Finite Group Databasebeta

← #37545432

Subgroups — #37545432

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×#19153835C211yesmaximal, characteristic, complemented
2C22×#15668437C211yesmaximal, characteristic, complemented
3#26041935C211yesmaximal, characteristic, complemented
4#26041923C211yesmaximal, characteristic, complemented
5#26041662C211yesmaximal, characteristic, complemented
6#26041507C211yesmaximal, characteristic, complemented
7#26041506C211yesmaximal, characteristic, complemented
8#26041500C211yesmaximal, characteristic, complemented
9#26041492C211yesmaximal, characteristic, complemented
10#26041442C211yesmaximal, characteristic, complemented
11#26041437C211yesmaximal, characteristic, complemented
12#26041421C211yesmaximal, characteristic, complemented
13#26041370C211yesmaximal, characteristic, complemented
14#26041342C211yesmaximal, characteristic, complemented
15#447×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×#15668448C2211yescharacteristic, complemented
18C22×Dic362C2211yescharacteristic
19#19153835—24no—
20C2×Dic724C2211yescharacteristic, complemented
21C2×Dic724C2211yescharacteristic, complemented
22C2×#15668437—24no—
23C2×#15668437C2211yescharacteristic
24C2×#15668437C2211yescharacteristic
25C2×#15668437C2211yescharacteristic
26C2×#15668437C2211yescharacteristic
27C2×#15668437C2211yescharacteristic, complemented
28C22×#13089678C2211yescharacteristic
29C2×C8×D181C2211yescharacteristic, complemented
30#19153803C2211yescharacteristic
31#19153803C2211yescharacteristic, complemented
32#19153802—21no—
33#19153797—21no—
34#19153796C2211yescharacteristic, complemented
35#19153795C2211yescharacteristic
36#19153793—21no—
37#19153788—21no—
38#19153788—21no—
39#19153787C2211yescharacteristic
40#19153785—21no—
41#19153781C2211yescharacteristic, complemented
42D4×Dic181—21no—
43D4×Dic181C2211yescharacteristic
44C4×C22⋊D181C2211yescharacteristic
45C2×C22⋊Dic181—21no—
46#19153747C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153741C2211yescharacteristic, complemented
50#19153739C2211yescharacteristic, complemented
51#19153737C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic, complemented
53C8×Dic181C2211yescharacteristic, complemented
54C22×D4×C181C2211yescharacteristic
55C2×D8×C181C2211yescharacteristic
56C2×D8×C181C2211yescharacteristic
57#121×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59#102×C181C2211yescharacteristic
60C22⋊C8×C181C2211yescharacteristic
61#15668448—22no—
62#15668448—22no—
63#15668448—41no—
64#15668448—41no—
65C2×Dic362—22no—
66C2×Dic362—21no—
67C2×Dic362—41no—
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70C2×Dic362C2311yescharacteristic
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C22⋊D181D411yescharacteristic
75C22×Dic181—21no—
76C22×Dic181—21no—
77C22×Dic181—21no—
78C22×Dic181—21no—
79C2×C4×D181C2311yescharacteristic
80Dic724—22no—
81Dic724—41no—
82#15668437—22no—
83#15668437—22no—
84#15668437—22no—
85#15668437—22no—
86#15668437D414yescomplemented
87C2×#13089678—22no—
88C2×#13089678C2311yescharacteristic
89C2×#13089678C2311yescharacteristic
90C8×D181—22no—
91C4⋊Dic181—21no—
92C4⋊Dic181C2311yescharacteristic
93#15668428—21no—
94#15668428—21no—
95#15668428D411yescharacteristic
96C22⋊Dic181—21no—
97C22⋊Dic181—21no—
98C22⋊Dic181—21no—
99C22⋊Dic181—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181—21no—
102C22⋊Dic181—21no—
103C22⋊Dic181D411yescharacteristic
104#15668426—21no—
105#15668426D411yescharacteristic
106C4×Dic181—21no—
107C4×Dic181C2311yescharacteristic
108C24×C181—21no—
109C2×D4×C181—22no—
110C2×D4×C181—21no—
111C2×D4×C181—21no—
112C2×D4×C181—41no—
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115C2×D4×C181C2311yescharacteristic
116C22×C4×C181C2311yescharacteristic
117D8×C181—22no—
118D8×C181—41no—
119C2×C8×C181C2311yescharacteristic
120C2×C8×C181C2311yescharacteristic
121C4⋊C4×C181C2311yescharacteristic
122C22⋊C4×C181—21no—
123C22×D181C2×D411yescharacteristic
124Dic362—21no—
125Dic362—21no—
126Dic362—21no—
127Dic362—21no—
128Dic362—21no—
129Dic362—42no—
130Dic362C2×D412yes—
131C22⋊D181—22no—
132C22⋊D181—42no—
133C22⋊D181—42no—
134C22⋊D181—81no—
135C2×Dic181—21no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C2×Dic181—42no—
140C2×Dic181—42no—
141C2×Dic181—42no—
142C2×Dic181—81no—
143C2×Dic181C2×D411yescharacteristic
144C2×Dic181C2×D411yescharacteristic
145C2×Dic181C2×D411yescharacteristic
146C4×D181—21no—
147C4×D181—21no—
148#13089678—21no—
149#13089678C2×D412yes—
150C23×C181—21no—
151C23×C181—21no—
152C23×C181—21no—
153C23×C181—42no—
154C23×C181—42no—
155C23×C181—81no—
156C23×C181C2×D411yescharacteristic
157D4×C181—21no—
158D4×C181—21no—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181—21no—
162D4×C181—42no—
163D4×C181—42no—
164D4×C181C2×D412yes—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181—21no—
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181—21no—
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—22no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181—41no—
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—21no—
187C2×D181—21no—
188C2×D181—21no—
189C4×C181—21no—
190C4×C181—41no—
191C4×C181C22×D411yescharacteristic
192C4×C181H(5,𝔽2)11yescharacteristic
193Dic181—21no—
194Dic181—22no—
195Dic181—21no—
196Dic181—21no—
197Dic181—41no—
198Dic181—41no—
199Dic181—41no—
200C2×C181—22no—
201C2×C181—42no—
202C2×C181—81no—
203C2×C181—81no—
204C2×C181D4211yescharacteristic
205C2×C181C2×#13511yescharacteristic
206C2×C181#57711yescharacteristic
207D181—41no—
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#449—1811no—
222#447D18111yescharacteristic, complemented
223C4×SD16—1811no—
224C8×D4—1811no—
225C2×#41—1811no—
226C2×#41—1811no—
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×D8C2×D18111yescharacteristic
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#103—1811no—
264#102—1811no—
265#102C2×D18111yescharacteristic
266#102C2×D18111yescharacteristic
267C22⋊C8—1811no—
268C22⋊C8C2×D18111yescharacteristic
269C4×C8—1811no—
270C24—21no—
271#41—3622no—
272#41—3622no—
273#41—7241no—
274#41—7241no—
275C2×Q8—3622no—
276C2×Q8—3621no—
277C2×Q8—1811no—
278C2×Q8—1811no—
279C2×Q8—1811no—
280C2×Q8—7241no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—3621no—
288C2×D4—1811no—
289C2×D4C22×D18111yescharacteristic
290C2×D4C22×D18111yescharacteristic
291C2×D4C22×D18111yescharacteristic
292C22×C4—3621no—
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—1811no—
297C22×C4C22×D18111yescharacteristic
298Q16—3622no—
299Q16—7241no—
300SD16—3622no—
301SD16—3622no—
302SD16—3622no—
303SD16—3622no—
304SD16—1814no—
305D8—22no—
306D8—41no—
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—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1811no—
339C23D4×D18111yescharacteristic
340Q8—3621no—
341Q8—3621no—
342Q8—3621no—
343Q8—3621no—
344Q8—3621no—
345Q8—1812no—
346Q8—7242no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—42no—
354D4—3622no—
355D4—14481no—
356D4—7242no—
357D4—7242no—
358D4D4×D18112yes—
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—3621no—
368C2×C4—14481no—
369C2×C4—1811no—
370C2×C4—1811no—
371C2×C4—1811no—
372C2×C4—7242no—
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—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—84no—
393C22—3621no—
394C22—3621no—
395C22—3621no—
396C22#1915383512yescomplemented
397C22C2×D4×D18111yescharacteristic
398C4—21no—
399C4—41no—
400C4—3621no—
401C4—3622no—
402C4—3621no—
403C4—3621no—
404C4—7241no—
405C4—7241no—
406C4—7241no—
407C4C2×D4×D18111yescharacteristic
408C4#1915386111yescharacteristic
409C2—22no—
410C2—42no—
411C2—81no—
412C2—81no—
413C2—7241no—
414C2#2604191111yescharacteristic
415C2C2×#1915383511yescharacteristic
416C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#447×C181C2
socleC22×C181C22×D4
hypercenter#447D181
hypocenterC181#3282
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#19153835C2semidirect
2. C22×#15668437C2semidirect
3. #26041935C2semidirect
4. #26041923C2semidirect
5. #26041662C2semidirect
6. #26041507C2semidirect
7. #26041506C2semidirect
8. #26041500C2semidirect
9. #26041492C2semidirect
10. #26041442C2semidirect
11. #26041437C2semidirect
12. #26041421C2semidirect
13. #26041370C2semidirect
14. #26041342C2semidirect
15. #447×C181C2semidirect
17. C2×#15668448C22semidirect
20. C2×Dic724C22semidirect
21. C2×Dic724C22semidirect
27. C2×#15668437C22semidirect
29. C2×C8×D181C22semidirect
31. #19153803C22semidirect
34. #19153796C22semidirect
41. #19153781C22semidirect
46. #19153747C22semidirect
47. #19153745C22semidirect
48. #19153741C22semidirect
49. #19153741C22semidirect
50. #19153739C22semidirect
51. #19153737C22semidirect
52. #19153728C22semidirect
53. C8×Dic181C22semidirect
86. #15668437D4semidirect
184. C22×C181#135semidirect
208. C181#3282semidirect
222. #447D181semidirect
396. C22#19153835semidirect

Maximal subgroups

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

subgroupindex
C2×#191538352
C22×#156684372
#260419352
#260419232
#260416622
#260415072
#260415062
#260415002
#260414922
#260414422
#260414372
#260414212
#260413702
#260413422
#447×C1812
#3282181