Finite Group Databasebeta

← #49135370

Subgroups — #49135370

Order 36224.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827198C213yesmaximal, complemented
2#31827191C211yesmaximal, characteristic, complemented
3#31827178C213yesmaximal, complemented
4#31827165C213yesmaximal, complemented
5#31827154C213yesmaximal, complemented
6#31827134C211yesmaximal, characteristic, complemented
7#31827127C211yesmaximal, characteristic, complemented
8#31827122C211yesmaximal, characteristic, complemented
9#31827080C213yesmaximal, complemented
10#31827066C213yesmaximal, complemented
11#31826999C213yesmaximal, complemented
12#31826999C213yesmaximal, complemented
13#31826947C211yesmaximal, characteristic, complemented
14#31826941C211yesmaximal, characteristic, complemented
15C4×#41×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×Q8×D283C2211yescharacteristic
18#22583609C2213yes—
19#22583606C2213yes—
20#22583604C2211yescharacteristic
21#22583604C2213yes—
22#22583603C2216yes—
23#22583602C2213yes—
24#22583602C2213yes—
25#22583601C2213yes—
26#22583598C2213yes—
27#22583598C2213yes—
28#22583597C2213yes—
29#22583596C2216yes—
30#22583593C2216yes—
31#22583593C2213yes—
32#22583593C2213yes—
33#22583592C2216yes—
34#22583590C2213yes—
35#22583589C2213yes—
36#22583589C2213yes—
37#22583585C2213yes—
38#22583584C2213yes—
39#22583584C2213yes—
40#22583583C2216yes—
41#22583581C2213yes—
42#22583581C2213yes—
43Q8×Dic283C2211yescharacteristic
44Q8×Dic283C2211yescharacteristic
45C4×Dic566C2213yes—
46C4×Dic566C2213yes—
47#22583575C2213yes—
48C4×D1132C2213yes—
49C4×D1132C2213yes—
50#22583572C2213yes—
51C4×C22⋊D283C2213yes—
52C4×C22⋊D283C2213yes—
53C4×C22⋊D283C2216yes—
54#22583568C2213yes—
55#22583567C2213yes—
56#22583567C2213yes—
57#22583567C2216yes—
58C4⋊C4×D283C2213yes—
59C2×#41×C283C2211yescharacteristic
60C4×Q8×C283C2211yescharacteristic
61C4×Q8×C283C2211yescharacteristic
62C4×D4×C283C2213yes—
63C4×D4×C283C2213yes—
64#117×C283C2213yes—
65C2×C42×C283C2213yes—
66#17985393—24no—
67C2×Dic566C2313yes—
68C2×Dic566C2313yes—
69Q8×D283—24no—
70C2×D1132C2313yes—
71C2×D1132C2313yes—
72C2×C22⋊D283C2313yes—
73C2×C22⋊D283C2316yes—
74C2×C22⋊D283C2313yes—
75C2×C4×D283C2313yes—
76C2×C4×D283C2313yes—
77C2×C4×D283C2316yes—
78C4⋊Dic283C2313yes—
79C4⋊Dic283C2313yes—
80C4⋊Dic283C2313yes—
81C4⋊Dic283C2313yes—
82Dic283⋊C4C2313yes—
83Dic283⋊C4C2311yescharacteristic
84Dic283⋊C4C2313yes—
85Dic283⋊C4C2313yes—
86Dic283⋊C4C2313yes—
87Dic283⋊C4C2316yes—
88Dic283⋊C4C2311yescharacteristic
89Dic283⋊C4C2313yes—
90Dic283⋊C4C2311yescharacteristic
91C22⋊Dic283C2313yes—
92C22⋊Dic283C2316yes—
93C22⋊Dic283C2313yes—
94(C2×D283)⋊C4C2311yescharacteristic
95(C2×D283)⋊C4C2313yes—
96(C2×D283)⋊C4C2313yes—
97(C2×D283)⋊C4C2313yes—
98(C2×D283)⋊C4C2313yes—
99(C2×D283)⋊C4C2316yes—
100(C2×D283)⋊C4C2311yescharacteristic
101(C2×D283)⋊C4C2313yes—
102(C2×D283)⋊C4C2311yescharacteristic
103C4×Dic283C2316yes—
104C4×Dic283C2313yes—
105C4×Dic283C2313yes—
106#41×C283—24no—
107C2×Q8×C283C2311yescharacteristic
108C2×D4×C283C2313yes—
109C22×C4×C283C2313yes—
110C22×C4×C283C2313yes—
111C22×C4×C283C2313yes—
112C4⋊C4×C283C2313yes—
113C4⋊C4×C283C2313yes—
114C22⋊C4×C283C2316yes—
115C42×C283C2313yes—
116C42×C283C2313yes—
117C42×C283C2313yes—
118C42×C283C2311yescharacteristic
119C22×D283C2411yescharacteristic
120C22×D283C2413yes—
121Dic566—26no—
122Dic566—43no—
123D1132—26no—
124D1132—43no—
125C22⋊D283—43no—
126C22⋊D283—43no—
127C22⋊D283—46no—
128C2×Dic283C2411yescharacteristic
129C2×Dic283C2413yes—
130C2×Dic283C2411yescharacteristic
131C2×Dic283C2413yes—
132C2×Dic283C2413yes—
133C2×Dic283C2411yescharacteristic
134C4×D283—26no—
135C4×D283—26no—
136C4×D283—46no—
137C23×C283C2413yes—
138Q8×C283#4114yes—
139D4×C283—26no—
140C2×C4×C283—26no—
141C2×C4×C283—43no—
142C2×C4×C283—43no—
143C2×C4×C283C2411yescharacteristic
144C2×C4×C283C2411yescharacteristic
145C2×C4×C283C2411yescharacteristic
146C2×C4×C283C2413yes—
147C2×C4×C283C2413yes—
148C2×C4×C283C2413yes—
149C22×C283—23no—
150C22×C283—23no—
151C22×C283—23no—
152C22×C283C2511yescharacteristic
153C2×D283—21no—
154C2×D283—23no—
155C2×D283—23no—
156C2×D283—21no—
157C2×D283—23no—
158C2×D283—21no—
159C4×C283—21no—
160C4×C283—21no—
161C4×C283—21no—
162C4×C283—23no—
163C4×C283—23no—
164C4×C283C2×#4116yes—
165Dic283—21no—
166Dic283—23no—
167Dic283—23no—
168Dic283—21no—
169Dic283—23no—
170Dic283—21no—
171C2×C283—43no—
172C2×C283C22×#4111yescharacteristic
173C2×C283#58511yescharacteristic
174C2×C283#58511yescharacteristic
175D283—41no—
176D283—43no—
177C283#352411yescharacteristic, complemented, Sylow
178#3524—2831nomaximal, Sylow
179#559—2833no—
180#557—2831no—
181#555—2833no—
182#553—2833no—
183#552—2833no—
184#550—2831no—
185#548—2831no—
186#548—2831no—
187#542—2833no—
188#540—2833no—
189#529—2833no—
190#529—2833no—
191#520—2831no—
192#519—2831no—
193C4×#41D28311yescharacteristic, complemented
194C2×#41—2831no—
195C2×#41C2×D28311yescharacteristic
196C22×Q8—2831no—
197C4⋊Q8—2833no—
198#127—2833no—
199#126—2836no—
200#126—2831no—
201#126—2833no—
202#125—2833no—
203#125—2833no—
204#125—2833no—
205#124—2833no—
206#124—2836no—
207#124—2833no—
208#124—2833no—
209#123—2836no—
210#123—2833no—
211#123—2833no—
212#123—2836no—
213C22⋊Q8—2833no—
214C22⋊Q8—2833no—
215C22⋊Q8—2833no—
216#121—2833no—
217#121—2833no—
218#121—2833no—
219#121—2833no—
220#121—2833no—
221#121—2836no—
222C4×Q8—2831no—
223C4×Q8—2831no—
224C4×Q8—2833no—
225C4×Q8—2833no—
226C4×Q8—2833no—
227C4×Q8C2×D28311yescharacteristic
228C4×Q8C2×D28311yescharacteristic
229C4×D4—2833no—
230C4×D4—2833no—
231C4×D4—2836no—
232C4×D4—2833no—
233C4×D4—2833no—
234C4×D4—2833no—
235C4×D4C2×D28313yes—
236C4×D4C2×D28313yes—
237#117—2833no—
238#117—2833no—
239#117—2833no—
240#117—2836no—
241#117C2×D28313yes—
242C2×C4⋊C4—2833no—
243C2×C42C2×D28313yes—
244#41—24no—
245#41—5664no—
246C2×Q8—2833no—
247C2×Q8—2833no—
248C2×Q8—5664no—
249C2×Q8C22×D28311yescharacteristic
250C2×D4—2833no—
251C2×D4—2836no—
252C2×D4—2833no—
253C2×D4—2833no—
254C2×D4—2833no—
255C2×D4C22×D28313yes—
256C22×C4—2833no—
257C22×C4—2833no—
258C22×C4—2836no—
259C22×C4C22×D28313yes—
260C22×C4C22×D28313yes—
261C22×C4C22×D28313yes—
262C4⋊C4—2833no—
263C4⋊C4—2831no—
264C4⋊C4—2833no—
265C4⋊C4—2833no—
266C4⋊C4—2833no—
267C4⋊C4—2836no—
268C4⋊C4—2831no—
269C4⋊C4—2833no—
270C4⋊C4—2831no—
271C4⋊C4—2833no—
272C4⋊C4—2833no—
273C4⋊C4—2833no—
274C4⋊C4—2833no—
275C4⋊C4C22×D28313yes—
276C4⋊C4C22×D28313yes—
277C22⋊C4—2833no—
278C22⋊C4—2836no—
279C22⋊C4—2833no—
280C22⋊C4—2831no—
281C22⋊C4—2833no—
282C22⋊C4—2833no—
283C22⋊C4—2833no—
284C22⋊C4—2833no—
285C22⋊C4—2836no—
286C22⋊C4—2831no—
287C22⋊C4—2833no—
288C22⋊C4—2831no—
289C22⋊C4C22×D28316yes—
290C42—2836no—
291C42—2833no—
292C42—2833no—
293C42C22×D28311yescharacteristic
294C42C22×D28313yes—
295C42C22×D28313yes—
296C42C22×D28313yes—
297C23—2831no—
298C23—2833no—
299C23C23×D28313yes—
300Q8—5666no—
301Q8—11323no—
302Q8#1798539214yes—
303D4—26no—
304D4—5666no—
305D4—11323no—
306D4—11323no—
307D4—11323no—
308D4—11326no—
309C2×C4—26no—
310C2×C4—43no—
311C2×C4—43no—
312C2×C4—2831no—
313C2×C4—2833no—
314C2×C4—2831no—
315C2×C4—2833no—
316C2×C4—2833no—
317C2×C4—2831no—
318C2×C4—5666no—
319C2×C4—5666no—
320C2×C4—11326no—
321C2×C4C23×D28313yes—
322C2×C4C23×D28311yescharacteristic
323C2×C4C23×D28311yescharacteristic
324C2×C4C23×D28311yescharacteristic
325C2×C4C23×D28313yes—
326C2×C4C23×D28313yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—5661no—
331C22—5663no—
332C22—5663no—
333C22—5661no—
334C22—5663no—
335C22—5661no—
336C22C24×D28311yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—23no—
341C4—23no—
342C4—5661no—
343C4—5663no—
344C4—5663no—
345C4—5661no—
346C4—5663no—
347C4—5661no—
348C4C2×#1798539216yes—
349C2—43no—
350C2—11321no—
351C2—11323no—
352C2C22×#1798539211yescharacteristic
353C2#3182730111yescharacteristic
354C2#3182730111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
FittingC4×#41×C283C2
hypercenterC4×#41D283
hypocenterC283#3524
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #31827198C2semidirect
2. #31827191C2semidirect
3. #31827178C2semidirect
4. #31827165C2semidirect
5. #31827154C2semidirect
6. #31827134C2semidirect
7. #31827127C2semidirect
8. #31827122C2semidirect
9. #31827080C2semidirect
10. #31827066C2semidirect
11. #31826999C2semidirect
12. #31826999C2semidirect
13. #31826947C2semidirect
14. #31826941C2semidirect
15. C4×#41×C283C2semidirect
177. C283#3524semidirect
193. C4×#41D283semidirect

Maximal subgroups

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

subgroupindex
#318271982
#318271912
#318271782
#318271652
#318271542
#318271342
#318271272
#318271222
#318270802
#318270662
#318269992
#318269472
#318269412
C4×#41×C2832
#3524283