Finite Group Databasebeta

← #49133342

Subgroups — #49133342

Order 36224.

410 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#31827172C211yesmaximal, characteristic, complemented
2#31826824C211yesmaximal, characteristic, complemented
3#31826798C211yesmaximal, characteristic, complemented
4#31826776C211yesmaximal, characteristic, complemented
5#31826745C211yesmaximal, characteristic, complemented
6#31826709C211yesmaximal, characteristic, complemented
7#31826704C211yesmaximal, characteristic, complemented
8#31826684C211yesmaximal, characteristic, complemented
9#31826627C211yesmaximal, characteristic, complemented
10#31826616C211yesmaximal, characteristic, complemented
11#31826565C211yesmaximal, characteristic, complemented
12#31826537C211yesmaximal, characteristic, complemented
13#31826466C211yesmaximal, characteristic, complemented
14C2×#22583531C211yesmaximal, characteristic, complemented
15#546×C283C211yesmaximal, characteristic
16C2×#17985382C2211yescharacteristic, complemented
17C2×#17985380C2211yescharacteristic
18C2×#17985380C2211yescharacteristic
19C2×#17985376C2211yescharacteristic, complemented
20C22×C283⋊C8C2211yescharacteristic
21#22583610C2211yescharacteristic, complemented
22#22583601—21no—
23#22583589C2211yescharacteristic
24#22583589C2211yescharacteristic, complemented
25#22583587—21no—
26#22583587—21no—
27C4×Dic566C2211yescharacteristic, complemented
28D4×Dic283—21no—
29C2×C4⋊Dic283C2211yescharacteristic
30C2×C4⋊Dic283C2211yescharacteristic, complemented
31#22583543C2211yescharacteristic, complemented
32#22583543C2211yescharacteristic, complemented
33#22583543C2211yescharacteristic, complemented
34C4⋊(C283⋊C8)C2211yescharacteristic, complemented
35#22583534C2211yescharacteristic, complemented
36#22583534C2211yescharacteristic, complemented
37#22583534C2211yescharacteristic, complemented
38#22583533C2211yescharacteristic, complemented
39#22583531C2211yescharacteristic
40#22583531C2211yescharacteristic
41#22583531C2211yescharacteristic
42#22583531C2211yescharacteristic
43#22583531C2211yescharacteristic
44#22583531C2211yescharacteristic
45C22⋊(C283⋊C8)C2211yescharacteristic
46C22⋊(C283⋊C8)C2211yescharacteristic, complemented
47#22583521C2211yescharacteristic, complemented
48C2×#41×C283C2211yescharacteristic
49C22×D4×C283C2211yescharacteristic
50#124×C283C2211yescharacteristic
51#123×C283—21no—
52C22⋊Q8×C283C2211yescharacteristic
53#121×C283—21no—
54#121×C283C2211yescharacteristic
55#120×C283—21no—
56C4×D4×C283C2211yescharacteristic
57C4×D4×C283C2211yescharacteristic
58C2×C22⋊C4×C283—21no—
59C2×Dic566—21no—
60C2×Dic566C2311yescharacteristic
61C22×Dic283—21no—
62C22×Dic283—21no—
63#17985382—41no—
64#17985380—22no—
65#17985380—41no—
66#17985376—41no—
67C2×C283⋊C8—41no—
68C2×C283⋊C8C2311yescharacteristic
69C2×C283⋊C8C2311yescharacteristic
70C2×C283⋊C8C2311yescharacteristic
71C2×C283⋊C8C2311yescharacteristic
72C4⋊Dic283—21no—
73C4⋊Dic283—21no—
74C4⋊Dic283—21no—
75C4⋊Dic283C2311yescharacteristic
76C4⋊Dic283C2311yescharacteristic
77C4⋊Dic283C2311yescharacteristic
78Dic283⋊C4—21no—
79Dic283⋊C4—21no—
80Dic283⋊C4—21no—
81C22⋊Dic283—21no—
82C22⋊Dic283—21no—
83C4×Dic283—21no—
84C24×C283—21no—
85#41×C283—22no—
86#41×C283—41no—
87C2×Q8×C283C2311yescharacteristic
88C2×D4×C283—22no—
89C2×D4×C283—21no—
90C2×D4×C283—21no—
91C2×D4×C283—21no—
92C2×D4×C283—41no—
93C2×D4×C283D411yescharacteristic
94C2×D4×C283C2311yescharacteristic
95C2×D4×C283C2311yescharacteristic
96C22×C4×C283—21no—
97C22×C4×C283—21no—
98C22×C4×C283C2311yescharacteristic
99C22×C4×C283C2311yescharacteristic
100C4⋊C4×C283—21no—
101C4⋊C4×C283D411yescharacteristic
102C4⋊C4×C283C2311yescharacteristic
103C22⋊C4×C283—21no—
104C22⋊C4×C283—21no—
105C22⋊C4×C283—21no—
106C22⋊C4×C283—21no—
107C22⋊C4×C283—21no—
108C22⋊C4×C283D411yescharacteristic
109C22⋊C4×C283D411yescharacteristic
110C42×C283C2311yescharacteristic
111Dic566—21no—
112Dic566—21no—
113Dic566—81no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283—21no—
117C2×Dic283—21no—
118C2×Dic283—81no—
119C2×Dic283—81no—
120C283⋊C8—21no—
121C283⋊C8—21no—
122C283⋊C8—21no—
123C283⋊C8—21no—
124C23×C283—21no—
125C23×C283—21no—
126C23×C283—42no—
127C23×C283—42no—
128C23×C283—81no—
129C23×C283C2×D411yescharacteristic
130C23×C283C2×D411yescharacteristic
131Q8×C283—21no—
132Q8×C283—21no—
133D4×C283—22no—
134D4×C283—21no—
135D4×C283—21no—
136D4×C283—21no—
137D4×C283—42no—
138D4×C283—42no—
139D4×C283—81no—
140D4×C283#4112yes—
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—42no—
148C2×C4×C283—81no—
149C2×C4×C283C2×D411yescharacteristic
150C2×C4×C283C2×D411yescharacteristic
151C2×C4×C283C2×D411yescharacteristic
152C2×C4×C283C2×D411yescharacteristic
153C2×C4×C283C2411yescharacteristic
154C22×C283—21no—
155C22×C283—21no—
156C22×C283—21no—
157C22×C283—22no—
158C22×C283—21no—
159C22×C283—21no—
160C22×C283—41no—
161C22×C283—41no—
162C22×C283—41no—
163C22×C283—41no—
164C22×C283—41no—
165C22×C283—84no—
166C22×C283#13712yes—
167C22×C283C22×D411yescharacteristic
168C4×C283—21no—
169C4×C283—21no—
170C4×C283—21no—
171C4×C283—21no—
172C4×C283—41no—
173C4×C283—41no—
174C4×C283C2×#4111yescharacteristic
175C4×C283H(5,𝔽2)11yescharacteristic
176Dic283—41no—
177Dic283—41no—
178Dic283—41no—
179Dic283—41no—
180C2×C283—22no—
181C2×C283—41no—
182C2×C283—42no—
183C2×C283—81no—
184C2×C283#54611yescharacteristic
185C2×C283C2×#13711yescharacteristic
186C2×C283#57611yescharacteristic
187C283#331411yescharacteristic, complemented, Sylow
188#3314—2831nomaximal, Sylow
189#554—2831no—
190#546D28311yescharacteristic
191#489—2831no—
192#484—2831no—
193#480—2831no—
194#474—2831no—
195#468—2831no—
196#467—2831no—
197#464—2831no—
198#452—2831no—
199#450—2831no—
200#440—2831no—
201#435—2831no—
202#417—2831no—
203C2×#102—2831no—
204C2×#41C2×D28311yescharacteristic
205C22×D4C2×D28311yescharacteristic
206C2×Q16—2831no—
207C2×SD16—2831no—
208C2×SD16—2831no—
209C2×M16—2831no—
210C22×C8—2831no—
211C4⋊Q8—2831no—
212#125—5661no—
213#124C2×D28311yescharacteristic
214#123—21no—
215C22⋊Q8—2831no—
216C22⋊Q8—2831no—
217C22⋊Q8—5661no—
218C22⋊Q8—5661no—
219C22⋊Q8C2×D28311yescharacteristic
220#121—21no—
221#121C2×D28311yescharacteristic
222#120—21no—
223C4×Q8—2831no—
224C4×D4—5661no—
225C4×D4C2×D28311yescharacteristic
226C4×D4C2×D28311yescharacteristic
227C2×C4⋊C4—2831no—
228C2×C4⋊C4—2831no—
229C2×C22⋊C4—21no—
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#102—2831no—
239#102—2831no—
240#102—2831no—
241#102—2831no—
242#102—2831no—
243#102—2831no—
244C22⋊C8—2831no—
245C22⋊C8—2831no—
246#97—2831no—
247C24—21no—
248#41—22no—
249#41—41no—
250C2×Q8—2831no—
251C2×Q8—5661no—
252C2×Q8C22×D28311yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4C22⋊D28311yescharacteristic
259C2×D4C22×D28311yescharacteristic
260C2×D4C22×D28311yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—5661no—
264C22×C4—5661no—
265C22×C4C22×D28311yescharacteristic
266C22×C4C22×D28311yescharacteristic
267Q16—11321no—
268SD16—5662no—
269SD16—11321no—
270M16—11321no—
271C2×C8—2831no—
272C2×C8—2831no—
273C2×C8—2831no—
274C2×C8—2831no—
275C2×C8—11321no—
276C4⋊C4—21no—
277C4⋊C4—2831no—
278C4⋊C4—2831no—
279C4⋊C4—2831no—
280C4⋊C4—5661no—
281C4⋊C4—5661no—
282C4⋊C4—5661no—
283C4⋊C4—5661no—
284C4⋊C4—5661no—
285C4⋊C4—5661no—
286C4⋊C4C22⋊D28311yescharacteristic
287C4⋊C4C22×D28311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—5661no—
294C22⋊C4—5661no—
295C22⋊C4C22⋊D28311yescharacteristic
296C22⋊C4C22⋊D28311yescharacteristic
297C42—5661no—
298C42C22×D28311yescharacteristic
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23C2×C22⋊D28311yescharacteristic
305C23C2×C22⋊D28311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—5661no—
309Q8—5661no—
310Q8—22641no—
311D4—22no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4#1798539112yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—5661no—
328C2×C4—5661no—
329C2×C4—5661no—
330C2×C4—5661no—
331C2×C4—22641no—
332C2×C4—22641no—
333C2×C4C2×C22⋊D28311yescharacteristic
334C2×C4C2×C22⋊D28311yescharacteristic
335C2×C4C2×C22⋊D28311yescharacteristic
336C2×C4C2×C22⋊D28311yescharacteristic
337C2×C4C23×D28311yescharacteristic
338C8—5661no—
339C8—5661no—
340C8—5661no—
341C8—5661no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—84no—
354C22#2258364012yes—
355C22C22×C22⋊D28311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—11321no—
363C4—11321no—
364C4—11321no—
365C4—11321no—
366C4C2×#1798539111yescharacteristic
367C4#2258365511yescharacteristic
368C2—22no—
369C2—41no—
370C2—42no—
371C2—81no—
372C2#3182710911yescharacteristic
373C2C2×#2258364011yescharacteristic
374C2#3182726811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting, prime radical#546×C283C2
socleC22×C283C22×D4
hypercenter#546D283
hypocenterC283#3314
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #31827172C2semidirect
2. #31826824C2semidirect
3. #31826798C2semidirect
4. #31826776C2semidirect
5. #31826745C2semidirect
6. #31826709C2semidirect
7. #31826704C2semidirect
8. #31826684C2semidirect
9. #31826627C2semidirect
10. #31826616C2semidirect
11. #31826565C2semidirect
12. #31826537C2semidirect
13. #31826466C2semidirect
14. C2×#22583531C2semidirect
16. C2×#17985382C22semidirect
19. C2×#17985376C22semidirect
21. #22583610C22semidirect
24. #22583589C22semidirect
27. C4×Dic566C22semidirect
30. C2×C4⋊Dic283C22semidirect
31. #22583543C22semidirect
32. #22583543C22semidirect
33. #22583543C22semidirect
34. C4⋊(C283⋊C8)C22semidirect
35. #22583534C22semidirect
36. #22583534C22semidirect
37. #22583534C22semidirect
38. #22583533C22semidirect
46. C22⋊(C283⋊C8)C22semidirect
47. #22583521C22semidirect
187. C283#3314semidirect

Maximal subgroups

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

subgroupindex
#318271722
#318268242
#318267982
#318267762
#318267452
#318267092
#318267042
#318266842
#318266272
#318266162
#318265652
#318265372
#318264662
C2×#225835312
#546×C2832
#3314283