Finite Group Databasebeta

← #37545232

Subgroups — #37545232

Order 23168.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153847C211yesmaximal, characteristic, complemented
2C22×#15668436C211yesmaximal, characteristic, complemented
3#26041925C211yesmaximal, characteristic
4#26041671C211yesmaximal, characteristic, complemented
5#26041527C211yesmaximal, characteristic, complemented
6#26041518C211yesmaximal, characteristic, complemented
7#26041501C211yesmaximal, characteristic, complemented
8#26041481C211yesmaximal, characteristic, complemented
9#26041439C211yesmaximal, characteristic, complemented
10#26041435C211yesmaximal, characteristic, complemented
11#26041429C211yesmaximal, characteristic, complemented
12#26041418C211yesmaximal, characteristic, complemented
13#26041377C211yesmaximal, characteristic, complemented
14#26041351C211yesmaximal, characteristic, complemented
15D4×Q8×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×D724C2211yescharacteristic
18#19153847—24no—
19C2×#15668439C2211yescharacteristic, complemented
20C2×#15668439C2211yescharacteristic, complemented
21C2×#15668439C2211yescharacteristic, complemented
22C2×#15668436—24no—
23C2×#15668436C2211yescharacteristic
24C2×#15668436C2211yescharacteristic
25C2×#15668436C2211yescharacteristic
26C2×#15668436C2211yescharacteristic
27C2×#15668436C2211yescharacteristic
28C2×#15668436C2211yescharacteristic
29C2×#15668433C2211yescharacteristic, complemented
30C22×#13089678C2211yescharacteristic
31#19153804C2211yescharacteristic
32#19153801—21no—
33#19153795C2211yescharacteristic
34#19153790C2211yescharacteristic
35#19153789—21no—
36#19153784—21no—
37C4×D724C2211yescharacteristic
38C22⋊C4×D181—21no—
39#19153749C2211yescharacteristic, complemented
40C4⋊#13089678C2211yescharacteristic, complemented
41#19153740C2211yescharacteristic, complemented
42#19153740C2211yescharacteristic, complemented
43#19153740C2211yescharacteristic, complemented
44#19153739C2211yescharacteristic, complemented
45#19153736C2211yescharacteristic
46#19153736C2211yescharacteristic
47C22⋊#13089678C2211yescharacteristic
48C22⋊#13089678C2211yescharacteristic, complemented
49#19153727C2211yescharacteristic, complemented
50C22×Q8×C181C2211yescharacteristic
51C22×Q8×C181C2211yescharacteristic, complemented
52C4⋊Q8×C181—21no—
53C4⋊Q8×C181C2211yescharacteristic, complemented
54C22⋊Q8×C181—21no—
55C22⋊Q8×C181—21no—
56C22⋊Q8×C181C2211yescharacteristic
57C22⋊Q8×C181C2211yescharacteristic, complemented
58C4×Q8×C181C2211yescharacteristic, complemented
59C4×D4×C181—21no—
60C4×D4×C181C2211yescharacteristic
61C23×D181—21no—
62#15668449—22no—
63#15668449—41no—
64C2×Dic362C2311yescharacteristic
65C2×D724—22no—
66C2×D724—21no—
67C2×D724—41no—
68C2×D724C2311yescharacteristic
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C4×D181—21no—
73C2×C4×D181—21no—
74#15668439—22no—
75#15668439—22no—
76#15668439—41no—
77#15668436—22no—
78#15668436—22no—
79#15668436—22no—
80#15668436—22no—
81#15668436—41no—
82#15668436D414yescomplemented
83#15668433—22no—
84C2×#13089678—22no—
85C2×#13089678C2311yescharacteristic
86C2×#13089678C2311yescharacteristic
87C2×#13089678C2311yescharacteristic
88C2×#13089678C2311yescharacteristic
89C4⋊Dic181C2311yescharacteristic
90#15668428—21no—
91C22⋊Dic181—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96C2×Q8×C181—21no—
97C2×Q8×C181—21no—
98C2×Q8×C181—22no—
99C2×Q8×C181—22no—
100C2×Q8×C181—41no—
101C2×Q8×C181—41no—
102C2×Q8×C181C2311yescharacteristic
103C2×Q8×C181C2311yescharacteristic
104C2×Q8×C181C2311yescharacteristic
105C2×D4×C181D411yescharacteristic
106C22×C4×C181—21no—
107C22×C4×C181—21no—
108C22×C4×C181C2311yescharacteristic
109C22×C4×C181C2311yescharacteristic
110C4⋊C4×C181—21no—
111C4⋊C4×C181—21no—
112C4⋊C4×C181—21no—
113C4⋊C4×C181—21no—
114C4⋊C4×C181—21no—
115C4⋊C4×C181D411yescharacteristic
116C4⋊C4×C181C2311yescharacteristic
117C22⋊C4×C181—21no—
118C22⋊C4×C181—21no—
119C22⋊C4×C181D411yescharacteristic
120C22⋊C4×C181D411yescharacteristic
121C42×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—42no—
126C22×D181—42no—
127C22×D181—81no—
128Dic362—21no—
129Dic362—21no—
130D724—21no—
131D724—21no—
132D724—21no—
133D724—42no—
134D724C2×D412yes—
135C22⋊D181—42no—
136C22⋊D181—81no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C4×D181—42no—
140C4×D181—81no—
141#13089678—21no—
142#13089678—21no—
143#13089678—21no—
144#13089678C2×D412yes—
145C23×C181C2×D411yescharacteristic
146C23×C181C2×D411yescharacteristic
147Q8×C181—21no—
148Q8×C181—21no—
149Q8×C181—21no—
150Q8×C181—21no—
151Q8×C181—21no—
152Q8×C181—42no—
153Q8×C181—42no—
154Q8×C181C2×D412yes—
155D4×C181—22no—
156C2×C4×C181—21no—
157C2×C4×C181—21no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—42no—
165C2×C4×C181—42no—
166C2×C4×C181C2×D411yescharacteristic
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2×D411yescharacteristic
170C2×C4×C181C2411yescharacteristic
171C22×C181—21no—
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—21no—
175C22×C181—21no—
176C22×C181#13712yescomplemented
177C22×C181C22×D411yescharacteristic
178C2×D181—22no—
179C2×D181—41no—
180C2×D181—41no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—41no—
184C2×D181—84no—
185C4×C181—21no—
186C4×C181—21no—
187C4×C181—22no—
188C4×C181—21no—
189C4×C181—21no—
190C4×C181—41no—
191C4×C181—41no—
192C4×C181—41no—
193C4×C181C22×D411yescharacteristic
194C4×C181H(5,𝔽2)11yescharacteristic
195Dic181—41no—
196Dic181—41no—
197C2×C181—22no—
198C2×C181—41no—
199C2×C181D4211yescharacteristic
200C2×C181C2×#13711yescharacteristic
201C2×C181#57711yescharacteristic
202D181—42no—
203D181—81no—
204C181#326611yescharacteristic, complemented, Sylow
205#3266—1811nomaximal, Sylow
206C2×#137—1811no—
207C22×SD16—1811no—
208D4×Q8D18111yescharacteristic, complemented
209#546—1811no—
210#497—1811no—
211#468—1811no—
212#467—1811no—
213#464—1811no—
214#461—1811no—
215#452—1811no—
216#451—1811no—
217#450—1811no—
218#448—1811no—
219#440—1811no—
220#435—1811no—
221C2×#41—1811no—
222C22×Q8C2×D18111yescharacteristic
223C22×Q8C2×D18111yescharacteristic, complemented
224C22×D4—1811no—
225#137—3624no—
226C2×Q16—1811no—
227C2×Q16—1811no—
228C2×Q16—1811no—
229C2×SD16—3624no—
230C2×SD16—1811no—
231C2×SD16—1811no—
232C2×SD16—1811no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×M16—1811no—
237C22×C8—1811no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D18111yescharacteristic, complemented
240#124—1811no—
241#123—3621no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1811no—
245C22⋊Q8C2×D18111yescharacteristic
246C22⋊Q8C2×D18111yescharacteristic, complemented
247#121—3621no—
248#121—1811no—
249#120—3621no—
250C4×Q8C2×D18111yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1811no—
253C4×D4C2×D18111yescharacteristic
254C2×C22⋊C4—3621no—
255#107—1811no—
256C4⋊C8—1811no—
257#103—1811no—
258#103—1811no—
259#103—1811no—
260#103—1811no—
261#102—1811no—
262#102—1811no—
263C22⋊C8—1811no—
264C22⋊C8—1811no—
265#97—1811no—
266C24—3621no—
267#41—3622no—
268#41—7241no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1811no—
276C2×Q8C22×D18111yescharacteristic
277C2×Q8C22×D18111yescharacteristic
278C2×Q8C22×D18111yescharacteristic
279C2×D4—3622no—
280C2×D4—3621no—
281C2×D4—3621no—
282C2×D4—3621no—
283C2×D4—1811no—
284C2×D4—1811no—
285C2×D4—7241no—
286C2×D4C22⋊D18111yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—3621no—
290C22×C4—3621no—
291C22×C4C22×D18111yescharacteristic
292C22×C4C22×D18111yescharacteristic
293Q16—3622no—
294Q16—3622no—
295Q16—7241no—
296SD16—3622no—
297SD16—3622no—
298SD16—3622no—
299SD16—3622no—
300SD16—1814no—
301SD16—7241no—
302M16—3622no—
303C2×C8—3622no—
304C2×C8—1811no—
305C2×C8—1811no—
306C2×C8—1811no—
307C2×C8—1811no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—3621no—
314C4⋊C4—1811no—
315C4⋊C4C22⋊D18111yescharacteristic
316C4⋊C4C22×D18111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4C22⋊D18111yescharacteristic
325C22⋊C4C22⋊D18111yescharacteristic
326C42—21no—
327C42C22×D18111yescharacteristic
328C23—3621no—
329C23—3621no—
330C23—14481no—
331C23—7242no—
332C23—7242no—
333C23C2×C22⋊D18111yescharacteristic
334C23C2×C22⋊D18111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—3621no—
343Q8—3621no—
344Q8D4×D18112yes—
345D4—22no—
346D4—3621no—
347D4—3621no—
348D4—3621no—
349D4—14481no—
350D4—1812no—
351D4—7242no—
352D4—7242no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—3621no—
364C2×C4—3621no—
365C2×C4—14481no—
366C2×C4—7242no—
367C2×C4C2×C22⋊D18111yescharacteristic
368C2×C4C2×C22⋊D18111yescharacteristic
369C2×C4C2×C22⋊D18111yescharacteristic
370C2×C4C2×C22⋊D18111yescharacteristic
371C2×C4C23×D18111yescharacteristic
372C8—3621no—
373C8—3621no—
374C8—3621no—
375C8—1812no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—3622no—
382C22—14484no—
383C22—7241no—
384C22—7241no—
385C22—7241no—
386C22—7241no—
387C22—7241no—
388C22#1915384712yescomplemented
389C22C22×C22⋊D18111yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—7241no—
399C4—7241no—
400C4C2×D4×D18111yescharacteristic
401C4#1915386111yescharacteristic
402C2—22no—
403C2—41no—
404C2—14481no—
405C2—7242no—
406C2D4×C22⋊D18111yescharacteristic
407C2C2×#1915384711yescharacteristic
408C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingD4×Q8×C181C2
socleC22×C181C22×D4
prime radical#26041925C2
hypercenterD4×Q8D181
hypocenterC181#3266
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#19153847C2semidirect
2. C22×#15668436C2semidirect
4. #26041671C2semidirect
5. #26041527C2semidirect
6. #26041518C2semidirect
7. #26041501C2semidirect
8. #26041481C2semidirect
9. #26041439C2semidirect
10. #26041435C2semidirect
11. #26041429C2semidirect
12. #26041418C2semidirect
13. #26041377C2semidirect
14. #26041351C2semidirect
15. D4×Q8×C181C2semidirect
19. C2×#15668439C22semidirect
20. C2×#15668439C22semidirect
21. C2×#15668439C22semidirect
29. C2×#15668433C22semidirect
39. #19153749C22semidirect
40. C4⋊#13089678C22semidirect
41. #19153740C22semidirect
42. #19153740C22semidirect
43. #19153740C22semidirect
44. #19153739C22semidirect
48. C22⋊#13089678C22semidirect
49. #19153727C22semidirect
51. C22×Q8×C181C22semidirect
53. C4⋊Q8×C181C22semidirect
57. C22⋊Q8×C181C22semidirect
58. C4×Q8×C181C22semidirect
82. #15668436D4semidirect
176. C22×C181#137semidirect
204. C181#3266semidirect
208. D4×Q8D181semidirect
223. C22×Q8C2×D181semidirect
239. C4⋊Q8C2×D181semidirect
246. C22⋊Q8C2×D181semidirect
250. C4×Q8C2×D181semidirect
388. C22#19153847semidirect

Maximal subgroups

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

subgroupindex
C2×#191538472
C22×#156684362
#260419252
#260416712
#260415272
#260415182
#260415012
#260414812
#260414392
#260414352
#260414292
#260414182
#260413772
#260413512
D4×Q8×C1812
#3266181