Finite Group Databasebeta

← #49132690

Subgroups — #49132690

Order 36224.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827088C211yesmaximal, characteristic
2#31827075C211yesmaximal, characteristic, complemented
3#31826770C211yesmaximal, characteristic, complemented
4#31826768C211yesmaximal, characteristic, complemented
5#31826759C211yesmaximal, characteristic, complemented
6#31826748C211yesmaximal, characteristic, complemented
7#31826721C211yesmaximal, characteristic, complemented
8#31826713C211yesmaximal, characteristic
9#31826694C211yesmaximal, characteristic
10#31826690C211yesmaximal, characteristic, complemented
11#31826689C211yesmaximal, characteristic, complemented
12#31826673C211yesmaximal, characteristic, complemented
13#31826669C211yesmaximal, characteristic, complemented
14#31826491C211yesmaximal, characteristic, complemented
15#467×C283C211yesmaximal, characteristic, complemented
16C2×#17985382C2211yescharacteristic
17C2×#17985381C2211yescharacteristic
18C2×#17985379C2211yescharacteristic
19C2×#17985376C2211yescharacteristic
20#22583609C2211yescharacteristic
21#22583608C2211yescharacteristic
22#22583605—21no—
23#22583603—21no—
24#22583603—21no—
25#22583601—21no—
26#22583601C2211yescharacteristic
27#22583600—21no—
28#22583599C2211yescharacteristic
29#22583596—21no—
30#22583592—21no—
31#22583590—21no—
32#22583589C2211yescharacteristic
33#22583584C2211yescharacteristic
34Q8×Dic283C2211yescharacteristic
35Q8×Dic283C2211yescharacteristic
36#22583575C2211yescharacteristic
37#22583572C2211yescharacteristic
38#22583569C2211yescharacteristic
39#22583543C2211yescharacteristic
40#22583542C2211yescharacteristic
41#22583541C2211yescharacteristic
42Dic283⋊C8C2211yescharacteristic
43Dic283⋊C8C2211yescharacteristic
44Dic283⋊C8C2211yescharacteristic
45Dic283⋊C8C2211yescharacteristic
46#22583535C2211yescharacteristic
47#22583534C2211yescharacteristic
48#22583534C2211yescharacteristic
49#22583533C2211yescharacteristic
50#22583532C2211yescharacteristic
51#22583530C2211yescharacteristic
52C2×Q16×C283C2211yescharacteristic
53C22×C8×C283C2211yescharacteristic
54C22⋊Q8×C283C2211yescharacteristic
55C22⋊Q8×C283C2211yescharacteristic
56#107×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58#103×C283C2211yescharacteristic
59C2×Dic566—21no—
60C2×Dic566C2311yescharacteristic
61C2×D1132C2311yescharacteristic
62C2×C22⋊D283—21no—
63C2×C4×D283—21no—
64#17985382—41no—
65#17985381—41no—
66#17985379—41no—
67#17985376—41no—
68C2×C283⋊C8C2311yescharacteristic
69C2×C283⋊C8C2311yescharacteristic
70C4⋊Dic283—21no—
71C4⋊Dic283—21no—
72C4⋊Dic283—21no—
73C4⋊Dic283D411yescharacteristic
74C4⋊Dic283D411yescharacteristic
75C4⋊Dic283C2311yescharacteristic
76C4⋊Dic283C2311yescharacteristic
77Dic283⋊C4—21no—
78Dic283⋊C4—21no—
79Dic283⋊C4—21no—
80Dic283⋊C4—21no—
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83C22⋊Dic283—21no—
84C22⋊Dic283D411yescharacteristic
85C22⋊Dic283D411yescharacteristic
86(C2×D283)⋊C4—21no—
87(C2×D283)⋊C4—21no—
88(C2×D283)⋊C4—21no—
89(C2×D283)⋊C4—21no—
90C4×Dic283—21no—
91C4×Dic283—21no—
92C4×Dic283—21no—
93C4×Dic283C2311yescharacteristic
94C4×Dic283C2311yescharacteristic
95C2×Q8×C283C2311yescharacteristic
96C2×Q8×C283C2311yescharacteristic
97C22×C4×C283C2311yescharacteristic
98Q16×C283—41no—
99C2×C8×C283—41no—
100C2×C8×C283C2311yescharacteristic
101C2×C8×C283C2311yescharacteristic
102C4⋊C4×C283—21no—
103C4⋊C4×C283—21no—
104C4⋊C4×C283C2311yescharacteristic
105C4⋊C4×C283C2311yescharacteristic
106C22⋊C4×C283—21no—
107C22⋊C4×C283—21no—
108C22×D283—21no—
109Dic566—21no—
110Dic566—21no—
111Dic566—81no—
112D1132—21no—
113D1132—21no—
114C22⋊D283—81no—
115C2×Dic283—21no—
116C2×Dic283—21no—
117C2×Dic283—21no—
118C2×Dic283C2×D411yescharacteristic
119C2×Dic283C2×D411yescharacteristic
120C2×Dic283C2×D411yescharacteristic
121C2×Dic283C2×D411yescharacteristic
122C4×D283—81no—
123C283⋊C8—21no—
124C283⋊C8—21no—
125C23×C283C2×D411yescharacteristic
126Q8×C283—21no—
127Q8×C283—21no—
128Q8×C283—21no—
129Q8×C283—21no—
130C2×C4×C283—21no—
131C2×C4×C283—21no—
132C2×C4×C283—21no—
133C2×C4×C283—21no—
134C2×C4×C283—21no—
135C2×C4×C283—21no—
136C2×C4×C283C2×D411yescharacteristic
137C2×C4×C283C2411yescharacteristic
138C8×C283—21no—
139C8×C283—21no—
140C22×C283—21no—
141C22×C283—21no—
142C22×C283—21no—
143C22×C283C22×D411yescharacteristic
144C2×D283—41no—
145C2×D283—41no—
146C2×D283—41no—
147C4×C283—21no—
148C4×C283—41no—
149C4×C283—41no—
150C4×C283—41no—
151C4×C283—41no—
152C4×C283H(5,𝔽2)11yescharacteristic
153C4×C2832−1+411yescharacteristic
154Dic283—21no—
155Dic283—21no—
156Dic283—21no—
157Dic283—21no—
158Dic283—41no—
159Dic283—41no—
160Dic283—41no—
161C2×C283—41no—
162C2×C283#53711yescharacteristic
163C2×C283#57711yescharacteristic
164C2×C283#57811yescharacteristic
165D283—81no—
166C283#325811yescharacteristic, complemented, Sylow
167#3258—2831nomaximal, Sylow
168#542—2831no—
169#541—2831no—
170#479—2831no—
171#478—2831no—
172#477—2831no—
173#475—2831no—
174#470—2831no—
175#468—2831no—
176#467D28311yescharacteristic, complemented
177#465—2831no—
178#464—2831no—
179#464—2831no—
180#462—2831no—
181#461—2831no—
182#424—2831no—
183C2×Q16—2831no—
184C2×Q16C2×D28311yescharacteristic
185C2×SD16—2831no—
186C2×SD16—2831no—
187C2×M16—2831no—
188C22×C8C2×D28311yescharacteristic
189C4⋊Q8—2831no—
190C4⋊Q8—2831no—
191#126—5661no—
192#126—5661no—
193#126—5661no—
194#125—2831no—
195#125—5661no—
196#125—5661no—
197#124—2831no—
198#124—5661no—
199#123—5661no—
200C22⋊Q8—2831no—
201C22⋊Q8—5661no—
202C22⋊Q8C2×D28311yescharacteristic
203C22⋊Q8C2×D28311yescharacteristic
204#121—2831no—
205C4×Q8—2831no—
206C4×Q8—2831no—
207C4×Q8—2831no—
208C4×D4—2831no—
209#117—2831no—
210#107—2831no—
211#107C2×D28311yescharacteristic
212#106—2831no—
213#106—2831no—
214C4⋊C8—2831no—
215C4⋊C8—2831no—
216C4⋊C8—2831no—
217C4⋊C8—2831no—
218#103—2831no—
219#103—2831no—
220#103—2831no—
221#103—2831no—
222#103C2×D28311yescharacteristic
223#103C2×D28311yescharacteristic
224#102—2831no—
225#102—2831no—
226C2×Q8—2831no—
227C2×Q8—5661no—
228C2×Q8C22×D28311yescharacteristic
229C2×Q8C22×D28311yescharacteristic
230C2×D4—2831no—
231C2×D4—5661no—
232C22×C4—5661no—
233C22×C4C22×D28311yescharacteristic
234Q16—41no—
235Q16—11321no—
236SD16—11321no—
237SD16—11321no—
238M16—11321no—
239C2×C8—41no—
240C2×C8—2831no—
241C2×C8—2831no—
242C2×C8C22×D28311yescharacteristic
243C2×C8C22×D28311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—2831no—
247C4⋊C4—2831no—
248C4⋊C4—2831no—
249C4⋊C4—2831no—
250C4⋊C4—5661no—
251C4⋊C4—5661no—
252C4⋊C4—5661no—
253C4⋊C4—5661no—
254C4⋊C4—5661no—
255C4⋊C4—5661no—
256C4⋊C4—5661no—
257C4⋊C4—5661no—
258C4⋊C4—5661no—
259C4⋊C4C22×D28311yescharacteristic
260C4⋊C4C22×D28311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—2831no—
264C22⋊C4—2831no—
265C22⋊C4—5661no—
266C22⋊C4—5661no—
267C22⋊C4—5661no—
268C22⋊C4—5661no—
269C22⋊C4—5661no—
270C42—2831no—
271C42—2831no—
272C42—5661no—
273C42—5661no—
274C42—5661no—
275C23—5661no—
276C23D4×D28311yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—5661no—
282Q8—5661no—
283Q8—22641no—
284D4—5661no—
285D4—5661no—
286D4—22641no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—2831no—
294C2×C4—2831no—
295C2×C4—2831no—
296C2×C4—2831no—
297C2×C4—5661no—
298C2×C4—5661no—
299C2×C4—5661no—
300C2×C4—22641no—
301C2×C4D4×D28311yescharacteristic
302C2×C4C23×D28311yescharacteristic
303C8—21no—
304C8—21no—
305C8—5661no—
306C8—5661no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—11321no—
311C22—11321no—
312C22—11321no—
313C22C2×D4×D28311yescharacteristic
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—5661no—
320C4—5661no—
321C4—5661no—
322C4—5661no—
323C4—11321no—
324C4—11321no—
325C4—11321no—
326C4#2258365511yescharacteristic
327C4#2258365711yescharacteristic
328C2—41no—
329C2—22641no—
330C2#3182704211yescharacteristic
331C2#3182727211yescharacteristic
332C2#3182727811yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#467×C283C2
socleC22×C283C22×D4
prime radical#22583584C22
hypercenter#467D283
hypocenterC283#3258
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #31827075C2semidirect
3. #31826770C2semidirect
4. #31826768C2semidirect
5. #31826759C2semidirect
6. #31826748C2semidirect
7. #31826721C2semidirect
10. #31826690C2semidirect
11. #31826689C2semidirect
12. #31826673C2semidirect
13. #31826669C2semidirect
14. #31826491C2semidirect
15. #467×C283C2semidirect
166. C283#3258semidirect
176. #467D283semidirect

Maximal subgroups

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

subgroupindex
#318270882
#318270752
#318267702
#318267682
#318267592
#318267482
#318267212
#318267132
#318266942
#318266902
#318266892
#318266732
#318266692
#318264912
#467×C2832
#3258283