Finite Group Databasebeta

← #49134851

Subgroups — #49134851

Order 36224.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827197C213yesmaximal, complemented
2#31827191C211yesmaximal, characteristic, complemented
3#31827186C213yesmaximal, complemented
4#31827167C213yesmaximal, complemented
5#31827161C213yesmaximal, complemented
6#31827134C211yesmaximal, characteristic, complemented
7#31827126C211yesmaximal, characteristic, complemented
8#31827010C213yesmaximal, complemented
9#31827001C213yesmaximal, complemented
10#31826994C213yesmaximal, complemented
11#41×Dic283C211yesmaximal, characteristic, complemented
12C4×#17985393C211yesmaximal, characteristic, complemented
13C4×#17985391C213yesmaximal, complemented
14C2×Q8×Dic283C211yesmaximal, characteristic, complemented
15#548×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985391C2213yes—
18#22583609C2213yes—
19#22583606C2213yes—
20#22583605C2216yes—
21#22583603C2216yes—
22#22583602C2213yes—
23#22583601C2213yes—
24#22583599C2213yes—
25#22583596C2216yes—
26#22583594C2216yes—
27#22583592C2216yes—
28#22583586C2213yes—
29#22583584C2213yes—
30#22583584C2213yes—
31#22583582C2216yes—
32#22583581C2213yes—
33Q8×Dic283C2213yes—
34Q8×Dic283C2211yescharacteristic
35Q8×Dic283C2211yescharacteristic
36Q8×Dic283C2213yes—
37Q8×Dic283C2211yescharacteristic
38C4×Dic566C2213yes—
39#22583575C2213yes—
40C4×D1132C2213yes—
41D4×Dic283C2213yes—
42D4×Dic283C2213yes—
43D4×Dic283C2213yes—
44#22583572C2213yes—
45C4×C22⋊D283C2213yes—
46C4×C22⋊D283C2213yes—
47(C22⋊D283)⋊C4C2216yes—
48(C22⋊D283)⋊C4C2213yes—
49(C22⋊D283)⋊C4C2213yes—
50#22583569C2213yes—
51#22583568C2213yes—
52#22583568C2213yes—
53#22583566C2213yes—
54#22583566C2211yescharacteristic
55C2×C4⋊Dic283C2213yes—
56C2×C4×Dic283C2213yes—
57C2×C4×Dic283C2213yes—
58C42×D283C2213yes—
59C2×#41×C283C2211yescharacteristic
60C22×Q8×C283C2211yescharacteristic
61#124×C283C2213yes—
62C22⋊Q8×C283C2213yes—
63#121×C283C2213yes—
64C4×Q8×C283C2211yescharacteristic
65C4×D4×C283C2213yes—
66#17985393—24no—
67#17985391—212no—
68C2×Dic566C2313yes—
69C2×D1132C2313yes—
70C2×C22⋊D283C2313yes—
71C2×C22⋊D283C2313yes—
72C2×C22⋊D283C2316yes—
73C22×Dic283C2313yes—
74C22×Dic283C2313yes—
75C22×Dic283C2313yes—
76C22×Dic283C2311yescharacteristic
77C2×C4×D283C2313yes—
78C2×C4×D283C2313yes—
79C2×C4×D283C2313yes—
80C4⋊Dic283C2313yes—
81C4⋊Dic283C2313yes—
82C4⋊Dic283C2316yes—
83C4⋊Dic283C2313yes—
84C4⋊Dic283C2313yes—
85Dic283⋊C4C2311yescharacteristic
86Dic283⋊C4C2313yes—
87Dic283⋊C4C2311yescharacteristic
88Dic283⋊C4C2313yes—
89Dic283⋊C4C2316yes—
90C22⋊Dic283C2311yescharacteristic
91C22⋊Dic283C2313yes—
92C22⋊Dic283C2316yes—
93(C2×D283)⋊C4C2313yes—
94(C2×D283)⋊C4C2313yes—
95(C2×D283)⋊C4C2316yes—
96(C2×D283)⋊C4C2316yes—
97C4×Dic283C2311yescharacteristic
98C4×Dic283C2313yes—
99C4×Dic283C2313yes—
100C4×Dic283C2311yescharacteristic
101C4×Dic283C2313yes—
102C4×Dic283C2313yes—
103C4×Dic283C2313yes—
104C4×Dic283C2313yes—
105C4×Dic283C2313yes—
106#41×C283—24no—
107C2×Q8×C283—24no—
108C2×Q8×C283C2313yes—
109C2×Q8×C283C2311yescharacteristic
110C2×D4×C283C2313yes—
111C2×D4×C283C2313yes—
112C22×C4×C283C2313yes—
113C22×C4×C283C2313yes—
114C4⋊C4×C283C2313yes—
115C4⋊C4×C283C2313yes—
116C22⋊C4×C283C2311yescharacteristic
117C22⋊C4×C283C2313yes—
118C22⋊C4×C283C2316yes—
119C42×C283C2313yes—
120C22×D283C2413yes—
121Dic566—26no—
122D1132—26no—
123C22⋊D283—26no—
124C22⋊D283—26no—
125C22⋊D283—46no—
126C2×Dic283—26no—
127C2×Dic283—43no—
128C2×Dic283—43no—
129C2×Dic283#4114yescomplemented
130C2×Dic283C2411yescharacteristic
131C2×Dic283C2413yes—
132C2×Dic283C2411yescharacteristic
133C2×Dic283C2413yes—
134C2×Dic283C2413yes—
135C2×Dic283C2411yescharacteristic
136C2×Dic283C2411yescharacteristic
137C4×D283—26no—
138C4×D283—26no—
139C4×D283—43no—
140C23×C283C2413yes—
141C23×C283C2411yescharacteristic
142Q8×C283—26no—
143Q8×C283#4114yescomplemented
144D4×C283—26no—
145D4×C283—26no—
146C2×C4×C283—26no—
147C2×C4×C283—26no—
148C2×C4×C283C2411yescharacteristic
149C2×C4×C283C2413yes—
150C2×C4×C283C2413yes—
151C2×C4×C283C2411yescharacteristic
152C2×C4×C283C2413yes—
153C22×C283—21no—
154C22×C283—21no—
155C22×C283—23no—
156C22×C283—23no—
157C22×C283—23no—
158C22×C283C2×#4112yes—
159C22×C283C2511yescharacteristic
160C2×D283—23no—
161C2×D283—23no—
162C2×D283—23no—
163C4×C283—21no—
164C4×C283—23no—
165C4×C283—23no—
166C4×C283—21no—
167C4×C283C2×#4116yes—
168Dic283—21no—
169Dic283—23no—
170Dic283—23no—
171Dic283—21no—
172Dic283—23no—
173Dic283C2×#4112yes—
174Dic283C2×#4112yes—
175C2×C283—22no—
176C2×C283—43no—
177C2×C283C22×#4111yescharacteristic
178C2×C283C22×#4111yescharacteristic
179C2×C283#58511yescharacteristic
180D283—43no—
181C283#348011yescharacteristic, complemented, Sylow
182#3480—2831nomaximal, Sylow
183#559—2833no—
184#557—2831no—
185#556—2833no—
186#553—2833no—
187#553—2833no—
188#550—2831no—
189#548—2831no—
190#548D28311yescharacteristic, complemented
191#532—2833no—
192#529—2833no—
193#529—2833no—
194C4×#41—2831no—
195C4×#41—2831no—
196C4×#41—2833no—
197C2×C4×Q8—2831no—
198C2×#41—2831no—
199C2×#41—2833no—
200C2×#41C2×D28311yescharacteristic
201C22×Q8C2×D28311yescharacteristic
202C4⋊Q8—2833no—
203#127—2833no—
204#126—2836no—
205#126—2836no—
206#125—2833no—
207#125—2833no—
208#124—2833no—
209#124—2836no—
210#124C2×D28313yes—
211#123—2836no—
212#123—2836no—
213C22⋊Q8—2833no—
214C22⋊Q8C2×D28313yes—
215#121—2833no—
216#121—2836no—
217#121—2833no—
218#121—2833no—
219#121C2×D28313yes—
220C4×Q8—2831no—
221C4×Q8—2833no—
222C4×Q8—2831no—
223C4×Q8—2833no—
224C4×Q8—2831no—
225C4×Q8—2833no—
226C4×Q8—2833no—
227C4×Q8C2×D28311yescharacteristic
228C4×D4—2833no—
229C4×D4—2833no—
230C4×D4—2833no—
231C4×D4—2833no—
232C4×D4—2833no—
233C4×D4—2833no—
234C4×D4—2833no—
235C4×D4—2833no—
236C4×D4—2836no—
237C4×D4—2833no—
238C4×D4C2×D28313yes—
239#117—2833no—
240#117—2833no—
241#117—2833no—
242#117—2831no—
243#117—2833no—
244C2×C4⋊C4—2833no—
245C2×C42—2833no—
246C2×C42—2833no—
247C2×C42—2833no—
248#41—24no—
249#41—56612no—
250#41—5664no—
251C2×Q8—24no—
252C2×Q8—2833no—
253C2×Q8C22×D28313yes—
254C2×Q8C22×D28311yescharacteristic
255C2×D4—2833no—
256C2×D4—2833no—
257C2×D4—2836no—
258C2×D4—2833no—
259C2×D4C22×D28313yes—
260C2×D4C22×D28313yes—
261C22×C4—2831no—
262C22×C4—2833no—
263C22×C4—2833no—
264C22×C4—2833no—
265C22×C4—2833no—
266C22×C4—2833no—
267C22×C4—2833no—
268C22×C4C22×D28313yes—
269C22×C4C22×D28313yes—
270C4⋊C4—2831no—
271C4⋊C4—2831no—
272C4⋊C4—2833no—
273C4⋊C4—2833no—
274C4⋊C4—2836no—
275C4⋊C4—2833no—
276C4⋊C4—2833no—
277C4⋊C4—2836no—
278C4⋊C4—2833no—
279C4⋊C4—2833no—
280C4⋊C4C22×D28313yes—
281C4⋊C4C22×D28313yes—
282C22⋊C4—2831no—
283C22⋊C4—2833no—
284C22⋊C4—2836no—
285C22⋊C4—2833no—
286C22⋊C4—2836no—
287C22⋊C4—2833no—
288C22⋊C4—2836no—
289C22⋊C4C22×D28311yescharacteristic
290C22⋊C4C22×D28313yes—
291C22⋊C4C22×D28316yes—
292C42—2831no—
293C42—2831no—
294C42—2833no—
295C42—2833no—
296C42—2833no—
297C42—2833no—
298C42—2833no—
299C42—2833no—
300C42—2833no—
301C42C22×D28313yes—
302C23—2833no—
303C23C23×D28313yes—
304C23C23×D28311yescharacteristic
305Q8—26no—
306Q8—5666no—
307Q8#1798539114yescomplemented
308D4—26no—
309D4—26no—
310D4—5666no—
311D4—5666no—
312D4—5666no—
313D4—11326no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—2834no—
317C2×C4—2831no—
318C2×C4—2831no—
319C2×C4—2831no—
320C2×C4—2831no—
321C2×C4—2833no—
322C2×C4—2833no—
323C2×C4—2833no—
324C2×C4—5666no—
325C2×C4—5666no—
326C2×C4—5666no—
327C2×C4—11323no—
328C2×C4—11323no—
329C2×C4—11323no—
330C2×C4C23×D28311yescharacteristic
331C2×C4C23×D28313yes—
332C2×C4C23×D28311yescharacteristic
333C2×C4C23×D28313yes—
334C2×C4C23×D28313yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—5663no—
341C22—5663no—
342C22—5663no—
343C22#41×D28312yes—
344C22C24×D28311yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—2832no—
350C4—2832no—
351C4—5663no—
352C4—5661no—
353C4—5663no—
354C4—5661no—
355C4—5663no—
356C4C2×#1798539116yes—
357C2—22no—
358C2—43no—
359C2—11323no—
360C2C2×#41×D28311yescharacteristic
361C2C22×#1798539111yescharacteristic
362C2#3182730211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#548×C283C2
hypercenter#548D283
hypocenterC283#3480
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31827197C2semidirect
2. #31827191C2semidirect
3. #31827186C2semidirect
4. #31827167C2semidirect
5. #31827161C2semidirect
6. #31827134C2semidirect
7. #31827126C2semidirect
8. #31827010C2semidirect
9. #31827001C2semidirect
10. #31826994C2semidirect
11. #41×Dic283C2semidirect
12. C4×#17985393C2semidirect
13. C4×#17985391C2semidirect
14. C2×Q8×Dic283C2semidirect
15. #548×C283C2semidirect
129. C2×Dic283#41semidirect
143. Q8×C283#41semidirect
181. C283#3480semidirect
190. #548D283semidirect
307. Q8#17985391semidirect

Maximal subgroups

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

subgroupindex
#318271972
#318271912
#318271862
#318271672
#318271612
#318271342
#318271262
#318270102
#318270012
#318269942
#41×Dic2832
C4×#179853932
C4×#179853912
C2×Q8×Dic2832
#548×C2832
#3480283