Finite Group Databasebeta

← #37545251

Subgroups — #37545251

Order 23168.

500 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×#19153844C211yesmaximal, characteristic, complemented
2C2×#19153838C211yesmaximal, characteristic, complemented
3#26041934C211yesmaximal, characteristic, complemented
4#26041925C211yesmaximal, characteristic, complemented
5#26041673C211yesmaximal, characteristic, complemented
6#26041537C211yesmaximal, characteristic, complemented
7#26041508C211yesmaximal, characteristic, complemented
8#26041495C211yesmaximal, characteristic, complemented
9#26041485C211yesmaximal, characteristic, complemented
10#26041443C211yesmaximal, characteristic, complemented
11#26041443C211yesmaximal, characteristic, complemented
12#26041432C211yesmaximal, characteristic, complemented
13#26041426C211yesmaximal, characteristic, complemented
14#26041392C211yesmaximal, characteristic, complemented
15#435×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C2×#15668449C2211yescharacteristic
18C2×#15668449C2211yescharacteristic, complemented
19C22×D724C2211yescharacteristic
20#19153844—24no—
21#19153838—24no—
22C2×Dic724C2211yescharacteristic, complemented
23C2×Dic724C2211yescharacteristic, complemented
24C2×#15668438C2211yescharacteristic
25C2×#15668438C2211yescharacteristic
26C2×#15668438C2211yescharacteristic
27C2×#15668438C2211yescharacteristic
28C2×#15668438C2211yescharacteristic, complemented
29C2×D1448C2211yescharacteristic
30C2×D1448C2211yescharacteristic
31#19153816C2211yescharacteristic, complemented
32#19153804C2211yescharacteristic
33#19153801—21no—
34#19153798—21no—
35#19153798—21no—
36#19153795C2211yescharacteristic
37#19153795C2211yescharacteristic
38#19153795C2211yescharacteristic, complemented
39#19153790C2211yescharacteristic
40#19153789—21no—
41#19153787—21no—
42#19153784—21no—
43C4×D724C2211yescharacteristic
44C4×D724C2211yescharacteristic, complemented
45C4⋊C4×D181—21no—
46C22⋊C4×D181—21no—
47#19153748C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153741C2211yescharacteristic, complemented
50#19153741C2211yescharacteristic, complemented
51#19153738C2211yescharacteristic
52#19153738C2211yescharacteristic
53#19153738C2211yescharacteristic, complemented
54C2×M16×C181C2211yescharacteristic
55C22×C8×C181C2211yescharacteristic, complemented
56C4×D4×C181C2211yescharacteristic
57C4⋊C8×C181C2211yescharacteristic, complemented
58C22⋊C8×C181C2211yescharacteristic
59C22⋊C8×C181C2211yescharacteristic, complemented
60#97×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668449—22no—
63#15668449—22no—
64#15668449—22no—
65#15668449—41no—
66#15668449—41no—
67#15668449—41no—
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70C2×Dic362C2311yescharacteristic
71C2×D724—22no—
72C2×D724—21no—
73C2×D724—41no—
74C2×D724C2311yescharacteristic
75C2×D724C2311yescharacteristic
76C2×D724C2311yescharacteristic
77C2×C22⋊D181—21no—
78C2×C22⋊D181—21no—
79C2×C22⋊D181—21no—
80C2×C22⋊D181—21no—
81C2×C4×D181—21no—
82C2×C4×D181—21no—
83C2×C4×D181—21no—
84C2×C4×D181—21no—
85C2×C4×D181—21no—
86Dic724—22no—
87Dic724—41no—
88#15668438—22no—
89#15668438—22no—
90#15668438—22no—
91#15668438—22no—
92#15668438—41no—
93D1448—22no—
94D1448D414yescomplemented
95C4⋊Dic181—21no—
96C4⋊Dic181—21no—
97C4⋊Dic181C2311yescharacteristic
98C4⋊Dic181C2311yescharacteristic
99#15668428—21no—
100#15668428—21no—
101#15668428—21no—
102C22⋊Dic181—21no—
103C22⋊Dic181—21no—
104C22⋊Dic181—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426—21no—
109#15668426—21no—
110C2×D4×C181D411yescharacteristic
111C22×C4×C181C2311yescharacteristic
112C22×C4×C181C2311yescharacteristic
113M16×C181—22no—
114C2×C8×C181—22no—
115C2×C8×C181C2311yescharacteristic
116C2×C8×C181C2311yescharacteristic
117C2×C8×C181C2311yescharacteristic
118C2×C8×C181C2311yescharacteristic
119C4⋊C4×C181D411yescharacteristic
120C22⋊C4×C181D411yescharacteristic
121C22⋊C4×C181D411yescharacteristic
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—21no—
126C22×D181—42no—
127C22×D181—42no—
128C22×D181—81no—
129Dic362—21no—
130Dic362—21no—
131Dic362—21no—
132Dic362—21no—
133Dic362—21no—
134Dic362—21no—
135D724—21no—
136D724—21no—
137D724—21no—
138D724—21no—
139D724—42no—
140D724C2×D412yes—
141D724C2×D412yes—
142C22⋊D181—42no—
143C22⋊D181—42no—
144C22⋊D181—42no—
145C22⋊D181—81no—
146C2×Dic181—21no—
147C2×Dic181—21no—
148C2×Dic181—21no—
149C2×Dic181—21no—
150C2×Dic181—21no—
151C4×D181—42no—
152C4×D181—42no—
153C4×D181—42no—
154C4×D181—42no—
155C4×D181—81no—
156C23×C181C2×D411yescharacteristic
157C23×C181C2×D411yescharacteristic
158D4×C181—41no—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181C2×D411yescharacteristic
164C2×C4×C181C2×D411yescharacteristic
165C2×C4×C181C2×D411yescharacteristic
166C2×C4×C181C2×D411yescharacteristic
167C2×C4×C181C2411yescharacteristic
168C8×C181—21no—
169C8×C181—21no—
170C8×C181—21no—
171C8×C181C2×D412yes—
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—21no—
175C22×C181—21no—
176C22×C181—21no—
177C22×C181—21no—
178C22×C181C22×D411yescharacteristic
179C2×D181—22no—
180C2×D181—22no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—41no—
184C2×D181—41no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—41no—
188C2×D181—84no—
189C4×C181—21no—
190C4×C181—21no—
191C4×C181—21no—
192C4×C181—21no—
193C4×C181C22×D411yescharacteristic
194C4×C181H(5,𝔽2)11yescharacteristic
195Dic181—41no—
196Dic181—41no—
197Dic181—41no—
198Dic181—41no—
199Dic181—41no—
200C2×C181—41no—
201C2×C181—41no—
202C2×C181D4211yescharacteristic
203C2×C181#57511yescharacteristic
204C2×C181#57711yescharacteristic
205D181—42no—
206D181—42no—
207D181—81no—
208C181#326711yescharacteristic, complemented, Sylow
209#3267—1811nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1811no—
211C2×#135—1811no—
212#547—1811no—
213#546—1811no—
214#497—1811no—
215#470—1811no—
216#465—1811no—
217#463—1811no—
218#461—1811no—
219#452—1811no—
220#452—1811no—
221#450—1811no—
222#449—1811no—
223#442—1811no—
224#435D18111yescharacteristic, complemented
225C2×#41—1811no—
226C2×#41—1811no—
227C2×#41—1811no—
228C22×D4—1811no—
229AGL(1,ℤ/8)—3624no—
230#135—3624no—
231C2×Q16—1811no—
232C2×Q16—1811no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16—1811no—
237C2×SD16—1811no—
238C2×D8—1811no—
239C2×D8—1811no—
240C2×M16C2×D18111yescharacteristic
241C22×C8C2×D18111yescharacteristic, complemented
242C4⋊Q8—1811no—
243#124—1811no—
244#123—3621no—
245#123—3621no—
246#123—3621no—
247C22⋊Q8—1811no—
248C22⋊Q8—1811no—
249C22⋊Q8—1811no—
250#121—3621no—
251#121—3621no—
252#121—1811no—
253#120—3621no—
254C4×D4—1811no—
255C4×D4—1811no—
256C4×D4C2×D18111yescharacteristic
257C2×C4⋊C4—3621no—
258C2×C22⋊C4—3621no—
259#106—1811no—
260C4⋊C8C2×D18111yescharacteristic, complemented
261#103—1811no—
262#103—1811no—
263#103—1811no—
264#102—1811no—
265#102—1811no—
266#102—1811no—
267C22⋊C8C2×D18111yescharacteristic
268C22⋊C8C2×D18111yescharacteristic, complemented
269#97C2×D18111yescharacteristic, complemented
270C24—3621no—
271#41—3622no—
272#41—3622no—
273#41—3622no—
274#41—7241no—
275#41—7241no—
276#41—7241no—
277C2×Q8—1811no—
278C2×Q8—1811no—
279C2×Q8—1811no—
280C2×D4—3622no—
281C2×D4—3621no—
282C2×D4—3621no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—1811no—
287C2×D4—1811no—
288C2×D4—1811no—
289C2×D4—7241no—
290C2×D4D72411yescharacteristic
291C22×C4—3621no—
292C22×C4—3621no—
293C22×C4—3621no—
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—7241no—
305D8—3622no—
306D8—1814no—
307M16—22no—
308C2×C8—22no—
309C2×C8C22×D18111yescharacteristic
310C2×C8C22×D18111yescharacteristic
311C2×C8C22×D18111yescharacteristic
312C2×C8C22×D18111yescharacteristic
313C4⋊C4—3621no—
314C4⋊C4—3621no—
315C4⋊C4—3621no—
316C4⋊C4—3621no—
317C4⋊C4—3621no—
318C4⋊C4—1811no—
319C4⋊C4—1811no—
320C4⋊C4D72411yescharacteristic
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—3621no—
329C22⋊C4D72411yescharacteristic
330C22⋊C4D72411yescharacteristic
331C42C22×D18111yescharacteristic
332C23—3621no—
333C23—3621no—
334C23—3621no—
335C23—14481no—
336C23—7242no—
337C23—7242no—
338C23C2×D72411yescharacteristic
339C23C2×D72411yescharacteristic
340Q8—3621no—
341Q8—3621no—
342Q8—3621no—
343Q8—3621no—
344Q8—3621no—
345Q8—3621no—
346D4—41no—
347D4—3621no—
348D4—3621no—
349D4—3621no—
350D4—3621no—
351D4—14481no—
352D4—1812no—
353D4—1812no—
354D4—7242no—
355D4—7242no—
356D4—7242no—
357D4—7242no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—3621no—
363C2×C4—3621no—
364C2×C4—3621no—
365C2×C4—3621no—
366C2×C4—3621no—
367C2×C4—14481no—
368C2×C4—7242no—
369C2×C4—7242no—
370C2×C4—7242no—
371C2×C4—7242no—
372C2×C4C2×D72411yescharacteristic
373C2×C4C2×D72411yescharacteristic
374C2×C4C2×D72411yescharacteristic
375C2×C4C2×D72411yescharacteristic
376C2×C4C23×D18111yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D18112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—3622no—
388C22—3622no—
389C22—14484no—
390C22—7241no—
391C22—7241no—
392C22—7241no—
393C22—7241no—
394C22—7241no—
395C22—7241no—
396C22—7241no—
397C22C22×D72411yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—7241no—
403C4—7241no—
404C4—7241no—
405C4—7241no—
406C4—7241no—
407C4C2×D4×D18111yescharacteristic
408C4#1915386211yescharacteristic
409C2—41no—
410C2—41no—
411C2—14481no—
412C2—7242no—
413C2—7242no—
414C2D4×D72411yescharacteristic
415C2#2604207911yescharacteristic
416C2#2604209111yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#435×C181C2
socleC22×C181C22×D4
hypercenter#435D181
hypocenterC181#3267
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19153844C2semidirect
2. C2×#19153838C2semidirect
3. #26041934C2semidirect
4. #26041925C2semidirect
5. #26041673C2semidirect
6. #26041537C2semidirect
7. #26041508C2semidirect
8. #26041495C2semidirect
9. #26041485C2semidirect
10. #26041443C2semidirect
11. #26041443C2semidirect
12. #26041432C2semidirect
13. #26041426C2semidirect
14. #26041392C2semidirect
15. #435×C181C2semidirect
18. C2×#15668449C22semidirect
22. C2×Dic724C22semidirect
23. C2×Dic724C22semidirect
28. C2×#15668438C22semidirect
31. #19153816C22semidirect
38. #19153795C22semidirect
44. C4×D724C22semidirect
47. #19153748C22semidirect
48. #19153741C22semidirect
49. #19153741C22semidirect
50. #19153741C22semidirect
53. #19153738C22semidirect
55. C22×C8×C181C22semidirect
57. C4⋊C8×C181C22semidirect
59. C22⋊C8×C181C22semidirect
60. #97×C181C22semidirect
94. D1448D4semidirect
208. C181#3267semidirect
224. #435D181semidirect
241. C22×C8C2×D181semidirect
260. C4⋊C8C2×D181semidirect
268. C22⋊C8C2×D181semidirect
269. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
C2×#191538382
#260419342
#260419252
#260416732
#260415372
#260415082
#260414952
#260414852
#260414432
#260414322
#260414262
#260413922
#435×C1812
#3267181