Finite Group Databasebeta

← #49133122

Subgroups — #49133122

Order 36224.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D283C211yesmaximal, characteristic, complemented
2#31827116C211yesmaximal, characteristic
3#31826814C211yesmaximal, characteristic, complemented
4#31826795C211yesmaximal, characteristic, complemented
5#31826794C211yesmaximal, characteristic, complemented
6#31826748C211yesmaximal, characteristic, complemented
7#31826697C211yesmaximal, characteristic, complemented
8#31826694C211yesmaximal, characteristic, complemented
9#31826669C211yesmaximal, characteristic, complemented
10#31826605C211yesmaximal, characteristic, complemented
11#31826604C211yesmaximal, characteristic, complemented
12#31826550C211yesmaximal, characteristic, complemented
13C8×C22⋊D283C211yesmaximal, characteristic, complemented
14#103×D283C211yesmaximal, characteristic, complemented
15C2×#103×C283C211yesmaximal, characteristic, complemented
16C2×#17985392—21no—
17C2×Q8×D283C2211yescharacteristic
18C2×#17985381C2211yescharacteristic
19C2×#17985379C2211yescharacteristic
20C2×#17985379C2211yescharacteristic
21C2×C8×D283C2211yescharacteristic
22#22583609C2211yescharacteristic
23#22583608—21no—
24#22583608C2211yescharacteristic
25#22583595—21no—
26#22583593—21no—
27#22583589—21no—
28#22583589C2211yescharacteristic
29#22583588—21no—
30#22583586—21no—
31#22583586C2211yescharacteristic
32#22583585C2211yescharacteristic
33#22583584C2211yescharacteristic
34Q8×Dic283C2211yescharacteristic
35#22583572C2211yescharacteristic
36C4×C22⋊D283—21no—
37C4×C22⋊D283C2211yescharacteristic
38C4⋊C4×D283C2211yescharacteristic
39#22583542C2211yescharacteristic
40#22583541C2211yescharacteristic
41#22583541C2211yescharacteristic
42Dic283⋊C8C2211yescharacteristic
43#22583535C2211yescharacteristic
44#22583534C2211yescharacteristic
45#22583534C2211yescharacteristic
46#22583533C2211yescharacteristic
47#22583533C2211yescharacteristic
48#22583532C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic
51C8×Dic283C2211yescharacteristic
52C22×Q8×C283C2211yescharacteristic
53C22×C8×C283C2211yescharacteristic
54C2×C4⋊C4×C283C2211yescharacteristic
55#103×C283C2211yescharacteristic
56#103×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58#103×C283C2211yescharacteristic
59#17985392—44no—
60C2×Dic566—21no—
61C2×Dic566—21no—
62C2×Dic566C2311yescharacteristic
63Q8×D283—22no—
64Q8×D283—41no—
65C2×D1132—21no—
66C2×D1132C2311yescharacteristic
67C2×C22⋊D283—21no—
68C2×C22⋊D283D411yescharacteristic
69C2×C4×D283—21no—
70C2×C4×D283—21no—
71C2×C4×D283—21no—
72C2×C4×D283C2311yescharacteristic
73#17985381—41no—
74#17985379—22no—
75#17985379—41no—
76C2×C283⋊C8C2311yescharacteristic
77C2×C283⋊C8C2311yescharacteristic
78C8×D283—41no—
79C4⋊Dic283—21no—
80C4⋊Dic283C2311yescharacteristic
81C4⋊Dic283C2311yescharacteristic
82Dic283⋊C4—21no—
83Dic283⋊C4—21no—
84Dic283⋊C4—21no—
85Dic283⋊C4—21no—
86Dic283⋊C4—21no—
87Dic283⋊C4—21no—
88Dic283⋊C4D411yescharacteristic
89C22⋊Dic283—21no—
90C22⋊Dic283D411yescharacteristic
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4—21no—
95(C2×D283)⋊C4D411yescharacteristic
96C4×Dic283—21no—
97C4×Dic283C2311yescharacteristic
98C2×Q8×C283—21no—
99C2×Q8×C283—22no—
100C2×Q8×C283—41no—
101C2×Q8×C283C2311yescharacteristic
102C2×Q8×C283C2311yescharacteristic
103C22×C4×C283—21no—
104C22×C4×C283—21no—
105C22×C4×C283C2311yescharacteristic
106C2×C8×C283—41no—
107C2×C8×C283C2311yescharacteristic
108C2×C8×C283C2311yescharacteristic
109C4⋊C4×C283—21no—
110C4⋊C4×C283C2311yescharacteristic
111C4⋊C4×C283C2311yescharacteristic
112C22×D283—21no—
113C22×D283C2×D411yescharacteristic
114Dic566—21no—
115Dic566—21no—
116Dic566—42no—
117Dic566—42no—
118D1132—21no—
119D1132—21no—
120D1132—42no—
121C22⋊D283—42no—
122C22⋊D283SD1614yescomplemented
123C2×Dic283—21no—
124C2×Dic283—21no—
125C2×Dic283—21no—
126C2×Dic283C2×D411yescharacteristic
127C2×Dic283C2×D411yescharacteristic
128C2×Dic283C2×D411yescharacteristic
129C4×D283—21no—
130C4×D283—21no—
131C4×D283—42no—
132C4×D283—42no—
133C4×D283—42no—
134C283⋊C8—21no—
135C283⋊C8—21no—
136C23×C283C2×D411yescharacteristic
137Q8×C283—21no—
138Q8×C283—21no—
139Q8×C283—21no—
140Q8×C283—42no—
141Q8×C283#4112yescomplemented
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283—42no—
149C2×C4×C283—42no—
150C2×C4×C283C2×D411yescharacteristic
151C2×C4×C283C2411yescharacteristic
152C8×C283—21no—
153C8×C283—21no—
154C22×C283—21no—
155C22×C283—21no—
156C22×C283C2×SD1612yes—
157C22×C283C22×D411yescharacteristic
158C2×D283—21no—
159C2×D283—21no—
160C2×D283—41no—
161C2×D283—41no—
162C2×D283—41no—
163C2×D283C2×SD1612yes—
164C4×C283—21no—
165C4×C283—22no—
166C4×C283—41no—
167C4×C283—41no—
168C4×C283—41no—
169C4×C283C2×#4111yescharacteristic
170C4×C283H(5,𝔽2)11yescharacteristic
171Dic283—21no—
172Dic283—21no—
173Dic283—41no—
174Dic283—41no—
175Dic283—41no—
176Dic283C2×SD1612yes—
177C2×C283—22no—
178C2×C283#54611yescharacteristic
179C2×C283C22×SD1611yescharacteristic
180C2×C283#57811yescharacteristic
181D283—22no—
182D283—81no—
183C283#329811yescharacteristic, complemented, Sylow
184#3298—2831nomaximal, Sylow
185D4×Q8—2831no—
186#547—2831no—
187#487—2831no—
188#483—2831no—
189#483—2831no—
190#475—2831no—
191#465—2831no—
192#465—2831no—
193#461—2831no—
194#448—2831no—
195#448—2831no—
196C4×SD16—2831no—
197C8×D4—2831no—
198C2×#103—2831no—
199C2×#103D28311yescharacteristic, complemented
200C2×#41—5661no—
201C22×Q8—2831no—
202C22×Q8C2×D28311yescharacteristic
203C2×SD16—2831no—
204C2×SD16—2831no—
205C2×SD16—2831no—
206C22×C8—2831no—
207C22×C8C2×D28311yescharacteristic
208C4⋊Q8—2831no—
209C4⋊Q8—2831no—
210C4⋊Q8—5661no—
211#123—5661no—
212#123—5661no—
213C22⋊Q8—2831no—
214C22⋊Q8—2831no—
215C22⋊Q8—5661no—
216C22⋊Q8—5661no—
217C22⋊Q8—5661no—
218#121—2831no—
219#121—2831no—
220C4×Q8—2831no—
221C4×D4—2831no—
222C4×D4—2831no—
223C4×D4—5661no—
224C2×C4⋊C4—2831no—
225C2×C4⋊C4C2×D28311yescharacteristic
226#106—2831no—
227#106—2831no—
228#106—2831no—
229C4⋊C8—2831no—
230#103—2831no—
231#103—2831no—
232#103—2831no—
233#103—2831no—
234#103—2831no—
235#103C2×D28311yescharacteristic
236#103C2×D28311yescharacteristic
237#103C2×D28311yescharacteristic
238#103C2×D28311yescharacteristic
239#102—2831no—
240C22⋊C8—2831no—
241C22⋊C8—2831no—
242C4×C8—2831no—
243#41—11324no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—2831no—
248C2×Q8—5661no—
249C2×Q8—5661no—
250C2×Q8—5662no—
251C2×Q8—11321no—
252C2×Q8C22×D28311yescharacteristic
253C2×Q8C22×D28311yescharacteristic
254C2×D4—2831no—
255C2×D4—2831no—
256C2×D4—5661no—
257C2×D4—5661no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2831no—
261C22×C4—5661no—
262C22×C4—5661no—
263C22×C4—5661no—
264C22×C4C22×D28311yescharacteristic
265SD16—5662no—
266SD16—11321no—
267SD16—11321no—
268C2×C8—41no—
269C2×C8—2831no—
270C2×C8—2831no—
271C2×C8—11321no—
272C2×C8C22×D28311yescharacteristic
273C2×C8C22×D28311yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—2831no—
276C4⋊C4—2831no—
277C4⋊C4—2831no—
278C4⋊C4—5661no—
279C4⋊C4—5661no—
280C4⋊C4—5661no—
281C4⋊C4—5661no—
282C4⋊C4—5661no—
283C4⋊C4—5661no—
284C4⋊C4—5661no—
285C4⋊C4C22×D28311yescharacteristic
286C4⋊C4C22×D28311yescharacteristic
287C22⋊C4—2831no—
288C22⋊C4—2831no—
289C22⋊C4—5661no—
290C22⋊C4—5661no—
291C22⋊C4—5661no—
292C22⋊C4—5661no—
293C22⋊C4—5661no—
294C42—2831no—
295C42—5661no—
296C23—2831no—
297C23—5661no—
298C23D4×D28311yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—5661no—
304Q8—5661no—
305Q8—11322no—
306Q8—11322no—
307Q8#1798539212yescomplemented
308D4—2834no—
309D4—5661no—
310D4—5661no—
311D4—11322no—
312D4—11322no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—2831no—
322C2×C4—2831no—
323C2×C4—2831no—
324C2×C4—5661no—
325C2×C4—5661no—
326C2×C4—5661no—
327C2×C4—5661no—
328C2×C4—5661no—
329C2×C4—11322no—
330C2×C4—11322no—
331C2×C4—11322no—
332C2×C4D4×D28311yescharacteristic
333C2×C4C23×D28311yescharacteristic
334C8—21no—
335C8—21no—
336C8—5661no—
337C8—5661no—
338C22—21no—
339C22—21no—
340C22—2832no—
341C22—5661no—
342C22—5661no—
343C22—11321no—
344C22—11321no—
345C22—11321no—
346C22SD16×D28312yes—
347C22C2×D4×D28311yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—2832no—
354C4—5661no—
355C4—5661no—
356C4—11321no—
357C4—11321no—
358C4—11321no—
359C4C2×#1798539211yescharacteristic
360C4#2258365511yescharacteristic
361C2—22no—
362C2—5662no—
363C2—22641no—
364C2#3182710511yescharacteristic
365C2C2×SD16×D28311yescharacteristic
366C2#3182727811yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×#103×C283C2
socleC22×C283C22×D4
prime radical#31827116C2
hypercenterC2×#103D283
hypocenterC283#3298
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D283C2semidirect
3. #31826814C2semidirect
4. #31826795C2semidirect
5. #31826794C2semidirect
6. #31826748C2semidirect
7. #31826697C2semidirect
8. #31826694C2semidirect
9. #31826669C2semidirect
10. #31826605C2semidirect
11. #31826604C2semidirect
12. #31826550C2semidirect
13. C8×C22⋊D283C2semidirect
14. #103×D283C2semidirect
15. C2×#103×C283C2semidirect
122. C22⋊D283SD16semidirect
141. Q8×C283#41semidirect
183. C283#3298semidirect
199. C2×#103D283semidirect
307. Q8#17985392semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D2832
#318271162
#318268142
#318267952
#318267942
#318267482
#318266972
#318266942
#318266692
#318266052
#318266042
#318265502
C8×C22⋊D2832
#103×D2832
C2×#103×C2832
#3298283