Finite Group Databasebeta

← #49133113

Subgroups — #49133113

Order 36224.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827161C211yesmaximal, characteristic, complemented
2#31827111C211yesmaximal, characteristic, complemented
3#31826820C211yesmaximal, characteristic, complemented
4#31826801C211yesmaximal, characteristic, complemented
5#31826790C211yesmaximal, characteristic, complemented
6#31826766C211yesmaximal, characteristic, complemented
7#31826701C211yesmaximal, characteristic, complemented
8#31826694C211yesmaximal, characteristic, complemented
9#31826681C211yesmaximal, characteristic, complemented
10#31826612C211yesmaximal, characteristic, complemented
11#31826601C211yesmaximal, characteristic, complemented
12C4×D2264C211yesmaximal, characteristic, complemented
13#31826462C211yesmaximal, characteristic, complemented
14C2×#22583532C211yesmaximal, characteristic, complemented
15C8×D4×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18C2×#17985381C2211yescharacteristic, complemented
19C2×D2264C2211yescharacteristic
20C2×D2264C2211yescharacteristic
21#22583605—21no—
22#22583602C2211yescharacteristic, complemented
23#22583598C2211yescharacteristic
24#22583595—21no—
25#22583592—21no—
26#22583589C2211yescharacteristic
27#22583584C2211yescharacteristic
28#22583584C2211yescharacteristic
29#22583584C2211yescharacteristic, complemented
30#22583583—21no—
31#22583582—21no—
32#22583578—21no—
33C4×D1132C2211yescharacteristic
34C4×D1132C2211yescharacteristic, complemented
35(C22⋊D283)⋊C4—21no—
36#22583569C2211yescharacteristic, complemented
37C2×C4⋊Dic283C2211yescharacteristic
38C22⋊C4×D283—21no—
39#22583544C2211yescharacteristic, complemented
40#22583544C2211yescharacteristic, complemented
41#22583542C2211yescharacteristic, complemented
42#22583535C2211yescharacteristic, complemented
43#22583535C2211yescharacteristic, complemented
44#22583535C2211yescharacteristic, complemented
45#22583532C2211yescharacteristic
46#22583532C2211yescharacteristic
47#22583532C2211yescharacteristic
48#22583532C2211yescharacteristic
49#22583532C2211yescharacteristic
50#22583532C2211yescharacteristic
51#22583532C2211yescharacteristic, complemented
52C22×C8×C283C2211yescharacteristic
53C22×C8×C283C2211yescharacteristic, complemented
54C4×D4×C283C2211yescharacteristic
55C4⋊C8×C283C2211yescharacteristic, complemented
56C22⋊C8×C283C2211yescharacteristic
57C22⋊C8×C283C2211yescharacteristic, complemented
58C4×C8×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60#17985392—22no—
61#17985392—41no—
62C2×Dic566C2311yescharacteristic
63C2×D1132—22no—
64C2×D1132—21no—
65C2×D1132—41no—
66C2×D1132C2311yescharacteristic
67C2×D1132C2311yescharacteristic
68C2×D1132C2311yescharacteristic
69C2×C22⋊D283—21no—
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C22×Dic283—21no—
74C2×C4×D283—21no—
75C2×C4×D283—21no—
76C2×C4×D283—21no—
77#17985381—41no—
78D2264—22no—
79D2264—41no—
80C4⋊Dic283—21no—
81C4⋊Dic283C2311yescharacteristic
82C4⋊Dic283C2311yescharacteristic
83C4⋊Dic283C2311yescharacteristic
84C4⋊Dic283C2311yescharacteristic
85Dic283⋊C4—21no—
86Dic283⋊C4—21no—
87Dic283⋊C4—21no—
88C22⋊Dic283—21no—
89C22⋊Dic283—21no—
90(C2×D283)⋊C4—21no—
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4—21no—
95(C2×D283)⋊C4—21no—
96(C2×D283)⋊C4—21no—
97C4×Dic283—21no—
98C2×D4×C283D411yescharacteristic
99C22×C4×C283C2311yescharacteristic
100C22×C4×C283C2311yescharacteristic
101C2×C8×C283—41no—
102C2×C8×C283—41no—
103C2×C8×C283C2311yescharacteristic
104C2×C8×C283C2311yescharacteristic
105C2×C8×C283C2311yescharacteristic
106C2×C8×C283C2311yescharacteristic
107C4⋊C4×C283D411yescharacteristic
108C22⋊C4×C283D411yescharacteristic
109C22⋊C4×C283D411yescharacteristic
110C42×C283C2311yescharacteristic
111C22×D283—21no—
112C22×D283—21no—
113C22×D283—21no—
114C22×D283—42no—
115C22×D283—42no—
116C22×D283—81no—
117Dic566—21no—
118Dic566—21no—
119D1132—21no—
120D1132—21no—
121D1132—21no—
122D1132—21no—
123D1132—21no—
124D1132—42no—
125D1132#4112yes—
126C22⋊D283—42no—
127C22⋊D283—42no—
128C22⋊D283—81no—
129C22⋊D283—81no—
130C2×Dic283—21no—
131C2×Dic283—21no—
132C2×Dic283—21no—
133C2×Dic283—21no—
134C2×Dic283—21no—
135C2×Dic283—81no—
136C4×D283—42no—
137C4×D283—81no—
138C4×D283—81no—
139C23×C283C2×D411yescharacteristic
140C23×C283C2×D411yescharacteristic
141D4×C283—22no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283C2×D411yescharacteristic
147C2×C4×C283C2×D411yescharacteristic
148C2×C4×C283C2×D411yescharacteristic
149C2×C4×C283C2×D411yescharacteristic
150C2×C4×C283C2411yescharacteristic
151C8×C283—21no—
152C8×C283—21no—
153C8×C283—21no—
154C8×C283—21no—
155C22×C283—21no—
156C22×C283—21no—
157C22×C283—21no—
158C22×C283—21no—
159C22×C283—21no—
160C22×C283#13512yes—
161C22×C283C22×D411yescharacteristic
162C2×D283—22no—
163C2×D283—41no—
164C2×D283—41no—
165C2×D283—41no—
166C2×D283—41no—
167C2×D283—41no—
168C2×D283—41no—
169C2×D283—41no—
170C2×D283—41no—
171C2×D283—84no—
172C4×C283—21no—
173C4×C283—21no—
174C4×C283—21no—
175C4×C283—21no—
176C4×C283C2×#4111yescharacteristic
177C4×C283H(5,𝔽2)11yescharacteristic
178Dic283—41no—
179Dic283—41no—
180Dic283—41no—
181Dic283—41no—
182Dic283—41no—
183C2×C283—22no—
184C2×C283—41no—
185C2×C283#54611yescharacteristic
186C2×C283C2×#13511yescharacteristic
187C2×C283#57611yescharacteristic
188D283—42no—
189D283—81no—
190D283—81no—
191C283#329711yescharacteristic, complemented, Sylow
192#3297—2831nomaximal, Sylow
193#553—2831no—
194#546—2831no—
195#488—2831no—
196#484—2831no—
197#482—2831no—
198#478—2831no—
199#466—2831no—
200#465—2831no—
201#463—2831no—
202#449—2831no—
203#447—2831no—
204C4×D8—2831no—
205C8×D4D28311yescharacteristic, complemented
206#416—2831no—
207C2×#102—2831no—
208C2×#41—2831no—
209C22×D4—2831no—
210C2×SD16—2831no—
211C2×D8—2831no—
212C2×D8—2831no—
213C22×C8C2×D28311yescharacteristic
214C22×C8C2×D28311yescharacteristic, complemented
215#126—5661no—
216#125—2831no—
217#124—2831no—
218#123—5661no—
219#123—5661no—
220C22⋊Q8—2831no—
221#121—2831no—
222#121—2831no—
223#121—2831no—
224#121—5661no—
225#121—5661no—
226#120—5661no—
227C4×D4—2831no—
228C4×D4—2831no—
229C4×D4—5661no—
230C4×D4C2×D28311yescharacteristic
231#117—2831no—
232C2×C4⋊C4—2831no—
233C2×C22⋊C4—5661no—
234#107—2831no—
235#107—2831no—
236#106—2831no—
237C4⋊C8C2×D28311yescharacteristic, complemented
238#103—2831no—
239#103—2831no—
240#103—2831no—
241#102—2831no—
242#102—2831no—
243#102—2831no—
244#102—2831no—
245#102—2831no—
246#102—2831no—
247#102—2831no—
248C22⋊C8C2×D28311yescharacteristic
249C22⋊C8C2×D28311yescharacteristic, complemented
250C4×C8C2×D28311yescharacteristic, complemented
251C24—5661no—
252#41—5662no—
253#41—11321no—
254C2×Q8—2831no—
255C2×D4—2831no—
256C2×D4—2831no—
257C2×D4—2831no—
258C2×D4—5662no—
259C2×D4—5661no—
260C2×D4—5661no—
261C2×D4—5661no—
262C2×D4—5661no—
263C2×D4—5661no—
264C2×D4—11321no—
265C2×D4D113211yescharacteristic
266C22×C4—5661no—
267C22×C4—5661no—
268C22×C4—5661no—
269C22×C4—5661no—
270C22×C4C22×D28311yescharacteristic
271C22×C4C22×D28311yescharacteristic
272SD16—11321no—
273D8—5662no—
274D8—11321no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D28311yescharacteristic
278C2×C8C22×D28311yescharacteristic
279C2×C8C22×D28311yescharacteristic
280C2×C8C22×D28311yescharacteristic
281C4⋊C4—2831no—
282C4⋊C4—2831no—
283C4⋊C4—2831no—
284C4⋊C4—2831no—
285C4⋊C4—5661no—
286C4⋊C4—5661no—
287C4⋊C4—5661no—
288C4⋊C4—5661no—
289C4⋊C4D113211yescharacteristic
290C22⋊C4—5661no—
291C22⋊C4—5661no—
292C22⋊C4—5661no—
293C22⋊C4—5661no—
294C22⋊C4—5661no—
295C22⋊C4—5661no—
296C22⋊C4—5661no—
297C22⋊C4—5661no—
298C22⋊C4—5661no—
299C22⋊C4D113211yescharacteristic
300C22⋊C4D113211yescharacteristic
301C42—5661no—
302C42C22×D28311yescharacteristic
303C23—5661no—
304C23—5661no—
305C23—5661no—
306C23—11322no—
307C23—11322no—
308C23—22641no—
309C23C2×D113211yescharacteristic
310C23C2×D113211yescharacteristic
311Q8—5661no—
312Q8—5661no—
313D4—22no—
314D4—2832no—
315D4—5661no—
316D4—5661no—
317D4—5661no—
318D4—5661no—
319D4—5661no—
320D4—11322no—
321D4—11322no—
322D4—11322no—
323D4—22641no—
324D4—22641no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—5661no—
330C2×C4—5661no—
331C2×C4—5661no—
332C2×C4—5661no—
333C2×C4—5661no—
334C2×C4—11322no—
335C2×C4—22641no—
336C2×C4—22641no—
337C2×C4—22641no—
338C2×C4C2×D113211yescharacteristic
339C2×C4C2×D113211yescharacteristic
340C2×C4C2×D113211yescharacteristic
341C2×C4C2×D113211yescharacteristic
342C2×C4C23×D28311yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—5662no—
353C22—11321no—
354C22—11321no—
355C22—11321no—
356C22—11321no—
357C22—11321no—
358C22—11321no—
359C22—11321no—
360C22—11321no—
361C22—22644no—
362C22#2258363212yes—
363C22C22×D113211yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—11321no—
369C4—11321no—
370C4—11321no—
371C4—11321no—
372C4—11321no—
373C4#41×D28311yescharacteristic
374C4#2258365511yescharacteristic
375C2—22no—
376C2—41no—
377C2—11322no—
378C2—22641no—
379C2—22641no—
380C2#3182711011yescharacteristic
381C2C2×#2258363211yescharacteristic
382C2#3182727011yescharacteristic
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC8×D4×C283C2
socleC22×C283C22×D4
hypercenterC8×D4D283
hypocenterC283#3297
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #31827161C2semidirect
2. #31827111C2semidirect
3. #31826820C2semidirect
4. #31826801C2semidirect
5. #31826790C2semidirect
6. #31826766C2semidirect
7. #31826701C2semidirect
8. #31826694C2semidirect
9. #31826681C2semidirect
10. #31826612C2semidirect
11. #31826601C2semidirect
12. C4×D2264C2semidirect
13. #31826462C2semidirect
14. C2×#22583532C2semidirect
15. C8×D4×C283C2semidirect
18. C2×#17985381C22semidirect
22. #22583602C22semidirect
29. #22583584C22semidirect
34. C4×D1132C22semidirect
36. #22583569C22semidirect
39. #22583544C22semidirect
40. #22583544C22semidirect
41. #22583542C22semidirect
42. #22583535C22semidirect
43. #22583535C22semidirect
44. #22583535C22semidirect
51. #22583532C22semidirect
53. C22×C8×C283C22semidirect
55. C4⋊C8×C283C22semidirect
57. C22⋊C8×C283C22semidirect
58. C4×C8×C283C22semidirect
191. C283#3297semidirect
205. C8×D4D283semidirect
214. C22×C8C2×D283semidirect
237. C4⋊C8C2×D283semidirect
249. C22⋊C8C2×D283semidirect
250. C4×C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318271612
#318271112
#318268202
#318268012
#318267902
#318267662
#318267012
#318266942
#318266812
#318266122
#318266012
C4×D22642
#318264622
C2×#225835322
C8×D4×C2832
#3297283