Finite Group Databasebeta

← #46822776

Subgroups — #46822776

Order 34432.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771297C211yesmaximal, characteristic, complemented
2#31771266C211yesmaximal, characteristic
3#31770961C211yesmaximal, characteristic, complemented
4#31770945C211yesmaximal, characteristic, complemented
5#31770919C211yesmaximal, characteristic, complemented
6#31770914C211yesmaximal, characteristic, complemented
7#31770843C211yesmaximal, characteristic, complemented
8#31770836C211yesmaximal, characteristic, complemented
9#31770829C211yesmaximal, characteristic, complemented
10#31770766C211yesmaximal, characteristic, complemented
11Q16×Dic269C211yesmaximal, characteristic, complemented
12#31770668C211yesmaximal, characteristic, complemented
13#31770600C211yesmaximal, characteristic, complemented
14C2×#22575377C211yesmaximal, characteristic, complemented
15#451×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864389C2211yescharacteristic
18C22×#14524130C2211yescharacteristic
19C2×C8×D269C2211yescharacteristic
20#22575448—21no—
21#22575444—21no—
22#22575444C2211yescharacteristic
23#22575442—21no—
24#22575442C2211yescharacteristic
25#22575437—21no—
26#22575429—21no—
27#22575429C2211yescharacteristic
28#22575429C2211yescharacteristic
29#22575427—21no—
30#22575427C2211yescharacteristic
31Q8×Dic269C2211yescharacteristic
32Q8×Dic269C2211yescharacteristic
33C4×C22⋊D269—21no—
34C4×C22⋊D269C2211yescharacteristic
35#22575411C2211yescharacteristic
36#22575409—21no—
37C2×C4⋊Dic269C2211yescharacteristic
38#22575387C2211yescharacteristic
39#22575387C2211yescharacteristic
40#22575384C2211yescharacteristic
41#22575382C2211yescharacteristic
42#22575377C2211yescharacteristic
43#22575377C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575377C2211yescharacteristic
47#22575375C2211yescharacteristic
48#22575375C2211yescharacteristic
49#22575373C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C22×Q8×C269C2211yescharacteristic
53C2×Q16×C269C2211yescharacteristic
54C2×Q16×C269C2211yescharacteristic
55C22⋊Q8×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic
59#16864403—22no—
60#16864403—41no—
61C2×D1076—21no—
62C2×D1076C2311yescharacteristic
63C2×C22⋊D269—21no—
64C2×C22⋊D269D411yescharacteristic
65C22×Dic269—21no—
66C2×C4×D269—21no—
67C2×C4×D269C2311yescharacteristic
68#16864389—41no—
69C2×#14524130—41no—
70C2×#14524130C2311yescharacteristic
71C2×#14524130C2311yescharacteristic
72C8×D269—41no—
73C4⋊Dic269—21no—
74C4⋊Dic269—21no—
75C4⋊Dic269—21no—
76C4⋊Dic269—21no—
77C4⋊Dic269C2311yescharacteristic
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83#16864381D411yescharacteristic
84C22⋊Dic269D411yescharacteristic
85#16864379—21no—
86#16864379—21no—
87#16864379—21no—
88#16864379—21no—
89#16864379—21no—
90#16864379—21no—
91#16864379D411yescharacteristic
92C4×Dic269—21no—
93C4×Dic269—21no—
94C4×Dic269—21no—
95C4×Dic269C2311yescharacteristic
96C2×Q8×C269—21no—
97C2×Q8×C269—22no—
98C2×Q8×C269—41no—
99C2×Q8×C269C2311yescharacteristic
100C2×Q8×C269C2311yescharacteristic
101C2×Q8×C269C2311yescharacteristic
102C22×C4×C269—21no—
103C22×C4×C269C2311yescharacteristic
104Q16×C269—22no—
105Q16×C269—41no—
106C2×C8×C269C2311yescharacteristic
107C2×C8×C269C2311yescharacteristic
108C4⋊C4×C269—21no—
109C4⋊C4×C269C2311yescharacteristic
110C22⋊C4×C269—21no—
111C22×D269—21no—
112C22×D269C2×D411yescharacteristic
113D1076—21no—
114D1076—21no—
115D1076—42no—
116C22⋊D269—22no—
117C22⋊D269—42no—
118C2×Dic269—21no—
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—42no—
122C2×Dic269C2×D411yescharacteristic
123C2×Dic269C2×D411yescharacteristic
124C2×Dic269C2×D411yescharacteristic
125C4×D269—21no—
126C4×D269—21no—
127C4×D269—42no—
128#14524130—21no—
129#14524130—21no—
130C23×C269C2×D411yescharacteristic
131Q8×C269—21no—
132Q8×C269—21no—
133Q8×C269—21no—
134Q8×C269—21no—
135Q8×C269—21no—
136Q8×C269—42no—
137Q8×C269#4112yes—
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—42no—
145C2×C4×C269C2×D411yescharacteristic
146C2×C4×C269C2411yescharacteristic
147C8×C269—21no—
148C8×C269—21no—
149C22×C269—21no—
150C22×C269—21no—
151C22×C269#13512yes—
152C22×C269C22×D411yescharacteristic
153C2×D269—21no—
154C2×D269—21no—
155C2×D269—21no—
156C2×D269—41no—
157C2×D269—41no—
158C2×D269—41no—
159C4×C269—21no—
160C4×C269—22no—
161C4×C269—41no—
162C4×C269—41no—
163C4×C269—41no—
164C4×C269C2×#4111yescharacteristic
165C4×C269H(5,𝔽2)11yescharacteristic
166Dic269—21no—
167Dic269—21no—
168Dic269—21no—
169Dic269—41no—
170Dic269—41no—
171Dic269—41no—
172C2×C269—22no—
173C2×C269#54611yescharacteristic
174C2×C269C2×#13511yescharacteristic
175C2×C269#57811yescharacteristic
176D269—41no—
177D269—81no—
178C269#330011yescharacteristic, complemented, Sylow
179#3300—2691nomaximal, Sylow
180#552—2691no—
181#548—2691no—
182#488—2691no—
183#485—2691no—
184#480—2691no—
185#479—2691no—
186#467—2691no—
187#465—2691no—
188#464—2691no—
189#452—2691no—
190#451D26911yescharacteristic, complemented
191C4×Q16—2691no—
192C8×D4—2691no—
193#416—2691no—
194C2×#103—2691no—
195C2×#41—2691no—
196C22×Q8C2×D26911yescharacteristic
197C2×Q16C2×D26911yescharacteristic
198C2×Q16C2×D26911yescharacteristic
199C2×SD16—2691no—
200C22×C8—2691no—
201C22×C8—2691no—
202#126—5381no—
203#125—2691no—
204#125—5381no—
205#124—2691no—
206#124—5381no—
207#123—5381no—
208C22⋊Q8—2691no—
209C22⋊Q8—2691no—
210C22⋊Q8—5381no—
211C22⋊Q8C2×D26911yescharacteristic
212#121—2691no—
213#121—5381no—
214C4×Q8—2691no—
215C4×Q8—2691no—
216C4×D4—2691no—
217C4×D4—5381no—
218#117—2691no—
219#117—5381no—
220C2×C4⋊C4—2691no—
221#107—2691no—
222#107—2691no—
223#106—2691no—
224C4⋊C8—2691no—
225#103—2691no—
226#103—2691no—
227#103—2691no—
228#103—2691no—
229#103—2691no—
230#103C2×D26911yescharacteristic
231#103C2×D26911yescharacteristic
232#102—2691no—
233#102—2691no—
234#102—2691no—
235C22⋊C8—2691no—
236C22⋊C8C2×D26911yescharacteristic
237C4×C8—2691no—
238#41—5382no—
239#41—10761no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8C22×D26911yescharacteristic
244C2×Q8C22×D26911yescharacteristic
245C2×Q8C22×D26911yescharacteristic
246C2×D4—2691no—
247C2×D4—2691no—
248C2×D4—5381no—
249C2×D4—5381no—
250C22×C4—21no—
251C22×C4—2691no—
252C22×C4—5381no—
253C22×C4—5381no—
254C22×C4C22×D26911yescharacteristic
255Q16—22no—
256Q16—41no—
257SD16—10761no—
258C2×C8—2691no—
259C2×C8—2691no—
260C2×C8—10761no—
261C2×C8—10761no—
262C2×C8C22×D26911yescharacteristic
263C2×C8C22×D26911yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—2691no—
266C4⋊C4—2691no—
267C4⋊C4—2691no—
268C4⋊C4—2691no—
269C4⋊C4—5381no—
270C4⋊C4—5381no—
271C4⋊C4—5381no—
272C4⋊C4—5381no—
273C4⋊C4—5381no—
274C4⋊C4—5381no—
275C4⋊C4—5381no—
276C4⋊C4C22×D26911yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2691no—
279C22⋊C4—2691no—
280C22⋊C4—5381no—
281C22⋊C4—5381no—
282C22⋊C4—5381no—
283C22⋊C4—5381no—
284C22⋊C4—5381no—
285C22⋊C4—5381no—
286C42—2691no—
287C42—5381no—
288C42—5381no—
289C42—5381no—
290C23—2691no—
291C23—5381no—
292C23D4×D26911yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8#1686440112yes—
300D4—5382no—
301D4—5381no—
302D4—5381no—
303D4—10762no—
304D4—10762no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—2691no—
313C2×C4—2691no—
314C2×C4—2691no—
315C2×C4—5381no—
316C2×C4—5381no—
317C2×C4—5381no—
318C2×C4—5381no—
319C2×C4—5381no—
320C2×C4—10762no—
321C2×C4—10762no—
322C2×C4D4×D26911yescharacteristic
323C2×C4C23×D26911yescharacteristic
324C8—21no—
325C8—21no—
326C8—5381no—
327C8—5381no—
328C22—21no—
329C22—21no—
330C22—5381no—
331C22—5381no—
332C22—5381no—
333C22—10761no—
334C22—10761no—
335C22—10761no—
336C22#2257547612yes—
337C22C2×D4×D26911yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—5381no—
344C4—5381no—
345C4—5381no—
346C4—10761no—
347C4—10761no—
348C4—10761no—
349C4C2×#1686440111yescharacteristic
350C4#2257549811yescharacteristic
351C2—22no—
352C2—10761no—
353C2—21521no—
354C2#3177124211yescharacteristic
355C2C2×#2257547611yescharacteristic
356C2#3177142111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#451×C269C2
socleC22×C269C22×D4
prime radical#31771266C2
hypercenter#451D269
hypocenterC269#3300
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #31771297C2semidirect
3. #31770961C2semidirect
4. #31770945C2semidirect
5. #31770919C2semidirect
6. #31770914C2semidirect
7. #31770843C2semidirect
8. #31770836C2semidirect
9. #31770829C2semidirect
10. #31770766C2semidirect
11. Q16×Dic269C2semidirect
12. #31770668C2semidirect
13. #31770600C2semidirect
14. C2×#22575377C2semidirect
15. #451×C269C2semidirect
178. C269#3300semidirect
190. #451D269semidirect

Maximal subgroups

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

subgroupindex
#317712972
#317712662
#317709612
#317709452
#317709192
#317709142
#317708432
#317708362
#317708292
#317707662
Q16×Dic2692
#317706682
#317706002
C2×#225753772
#451×C2692
#3300269