Finite Group Databasebeta

← #49133306

Subgroups — #49133306

Order 36224.

376 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
1D4×Dic566C211yesmaximal, characteristic
2#31826835C211yesmaximal, characteristic, complemented
3#31826806C211yesmaximal, characteristic, complemented
4#31826795C211yesmaximal, characteristic, complemented
5#31826769C211yesmaximal, characteristic, complemented
6#31826721C211yesmaximal, characteristic, complemented
7#31826704C211yesmaximal, characteristic, complemented
8#31826675C211yesmaximal, characteristic, complemented
9#31826620C211yesmaximal, characteristic, complemented
10#31826620C211yesmaximal, characteristic, complemented
11#31826570C211yesmaximal, characteristic, complemented
12#31826537C211yesmaximal, characteristic
13#31826473C211yesmaximal, characteristic, complemented
14C2×#22583533C211yesmaximal, characteristic, complemented
15#552×C283C211yesmaximal, characteristic
16C22×Dic566C2211yescharacteristic
17C22×Dic566C2211yescharacteristic
18C2×#17985382C2211yescharacteristic
19C2×#17985382C2211yescharacteristic
20C2×#17985382C2211yescharacteristic
21C2×#17985376C2211yescharacteristic
22C22×C283⋊C8C2211yescharacteristic
23#22583610C2211yescharacteristic
24#22583609—21no—
25#22583589C2211yescharacteristic
26#22583589C2211yescharacteristic
27#22583587—21no—
28#22583587—21no—
29C4×Dic566C2211yescharacteristic
30D4×Dic283—21no—
31#22583543C2211yescharacteristic
32#22583541C2211yescharacteristic
33#22583541C2211yescharacteristic
34C4⋊(C283⋊C8)C2211yescharacteristic
35#22583534C2211yescharacteristic
36#22583533C2211yescharacteristic
37#22583533C2211yescharacteristic
38#22583533C2211yescharacteristic
39#22583533C2211yescharacteristic
40#22583533C2211yescharacteristic
41#22583533C2211yescharacteristic
42#22583533C2211yescharacteristic
43#22583533C2211yescharacteristic
44#22583533C2211yescharacteristic
45C22⋊(C283⋊C8)C2211yescharacteristic
46C22⋊(C283⋊C8)C2211yescharacteristic
47#22583521C2211yescharacteristic
48#126×C283—21no—
49#125×C283—21no—
50#125×C283C2211yescharacteristic
51#123×C283—21no—
52C22⋊Q8×C283C2211yescharacteristic
53C22⋊Q8×C283C2211yescharacteristic
54C4×Q8×C283C2211yescharacteristic
55C4×D4×C283C2211yescharacteristic
56#117×C283—21no—
57#117×C283C2211yescharacteristic
58C2×C4⋊C4×C283C2211yescharacteristic
59C2×Dic566—22no—
60C2×Dic566—22no—
61C2×Dic566—21no—
62C2×Dic566—21no—
63C2×Dic566—41no—
64C2×Dic566—41no—
65C2×Dic566C2311yescharacteristic
66C2×Dic566C2311yescharacteristic
67C2×Dic566C2311yescharacteristic
68C22×Dic283—21no—
69C22×Dic283—21no—
70#17985382—22no—
71#17985382—41no—
72#17985382—41no—
73#17985376—41no—
74C2×C283⋊C8—41no—
75C2×C283⋊C8C2311yescharacteristic
76C2×C283⋊C8C2311yescharacteristic
77C2×C283⋊C8C2311yescharacteristic
78C2×C283⋊C8C2311yescharacteristic
79C4⋊Dic283—21no—
80C4⋊Dic283—21no—
81C4⋊Dic283C2311yescharacteristic
82Dic283⋊C4—21no—
83Dic283⋊C4—21no—
84Dic283⋊C4—21no—
85C22⋊Dic283—21no—
86C22⋊Dic283—21no—
87C4×Dic283—21no—
88C2×Q8×C283C2311yescharacteristic
89C2×D4×C283D411yescharacteristic
90C22×C4×C283—21no—
91C22×C4×C283C2311yescharacteristic
92C22×C4×C283C2311yescharacteristic
93C4⋊C4×C283—21no—
94C4⋊C4×C283—21no—
95C4⋊C4×C283—21no—
96C4⋊C4×C283—21no—
97C4⋊C4×C283—21no—
98C4⋊C4×C283—21no—
99C4⋊C4×C283D411yescharacteristic
100C4⋊C4×C283C2311yescharacteristic
101C4⋊C4×C283C2311yescharacteristic
102C4⋊C4×C283C2311yescharacteristic
103C22⋊C4×C283—21no—
104C22⋊C4×C283—21no—
105C22⋊C4×C283—21no—
106C22⋊C4×C283D411yescharacteristic
107C22⋊C4×C283D411yescharacteristic
108C42×C283—21no—
109C42×C283—21no—
110C42×C283C2311yescharacteristic
111Dic566—21no—
112Dic566—21no—
113Dic566—21no—
114Dic566—21no—
115Dic566—21no—
116Dic566—42no—
117Dic566—42no—
118Dic566#4112yes—
119C2×Dic283—21no—
120C2×Dic283—21no—
121C2×Dic283—21no—
122C2×Dic283—21no—
123C2×Dic283—42no—
124C2×Dic283—42no—
125C283⋊C8—21no—
126C283⋊C8—21no—
127C283⋊C8—21no—
128C283⋊C8—21no—
129C23×C283C2×D411yescharacteristic
130C23×C283C2×D411yescharacteristic
131Q8×C283—21no—
132Q8×C283—21no—
133D4×C283—22no—
134C2×C4×C283—21no—
135C2×C4×C283—21no—
136C2×C4×C283—21no—
137C2×C4×C283—21no—
138C2×C4×C283—21no—
139C2×C4×C283—21no—
140C2×C4×C283—21no—
141C2×C4×C283—21no—
142C2×C4×C283—42no—
143C2×C4×C283C2×D411yescharacteristic
144C2×C4×C283C2×D411yescharacteristic
145C2×C4×C283C2×D411yescharacteristic
146C2×C4×C283C2×D411yescharacteristic
147C2×C4×C283C2411yescharacteristic
148C22×C283—21no—
149C22×C283—21no—
150C22×C283—21no—
151C22×C283—21no—
152C22×C283—21no—
153C22×C283#13712yes—
154C22×C283C22×D411yescharacteristic
155C4×C283—21no—
156C4×C283—21no—
157C4×C283—21no—
158C4×C283—21no—
159C4×C283—41no—
160C4×C283—41no—
161C4×C283—41no—
162C4×C283—41no—
163C4×C283C2×#4111yescharacteristic
164C4×C283H(5,𝔽2)11yescharacteristic
165Dic283—22no—
166Dic283—41no—
167Dic283—41no—
168Dic283—41no—
169C2×C283—22no—
170C2×C283—41no—
171C2×C283#54611yescharacteristic
172C2×C283C2×#13711yescharacteristic
173C2×C283#57811yescharacteristic
174C283#331211yescharacteristic, complemented, Sylow
175#3312—2831nomaximal, Sylow
176#552D28311yescharacteristic
177D4×Q8—2831no—
178#491—2831no—
179#485—2831no—
180#483—2831no—
181#479—2831no—
182#470—2831no—
183#467—2831no—
184#462—2831no—
185#451—2831no—
186#451—2831no—
187#441—2831no—
188#435—2831no—
189#419—2831no—
190C2×#103—2831no—
191C22×Q8—2831no—
192C22×Q8—2831no—
193C2×Q16—2831no—
194C2×Q16—2831no—
195C2×Q16—2831no—
196C2×M16—2831no—
197C22×C8—2831no—
198C4⋊Q8—2831no—
199C4⋊Q8—5661no—
200#126—21no—
201#125—21no—
202#125C2×D28311yescharacteristic
203#123—21no—
204C22⋊Q8—2831no—
205C22⋊Q8—2831no—
206C22⋊Q8—5661no—
207C22⋊Q8—5661no—
208C22⋊Q8C2×D28311yescharacteristic
209C22⋊Q8C2×D28311yescharacteristic
210C4×Q8—2831no—
211C4×Q8C2×D28311yescharacteristic
212C4×D4—5661no—
213C4×D4C2×D28311yescharacteristic
214#117—21no—
215#117C2×D28311yescharacteristic
216C2×C4⋊C4C2×D28311yescharacteristic
217#107—2831no—
218#106—2831no—
219#106—2831no—
220C4⋊C8—2831no—
221#103—2831no—
222#103—2831no—
223#103—2831no—
224#103—2831no—
225#103—2831no—
226#103—2831no—
227#103—2831no—
228#103—2831no—
229#103—2831no—
230#103—2831no—
231C22⋊C8—2831no—
232C22⋊C8—2831no—
233#97—2831no—
234C2×Q8—2831no—
235C2×Q8—2831no—
236C2×Q8—2831no—
237C2×Q8—5662no—
238C2×Q8—5662no—
239C2×Q8—5661no—
240C2×Q8—5661no—
241C2×Q8—11321no—
242C2×Q8—11321no—
243C2×Q8C22×D28311yescharacteristic
244C2×D4C22⋊D28311yescharacteristic
245C22×C4—21no—
246C22×C4—5661no—
247C22×C4—5661no—
248C22×C4C22×D28311yescharacteristic
249C22×C4C22×D28311yescharacteristic
250Q16—5662no—
251Q16—11321no—
252Q16—11321no—
253M16—11321no—
254C2×C8—2831no—
255C2×C8—2831no—
256C2×C8—2831no—
257C2×C8—2831no—
258C2×C8—11321no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—2831no—
266C4⋊C4—5661no—
267C4⋊C4—5661no—
268C4⋊C4—5661no—
269C4⋊C4—5661no—
270C4⋊C4—5661no—
271C4⋊C4C22⋊D28311yescharacteristic
272C4⋊C4C22×D28311yescharacteristic
273C4⋊C4C22×D28311yescharacteristic
274C4⋊C4C22×D28311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—5661no—
279C22⋊C4—5661no—
280C22⋊C4C22⋊D28311yescharacteristic
281C22⋊C4C22⋊D28311yescharacteristic
282C42—21no—
283C42—21no—
284C42—5661no—
285C42C22×D28311yescharacteristic
286C23C2×C22⋊D28311yescharacteristic
287C23C2×C22⋊D28311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2832no—
291Q8—5661no—
292Q8—5661no—
293Q8—5661no—
294Q8—5661no—
295Q8—5661no—
296Q8—11322no—
297Q8—11322no—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—5661no—
309C2×C4—5661no—
310C2×C4—5661no—
311C2×C4—5661no—
312C2×C4—11322no—
313C2×C4—11322no—
314C2×C4C2×C22⋊D28311yescharacteristic
315C2×C4C2×C22⋊D28311yescharacteristic
316C2×C4C2×C22⋊D28311yescharacteristic
317C2×C4C2×C22⋊D28311yescharacteristic
318C2×C4C23×D28311yescharacteristic
319C8—5661no—
320C8—5661no—
321C8—5661no—
322C8—5661no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#2258364012yes—
329C22C22×C22⋊D28311yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—5662no—
339C4—11321no—
340C4—11321no—
341C4—11321no—
342C4#41×D28311yescharacteristic
343C4#2258365611yescharacteristic
344C2—22no—
345C2—41no—
346C2#3182710311yescharacteristic
347C2C2×#2258364011yescharacteristic
348C2#3182728011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#552×C283C2
socleC22×C283C22×D4
prime radicalC2×D4×C283D4
hypercenter#552D283
hypocenterC283#3312
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #31826835C2semidirect
3. #31826806C2semidirect
4. #31826795C2semidirect
5. #31826769C2semidirect
6. #31826721C2semidirect
7. #31826704C2semidirect
8. #31826675C2semidirect
9. #31826620C2semidirect
10. #31826620C2semidirect
11. #31826570C2semidirect
13. #31826473C2semidirect
14. C2×#22583533C2semidirect
174. C283#3312semidirect

Maximal subgroups

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

subgroupindex
D4×Dic5662
#318268352
#318268062
#318267952
#318267692
#318267212
#318267042
#318266752
#318266202
#318265702
#318265372
#318264732
C2×#225835332
#552×C2832
#3312283