Finite Group Databasebeta

← #46822772

Subgroups — #46822772

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#31771295C211yesmaximal, characteristic, complemented
2#31770959C211yesmaximal, characteristic, complemented
3#31770943C211yesmaximal, characteristic, complemented
4#31770916C211yesmaximal, characteristic, complemented
5#31770912C211yesmaximal, characteristic, complemented
6#31770844C211yesmaximal, characteristic, complemented
7#31770832C211yesmaximal, characteristic, complemented
8#31770824C211yesmaximal, characteristic, complemented
9#31770767C211yesmaximal, characteristic, complemented
10#31770761C211yesmaximal, characteristic, complemented
11C4×#16864392C211yesmaximal, characteristic, complemented
12D4×#14524130C211yesmaximal, characteristic, complemented
13#31770601C211yesmaximal, characteristic, complemented
14C2×#22575377C211yesmaximal, characteristic, complemented
15#548×C269C211yesmaximal, characteristic
16C2×#16864392C2211yescharacteristic
17C2×#16864392C2211yescharacteristic
18C2×#16864390C2211yescharacteristic
19C22×#14524130C2211yescharacteristic
20C22×#14524130C2211yescharacteristic
21#22575446—21no—
22#22575445C2211yescharacteristic
23#22575443—21no—
24#22575434—21no—
25#22575432C2211yescharacteristic
26#22575432C2211yescharacteristic
27C4×Dic538C2211yescharacteristic
28#22575412C2211yescharacteristic
29#22575409—21no—
30C2×C4⋊Dic269C2211yescharacteristic
31#22575386C2211yescharacteristic
32#22575386C2211yescharacteristic
33#22575384C2211yescharacteristic
34C4⋊#14524130C2211yescharacteristic
35#22575377C2211yescharacteristic
36#22575377C2211yescharacteristic
37#22575377C2211yescharacteristic
38#22575377C2211yescharacteristic
39#22575377C2211yescharacteristic
40#22575377C2211yescharacteristic
41#22575376C2211yescharacteristic
42#22575374C2211yescharacteristic
43#22575374C2211yescharacteristic
44#22575374C2211yescharacteristic
45C22⋊#14524130C2211yescharacteristic
46C22⋊#14524130C2211yescharacteristic
47C4×#14524130C2211yescharacteristic
48C2×#41×C269C2211yescharacteristic
49C22×Q8×C269C2211yescharacteristic
50#124×C269—21no—
51#124×C269C2211yescharacteristic
52C22⋊Q8×C269—21no—
53C22⋊Q8×C269C2211yescharacteristic
54#121×C269—21no—
55#121×C269C2211yescharacteristic
56C4×Q8×C269C2211yescharacteristic
57C4×D4×C269—21no—
58C4×D4×C269C2211yescharacteristic
59C2×Dic538C2311yescharacteristic
60C22×Dic269—21no—
61#16864392—22no—
62#16864392—41no—
63#16864390—41no—
64C2×#14524130—41no—
65C2×#14524130—41no—
66C2×#14524130C2311yescharacteristic
67C2×#14524130C2311yescharacteristic
68C2×#14524130C2311yescharacteristic
69C2×#14524130C2311yescharacteristic
70C4⋊Dic269—21no—
71C4⋊Dic269C2311yescharacteristic
72C4⋊Dic269C2311yescharacteristic
73C4⋊Dic269C2311yescharacteristic
74#16864381—21no—
75#16864381—21no—
76#16864381—21no—
77#16864381—21no—
78#16864381—21no—
79C22⋊Dic269—21no—
80C22⋊Dic269—21no—
81C22⋊Dic269—21no—
82C4×Dic269—21no—
83C4×Dic269—21no—
84#41×C269—22no—
85#41×C269—41no—
86C2×Q8×C269—21no—
87C2×Q8×C269—22no—
88C2×Q8×C269—41no—
89C2×Q8×C269C2311yescharacteristic
90C2×Q8×C269C2311yescharacteristic
91C2×D4×C269—21no—
92C2×D4×C269—21no—
93C2×D4×C269D411yescharacteristic
94C2×D4×C269C2311yescharacteristic
95C22×C4×C269—21no—
96C22×C4×C269—21no—
97C22×C4×C269C2311yescharacteristic
98C22×C4×C269C2311yescharacteristic
99C4⋊C4×C269—21no—
100C4⋊C4×C269—21no—
101C4⋊C4×C269D411yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C22⋊C4×C269—21no—
107C22⋊C4×C269D411yescharacteristic
108C22⋊C4×C269D411yescharacteristic
109C42×C269—21no—
110C42×C269C2311yescharacteristic
111Dic538—21no—
112Dic538—21no—
113C2×Dic269—21no—
114C2×Dic269—21no—
115C2×Dic269—21no—
116C2×Dic269—21no—
117C2×Dic269—42no—
118#14524130—21no—
119#14524130—21no—
120#14524130—21no—
121#14524130—21no—
122C23×C269—21no—
123C23×C269C2×D411yescharacteristic
124C23×C269C2×D411yescharacteristic
125Q8×C269—21no—
126Q8×C269—21no—
127Q8×C269—21no—
128Q8×C269—42no—
129Q8×C269#4112yes—
130D4×C269—22no—
131D4×C269—21no—
132D4×C269—21no—
133D4×C269—42no—
134D4×C269—42no—
135C2×C4×C269—21no—
136C2×C4×C269—21no—
137C2×C4×C269—21no—
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—42no—
143C2×C4×C269—42no—
144C2×C4×C269C2×D411yescharacteristic
145C2×C4×C269C2×D411yescharacteristic
146C2×C4×C269C2×D411yescharacteristic
147C2×C4×C269C2×D411yescharacteristic
148C2×C4×C269C2411yescharacteristic
149C22×C269—21no—
150C22×C269—21no—
151C22×C269—21no—
152C22×C269—21no—
153C22×C269—21no—
154C22×C269—41no—
155C22×C269—41no—
156C22×C269—41no—
157C22×C269#13512yes—
158C22×C269C22×D411yescharacteristic
159C4×C269—21no—
160C4×C269—21no—
161C4×C269—22no—
162C4×C269—21no—
163C4×C269—21no—
164C4×C269—41no—
165C4×C269—41no—
166C4×C269C2×#4111yescharacteristic
167C4×C269H(5,𝔽2)11yescharacteristic
168Dic269—41no—
169Dic269—41no—
170Dic269—41no—
171Dic269—41no—
172C2×C269—22no—
173C2×C269—41no—
174C2×C269—81no—
175C2×C269#54611yescharacteristic
176C2×C269C2×#13511yescharacteristic
177C2×C269#57811yescharacteristic
178C269#330011yescharacteristic, complemented, Sylow
179#3300—2691nomaximal, Sylow
180#552—2691no—
181#548D26911yescharacteristic
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#451—2691no—
191C4×Q16—2691no—
192C8×D4—2691no—
193#416—2691no—
194C2×#103—2691no—
195C2×#41C2×D26911yescharacteristic
196C22×Q8C2×D26911yescharacteristic
197C2×Q16—2691no—
198C2×Q16—2691no—
199C2×SD16—2691no—
200C22×C8—2691no—
201C22×C8—2691no—
202#126—5381no—
203#125—2691no—
204#125—5381no—
205#124—21no—
206#124C2×D26911yescharacteristic
207#123—5381no—
208C22⋊Q8—21no—
209C22⋊Q8—2691no—
210C22⋊Q8—2691no—
211C22⋊Q8C2×D26911yescharacteristic
212#121—21no—
213#121C2×D26911yescharacteristic
214C4×Q8—2691no—
215C4×Q8C2×D26911yescharacteristic
216C4×D4—21no—
217C4×D4C2×D26911yescharacteristic
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#103—2691no—
231#103—2691no—
232#102—2691no—
233#102—2691no—
234#102—2691no—
235C22⋊C8—2691no—
236C22⋊C8—2691no—
237C4×C8—2691no—
238#41—22no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—2691no—
244C2×Q8C22×D26911yescharacteristic
245C2×Q8C22×D26911yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22⋊D26911yescharacteristic
249C2×D4C22×D26911yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—5381no—
253C22×C4C22×D26911yescharacteristic
254C22×C4C22×D26911yescharacteristic
255Q16—5382no—
256Q16—10761no—
257SD16—10761no—
258C2×C8—2691no—
259C2×C8—2691no—
260C2×C8—2691no—
261C2×C8—2691no—
262C2×C8—10761no—
263C2×C8—10761no—
264C4⋊C4—21no—
265C4⋊C4—21no—
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⋊C4C22⋊D26911yescharacteristic
276C4⋊C4C22×D26911yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—5381no—
282C22⋊C4—5381no—
283C22⋊C4—5381no—
284C22⋊C4C22⋊D26911yescharacteristic
285C22⋊C4C22⋊D26911yescharacteristic
286C42—21no—
287C42—5381no—
288C42—5381no—
289C42C22×D26911yescharacteristic
290C23—21no—
291C23C2×C22⋊D26911yescharacteristic
292C23C2×C22⋊D26911yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—5381no—
298Q8—5381no—
299Q8#1686440112yes—
300D4—22no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—5381no—
315C2×C4—5381no—
316C2×C4—5381no—
317C2×C4—5381no—
318C2×C4—10762no—
319C2×C4C2×C22⋊D26911yescharacteristic
320C2×C4C2×C22⋊D26911yescharacteristic
321C2×C4C2×C22⋊D26911yescharacteristic
322C2×C4C2×C22⋊D26911yescharacteristic
323C2×C4C23×D26911yescharacteristic
324C8—5381no—
325C8—5381no—
326C8—5381no—
327C8—5381no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#2257547312yes—
337C22C22×C22⋊D26911yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—22no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—10761no—
346C4—10761no—
347C4—10761no—
348C4—10761no—
349C4C2×#1686440111yescharacteristic
350C4#2257549811yescharacteristic
351C2—22no—
352C2—41no—
353C2—81no—
354C2#3177124911yescharacteristic
355C2C2×#2257547311yescharacteristic
356C2#3177142011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting, prime radical#548×C269C2
socleC22×C269C22×D4
hypercenter#548D269
hypocenterC269#3300
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31771295C2semidirect
2. #31770959C2semidirect
3. #31770943C2semidirect
4. #31770916C2semidirect
5. #31770912C2semidirect
6. #31770844C2semidirect
7. #31770832C2semidirect
8. #31770824C2semidirect
9. #31770767C2semidirect
10. #31770761C2semidirect
11. C4×#16864392C2semidirect
12. D4×#14524130C2semidirect
13. #31770601C2semidirect
14. C2×#22575377C2semidirect
178. C269#3300semidirect

Maximal subgroups

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

subgroupindex
#317712952
#317709592
#317709432
#317709162
#317709122
#317708442
#317708322
#317708242
#317707672
#317707612
C4×#168643922
D4×#145241302
#317706012
C2×#225753772
#548×C2692
#3300269