Finite Group Databasebeta

← #37545431

Subgroups — #37545431

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
2C2×SD16×D181C211yesmaximal, characteristic, complemented
3#26041936C211yesmaximal, characteristic, complemented
4#26041926C211yesmaximal, characteristic, complemented
5#26041662C211yesmaximal, characteristic, complemented
6#26041507C211yesmaximal, characteristic, complemented
7#26041506C211yesmaximal, characteristic, complemented
8#26041499C211yesmaximal, characteristic, complemented
9#26041493C211yesmaximal, characteristic, complemented
10#26041443C211yesmaximal, characteristic, complemented
11#26041436C211yesmaximal, characteristic, complemented
12#26041411C211yesmaximal, characteristic, complemented
13#26041364C211yesmaximal, characteristic, complemented
14#26041342C211yesmaximal, characteristic, complemented
15#449×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19#19153836—24no—
20C2×Dic724C2211yescharacteristic, complemented
21C2×#15668439C2211yescharacteristic, complemented
22C2×#15668438C2211yescharacteristic
23C2×#15668437C2211yescharacteristic
24C2×#15668437C2211yescharacteristic, complemented
25C2×#15668436C2211yescharacteristic
26SD16×D181—24no—
27C2×#15668434C2211yescharacteristic
28C22×#13089678C2211yescharacteristic, complemented
29C2×C8×D181C2211yescharacteristic
30#19153803—21no—
31#19153803C2211yescharacteristic
32#19153803C2211yescharacteristic, complemented
33#19153798—21no—
34#19153796C2211yescharacteristic
35#19153795—21no—
36#19153795C2211yescharacteristic, complemented
37#19153787—21no—
38#19153787—21no—
39#19153787C2211yescharacteristic
40#19153784—21no—
41Q8×Dic181C2211yescharacteristic, complemented
42#19153778C2211yescharacteristic
43C4×C22⋊D181—21no—
44C4×C22⋊D181C2211yescharacteristic
45C22⋊C4×D181—21no—
46#19153747C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153739C2211yescharacteristic, complemented
50#19153738C2211yescharacteristic
51#19153737C2211yescharacteristic
52#19153728C2211yescharacteristic
53C8×Dic181C2211yescharacteristic, complemented
54C2×#41×C181C2211yescharacteristic, complemented
55C2×SD16×C181C2211yescharacteristic
56C2×D8×C181C2211yescharacteristic
57#121×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic, complemented
59#102×C181C2211yescharacteristic, complemented
60C22⋊C8×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668449—22no—
63#15668449—41no—
64C2×Dic362—21no—
65C2×Dic362C2311yescharacteristic
66C2×Dic362C2311yescharacteristic
67Q8×D181—22no—
68Q8×D181—41no—
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181D411yescharacteristic
74D4×D181—22no—
75D4×D181—41no—
76C2×C4×D181—21no—
77C2×C4×D181—21no—
78C2×C4×D181—21no—
79C2×C4×D181C2311yescharacteristic
80Dic724—41no—
81#15668439—22no—
82#15668438—22no—
83#15668437—22no—
84#15668437—22no—
85#15668436D414yescomplemented
86#15668434—22no—
87C2×#13089678—22no—
88C2×#13089678C2311yescharacteristic
89C2×#13089678C2311yescharacteristic
90C8×D181—22no—
91C4⋊Dic181—21no—
92C4⋊Dic181—21no—
93C4⋊Dic181C2311yescharacteristic
94#15668428—21no—
95#15668428D411yescharacteristic
96C22⋊Dic181—21no—
97C22⋊Dic181—21no—
98C22⋊Dic181—21no—
99C22⋊Dic181—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181—21no—
102C22⋊Dic181D411yescharacteristic
103#15668426—21no—
104#15668426—21no—
105#15668426D411yescharacteristic
106C4×Dic181—21no—
107C4×Dic181C2311yescharacteristic
108#41×C181—22no—
109#41×C181—41no—
110C2×Q8×C181C2311yescharacteristic
111C2×D4×C181—21no—
112C2×D4×C181—21no—
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115C22×C4×C181—21no—
116C22×C4×C181C2311yescharacteristic
117SD16×C181—22no—
118D8×C181—41no—
119C2×C8×C181C2311yescharacteristic
120C2×C8×C181C2311yescharacteristic
121C4⋊C4×C181C2311yescharacteristic
122C22⋊C4×C181—21no—
123C22×D181—21no—
124C22×D181—42no—
125C22×D181—42no—
126C22×D181—81no—
127C22×D181C2×D411yescharacteristic
128Dic362—21no—
129Dic362—21no—
130Dic362—21no—
131Dic362—21no—
132Dic362—42no—
133D724—21no—
134D724C2×D412yes—
135C22⋊D181—22no—
136C22⋊D181—42no—
137C22⋊D181—42no—
138C22⋊D181—42no—
139C2×Dic181—21no—
140C2×Dic181—21no—
141C2×Dic181—21no—
142C2×Dic181C2×D411yescharacteristic
143C2×Dic181C2×D411yescharacteristic
144C2×Dic181C2×D411yescharacteristic
145C4×D181—21no—
146C4×D181—21no—
147C4×D181—42no—
148C4×D181—42no—
149C4×D181—81no—
150#13089678—21no—
151#13089678C2×D412yes—
152C23×C181—21no—
153C23×C181—21no—
154C23×C181C2×D411yescharacteristic
155Q8×C181—21no—
156Q8×C181C2×D412yes—
157D4×C181—21no—
158D4×C181—21no—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181—42no—
162D4×C181—81no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181—42no—
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181—21no—
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—21no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181—41no—
178C22×C181—41no—
179C22×C181—41no—
180C22×C181—41no—
181C22×C181C22×D411yescharacteristic
182C2×D181—21no—
183C2×D181—21no—
184C2×D181—22no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—84no—
188C2×D181#13512yescomplemented
189C4×C181—21no—
190C4×C181—22no—
191C4×C181—41no—
192C4×C181C22×D411yescharacteristic
193C4×C181H(5,𝔽2)11yescharacteristic
194Dic181—21no—
195Dic181—21no—
196Dic181—21no—
197Dic181—41no—
198Dic181—41no—
199Dic181—41no—
200C2×C181—41no—
201C2×C181—81no—
202C2×C181—81no—
203C2×C181D4211yescharacteristic
204C2×C181C2×#13511yescharacteristic
205C2×C181#57711yescharacteristic
206D181—22no—
207D181—42no—
208C181#328211yescharacteristic, complemented, Sylow
209#3282—1811nomaximal, Sylow
210C2×#135—1811no—
211C22×SD16—1811no—
212#548—1811no—
213#546—1811no—
214#495—1811no—
215#465—1811no—
216#465—1811no—
217#464—1811no—
218#463—1811no—
219#452—1811no—
220#451—1811no—
221#449D18111yescharacteristic, complemented
222#447—1811no—
223C4×SD16—1811no—
224C8×D4—1811no—
225C2×#41—1811no—
226C2×#41C2×D18111yescharacteristic, complemented
227C22×Q8—1811no—
228C22×D4—1811no—
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×SD16C2×D18111yescharacteristic
238C2×D8—1811no—
239C2×D8C2×D18111yescharacteristic
240C22×C8—1811no—
241C22×C8—1811no—
242#124—3621no—
243#124—1811no—
244#124—1811no—
245#123—3621no—
246C22⋊Q8—3621no—
247C22⋊Q8—1811no—
248C22⋊Q8—1811no—
249#121—3621no—
250#121—3621no—
251#121—1811no—
252#121C2×D18111yescharacteristic
253#120—3621no—
254C4×Q8—1811no—
255C4×D4—3621no—
256C4×D4—1811no—
257C4×D4—1811no—
258C2×C22⋊C4—3621no—
259#106—1811no—
260C4⋊C8—1811no—
261#103—1811no—
262#103—1811no—
263#103C2×D18111yescharacteristic, complemented
264#102—1811no—
265#102—1811no—
266#102C2×D18111yescharacteristic, complemented
267C22⋊C8—1811no—
268C22⋊C8C2×D18111yescharacteristic, complemented
269C4×C8—1811no—
270C24—3621no—
271#41—22no—
272#41—41no—
273#41—3622no—
274#41—7241no—
275C2×Q8—3621no—
276C2×Q8—3622no—
277C2×Q8—1811no—
278C2×Q8—1811no—
279C2×Q8—7241no—
280C2×Q8C22×D18111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—3622no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—1811no—
288C2×D4—1811no—
289C2×D4—7241no—
290C2×D4C22×D18111yescharacteristic
291C2×D4C22×D18111yescharacteristic
292C22×C4—21no—
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—1811no—
297C22×C4C22×D18111yescharacteristic
298Q16—3622no—
299Q16—7241no—
300SD16—22no—
301SD16—3622no—
302SD16—3622no—
303SD16—3622no—
304SD16—1814no—
305D8—41no—
306D8—3622no—
307C2×C8—3622no—
308C2×C8—3622no—
309C2×C8—1811no—
310C2×C8—1811no—
311C2×C8C22×D18111yescharacteristic
312C2×C8C22×D18111yescharacteristic
313C4⋊C4—3621no—
314C4⋊C4—3621no—
315C4⋊C4—3621no—
316C4⋊C4—1811no—
317C4⋊C4—1811no—
318C4⋊C4C22×D18111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—1811no—
329C22⋊C4—1811no—
330C42—3621no—
331C42—1811no—
332C23—21no—
333C23—21no—
334C23—3621no—
335C23—14481no—
336C23—1811no—
337C23—7242no—
338C23—7242no—
339C23D4×D18111yescharacteristic
340Q8—21no—
341Q8—3621no—
342Q8—3621no—
343Q8—3621no—
344Q8—3621no—
345Q8—7242no—
346Q8D4×D18112yes—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—81no—
353D4—3622no—
354D4—3621no—
355D4—1812no—
356D4—7242no—
357D4—7242no—
358D4—7242no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—3621no—
365C2×C4—3621no—
366C2×C4—3621no—
367C2×C4—3621no—
368C2×C4—3621no—
369C2×C4—14481no—
370C2×C4—1811no—
371C2×C4—1811no—
372C2×C4—1811no—
373C2×C4—7242no—
374C2×C4—7242no—
375C2×C4D4×D18111yescharacteristic
376C2×C4C23×D18111yescharacteristic
377C8—21no—
378C8—21no—
379C8—3621no—
380C8—1812no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—3621no—
391C22—3621no—
392C22—3622no—
393C22—14484no—
394C22—1812no—
395C22—7241no—
396C22—7241no—
397C22C2×D4×D18111yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—3621no—
402C4—3621no—
403C4—3621no—
404C4—7241no—
405C4—7241no—
406C4—7241no—
407C4C2×D4×D18111yescharacteristic
408C4#1915386111yescharacteristic
409C2—41no—
410C2—81no—
411C2—81no—
412C2—3622no—
413C2—7242no—
414C2#2604191111yescharacteristic
415C2#135×D18111yescharacteristic
416C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#449×C181C2
socleC22×C181C22×D4
hypercenter#449D181
hypocenterC181#3282
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#19153836C2semidirect
2. C2×SD16×D181C2semidirect
3. #26041936C2semidirect
4. #26041926C2semidirect
5. #26041662C2semidirect
6. #26041507C2semidirect
7. #26041506C2semidirect
8. #26041499C2semidirect
9. #26041493C2semidirect
10. #26041443C2semidirect
11. #26041436C2semidirect
12. #26041411C2semidirect
13. #26041364C2semidirect
14. #26041342C2semidirect
15. #449×C181C2semidirect
20. C2×Dic724C22semidirect
21. C2×#15668439C22semidirect
24. C2×#15668437C22semidirect
28. C22×#13089678C22semidirect
32. #19153803C22semidirect
36. #19153795C22semidirect
41. Q8×Dic181C22semidirect
46. #19153747C22semidirect
47. #19153745C22semidirect
48. #19153741C22semidirect
49. #19153739C22semidirect
53. C8×Dic181C22semidirect
54. C2×#41×C181C22semidirect
58. #103×C181C22semidirect
59. #102×C181C22semidirect
60. C22⋊C8×C181C22semidirect
85. #15668436D4semidirect
188. C2×D181#135semidirect
208. C181#3282semidirect
221. #449D181semidirect
226. C2×#41C2×D181semidirect
263. #103C2×D181semidirect
266. #102C2×D181semidirect
268. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538362
C2×SD16×D1812
#260419362
#260419262
#260416622
#260415072
#260415062
#260414992
#260414932
#260414432
#260414362
#260414112
#260413642
#260413422
#449×C1812
#3282181