Finite Group Databasebeta

← #46824995

Subgroups — #46824995

Order 34432.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771337C213yesmaximal, complemented
2#31771334C211yesmaximal, characteristic, complemented
3#31771320C213yesmaximal, complemented
4#31771302C213yesmaximal, complemented
5#31771289C213yesmaximal, complemented
6#31771274C211yesmaximal, characteristic, complemented
7#31771266C211yesmaximal, characteristic, complemented
8#31771224C213yesmaximal, complemented
9#31771208C213yesmaximal, complemented
10#31771141C213yesmaximal, complemented
11#31771135C213yesmaximal, complemented
12#31771087C211yesmaximal, characteristic, complemented
13#31771083C211yesmaximal, characteristic, complemented
14#41×Dic269C211yesmaximal, characteristic, complemented
15#548×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17#22575453C2213yes—
18#22575449C2213yes—
19#22575448C2213yes—
20#22575446C2216yes—
21#22575446C2211yescharacteristic
22#22575444C2213yes—
23#22575444C2213yes—
24#22575443C2213yes—
25#22575442C2213yes—
26#22575439C2216yes—
27#22575439C2213yes—
28#22575437C2216yes—
29#22575437C2213yes—
30#22575435C2216yes—
31#22575434C2213yes—
32#22575430C2213yes—
33#22575429C2213yes—
34#22575427C2213yes—
35#22575427C2213yes—
36#22575425C2213yes—
37#22575424C2216yes—
38#22575424C2213yes—
39Q8×Dic269C2213yes—
40Q8×Dic269C2211yescharacteristic
41Q8×Dic269C2211yescharacteristic
42Q8×Dic269C2213yes—
43Q8×Dic269C2211yescharacteristic
44C4×Dic538C2213yes—
45C4×D1076C2213yes—
46D4×Dic269C2213yes—
47D4×Dic269C2213yes—
48#22575415C2213yes—
49C4×C22⋊D269C2213yes—
50C4×C22⋊D269C2213yes—
51#22575413C2216yes—
52#22575412C2213yes—
53#22575412C2213yes—
54#22575412C2213yes—
55#22575411C2216yes—
56#22575410C2213yes—
57C4⋊C4×D269C2213yes—
58C2×C4×Dic269C2213yes—
59C2×#41×C269C2211yescharacteristic
60C22×Q8×C269C2211yescharacteristic
61#124×C269C2213yes—
62C22⋊Q8×C269C2213yes—
63#121×C269C2213yes—
64C4×Q8×C269C2211yescharacteristic
65C4×D4×C269C2213yes—
66#16864403—24no—
67C2×Dic538C2313yes—
68C2×D1076C2313yes—
69C2×C22⋊D269C2316yes—
70C2×C22⋊D269C2313yes—
71C2×C22⋊D269C2313yes—
72C22×Dic269C2313yes—
73C22×Dic269C2313yes—
74C2×C4×D269C2313yes—
75C2×C4×D269C2316yes—
76C4⋊Dic269C2313yes—
77C4⋊Dic269C2316yes—
78C4⋊Dic269C2313yes—
79C4⋊Dic269C2313yes—
80C4⋊Dic269C2313yes—
81C4⋊Dic269C2313yes—
82C4⋊Dic269C2311yescharacteristic
83#16864381C2313yes—
84#16864381C2311yescharacteristic
85#16864381C2313yes—
86#16864381C2311yescharacteristic
87#16864381C2313yes—
88#16864381C2313yes—
89C22⋊Dic269C2313yes—
90C22⋊Dic269C2311yescharacteristic
91C22⋊Dic269C2316yes—
92C22⋊Dic269C2311yescharacteristic
93C22⋊Dic269C2313yes—
94#16864379C2313yes—
95#16864379C2313yes—
96#16864379C2313yes—
97#16864379C2316yes—
98#16864379C2313yes—
99C4×Dic269C2311yescharacteristic
100C4×Dic269C2313yes—
101C4×Dic269C2313yes—
102C4×Dic269C2313yes—
103C4×Dic269C2316yes—
104C4×Dic269C2313yes—
105#41×C269—24no—
106C2×Q8×C269—24no—
107C2×Q8×C269C2313yes—
108C2×Q8×C269C2311yescharacteristic
109C2×D4×C269C2313yes—
110C2×D4×C269C2313yes—
111C22×C4×C269C2313yes—
112C22×C4×C269C2313yes—
113C4⋊C4×C269C2313yes—
114C4⋊C4×C269C2313yes—
115C22⋊C4×C269C2316yes—
116C22⋊C4×C269C2313yes—
117C22⋊C4×C269C2311yescharacteristic
118C42×C269C2313yes—
119C22×D269C2413yes—
120Dic538—43no—
121D1076—26no—
122C22⋊D269—43no—
123C22⋊D269—43no—
124C22⋊D269—46no—
125C2×Dic269—43no—
126C2×Dic269—43no—
127C2×Dic269C2411yescharacteristic
128C2×Dic269C2411yescharacteristic
129C2×Dic269C2413yes—
130C2×Dic269C2413yes—
131C2×Dic269C2411yescharacteristic
132C2×Dic269C2413yes—
133C2×Dic269C2411yescharacteristic
134C4×D269—26no—
135C4×D269—46no—
136C23×C269C2413yes—
137C23×C269C2411yescharacteristic
138Q8×C269—26no—
139Q8×C269#4114yes—
140D4×C269—26no—
141D4×C269—43no—
142C2×C4×C269—26no—
143C2×C4×C269—26no—
144C2×C4×C269C2411yescharacteristic
145C2×C4×C269C2413yes—
146C2×C4×C269C2413yes—
147C2×C4×C269C2411yescharacteristic
148C2×C4×C269C2413yes—
149C22×C269—21no—
150C22×C269—23no—
151C22×C269—21no—
152C22×C269—23no—
153C22×C269—23no—
154C22×C269—21no—
155C22×C269C2511yescharacteristic
156C2×D269—23no—
157C2×D269—23no—
158C2×D269—23no—
159C4×C269—21no—
160C4×C269—23no—
161C4×C269—23no—
162C4×C269—21no—
163C4×C269C2×#4116yes—
164Dic269—21no—
165Dic269—23no—
166Dic269—21no—
167Dic269—21no—
168Dic269—23no—
169Dic269—21no—
170Dic269—23no—
171C2×C269—41no—
172C2×C269—43no—
173C2×C269C22×#4111yescharacteristic
174C2×C269#58511yescharacteristic
175C2×C269#58511yescharacteristic
176D269—43no—
177C269#352411yescharacteristic, complemented, Sylow
178#3524—2691nomaximal, Sylow
179#559—2693no—
180#557—2691no—
181#555—2693no—
182#553—2693no—
183#552—2693no—
184#550—2691no—
185#548—2691no—
186#548D26911yescharacteristic, complemented
187#542—2693no—
188#540—2693no—
189#529—2693no—
190#529—2693no—
191#520—2691no—
192#519—2691no—
193C4×#41—2691no—
194C2×#41—2691no—
195C2×#41C2×D26911yescharacteristic
196C22×Q8C2×D26911yescharacteristic
197C4⋊Q8—2693no—
198#127—2693no—
199#126—2693no—
200#126—2691no—
201#126—2696no—
202#125—2693no—
203#125—2693no—
204#125—2693no—
205#124—2693no—
206#124—2693no—
207#124—2696no—
208#124C2×D26913yes—
209#123—2693no—
210#123—2696no—
211#123—2693no—
212#123—2696no—
213C22⋊Q8—2693no—
214C22⋊Q8—2693no—
215C22⋊Q8C2×D26913yes—
216#121—2693no—
217#121—2696no—
218#121—2693no—
219#121—2693no—
220#121—2693no—
221#121C2×D26913yes—
222C4×Q8—2691no—
223C4×Q8—2691no—
224C4×Q8—2693no—
225C4×Q8—2691no—
226C4×Q8—2693no—
227C4×Q8—2693no—
228C4×Q8C2×D26911yescharacteristic
229C4×D4—2693no—
230C4×D4—2693no—
231C4×D4—2693no—
232C4×D4—2693no—
233C4×D4—2693no—
234C4×D4—2696no—
235C4×D4—2693no—
236C4×D4C2×D26913yes—
237#117—2693no—
238#117—2693no—
239#117—2693no—
240#117—2696no—
241#117—2693no—
242C2×C4⋊C4—2693no—
243C2×C42—2693no—
244#41—24no—
245#41—5384no—
246C2×Q8—24no—
247C2×Q8—2693no—
248C2×Q8C22×D26913yes—
249C2×Q8C22×D26911yescharacteristic
250C2×D4—2693no—
251C2×D4—2696no—
252C2×D4—2693no—
253C2×D4—2693no—
254C2×D4C22×D26913yes—
255C2×D4C22×D26913yes—
256C22×C4—2693no—
257C22×C4—2693no—
258C22×C4—2693no—
259C22×C4—2696no—
260C22×C4C22×D26913yes—
261C22×C4C22×D26913yes—
262C4⋊C4—2693no—
263C4⋊C4—2693no—
264C4⋊C4—2691no—
265C4⋊C4—2693no—
266C4⋊C4—2693no—
267C4⋊C4—2693no—
268C4⋊C4—2693no—
269C4⋊C4—2696no—
270C4⋊C4—2691no—
271C4⋊C4—2693no—
272C4⋊C4—2693no—
273C4⋊C4—2691no—
274C4⋊C4—2693no—
275C4⋊C4C22×D26913yes—
276C4⋊C4C22×D26913yes—
277C22⋊C4—2691no—
278C22⋊C4—2693no—
279C22⋊C4—2696no—
280C22⋊C4—2691no—
281C22⋊C4—2693no—
282C22⋊C4—2693no—
283C22⋊C4—2693no—
284C22⋊C4—2693no—
285C22⋊C4—2696no—
286C22⋊C4—2693no—
287C22⋊C4C22×D26916yes—
288C22⋊C4C22×D26911yescharacteristic
289C22⋊C4C22×D26913yes—
290C42—2691no—
291C42—2693no—
292C42—2693no—
293C42—2693no—
294C42—2696no—
295C42—2693no—
296C42C22×D26913yes—
297C23—2693no—
298C23C23×D26911yescharacteristic
299C23C23×D26913yes—
300Q8—26no—
301Q8—10763no—
302Q8#1686440114yes—
303D4—26no—
304D4—43no—
305D4—5386no—
306D4—10763no—
307D4—10763no—
308D4—10766no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—2691no—
312C2×C4—2693no—
313C2×C4—2691no—
314C2×C4—2693no—
315C2×C4—2691no—
316C2×C4—2693no—
317C2×C4—2691no—
318C2×C4—5386no—
319C2×C4—10766no—
320C2×C4—10763no—
321C2×C4—10763no—
322C2×C4C23×D26911yescharacteristic
323C2×C4C23×D26913yes—
324C2×C4C23×D26911yescharacteristic
325C2×C4C23×D26913yes—
326C2×C4C23×D26913yes—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—21no—
331C22—23no—
332C22—21no—
333C22—5383no—
334C22—5383no—
335C22—5383no—
336C22C24×D26911yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—5381no—
342C4—5383no—
343C4—5381no—
344C4—5381no—
345C4—5381no—
346C4—5383no—
347C4—5383no—
348C4C2×#1686440116yes—
349C2—41no—
350C2—43no—
351C2—10763no—
352C2C22×#1686440111yescharacteristic
353C2#3177144111yescharacteristic
354C2#3177144211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#548×C269C2
hypercenter#548D269
hypocenterC269#3524
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31771337C2semidirect
2. #31771334C2semidirect
3. #31771320C2semidirect
4. #31771302C2semidirect
5. #31771289C2semidirect
6. #31771274C2semidirect
7. #31771266C2semidirect
8. #31771224C2semidirect
9. #31771208C2semidirect
10. #31771141C2semidirect
11. #31771135C2semidirect
12. #31771087C2semidirect
13. #31771083C2semidirect
14. #41×Dic269C2semidirect
15. #548×C269C2semidirect
177. C269#3524semidirect
186. #548D269semidirect

Maximal subgroups

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

subgroupindex
#317713372
#317713342
#317713202
#317713022
#317712892
#317712742
#317712662
#317712242
#317712082
#317711412
#317711352
#317710872
#317710832
#41×Dic2692
#548×C2692
#3524269