Finite Group Databasebeta

← #37547338

Subgroups — #37547338

Order 23168.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042005C211yesmaximal, characteristic, complemented
2#26042001C213yesmaximal, complemented
3#26041970C213yesmaximal, complemented
4#26041964C213yesmaximal, complemented
5Q8×C22⋊D181C211yesmaximal, characteristic, complemented
6#26041941C211yesmaximal, characteristic, complemented
7#26041940C211yesmaximal, characteristic, complemented
8#26041930C213yesmaximal, complemented
9#26041833C213yesmaximal, complemented
10#26041816C213yesmaximal, complemented
11#26041813C213yesmaximal, complemented
12C4×#15668450C211yesmaximal, characteristic, complemented
13C4×#15668449C213yesmaximal, complemented
14C4×Q8×D181C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668449C2213yes—
18C2×Q8×D181C2211yescharacteristic
19#19153815C2213yes—
20#19153814C2213yes—
21#19153811C2216yes—
22#19153810C2216yes—
23#19153808C2213yes—
24#19153807C2213yes—
25#19153806C2216yes—
26#19153805C2213yes—
27#19153804C2213yes—
28#19153801C2216yes—
29#19153799C2216yes—
30#19153796C2213yes—
31#19153795C2213yes—
32#19153794C2216yes—
33#19153792C2213yes—
34#19153792C2213yes—
35#19153791C2213yes—
36#19153790C2213yes—
37Q8×Dic181C2211yescharacteristic
38Q8×Dic181C2211yescharacteristic
39C4×Dic362C2213yes—
40C4×Dic362C2213yes—
41C4×Dic362C2213yes—
42#19153781C2213yes—
43C4×D724C2213yes—
44C4×D724C2213yes—
45C4×D724C2213yes—
46#19153778C2213yes—
47C4×C22⋊D181C2213yes—
48C4×C22⋊D181C2213yes—
49C4×C22⋊D181C2211yescharacteristic
50C4×C22⋊D181C2213yes—
51#19153775C2213yes—
52#19153774C2213yes—
53#19153773C2213yes—
54#19153773C2213yes—
55#19153773C2216yes—
56C4⋊C4×D181C2213yes—
57C42×D181C2213yes—
58C42×D181C2213yes—
59C22×Q8×C181C2211yescharacteristic
60C4×Q8×C181C2211yescharacteristic
61C4×Q8×C181C2213yes—
62C4×Q8×C181C2211yescharacteristic
63C4×Q8×C181C2213yes—
64C2×C4⋊C4×C181C2213yes—
65C2×C42×C181C2213yes—
66#15668450—24no—
67#15668449—212no—
68C2×Dic362C2313yes—
69C2×Dic362C2313yes—
70Q8×D181—24no—
71C2×D724C2313yes—
72C2×D724C2313yes—
73C2×C22⋊D181C2313yes—
74C2×C22⋊D181C2311yescharacteristic
75C2×C4×D181C2311yescharacteristic
76C2×C4×D181C2313yes—
77C2×C4×D181C2313yes—
78C2×C4×D181C2313yes—
79C2×C4×D181C2313yes—
80C2×C4×D181C2313yes—
81C4⋊Dic181C2313yes—
82C4⋊Dic181C2313yes—
83C4⋊Dic181C2313yes—
84C4⋊Dic181C2313yes—
85#15668428C2316yes—
86#15668428C2311yescharacteristic
87#15668428C2313yes—
88#15668428C2313yes—
89#15668428C2313yes—
90#15668428C2316yes—
91#15668428C2316yes—
92C22⋊Dic181C2311yescharacteristic
93C22⋊Dic181C2313yes—
94#15668426C2311yescharacteristic
95#15668426C2313yes—
96#15668426C2313yes—
97#15668426C2313yes—
98#15668426C2316yes—
99#15668426C2316yes—
100#15668426C2316yes—
101C4×Dic181C2311yescharacteristic
102C4×Dic181C2313yes—
103C4×Dic181C2313yes—
104C4×Dic181C2313yes—
105C4×Dic181C2313yes—
106C4×Dic181C2313yes—
107C2×Q8×C181—24no—
108C2×Q8×C181C2313yes—
109C2×Q8×C181C2311yescharacteristic
110C22×C4×C181C2311yescharacteristic
111C22×C4×C181C2313yes—
112C22×C4×C181C2313yes—
113C4⋊C4×C181C2313yes—
114C4⋊C4×C181C2313yes—
115C4⋊C4×C181C2316yes—
116C42×C181C2313yes—
117C42×C181C2313yes—
118C42×C181C2313yes—
119C42×C181C2313yes—
120C22×D181C2413yes—
121C22×D181C2411yescharacteristic
122Dic362—26no—
123Dic362—26no—
124D724—26no—
125D724—26no—
126C22⋊D181—26no—
127C22⋊D181#4114yescomplemented
128C2×Dic181C2411yescharacteristic
129C2×Dic181C2411yescharacteristic
130C2×Dic181C2413yes—
131C2×Dic181C2413yes—
132C2×Dic181C2413yes—
133C2×Dic181C2411yescharacteristic
134C4×D181—26no—
135C4×D181—26no—
136C4×D181—26no—
137C4×D181—22no—
138C4×D181—26no—
139C4×D181—26no—
140C23×C181C2411yescharacteristic
141Q8×C181—26no—
142Q8×C181#4114yescomplemented
143C2×C4×C181—26no—
144C2×C4×C181—26no—
145C2×C4×C181—22no—
146C2×C4×C181C2411yescharacteristic
147C2×C4×C181C2413yes—
148C2×C4×C181C2413yes—
149C2×C4×C181C2413yes—
150C2×C4×C181C2413yes—
151C2×C4×C181C2411yescharacteristic
152C22×C181—21no—
153C22×C181—21no—
154C22×C181C2×#4112yes—
155C22×C181C2511yescharacteristic
156C2×D181—21no—
157C2×D181—21no—
158C2×D181—23no—
159C2×D181—23no—
160C2×D181—23no—
161C2×D181C2×#4112yes—
162C4×C181—23no—
163C4×C181—23no—
164C4×C181—23no—
165C4×C181—21no—
166C4×C181C2×#4116yes—
167C4×C1812−1+412yes—
168Dic181—21no—
169Dic181—23no—
170Dic181—23no—
171Dic181—21no—
172Dic181—23no—
173Dic181C2×#4112yes—
174C2×C181—22no—
175C2×C181C22×#4111yescharacteristic
176C2×C181C22×#4111yescharacteristic
177C2×C181C2×2−1+411yescharacteristic
178D181—22no—
179D181—43no—
180C181#348211yescharacteristic, complemented, Sylow
181#3482—1811nomaximal, Sylow
182#557—1811no—
183#556—1813no—
184#552—1813no—
185#552—1813no—
186D4×Q8—1811no—
187#548—1811no—
188#548—1811no—
189#547—1813no—
190#533—1813no—
191#529—1813no—
192#529—1813no—
193C4×#41—1811no—
194C4×#41—1813no—
195C2×C4×Q8—1811no—
196C2×C4×Q8D18111yescharacteristic, complemented
197C2×#41—1811no—
198C2×#41—1813no—
199C22×Q8—1811no—
200C22×Q8C2×D18111yescharacteristic
201C4⋊Q8—1813no—
202C4⋊Q8—1813no—
203#126—1816no—
204#126—1816no—
205#125—1813no—
206#125—1816no—
207#125—1813no—
208#124—1813no—
209#124—1813no—
210#123—1816no—
211#123—1816no—
212C22⋊Q8—1813no—
213C22⋊Q8—1813no—
214C22⋊Q8—1813no—
215C22⋊Q8—1816no—
216C22⋊Q8—1813no—
217#121—1813no—
218#121—1813no—
219C4×Q8—1811no—
220C4×Q8—1811no—
221C4×Q8—1813no—
222C4×Q8—1813no—
223C4×Q8—1813no—
224C4×Q8—1813no—
225C4×Q8C2×D18111yescharacteristic
226C4×Q8C2×D18113yes—
227C4×Q8C2×D18111yescharacteristic
228C4×Q8C2×D18113yes—
229C4×D4—1811no—
230C4×D4—1813no—
231C4×D4—1813no—
232C4×D4—1813no—
233C4×D4—1813no—
234C4×D4—1813no—
235C4×D4—1813no—
236C4×D4—1813no—
237#117—1813no—
238#117—1813no—
239#117—1813no—
240#117—1813no—
241#117—1816no—
242C2×C4⋊C4—1813no—
243C2×C4⋊C4C2×D18113yes—
244C2×C42—1813no—
245C2×C42—1813no—
246C2×C42C2×D18113yes—
247#41—36212no—
248#41—3624no—
249C2×Q8—24no—
250C2×Q8—3624no—
251C2×Q8—1813no—
252C2×Q8—1813no—
253C2×Q8C22×D18113yes—
254C2×Q8C22×D18111yescharacteristic
255C2×D4—1813no—
256C2×D4—1811no—
257C2×D4—1813no—
258C2×D4—1813no—
259C22×C4—1813no—
260C22×C4—1813no—
261C22×C4—1813no—
262C22×C4—1813no—
263C22×C4—1811no—
264C22×C4—1813no—
265C22×C4C22×D18113yes—
266C22×C4C22×D18113yes—
267C22×C4C22×D18111yescharacteristic
268C4⋊C4—1813no—
269C4⋊C4—1816no—
270C4⋊C4—1811no—
271C4⋊C4—1813no—
272C4⋊C4—1813no—
273C4⋊C4—1813no—
274C4⋊C4—1813no—
275C4⋊C4—1813no—
276C4⋊C4—1813no—
277C4⋊C4—1816no—
278C4⋊C4—1816no—
279C4⋊C4C22×D18113yes—
280C4⋊C4C22×D18113yes—
281C4⋊C4C22×D18116yes—
282C22⋊C4—1811no—
283C22⋊C4—1813no—
284C22⋊C4—1811no—
285C22⋊C4—1813no—
286C22⋊C4—1816no—
287C22⋊C4—1813no—
288C22⋊C4—1813no—
289C22⋊C4—1816no—
290C22⋊C4—1816no—
291C42—1813no—
292C42—1813no—
293C42—1813no—
294C42—1813no—
295C42—1811no—
296C42—1813no—
297C42C22×D18113yes—
298C42C22×D18113yes—
299C42C22×D18113yes—
300C42C22×D18113yes—
301C23—1813no—
302C23—1811no—
303C23C23×D18111yescharacteristic
304Q8—26no—
305Q8—3626no—
306Q8—3626no—
307Q8#1566844914yescomplemented
308D4—3626no—
309D4—3626no—
310D4—3626no—
311D4—1814no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—3626no—
316C2×C4—3626no—
317C2×C4—3626no—
318C2×C4—3626no—
319C2×C4—3622no—
320C2×C4—3626no—
321C2×C4—1811no—
322C2×C4—1811no—
323C2×C4—1813no—
324C2×C4—1813no—
325C2×C4—1813no—
326C2×C4—1811no—
327C2×C4C23×D18113yes—
328C2×C4C23×D18111yescharacteristic
329C2×C4C23×D18113yes—
330C2×C4C23×D18113yes—
331C2×C4C23×D18113yes—
332C2×C4C23×D18111yescharacteristic
333C22—21no—
334C22—21no—
335C22—3621no—
336C22—3621no—
337C22—3623no—
338C22—3623no—
339C22—3623no—
340C22—1812no—
341C22#41×D18112yes—
342C22C24×D18111yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—23no—
347C4—3621no—
348C4—3623no—
349C4—3623no—
350C4—3621no—
351C4—3623no—
352C4—1812no—
353C4C2×#1566844916yes—
354C4#1915386412yes—
355C2—22no—
356C2—3622no—
357C2—7243no—
358C2C2×#41×D18111yescharacteristic
359C2C22×#1566844911yescharacteristic
360C2C2×#1915386411yescharacteristic
subgroupG/H
centerC2×C4C23×D181
commutator, socle, cosocleC22×C181C25
FrattiniC22C24×D181
FittingC2×C4×Q8×C181C2
hypercenterC2×C4×Q8D181
hypocenterC181#3482
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #26042005C2semidirect
2. #26042001C2semidirect
3. #26041970C2semidirect
4. #26041964C2semidirect
5. Q8×C22⋊D181C2semidirect
6. #26041941C2semidirect
7. #26041940C2semidirect
8. #26041930C2semidirect
9. #26041833C2semidirect
10. #26041816C2semidirect
11. #26041813C2semidirect
12. C4×#15668450C2semidirect
13. C4×#15668449C2semidirect
14. C4×Q8×D181C2semidirect
15. C2×C4×Q8×C181C2semidirect
127. C22⋊D181#41semidirect
142. Q8×C181#41semidirect
180. C181#3482semidirect
196. C2×C4×Q8D181semidirect
307. Q8#15668449semidirect

Maximal subgroups

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

subgroupindex
#260420052
#260420012
#260419702
#260419642
Q8×C22⋊D1812
#260419412
#260419402
#260419302
#260418332
#260418162
#260418132
C4×#156684502
C4×#156684492
C4×Q8×D1812
C2×C4×Q8×C1812
#3482181