Finite Group Databasebeta

← #49131520

Subgroups — #49131520

Order 36224.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827084C211yesmaximal, characteristic, complemented
2#31826944C211yesmaximal, characteristic, complemented
3#31826799C211yesmaximal, characteristic, complemented
4#31826777C211yesmaximal, characteristic, complemented
5#31826761C211yesmaximal, characteristic, complemented
6#31826745C211yesmaximal, characteristic, complemented
7#31826684C211yesmaximal, characteristic, complemented
8#31826624C211yesmaximal, characteristic, complemented
9#31826607C211yesmaximal, characteristic
10#31826566C211yesmaximal, characteristic, complemented
11#31826562C211yesmaximal, characteristic, complemented
12#31826557C211yesmaximal, characteristic, complemented
13D8×Dic283C211yesmaximal, characteristic, complemented
14#31826525C211yesmaximal, characteristic, complemented
15#463×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×#17985382C2211yescharacteristic
18C2×#17985380C2211yescharacteristic
19#22583610C2211yescharacteristic
20#22583603—21no—
21#22583596—21no—
22#22583594—21no—
23#22583587—21no—
24#22583586C2211yescharacteristic
25#22583581C2211yescharacteristic
26Q8×Dic283C2211yescharacteristic
27C4×Dic566C2211yescharacteristic
28#22583575C2211yescharacteristic
29D4×Dic283C2211yescharacteristic
30D4×Dic283C2211yescharacteristic
31C4×C22⋊D283—21no—
32(C22⋊D283)⋊C4—21no—
33#22583567C2211yescharacteristic
34#22583566—21no—
35C2×Dic283⋊C4—21no—
36C4⋊C4×D283C2211yescharacteristic
37#22583544C2211yescharacteristic
38#22583544C2211yescharacteristic
39#22583543C2211yescharacteristic
40#22583543C2211yescharacteristic
41C4⋊(C283⋊C8)C2211yescharacteristic
42#22583535C2211yescharacteristic
43#22583535C2211yescharacteristic
44#22583534C2211yescharacteristic
45#22583531C2211yescharacteristic
46#22583531C2211yescharacteristic
47#22583531C2211yescharacteristic
48(C2×D283)⋊C8C2211yescharacteristic
49(C2×D283)⋊C8C2211yescharacteristic
50#22583520C2211yescharacteristic
51C8×Dic283C2211yescharacteristic
52C2×SD16×C283C2211yescharacteristic
53C2×D8×C283C2211yescharacteristic
54#124×C283C2211yescharacteristic
55C4×D4×C283C2211yescharacteristic
56C4⋊C8×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59#17985391—22no—
60#17985391—22no—
61C2×Dic566—21no—
62C2×Dic566C2311yescharacteristic
63C2×C22⋊D283—21no—
64C2×C22⋊D283—21no—
65C22×Dic283—21no—
66C22×Dic283—21no—
67C22×Dic283—21no—
68C2×C4×D283—21no—
69C2×C4×D283C2311yescharacteristic
70#17985382—22no—
71#17985380—22no—
72C2×C283⋊C8C2311yescharacteristic
73C2×C283⋊C8C2311yescharacteristic
74C4⋊Dic283—21no—
75C4⋊Dic283—21no—
76C4⋊Dic283C2311yescharacteristic
77C4⋊Dic283C2311yescharacteristic
78C4⋊Dic283C2311yescharacteristic
79Dic283⋊C4—21no—
80Dic283⋊C4—21no—
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83Dic283⋊C4D411yescharacteristic
84Dic283⋊C4D411yescharacteristic
85C22⋊Dic283—21no—
86C22⋊Dic283—21no—
87C22⋊Dic283—21no—
88(C2×D283)⋊C4—21no—
89(C2×D283)⋊C4D411yescharacteristic
90(C2×D283)⋊C4D411yescharacteristic
91C4×Dic283—21no—
92C4×Dic283—21no—
93C4×Dic283—21no—
94C4×Dic283C2311yescharacteristic
95C2×Q8×C283C2311yescharacteristic
96C2×D4×C283C2311yescharacteristic
97C2×D4×C283C2311yescharacteristic
98C22×C4×C283—21no—
99SD16×C283—22no—
100D8×C283—22no—
101C2×C8×C283C2311yescharacteristic
102C2×C8×C283C2311yescharacteristic
103C4⋊C4×C283C2311yescharacteristic
104C22⋊C4×C283—21no—
105C22⋊C4×C283—21no—
106C22⋊C4×C283—21no—
107C42×C283C2311yescharacteristic
108C22×D283C2×D411yescharacteristic
109Dic566—21no—
110Dic566—81no—
111Dic566#4112yes—
112C22⋊D283—24no—
113C22⋊D283—81no—
114C2×Dic283—24no—
115C2×Dic283—21no—
116C2×Dic283—21no—
117C2×Dic283—21no—
118C2×Dic283—21no—
119C2×Dic283—81no—
120C2×Dic283—81no—
121C2×Dic283C2×D411yescharacteristic
122C2×Dic283C2×D411yescharacteristic
123C2×Dic283C2×D411yescharacteristic
124C4×D283—21no—
125C4×D283—21no—
126C4×D283—81no—
127C283⋊C8—21no—
128C283⋊C8—21no—
129C23×C283—21no—
130C23×C283—21no—
131Q8×C283—21no—
132Q8×C283—21no—
133D4×C283—21no—
134D4×C283—21no—
135D4×C283—21no—
136D4×C283#4112yes—
137C2×C4×C283—21no—
138C2×C4×C283—21no—
139C2×C4×C283—81no—
140C2×C4×C283C2×D411yescharacteristic
141C2×C4×C283C2×D411yescharacteristic
142C2×C4×C283C2411yescharacteristic
143C8×C283—21no—
144C8×C283—21no—
145C22×C283—22no—
146C22×C283—41no—
147C22×C283—41no—
148C22×C283—41no—
149C22×C283—41no—
150C22×C283—41no—
151C22×C283C22×D411yescharacteristic
152C2×D283—21no—
153C2×D283—21no—
154C2×D283—21no—
155C4×C283—21no—
156C4×C283—21no—
157C4×C283—41no—
158C4×C283—41no—
159C4×C283C2×#4111yescharacteristic
160C4×C283C2×#4111yescharacteristic
161Dic283—21no—
162Dic283—22no—
163Dic283—21no—
164Dic283—21no—
165Dic283—41no—
166Dic283—41no—
167Dic283—41no—
168C2×C283—42no—
169C2×C283—81no—
170C2×C283#52511yescharacteristic
171C2×C283#57611yescharacteristic
172C2×C283#57811yescharacteristic
173D283—41no—
174C283#312411yescharacteristic, complemented, Sylow
175#3124—2831nomaximal, Sylow
176#542—2831no—
177#520—2831no—
178#484—2831no—
179#480—2831no—
180#477—2831no—
181#474—2831no—
182#464—2831no—
183#463D28311yescharacteristic, complemented
184#452—2831no—
185#449—2831no—
186#440—2831no—
187#440—2831no—
188C4×Q16—2831no—
189C4×D8—2831no—
190#433—2831no—
191C2×#41—2831no—
192C2×Q16—2831no—
193C2×SD16—2831no—
194C2×SD16C2×D28311yescharacteristic
195C2×D8C2×D28311yescharacteristic
196C4⋊Q8—2831no—
197#126—5661no—
198#124—5661no—
199#124C2×D28311yescharacteristic
200#123—5661no—
201C22⋊Q8—2831no—
202C22⋊Q8—5661no—
203#121—2831no—
204C4×Q8—2831no—
205C4×Q8—2831no—
206C4×Q8—2831no—
207C4×D4—2831no—
208C4×D4—2831no—
209C4×D4—5661no—
210C4×D4—5661no—
211C4×D4C2×D28311yescharacteristic
212#117—2831no—
213#117—5661no—
214C2×C4⋊C4—2831no—
215C2×C4⋊C4—5661no—
216#107—2831no—
217#107—2831no—
218#107—2831no—
219#107—2831no—
220C4⋊C8—2831no—
221C4⋊C8C2×D28311yescharacteristic
222#103—2831no—
223#103—2831no—
224#103—2831no—
225#103C2×D28311yescharacteristic
226#102—2831no—
227#102—2831no—
228#102—2831no—
229#102C2×D28311yescharacteristic
230C22⋊C8—2831no—
231C22⋊C8—2831no—
232#97—2831no—
233C4×C8—2831no—
234#41—5662no—
235#41—5662no—
236C2×Q8—2831no—
237C2×Q8—5661no—
238C2×Q8C22×D28311yescharacteristic
239C2×D4—5661no—
240C2×D4—5661no—
241C2×D4C22×D28311yescharacteristic
242C2×D4C22×D28311yescharacteristic
243C22×C4—21no—
244C22×C4—2831no—
245C22×C4—5661no—
246C22×C4—5661no—
247C22×C4—5661no—
248C22×C4—5661no—
249Q16—5662no—
250SD16—22no—
251SD16—5662no—
252D8—22no—
253C2×C8—2831no—
254C2×C8—2831no—
255C2×C8C22×D28311yescharacteristic
256C2×C8C22×D28311yescharacteristic
257C4⋊C4—2831no—
258C4⋊C4—2831no—
259C4⋊C4—2831no—
260C4⋊C4—2831no—
261C4⋊C4—2831no—
262C4⋊C4—5661no—
263C4⋊C4—5661no—
264C4⋊C4—5661no—
265C4⋊C4—5661no—
266C4⋊C4—5661no—
267C4⋊C4—5661no—
268C4⋊C4C22×D28311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—2831no—
273C22⋊C4—2831no—
274C22⋊C4—5661no—
275C22⋊C4—5661no—
276C22⋊C4—5661no—
277C22⋊C4—5661no—
278C42—2831no—
279C42—5661no—
280C42—5661no—
281C42—5661no—
282C42C22×D28311yescharacteristic
283C23—21no—
284C23—21no—
285C23—2831no—
286Q8—21no—
287Q8—21no—
288Q8—2832no—
289Q8—5661no—
290Q8—22641no—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—5664no—
295D4—22641no—
296D4#1798539112yes—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—2831no—
301C2×C4—2831no—
302C2×C4—2831no—
303C2×C4—5664no—
304C2×C4—5661no—
305C2×C4—5661no—
306C2×C4—5661no—
307C2×C4—5661no—
308C2×C4—5661no—
309C2×C4—5661no—
310C2×C4—22641no—
311C2×C4—22641no—
312C2×C4—22641no—
313C2×C4D4×D28311yescharacteristic
314C2×C4D4×D28311yescharacteristic
315C2×C4C23×D28311yescharacteristic
316C8—21no—
317C8—21no—
318C8—5661no—
319C8—5661no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—5661no—
327C22—5661no—
328C22—5661no—
329C22C2×D4×D28311yescharacteristic
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—5661no—
335C4—5662no—
336C4—5661no—
337C4—5661no—
338C4—11321no—
339C4—11321no—
340C4—11321no—
341C4#41×D28311yescharacteristic
342C4C2×#1798539111yescharacteristic
343C2—42no—
344C2—81no—
345C2—11321no—
346C2#3182697511yescharacteristic
347C2#3182726911yescharacteristic
348C2#3182727911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#463×C283C2
socleC22×C283C22×D4
prime radical#31826607C2
hypercenter#463D283
hypocenterC283#3124
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #31827084C2semidirect
2. #31826944C2semidirect
3. #31826799C2semidirect
4. #31826777C2semidirect
5. #31826761C2semidirect
6. #31826745C2semidirect
7. #31826684C2semidirect
8. #31826624C2semidirect
10. #31826566C2semidirect
11. #31826562C2semidirect
12. #31826557C2semidirect
13. D8×Dic283C2semidirect
14. #31826525C2semidirect
15. #463×C283C2semidirect
174. C283#3124semidirect
183. #463D283semidirect

Maximal subgroups

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

subgroupindex
#318270842
#318269442
#318267992
#318267772
#318267612
#318267452
#318266842
#318266242
#318266072
#318265662
#318265622
#318265572
D8×Dic2832
#318265252
#463×C2832
#3124283