
264#120—1736no—
265#120—1736no—
266C4×Q8—1731no—
267C4×Q8C2×D17311yescharacteristic, complemented
268C4×D4—1733no—
269C4×D4—1733no—
270C4×D4—1733no—
271C4×D4—1733no—
272C4×D4—1733no—
273C4×D4—1731no—
274C4×D4—1733no—
275C4×D4—1736no—
276C4×D4—1733no—
277C4×D4—1733no—
278C4×D4C2×D17313yescomplemented
279#117—1733no—
280C2×C4⋊C4—1733no—
281C2×C22⋊C4—1733no—
282C2×C22⋊C4—1733no—
283C2×C42—1733no—
284C24—1733no—
285C24—1733no—
286#41—24no—
287#41—34612no—
288#41—3464no—
289#41—34612no—
290#41—34612no—
291#41—34612no—
292#41—1738no—
293C2×Q8—24no—
294C2×Q8—3464no—
295C2×Q8—1733no—
296C2×Q8C22×D17313yes—
297C2×Q8C22×D17311yescharacteristic
298C2×D4—34612no—
299C2×D4—34612no—
300C2×D4—34612no—
301C2×D4—1733no—
302C2×D4—1736no—
303C2×D4—1731no—
304C2×D4—1736no—
305C2×D4—1733no—
306C2×D4—1736no—
307C2×D4—1733no—
308C2×D4—1733no—
309C2×D4C22×D17313yes—
310C2×D4C22×D17313yes—
311C22×C4—34612no—
312C22×C4—34612no—
313C22×C4—1731no—
314C22×C4—1733no—
315C22×C4—1733no—
316C22×C4—1733no—
317C22×C4—1733no—
318C22×C4—1731no—
319C22×C4—1733no—
320C22×C4—1733no—
321C22×C4—1733no—
322C22×C4—1733no—
323C22×C4—1733no—
324C22×C4C22×D17313yes—
325C22×C4C22×D17313yes—
326C4⋊C4—1731no—
327C4⋊C4—1733no—
328C4⋊C4—1733no—
329C4⋊C4—1733no—
330C4⋊C4—1736no—
331C4⋊C4C22×D17313yes—
332C4⋊C4C22×D17313yes—
333C22⋊C4—1733no—
334C22⋊C4—1733no—
335C22⋊C4—1731no—
336C22⋊C4—1733no—
337C22⋊C4—1733no—
338C22⋊C4—1733no—
339C22⋊C4—1736no—
340C22⋊C4—1736no—
341C22⋊C4C22×D17311yescharacteristic
342C22⋊C4C22×D17313yes—
343C22⋊C4C22×D17316yes—
344C42—1731no—
345C42—1733no—
346C42—1733no—
347C42C22×D17313yes—
348C23—3466no—
349C23—3466no—
350C23—3466no—
351C23—3466no—
352C23—1733no—
353C23—1731no—
354C23—1733no—
355C23—6923no—
356C23—6923no—
357C23C23×D17313yes—
358C23C23×D17311yescharacteristic
359Q8—26no—
360Q8—3466no—
361Q8D4×D17314yescomplemented
362D4—26no—
363D4—26no—
364D4—3466no—
365D4—34612no—
366D4—34612no—
367D4—3466no—
368D4—3466no—
369D4—1734no—
370D4—17312no—
371D4—6926no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—3466no—
375C2×C4—3466no—
376C2×C4—3462no—
377C2×C4—3466no—
378C2×C4—3466no—
379C2×C4—3462no—
380C2×C4—3466no—
381C2×C4—3466no—
382C2×C4—3466no—
383C2×C4—1733no—
384C2×C4—1731no—
385C2×C4—1733no—
386C2×C4—1731no—
387C2×C4—1731no—
388C2×C4—17312no—
389C2×C4—6923no—
390C2×C4C23×D17311yescharacteristic
391C2×C4C23×D17313yes—
392C2×C4C23×D17311yescharacteristic
393C2×C4C23×D17313yes—
394C2×C4C23×D17313yes—
395C22—21no—
396C22—21no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—3461no—
401C22—3463no—
402C22—3463no—
403C22—3461no—
404C22—3463no—
405C22—3463no—
406C22—3463no—
407C22—1736no—
408C22—1732no—
409C22—69212no—
410C22—69212no—
411C22#41×D17312yescomplemented
412C22C24×D17311yescharacteristic
413C4—21no—
414C4—23no—
415C4—23no—
416C4—21no—
417C4—3461no—
418C4—3463no—
419C4—3463no—
420C4—1732no—
421C4—1732no—
422C4C2×D4×D17316yes—
423C2—22no—
424C2—43no—
425C2—3466no—
426C2—3462no—
427C2—6923no—
428C2C22×D4×D17311yescharacteristic
429C2C2×#41×D17311yescharacteristic
430C2#2601232811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#548×C173C2
hypercenter#548D173
hypocenterC173D4×#41
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C22×#14583827C2semidirect
2. C2×#41×D173C2semidirect
3. #26012160C2semidirect
4. Q8×C22⋊D173C2semidirect
5. #26012153C2semidirect
6. #26012143C2semidirect
7. #26012137C2semidirect
8. #26012134C2semidirect
9. #26012125C2semidirect
10. #26012040C2semidirect
11. #26012004C2semidirect
12. #26012001C2semidirect
13. C4×#14583827C2semidirect
14. C4×D4×D173C2semidirect
15. #548×C173C2semidirect
18. C2×#14583827C22semidirect
19. C2×#14583827C22semidirect
20. C2×#14583827C22semidirect
21. C2×#14583826C22semidirect
22. C2×#14583825C22semidirect
24. C2×Q8×D173C22semidirect
25. C22×D692C22semidirect
27. C2×D4×D173C22semidirect
28. C22×C4×D173C22semidirect
29. C22×C4×D173C22semidirect
30. #19149677C22semidirect
31. #19149675C22semidirect
32. #19149667C22semidirect
33. #19149664C22semidirect
34. #19149661C22semidirect
35. #19149659C22semidirect
36. #19149658C22semidirect
37. #19149655C22semidirect
38. #19149654C22semidirect
39. #19149653C22semidirect
40. #19149652C22semidirect
41. #19149652C22semidirect
42. #19149650C22semidirect
43. #19149649C22semidirect
44. #19149647C22semidirect
45. Q8×Dic173C22semidirect
46. C4×D692C22semidirect
47. C4×D692C22semidirect
48. D4×Dic173C22semidirect
49. #19149641C22semidirect
50. #19149641C22semidirect
52. C4×C22⋊D173C22semidirect
54. #19149639C22semidirect
55. #19149639C22semidirect
56. #19149637C22semidirect
57. C4⋊C4×D173C22semidirect
58. C22⋊C4×D173C22semidirect
59. C22⋊C4×D173C22semidirect
60. C42×D173C22semidirect
61. C2×#41×C173C22semidirect
62. C22×Q8×C173C22semidirect
63. #124×C173C22semidirect
64. C22⋊Q8×C173C22semidirect
65. #121×C173C22semidirect
66. C4×Q8×C173C22semidirect
67. C4×D4×C173C22semidirect
73. #14583827D4semidirect
145. D692C2×D4semidirect
149. C22⋊D173#41semidirect
165. C4×D173C2×D4semidirect
169. Q8×C173C2×D4semidirect
184. C22×C173C2×#41semidirect
196. C2×D173C2×#41semidirect
206. Dic173C2×#41semidirect
215. C173D4×#41semidirect
222. #548D173semidirect
240. C2×#41C2×D173semidirect
242. C22×Q8C2×D173semidirect
251. #124C2×D173semidirect
257. C22⋊Q8C2×D173semidirect
263. #121C2×D173semidirect
267. C4×Q8C2×D173semidirect
278. C4×D4C2×D173semidirect
361. Q8D4×D173semidirect
411. C22#41×D173semidirect

Maximal subgroups

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

subgroupindex
C22×#145838272
C2×#41×D1732
#260121602
Q8×C22⋊D1732
#260121532
#260121432
#260121372
#260121342
#260121252
#260120402
#260120042
#260120012
C4×#145838272
C4×D4×D1732
#548×C1732
D4×#41173