Finite Group Databasebeta

← #49131416

Subgroups — #49131416

Order 36224.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827017C211yesmaximal, characteristic, complemented
2#41×Dic283C211yesmaximal, characteristic
3#31826807C211yesmaximal, characteristic, complemented
4#31826803C211yesmaximal, characteristic, complemented
5#31826758C211yesmaximal, characteristic, complemented
6#31826753C211yesmaximal, characteristic, complemented
7#31826673C211yesmaximal, characteristic, complemented
8#31826668C211yesmaximal, characteristic, complemented
9#31826627C211yesmaximal, characteristic
10#31826572C211yesmaximal, characteristic, complemented
11#31826569C211yesmaximal, characteristic, complemented
12#31826566C211yesmaximal, characteristic, complemented
13#31826562C211yesmaximal, characteristic, complemented
14#31826519C211yesmaximal, characteristic, complemented
15#452×C283C211yesmaximal, characteristic
16C2×#17985382C2211yescharacteristic
17C2×#17985380C2211yescharacteristic
18#22583609C2211yescharacteristic
19#22583608C2211yescharacteristic
20#22583600—21no—
21#22583589C2211yescharacteristic
22#22583587—21no—
23Q8×Dic283C2211yescharacteristic
24Q8×Dic283C2211yescharacteristic
25Q8×Dic283C2211yescharacteristic
26#22583575—21no—
27#22583575C2211yescharacteristic
28D4×Dic283—21no—
29D4×Dic283—21no—
30D4×Dic283C2211yescharacteristic
31D4×Dic283C2211yescharacteristic
32#22583569—21no—
33#22583569C2211yescharacteristic
34#22583566—21no—
35C2×C4×Dic283—21no—
36C2×C4×Dic283C2211yescharacteristic
37#22583544C2211yescharacteristic
38#22583543C2211yescharacteristic
39#22583542C2211yescharacteristic
40#22583541C2211yescharacteristic
41Dic283⋊C8C2211yescharacteristic
42Dic283⋊C8C2211yescharacteristic
43#22583535C2211yescharacteristic
44#22583535C2211yescharacteristic
45#22583534C2211yescharacteristic
46#22583534C2211yescharacteristic
47#22583534C2211yescharacteristic
48#22583531C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50#22583520C2211yescharacteristic
51#22583520C2211yescharacteristic
52C2×#41×C283C2211yescharacteristic
53C2×Q16×C283C2211yescharacteristic
54C2×SD16×C283C2211yescharacteristic
55C22⋊Q8×C283C2211yescharacteristic
56#103×C283C2211yescharacteristic
57#102×C283C2211yescharacteristic
58C22⋊C8×C283C2211yescharacteristic
59C2×Dic566—21no—
60C2×Dic566C2311yescharacteristic
61C22×Dic283—21no—
62C22×Dic283—21no—
63C22×Dic283D411yescharacteristic
64C22×Dic283D411yescharacteristic
65#17985382—22no—
66#17985380—22no—
67C2×C283⋊C8C2311yescharacteristic
68C2×C283⋊C8C2311yescharacteristic
69C4⋊Dic283—21no—
70C4⋊Dic283—21no—
71C4⋊Dic283—21no—
72C4⋊Dic283C2311yescharacteristic
73C4⋊Dic283C2311yescharacteristic
74C4⋊Dic283C2311yescharacteristic
75Dic283⋊C4—21no—
76Dic283⋊C4—21no—
77Dic283⋊C4—21no—
78Dic283⋊C4—21no—
79Dic283⋊C4—21no—
80C22⋊Dic283—21no—
81C22⋊Dic283—21no—
82C22⋊Dic283—21no—
83C22⋊Dic283—21no—
84C4×Dic283—21no—
85C4×Dic283—21no—
86C4×Dic283—21no—
87C4×Dic283—21no—
88C4×Dic283—21no—
89C4×Dic283D411yescharacteristic
90C4×Dic283D411yescharacteristic
91C4×Dic283C2311yescharacteristic
92C4×Dic283C2311yescharacteristic
93#41×C283—22no—
94#41×C283—22no—
95C2×Q8×C283C2311yescharacteristic
96C2×Q8×C283C2311yescharacteristic
97C2×D4×C283—21no—
98C2×D4×C283C2311yescharacteristic
99C22×C4×C283—21no—
100C22×C4×C283C2311yescharacteristic
101Q16×C283—22no—
102SD16×C283—22no—
103C2×C8×C283C2311yescharacteristic
104C2×C8×C283C2311yescharacteristic
105C4⋊C4×C283—21no—
106C4⋊C4×C283C2311yescharacteristic
107C22⋊C4×C283—21no—
108Dic566—21no—
109Dic566—21no—
110Dic566—42no—
111C2×Dic283—22no—
112C2×Dic283—21no—
113C2×Dic283—21no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283—41no—
117C2×Dic283—42no—
118C2×Dic283—42no—
119C2×Dic283C2×D411yescharacteristic
120C2×Dic283C2×D411yescharacteristic
121C2×Dic283C2×D411yescharacteristic
122C2×Dic283C2×D411yescharacteristic
123C283⋊C8—21no—
124C283⋊C8—21no—
125C23×C283—21no—
126C23×C283C2×D411yescharacteristic
127Q8×C283—21no—
128Q8×C283—21no—
129Q8×C283—21no—
130Q8×C283#4112yes—
131D4×C283—24no—
132D4×C283—21no—
133D4×C283#4112yes—
134C2×C4×C283—24no—
135C2×C4×C283—21no—
136C2×C4×C283—21no—
137C2×C4×C283—21no—
138C2×C4×C283—21no—
139C2×C4×C283—21no—
140C2×C4×C283C2×D411yescharacteristic
141C2×C4×C283C2411yescharacteristic
142C8×C283—21no—
143C8×C283—21no—
144C22×C283—21no—
145C22×C283—22no—
146C22×C283—21no—
147C22×C283—21no—
148C22×C283—41no—
149C22×C283—41no—
150C22×C283C22×D411yescharacteristic
151C4×C283—21no—
152C4×C283—22no—
153C4×C283—41no—
154C4×C283—41no—
155C4×C283C2×#4111yescharacteristic
156C4×C283C2×#4111yescharacteristic
157Dic283—21no—
158Dic283—21no—
159Dic283—21no—
160Dic283—41no—
161Dic283—41no—
162Dic283—41no—
163Dic283—41no—
164Dic283#13712yes—
165C2×C283—41no—
166C2×C283—42no—
167C2×C283#52511yescharacteristic
168C2×C283C2×#13711yescharacteristic
169C2×C283#57511yescharacteristic
170C283#311611yescharacteristic, complemented, Sylow
171#3116—2831nomaximal, Sylow
172#533—2831no—
173C4×#41—2831no—
174#485—2831no—
175#485—2831no—
176#477—2831no—
177#476—2831no—
178#462—2831no—
179#461—2831no—
180#452—2831no—
181#452D28311yescharacteristic
182#441—2831no—
183#441—2831no—
184#440—2831no—
185#440—2831no—
186#432—2831no—
187C2×#41C2×D28311yescharacteristic
188C2×Q16—2831no—
189C2×Q16C2×D28311yescharacteristic
190C2×SD16—2831no—
191C2×SD16C2×D28311yescharacteristic
192C4⋊Q8—2831no—
193C4⋊Q8—2831no—
194#125—5661no—
195C22⋊Q8—2831no—
196C22⋊Q8—5661no—
197C22⋊Q8C2×D28311yescharacteristic
198C4×Q8—2831no—
199C4×Q8—2831no—
200C4×Q8—2831no—
201C4×Q8—2831no—
202C4×Q8—5661no—
203C4×D4—2831no—
204C4×D4—2831no—
205C4×D4—5661no—
206C4×D4—5661no—
207#117—2831no—
208#117—5661no—
209#117—5661no—
210C2×C42—2831no—
211C2×C42—5661no—
212#107—2831no—
213#107—2831no—
214#106—2831no—
215#106—2831no—
216C4⋊C8—2831no—
217C4⋊C8—2831no—
218#103—2831no—
219#103—2831no—
220#103—2831no—
221#103—2831no—
222#103—2831no—
223#103C2×D28311yescharacteristic
224#102—2831no—
225#102C2×D28311yescharacteristic
226C22⋊C8—2831no—
227C22⋊C8C2×D28311yescharacteristic
228#97—2831no—
229#97—2831no—
230#41—22no—
231#41—22no—
232C2×Q8—2831no—
233C2×Q8—5661no—
234C2×Q8C22×D28311yescharacteristic
235C2×Q8C22×D28311yescharacteristic
236C2×D4—21no—
237C2×D4C22×D28311yescharacteristic
238C22×C4—21no—
239C22×C4—2831no—
240C22×C4—2831no—
241C22×C4—5661no—
242C22×C4—5661no—
243C22×C4C22×D28311yescharacteristic
244Q16—22no—
245Q16—5662no—
246SD16—22no—
247SD16—5662no—
248C2×C8—2831no—
249C2×C8—2831no—
250C2×C8C22×D28311yescharacteristic
251C2×C8C22×D28311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—2831no—
254C4⋊C4—2831no—
255C4⋊C4—2831no—
256C4⋊C4—5661no—
257C4⋊C4—5661no—
258C4⋊C4—5661no—
259C4⋊C4—5661no—
260C4⋊C4—5661no—
261C4⋊C4—5661no—
262C4⋊C4—5661no—
263C4⋊C4—5661no—
264C4⋊C4C22×D28311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—5661no—
267C22⋊C4—5661no—
268C22⋊C4—5661no—
269C22⋊C4—5661no—
270C42—2831no—
271C42—2831no—
272C42—2831no—
273C42—2831no—
274C42—5661no—
275C42—5661no—
276C42—5661no—
277C42—5661no—
278C42—5661no—
279C23—21no—
280C23D4×D28311yescharacteristic
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—5661no—
285Q8—5661no—
286Q8—11322no—
287Q8#1798539112yes—
288D4—24no—
289D4—21no—
290D4#1798539112yes—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—2831no—
298C2×C4—2831no—
299C2×C4—2831no—
300C2×C4—2831no—
301C2×C4—5662no—
302C2×C4—5661no—
303C2×C4—5661no—
304C2×C4—5661no—
305C2×C4—5661no—
306C2×C4—11322no—
307C2×C4—11321no—
308C2×C4—11322no—
309C2×C4D4×D28311yescharacteristic
310C2×C4C23×D28311yescharacteristic
311C8—21no—
312C8—21no—
313C8—5661no—
314C8—5661no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22C2×D4×D28311yescharacteristic
322C4—21no—
323C4—22no—
324C4—41no—
325C4—41no—
326C4—2832no—
327C4—5661no—
328C4—5661no—
329C4—5661no—
330C4—11321no—
331C4—11321no—
332C4—11321no—
333C4—11321no—
334C4C2×#1798539111yescharacteristic
335C4C2×#1798539111yescharacteristic
336C2—41no—
337C2—42no—
338C2#3182697111yescharacteristic
339C2#137×D28311yescharacteristic
340C2#3182726211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#452×C283C2
socleC22×C283C22×D4
prime radicalC2×#41×C283C22
hypercenter#452D283
hypocenterC283#3116
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #31827017C2semidirect
3. #31826807C2semidirect
4. #31826803C2semidirect
5. #31826758C2semidirect
6. #31826753C2semidirect
7. #31826673C2semidirect
8. #31826668C2semidirect
10. #31826572C2semidirect
11. #31826569C2semidirect
12. #31826566C2semidirect
13. #31826562C2semidirect
14. #31826519C2semidirect
170. C283#3116semidirect

Maximal subgroups

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

subgroupindex
#318270172
#41×Dic2832
#318268072
#318268032
#318267582
#318267532
#318266732
#318266682
#318266272
#318265722
#318265692
#318265662
#318265622
#318265192
#452×C2832
#3116283