Finite Group Databasebeta

← #37545424

Subgroups — #37545424

Order 23168.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153836C211yesmaximal, characteristic, complemented
2C22×#15668437C211yesmaximal, characteristic, complemented
3#26041938C211yesmaximal, characteristic, complemented
4#26041661C211yesmaximal, characteristic, complemented
5#26041510C211yesmaximal, characteristic, complemented
6#26041506C211yesmaximal, characteristic, complemented
7#26041498C211yesmaximal, characteristic, complemented
8#26041494C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041434C211yesmaximal, characteristic, complemented
11#26041422C211yesmaximal, characteristic, complemented
12#26041413C211yesmaximal, characteristic, complemented
13C4×#15668437C211yesmaximal, characteristic, complemented
14D4×#13089678C211yesmaximal, characteristic, complemented
15#546×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic, complemented
17C22×Dic362C2211yescharacteristic
18#19153836—24no—
19C2×#15668439C2211yescharacteristic, complemented
20C2×#15668439C2211yescharacteristic, complemented
21C2×#15668437—24no—
22C2×#15668437C2211yescharacteristic
23C2×#15668437C2211yescharacteristic
24C2×#15668437C2211yescharacteristic
25C2×#15668437C2211yescharacteristic
26C2×#15668436C2211yescharacteristic, complemented
27C2×#15668434C2211yescharacteristic
28C2×#15668434C2211yescharacteristic
29C22×#13089678C2211yescharacteristic
30C22×#13089678C2211yescharacteristic, complemented
31#19153804C2211yescharacteristic, complemented
32#19153802—21no—
33#19153795C2211yescharacteristic, complemented
34#19153794—21no—
35#19153790C2211yescharacteristic
36#19153788—21no—
37C4×Dic362C2211yescharacteristic, complemented
38#19153776—21no—
39#19153747C2211yescharacteristic, complemented
40C4⋊#13089678C2211yescharacteristic, complemented
41#19153740C2211yescharacteristic, complemented
42#19153739C2211yescharacteristic, complemented
43#19153739C2211yescharacteristic, complemented
44#19153737C2211yescharacteristic
45#19153737C2211yescharacteristic
46#19153736C2211yescharacteristic, complemented
47C22⋊#13089678C2211yescharacteristic
48C22⋊#13089678C2211yescharacteristic, complemented
49C4×#13089678C2211yescharacteristic, complemented
50C2×#41×C181C2211yescharacteristic
51C22×D4×C181C2211yescharacteristic
52#124×C181C2211yescharacteristic
53#123×C181—21no—
54C22⋊Q8×C181C2211yescharacteristic
55#121×C181—21no—
56#121×C181C2211yescharacteristic
57#120×C181—21no—
58C4×D4×C181C2211yescharacteristic
59C4×D4×C181C2211yescharacteristic
60C2×C22⋊C4×C181—21no—
61#15668449—22no—
62#15668449—41no—
63C2×Dic362—22no—
64C2×Dic362—21no—
65C2×Dic362—41no—
66C2×Dic362C2311yescharacteristic
67C2×Dic362C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C22×Dic181—21no—
72C2×C4×D181—21no—
73#15668439—22no—
74#15668439—41no—
75#15668437—22no—
76#15668437—22no—
77#15668437—22no—
78#15668437D414yescomplemented
79#15668436—22no—
80#15668434—22no—
81#15668434—41no—
82C2×#13089678—22no—
83C2×#13089678—22no—
84C2×#13089678C2311yescharacteristic
85C2×#13089678C2311yescharacteristic
86C2×#13089678C2311yescharacteristic
87C2×#13089678C2311yescharacteristic
88C4⋊Dic181C2311yescharacteristic
89#15668428—21no—
90#15668428—21no—
91C22⋊Dic181—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95C4×Dic181—21no—
96C24×C181—21no—
97#41×C181—22no—
98#41×C181—41no—
99C2×Q8×C181C2311yescharacteristic
100C2×D4×C181—22no—
101C2×D4×C181—21no—
102C2×D4×C181—21no—
103C2×D4×C181—21no—
104C2×D4×C181—41no—
105C2×D4×C181D411yescharacteristic
106C2×D4×C181C2311yescharacteristic
107C2×D4×C181C2311yescharacteristic
108C22×C4×C181—21no—
109C22×C4×C181—21no—
110C22×C4×C181C2311yescharacteristic
111C22×C4×C181C2311yescharacteristic
112C4⋊C4×C181—21no—
113C4⋊C4×C181D411yescharacteristic
114C4⋊C4×C181C2311yescharacteristic
115C22⋊C4×C181—21no—
116C22⋊C4×C181—21no—
117C22⋊C4×C181—21no—
118C22⋊C4×C181—21no—
119C22⋊C4×C181—21no—
120C22⋊C4×C181D411yescharacteristic
121C22⋊C4×C181D411yescharacteristic
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124Dic362—21no—
125Dic362—21no—
126Dic362—21no—
127Dic362—42no—
128Dic362C2×D412yes—
129D724—21no—
130D724—21no—
131C22⋊D181—42no—
132C22⋊D181—42no—
133C2×Dic181—21no—
134C2×Dic181—21no—
135C2×Dic181—21no—
136C2×Dic181—42no—
137C4×D181—42no—
138#13089678—21no—
139#13089678—21no—
140#13089678—21no—
141#13089678C2×D412yes—
142C23×C181—21no—
143C23×C181—21no—
144C23×C181—42no—
145C23×C181—42no—
146C23×C181—81no—
147C23×C181C2×D411yescharacteristic
148C23×C181C2×D411yescharacteristic
149Q8×C181—21no—
150Q8×C181—21no—
151D4×C181—22no—
152D4×C181—21no—
153D4×C181—21no—
154D4×C181—21no—
155D4×C181—42no—
156D4×C181—42no—
157D4×C181—81no—
158D4×C181C2×D412yes—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—42no—
166C2×C4×C181—81no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2×D411yescharacteristic
170C2×C4×C181C2×D411yescharacteristic
171C2×C4×C181C2411yescharacteristic
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—21no—
175C22×C181—22no—
176C22×C181—21no—
177C22×C181—21no—
178C22×C181—41no—
179C22×C181—41no—
180C22×C181—41no—
181C22×C181—41no—
182C22×C181—41no—
183C22×C181—84no—
184C22×C181#13512yescomplemented
185C22×C181C22×D411yescharacteristic
186C2×D181—41no—
187C2×D181—41no—
188C2×D181—41no—
189C4×C181—21no—
190C4×C181—21no—
191C4×C181—21no—
192C4×C181—21no—
193C4×C181—41no—
194C4×C181—41no—
195C4×C181C22×D411yescharacteristic
196C4×C181H(5,𝔽2)11yescharacteristic
197Dic181—22no—
198Dic181—41no—
199Dic181—41no—
200C2×C181—22no—
201C2×C181—41no—
202C2×C181—42no—
203C2×C181—81no—
204C2×C181D4211yescharacteristic
205C2×C181C2×#13511yescharacteristic
206C2×C181#57711yescharacteristic
207D181—81no—
208C181#328211yescharacteristic, complemented, Sylow
209#3282—1811nomaximal, Sylow
210C2×#135—1811no—
211C22×SD16—1811no—
212#548—1811no—
213#546D18111yescharacteristic, complemented
214#495—1811no—
215#465—1811no—
216#465—1811no—
217#464—1811no—
218#463—1811no—
219#452—1811no—
220#451—1811no—
221#449—1811no—
222#447—1811no—
223C4×SD16—1811no—
224C8×D4—1811no—
225C2×#41—1811no—
226C2×#41C2×D18111yescharacteristic
227C22×Q8—1811no—
228C22×D4C2×D18111yescharacteristic
229#135—3624no—
230C2×Q16—1811no—
231C2×Q16—1811no—
232C2×SD16—3624no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16—1811no—
237C2×SD16—1811no—
238C2×D8—1811no—
239C2×D8—1811no—
240C22×C8—1811no—
241C22×C8—1811no—
242#124—3621no—
243#124—1811no—
244#124C2×D18111yescharacteristic
245#123—21no—
246C22⋊Q8—3621no—
247C22⋊Q8—1811no—
248C22⋊Q8C2×D18111yescharacteristic
249#121—21no—
250#121—3621no—
251#121—1811no—
252#121C2×D18111yescharacteristic
253#120—21no—
254C4×Q8—1811no—
255C4×D4—3621no—
256C4×D4C2×D18111yescharacteristic
257C4×D4C2×D18111yescharacteristic
258C2×C22⋊C4—21no—
259#106—1811no—
260C4⋊C8—1811no—
261#103—1811no—
262#103—1811no—
263#103—1811no—
264#102—1811no—
265#102—1811no—
266#102—1811no—
267C22⋊C8—1811no—
268C22⋊C8—1811no—
269C4×C8—1811no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—3622no—
274#41—7241no—
275C2×Q8—3622no—
276C2×Q8—3621no—
277C2×Q8—1811no—
278C2×Q8—1811no—
279C2×Q8—7241no—
280C2×Q8C22×D18111yescharacteristic
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—3621no—
287C2×D4—3621no—
288C2×D4—1811no—
289C2×D4C22⋊D18111yescharacteristic
290C2×D4C22×D18111yescharacteristic
291C2×D4C22×D18111yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4C22×D18111yescharacteristic
297C22×C4C22×D18111yescharacteristic
298Q16—3622no—
299Q16—7241no—
300SD16—3622no—
301SD16—3622no—
302SD16—3622no—
303SD16—3622no—
304SD16—1814no—
305D8—3622no—
306D8—7241no—
307C2×C8—3622no—
308C2×C8—3622no—
309C2×C8—1811no—
310C2×C8—1811no—
311C2×C8—1811no—
312C2×C8—1811no—
313C4⋊C4—21no—
314C4⋊C4—3621no—
315C4⋊C4—3621no—
316C4⋊C4—1811no—
317C4⋊C4C22⋊D18111yescharacteristic
318C4⋊C4C22×D18111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4C22⋊D18111yescharacteristic
329C22⋊C4C22⋊D18111yescharacteristic
330C42—3621no—
331C42C22×D18111yescharacteristic
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—3621no—
338C23C2×C22⋊D18111yescharacteristic
339C23C2×C22⋊D18111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3621no—
343Q8—3621no—
344Q8—3621no—
345Q8—1812no—
346Q8—7242no—
347D4—22no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—42no—
353D4—81no—
354D4—3621no—
355D4—3621no—
356D4—7242no—
357D4—7242no—
358D4D4×D18112yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—3621no—
368C2×C4—3621no—
369C2×C4—3621no—
370C2×C4—7242no—
371C2×C4—7242no—
372C2×C4C2×C22⋊D18111yescharacteristic
373C2×C4C2×C22⋊D18111yescharacteristic
374C2×C4C2×C22⋊D18111yescharacteristic
375C2×C4C2×C22⋊D18111yescharacteristic
376C2×C4C23×D18111yescharacteristic
377C8—3621no—
378C8—3621no—
379C8—3621no—
380C8—1812no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—21no—
386C22—21no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—84no—
393C22—7241no—
394C22—7241no—
395C22—7241no—
396C22#1915383612yescomplemented
397C22C22×C22⋊D18111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—3622no—
405C4—7241no—
406C4—7241no—
407C4C2×D4×D18111yescharacteristic
408C4#1915386111yescharacteristic
409C2—22no—
410C2—41no—
411C2—42no—
412C2—81no—
413C2—14481no—
414C2D4×C22⋊D18111yescharacteristic
415C2C2×#1915383611yescharacteristic
416C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#546×C181C2
socleC22×C181C22×D4
hypercenter#546D181
hypocenterC181#3282
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#19153836C2semidirect
2. C22×#15668437C2semidirect
3. #26041938C2semidirect
4. #26041661C2semidirect
5. #26041510C2semidirect
6. #26041506C2semidirect
7. #26041498C2semidirect
8. #26041494C2semidirect
9. #26041441C2semidirect
10. #26041434C2semidirect
11. #26041422C2semidirect
12. #26041413C2semidirect
13. C4×#15668437C2semidirect
14. D4×#13089678C2semidirect
15. #546×C181C2semidirect
16. C2×#15668449C22semidirect
19. C2×#15668439C22semidirect
20. C2×#15668439C22semidirect
26. C2×#15668436C22semidirect
30. C22×#13089678C22semidirect
31. #19153804C22semidirect
33. #19153795C22semidirect
37. C4×Dic362C22semidirect
39. #19153747C22semidirect
40. C4⋊#13089678C22semidirect
41. #19153740C22semidirect
42. #19153739C22semidirect
43. #19153739C22semidirect
46. #19153736C22semidirect
48. C22⋊#13089678C22semidirect
49. C4×#13089678C22semidirect
78. #15668437D4semidirect
184. C22×C181#135semidirect
208. C181#3282semidirect
213. #546D181semidirect
396. C22#19153836semidirect

Maximal subgroups

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

subgroupindex
C2×#191538362
C22×#156684372
#260419382
#260416612
#260415102
#260415062
#260414982
#260414942
#260414412
#260414342
#260414222
#260414132
C4×#156684372
D4×#130896782
#546×C1812
#3282181