Finite Group Databasebeta

← #49133252

Subgroups — #49133252

Order 36224.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827119C211yesmaximal, characteristic, complemented
2#31826824C211yesmaximal, characteristic, complemented
3#31826798C211yesmaximal, characteristic, complemented
4#31826776C211yesmaximal, characteristic, complemented
5#31826760C211yesmaximal, characteristic, complemented
6#31826713C211yesmaximal, characteristic, complemented
7#31826699C211yesmaximal, characteristic, complemented
8#31826677C211yesmaximal, characteristic, complemented
9#31826610C211yesmaximal, characteristic, complemented
10#31826604C211yesmaximal, characteristic, complemented
11#31826563C211yesmaximal, characteristic, complemented
12#31826537C211yesmaximal, characteristic, complemented
13#31826466C211yesmaximal, characteristic, complemented
14C2×#22583534C211yesmaximal, characteristic, complemented
15#548×C283C211yesmaximal, characteristic, complemented
16C2×#17985391—21no—
17C2×#17985379C2211yescharacteristic
18C2×#17985379C2211yescharacteristic
19C2×#17985377C2211yescharacteristic
20C2×#17985376C2211yescharacteristic
21C22×C283⋊C8C2211yescharacteristic
22#22583610C2211yescharacteristic
23#22583594—21no—
24#22583594—21no—
25#22583590—21no—
26#22583584C2211yescharacteristic
27#22583584C2211yescharacteristic
28C4×D1132C2211yescharacteristic
29C2×C4⋊Dic283C2211yescharacteristic
30C2×C4⋊Dic283C2211yescharacteristic
31#22583543C2211yescharacteristic
32#22583543C2211yescharacteristic
33#22583543C2211yescharacteristic
34C4⋊(C283⋊C8)C2211yescharacteristic
35#22583534C2211yescharacteristic
36#22583534C2211yescharacteristic
37#22583534C2211yescharacteristic
38#22583534C2211yescharacteristic
39#22583534C2211yescharacteristic
40#22583534C2211yescharacteristic
41#22583531C2211yescharacteristic
42#22583531C2211yescharacteristic
43#22583531C2211yescharacteristic
44#22583530C2211yescharacteristic
45C22⋊(C283⋊C8)C2211yescharacteristic
46C22⋊(C283⋊C8)C2211yescharacteristic
47#22583521C2211yescharacteristic
48C2×#41×C283C2211yescharacteristic
49C22×Q8×C283C2211yescharacteristic
50#124×C283—21no—
51#124×C283C2211yescharacteristic
52C22⋊Q8×C283—21no—
53C22⋊Q8×C283C2211yescharacteristic
54#121×C283—21no—
55#121×C283C2211yescharacteristic
56C4×Q8×C283C2211yescharacteristic
57C4×D4×C283—21no—
58C4×D4×C283C2211yescharacteristic
59#17985391—44no—
60C2×Dic566—21no—
61C2×D1132C2311yescharacteristic
62C2×C22⋊D283—21no—
63C2×C22⋊D283—21no—
64C22×Dic283—21no—
65C22×Dic283—21no—
66C2×C4×D283—21no—
67#17985379—22no—
68#17985379—41no—
69#17985377—41no—
70#17985376—41no—
71C2×C283⋊C8—41no—
72C2×C283⋊C8C2311yescharacteristic
73C2×C283⋊C8C2311yescharacteristic
74C2×C283⋊C8C2311yescharacteristic
75C2×C283⋊C8C2311yescharacteristic
76C4⋊Dic283—21no—
77C4⋊Dic283—21no—
78C4⋊Dic283—21no—
79C4⋊Dic283C2311yescharacteristic
80C4⋊Dic283C2311yescharacteristic
81C4⋊Dic283C2311yescharacteristic
82(C2×D283)⋊C4—21no—
83(C2×D283)⋊C4—21no—
84(C2×D283)⋊C4—21no—
85#41×C283—22no—
86#41×C283—41no—
87C2×Q8×C283—21no—
88C2×Q8×C283—22no—
89C2×Q8×C283—41no—
90C2×Q8×C283C2311yescharacteristic
91C2×Q8×C283C2311yescharacteristic
92C2×D4×C283—21no—
93C2×D4×C283—21no—
94C2×D4×C283D411yescharacteristic
95C2×D4×C283C2311yescharacteristic
96C22×C4×C283—21no—
97C22×C4×C283—21no—
98C22×C4×C283C2311yescharacteristic
99C22×C4×C283C2311yescharacteristic
100C4⋊C4×C283—21no—
101C4⋊C4×C283—21no—
102C4⋊C4×C283D411yescharacteristic
103C4⋊C4×C283C2311yescharacteristic
104C22⋊C4×C283—21no—
105C22⋊C4×C283—21no—
106C22⋊C4×C283—21no—
107C22⋊C4×C283—21no—
108C22⋊C4×C283D411yescharacteristic
109C22⋊C4×C283D411yescharacteristic
110C42×C283—21no—
111C42×C283C2311yescharacteristic
112C22×D283—21no—
113Dic566—42no—
114D1132—21no—
115D1132—21no—
116C22⋊D283—42no—
117C22⋊D283—42no—
118C2×Dic283—21no—
119C2×Dic283—21no—
120C2×Dic283—21no—
121C2×Dic283—42no—
122C2×Dic283—42no—
123C4×D283—42no—
124C283⋊C8—21no—
125C283⋊C8—21no—
126C283⋊C8—21no—
127C283⋊C8—21no—
128C23×C283—21no—
129C23×C283C2×D411yescharacteristic
130C23×C283C2×D411yescharacteristic
131Q8×C283—21no—
132Q8×C283—21no—
133Q8×C283—21no—
134Q8×C283—42no—
135Q8×C283#4112yescomplemented
136D4×C283—22no—
137D4×C283—21no—
138D4×C283—21no—
139D4×C283—42no—
140D4×C283—42no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283—42no—
149C2×C4×C283—42no—
150C2×C4×C283C2×D411yescharacteristic
151C2×C4×C283C2×D411yescharacteristic
152C2×C4×C283C2×D411yescharacteristic
153C2×C4×C283C2×D411yescharacteristic
154C2×C4×C283C2411yescharacteristic
155C22×C283—21no—
156C22×C283—21no—
157C22×C283—21no—
158C22×C283—21no—
159C22×C283—21no—
160C22×C283—41no—
161C22×C283—41no—
162C22×C283—41no—
163C22×C283AGL(1,ℤ/8)12yes—
164C22×C283C22×D411yescharacteristic
165C2×D283—41no—
166C2×D283—41no—
167C2×D283—41no—
168C4×C283—21no—
169C4×C283—21no—
170C4×C283—22no—
171C4×C283—21no—
172C4×C283—21no—
173C4×C283—41no—
174C4×C283—41no—
175C4×C283C2×#4111yescharacteristic
176C4×C283H(5,𝔽2)11yescharacteristic
177Dic283—41no—
178Dic283—41no—
179Dic283—41no—
180C2×C283—22no—
181C2×C283—41no—
182C2×C283—81no—
183C2×C283#54611yescharacteristic
184C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C283#57811yescharacteristic
186D283—81no—
187C283#330811yescharacteristic, complemented, Sylow
188#3308—2831nomaximal, Sylow
189#548D28311yescharacteristic, complemented
190#547—2831no—
191#489—2831no—
192#484—2831no—
193#480—2831no—
194#477—2831no—
195#468—2831no—
196#466—2831no—
197#463—2831no—
198#449—2831no—
199#448—2831no—
200#440—2831no—
201#435—2831no—
202#417—2831no—
203C2×#103—2831no—
204C2×#41—5661no—
205C2×#41C2×D28311yescharacteristic
206C22×Q8C2×D28311yescharacteristic
207C2×SD16—2831no—
208C2×SD16—2831no—
209C2×D8—2831no—
210C2×M16—2831no—
211C22×C8—2831no—
212C4⋊Q8—2831no—
213#124—21no—
214#124C2×D28311yescharacteristic
215#123—5661no—
216#123—5661no—
217C22⋊Q8—21no—
218C22⋊Q8—5661no—
219C22⋊Q8C2×D28311yescharacteristic
220#121—21no—
221#121—2831no—
222#121—2831no—
223#121C2×D28311yescharacteristic
224C4×Q8C2×D28311yescharacteristic
225C4×D4—21no—
226C4×D4—2831no—
227C4×D4C2×D28311yescharacteristic
228C2×C4⋊C4—2831no—
229C2×C4⋊C4—2831no—
230#107—2831no—
231#107—2831no—
232#107—2831no—
233C4⋊C8—2831no—
234#103—2831no—
235#103—2831no—
236#103—2831no—
237#103—2831no—
238#103—2831no—
239#103—2831no—
240#102—2831no—
241#102—2831no—
242#102—2831no—
243#102—2831no—
244C22⋊C8—2831no—
245C22⋊C8—2831no—
246#97—2831no—
247#41—22no—
248#41—41no—
249#41—11324no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—5661no—
254C2×Q8C22×D28311yescharacteristic
255C2×Q8C22×D28311yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—2831no—
259C2×D4—5661no—
260C2×D4—5661no—
261C2×D4C22⋊D28311yescharacteristic
262C2×D4C22×D28311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—5661no—
266C22×C4—5661no—
267C22×C4—5661no—
268C22×C4C22×D28311yescharacteristic
269C22×C4C22×D28311yescharacteristic
270SD16—5662no—
271SD16—11321no—
272D8—11321no—
273M16—11321no—
274C2×C8—2831no—
275C2×C8—2831no—
276C2×C8—2831no—
277C2×C8—2831no—
278C2×C8—11321no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—2831no—
282C4⋊C4—2831no—
283C4⋊C4—2831no—
284C4⋊C4—5661no—
285C4⋊C4—5661no—
286C4⋊C4—5661no—
287C4⋊C4C22⋊D28311yescharacteristic
288C4⋊C4C22×D28311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—5661no—
294C22⋊C4—5661no—
295C22⋊C4—5661no—
296C22⋊C4C22⋊D28311yescharacteristic
297C22⋊C4C22⋊D28311yescharacteristic
298C42—21no—
299C42C22×D28311yescharacteristic
300C23—21no—
301C23—5661no—
302C23C2×C22⋊D28311yescharacteristic
303C23C2×C22⋊D28311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—11322no—
309Q8#1798539112yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—5661no—
316D4—5661no—
317D4—11322no—
318D4—11322no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—5661no—
329C2×C4—5661no—
330C2×C4—5661no—
331C2×C4—11322no—
332C2×C4—11322no—
333C2×C4—11322no—
334C2×C4C2×C22⋊D28311yescharacteristic
335C2×C4C2×C22⋊D28311yescharacteristic
336C2×C4C2×C22⋊D28311yescharacteristic
337C2×C4C2×C22⋊D28311yescharacteristic
338C2×C4C23×D28311yescharacteristic
339C8—5661no—
340C8—5661no—
341C8—5661no—
342C8—5661no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—11321no—
352C22—11321no—
353C22—11321no—
354C22#2258363512yes—
355C22C22×C22⋊D28311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—11321no—
364C4—11321no—
365C4—11321no—
366C4C2×#1798539111yescharacteristic
367C4#2258365511yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—22641no—
372C2#3182710911yescharacteristic
373C2C2×#2258363511yescharacteristic
374C2#3182728011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#548×C283C2
socleC22×C283C22×D4
hypercenter#548D283
hypocenterC283#3308
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31827119C2semidirect
2. #31826824C2semidirect
3. #31826798C2semidirect
4. #31826776C2semidirect
5. #31826760C2semidirect
6. #31826713C2semidirect
7. #31826699C2semidirect
8. #31826677C2semidirect
9. #31826610C2semidirect
10. #31826604C2semidirect
11. #31826563C2semidirect
12. #31826537C2semidirect
13. #31826466C2semidirect
14. C2×#22583534C2semidirect
15. #548×C283C2semidirect
135. Q8×C283#41semidirect
187. C283#3308semidirect
189. #548D283semidirect
309. Q8#17985391semidirect

Maximal subgroups

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

subgroupindex
#318271192
#318268242
#318267982
#318267762
#318267602
#318267132
#318266992
#318266772
#318266102
#318266042
#318265632
#318265372
#318264662
C2×#225835342
#548×C2832
#3308283