Finite Group Databasebeta

← #46824997

Subgroups — #46824997

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#31771340C213yesmaximal, complemented
2#31771321C213yesmaximal, complemented
3#31771308C213yesmaximal, complemented
4#31771293C213yesmaximal, complemented
5#31771277C211yesmaximal, characteristic, complemented
6#31771267C211yesmaximal, characteristic, complemented
7#31771267C211yesmaximal, characteristic, complemented
8#31771229C213yesmaximal, complemented
9#31771211C213yesmaximal, complemented
10#31771142C213yesmaximal, complemented
11#31771140C213yesmaximal, complemented
12#31771087C211yesmaximal, characteristic, complemented
13#31771081C211yesmaximal, characteristic, complemented
14C4×#16864403C211yesmaximal, characteristic, complemented
15#557×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864403C2211yescharacteristic
18C2×Q8×D269C2211yescharacteristic
19#22575450C2213yes—
20#22575448C2213yes—
21#22575448C2211yescharacteristic
22#22575448C2216yes—
23#22575444C2213yes—
24#22575442C2213yes—
25#22575442C2216yes—
26#22575441C2213yes—
27#22575441C2213yes—
28#22575438C2216yes—
29#22575438C2213yes—
30#22575438C2213yes—
31#22575438C2216yes—
32#22575433C2213yes—
33#22575433C2213yes—
34#22575429C2213yes—
35#22575428C2213yes—
36#22575428C2213yes—
37#22575428C2213yes—
38#22575428C2213yes—
39#22575428C2216yes—
40#22575428C2213yes—
41Q8×Dic269C2211yescharacteristic
42C4×Dic538C2213yes—
43#22575418C2213yes—
44C4×D1076C2213yes—
45C4×D1076C2213yes—
46C4×D1076C2213yes—
47C4×D1076C2213yes—
48C4×D1076C2213yes—
49#22575415C2213yes—
50#22575415C2216yes—
51#22575415C2213yes—
52#22575411C2216yes—
53#22575411C2213yes—
54#22575411C2213yes—
55#22575410C2213yes—
56#22575410C2213yes—
57C4⋊C4×D269C2213yes—
58C42×D269C2213yes—
59C4⋊Q8×C269C2213yes—
60#125×C269C2213yes—
61#125×C269C2213yes—
62C4×Q8×C269C2213yes—
63C4×Q8×C269C2211yescharacteristic
64C4×Q8×C269C2211yescharacteristic
65C4×Q8×C269C2211yescharacteristic
66#16864403—24no—
67#16864403—24no—
68C2×Dic538C2313yes—
69Q8×D269—24no—
70C2×D1076C2313yes—
71C2×D1076C2313yes—
72C2×D1076C2316yes—
73C2×D1076C2313yes—
74C2×D1076C2313yes—
75C2×D1076C2313yes—
76C2×C4×D269C2313yes—
77C2×C4×D269C2313yes—
78C2×C4×D269C2313yes—
79C2×C4×D269C2313yes—
80C2×C4×D269C2313yes—
81C2×C4×D269C2316yes—
82C4⋊Dic269C2313yes—
83C4⋊Dic269C2313yes—
84C4⋊Dic269C2311yescharacteristic
85C4⋊Dic269C2313yes—
86#16864381C2313yes—
87#16864381C2311yescharacteristic
88#16864381C2313yes—
89#16864381C2313yes—
90#16864379C2311yescharacteristic
91#16864379C2313yes—
92#16864379C2313yes—
93#16864379C2313yes—
94#16864379C2313yes—
95#16864379C2316yes—
96#16864379C2316yes—
97#16864379C2313yes—
98#16864379C2316yes—
99#16864379C2311yescharacteristic
100#16864379C2311yescharacteristic
101#16864379C2313yes—
102#16864379C2313yes—
103C4×Dic269C2311yescharacteristic
104C4×Dic269C2313yes—
105C4×Dic269C2313yes—
106C4×Dic269C2316yes—
107C2×Q8×C269C2313yes—
108C2×Q8×C269C2311yescharacteristic
109C4⋊C4×C269C2313yes—
110C4⋊C4×C269C2313yes—
111C4⋊C4×C269C2313yes—
112C4⋊C4×C269C2316yes—
113C4⋊C4×C269C2313yes—
114C4⋊C4×C269C2313yes—
115C4⋊C4×C269C2311yescharacteristic
116C42×C269C2313yes—
117C42×C269C2313yes—
118C42×C269C2313yes—
119C22×D269C2411yescharacteristic
120C22×D269C2413yes—
121C22×D269C2413yes—
122Dic538—26no—
123D1076—26no—
124D1076—26no—
125D1076—43no—
126D1076—46no—
127D1076—43no—
128D1076—43no—
129C2×Dic269C2411yescharacteristic
130C2×Dic269C2413yes—
131C2×Dic269C2411yescharacteristic
132C2×Dic269C2413yes—
133C2×Dic269C2411yescharacteristic
134C4×D269—26no—
135C4×D269—26no—
136C4×D269—26no—
137C4×D269—43no—
138C4×D269—43no—
139C4×D269—46no—
140Q8×C269—43no—
141Q8×C269#4114yes—
142C2×C4×C269C2411yescharacteristic
143C2×C4×C269C2411yescharacteristic
144C2×C4×C269C2411yescharacteristic
145C2×C4×C269C2413yes—
146C2×C4×C269C2413yes—
147C2×C4×C269C2413yes—
148C2×C4×C269C2413yes—
149C22×C269C2511yescharacteristic
150C2×D269—21no—
151C2×D269—23no—
152C2×D269—23no—
153C2×D269—23no—
154C2×D269—23no—
155C2×D269—23no—
156C2×D269—21no—
157C2×D269—23no—
158C2×D269—21no—
159C4×C269—21no—
160C4×C269—23no—
161C4×C269—23no—
162C4×C269—21no—
163C4×C269—23no—
164C4×C269—21no—
165C4×C269C2×#4116yes—
166Dic269—21no—
167Dic269—21no—
168Dic269—23no—
169Dic269—21no—
170Dic269—23no—
171C2×C269C22×#4111yescharacteristic
172C2×C269#58511yescharacteristic
173C2×C269#58511yescharacteristic
174D269—41no—
175D269—43no—
176D269—43no—
177C269#352411yescharacteristic, complemented, Sylow
178#3524—2691nomaximal, Sylow
179#559—2693no—
180#557D26911yescharacteristic, complemented
181#555—2693no—
182#553—2693no—
183#552—2693no—
184#550—2691no—
185#548—2691no—
186#548—2691no—
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×#41—2691no—
196C22×Q8—2691no—
197C4⋊Q8C2×D26913yes—
198#127—2693no—
199#126—2691no—
200#126—2693no—
201#126—2696no—
202#125—2693no—
203#125C2×D26913yes—
204#125C2×D26913yes—
205#124—2693no—
206#124—2696no—
207#124—2693no—
208#124—2693no—
209#123—2696no—
210#123—2693no—
211#123—2693no—
212#123—2696no—
213C22⋊Q8—2693no—
214C22⋊Q8—2693no—
215C22⋊Q8—2693no—
216#121—2693no—
217#121—2693no—
218#121—2693no—
219#121—2693no—
220#121—2696no—
221#121—2693no—
222C4×Q8—2691no—
223C4×Q8—2693no—
224C4×Q8—2693no—
225C4×Q8C2×D26911yescharacteristic
226C4×Q8C2×D26911yescharacteristic
227C4×Q8C2×D26911yescharacteristic
228C4×Q8C2×D26913yes—
229C4×D4—2693no—
230C4×D4—2693no—
231C4×D4—2696no—
232C4×D4—2693no—
233C4×D4—2693no—
234C4×D4—2693no—
235C4×D4—2693no—
236C4×D4—2693no—
237#117—2693no—
238#117—2693no—
239#117—2696no—
240#117—2693no—
241#117—2693no—
242C2×C4⋊C4—2693no—
243C2×C42—2693no—
244#41—5384no—
245#41—5384no—
246C2×Q8—2693no—
247C2×Q8—5384no—
248C2×Q8C22×D26913yes—
249C2×Q8C22×D26911yescharacteristic
250C2×D4—2693no—
251C2×D4—2693no—
252C2×D4—2693no—
253C2×D4—2693no—
254C2×D4—2696no—
255C2×D4—2693no—
256C22×C4—2693no—
257C22×C4—2693no—
258C22×C4—2693no—
259C22×C4—2693no—
260C22×C4—2693no—
261C22×C4—2696no—
262C4⋊C4—2693no—
263C4⋊C4—2693no—
264C4⋊C4—2691no—
265C4⋊C4—2693no—
266C4⋊C4—2693no—
267C4⋊C4—2693no—
268C4⋊C4—2691no—
269C4⋊C4—2693no—
270C4⋊C4C22×D26913yes—
271C4⋊C4C22×D26913yes—
272C4⋊C4C22×D26913yes—
273C4⋊C4C22×D26916yes—
274C4⋊C4C22×D26913yes—
275C4⋊C4C22×D26911yescharacteristic
276C4⋊C4C22×D26913yes—
277C22⋊C4—2691no—
278C22⋊C4—2693no—
279C22⋊C4—2693no—
280C22⋊C4—2693no—
281C22⋊C4—2696no—
282C22⋊C4—2693no—
283C22⋊C4—2696no—
284C22⋊C4—2691no—
285C22⋊C4—2693no—
286C22⋊C4—2691no—
287C22⋊C4—2693no—
288C22⋊C4—2696no—
289C22⋊C4—2693no—
290C42—2691no—
291C42—2693no—
292C42—2693no—
293C42—2696no—
294C42C22×D26913yes—
295C42C22×D26913yes—
296C42C22×D26913yes—
297C23—2691no—
298C23—2693no—
299C23—2693no—
300Q8—43no—
301Q8—5386no—
302Q8#1686440314yes—
303D4—5386no—
304D4—5386no—
305D4—10763no—
306D4—10766no—
307D4—10763no—
308D4—10763no—
309C2×C4—2691no—
310C2×C4—2691no—
311C2×C4—2693no—
312C2×C4—2693no—
313C2×C4—2691no—
314C2×C4—5386no—
315C2×C4—5386no—
316C2×C4—5386no—
317C2×C4—10763no—
318C2×C4—10763no—
319C2×C4—10766no—
320C2×C4C23×D26913yes—
321C2×C4C23×D26911yescharacteristic
322C2×C4C23×D26913yes—
323C2×C4C23×D26911yescharacteristic
324C2×C4C23×D26911yescharacteristic
325C2×C4C23×D26913yes—
326C2×C4C23×D26913yes—
327C22—5381no—
328C22—5383no—
329C22—5383no—
330C22—5383no—
331C22—5381no—
332C22—5383no—
333C22—5383no—
334C22—5383no—
335C22—5381no—
336C22C24×D26911yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—5381no—
344C4—5381no—
345C4—5383no—
346C4—5381no—
347C4—5383no—
348C4C2×#1686440316yes—
349C2—10761no—
350C2—10763no—
351C2—10763no—
352C2C22×#1686440311yescharacteristic
353C2#3177144111yescharacteristic
354C2#3177144211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#557×C269C2
hypercenter#557D269
hypocenterC269#3524
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #31771340C2semidirect
2. #31771321C2semidirect
3. #31771308C2semidirect
4. #31771293C2semidirect
5. #31771277C2semidirect
6. #31771267C2semidirect
7. #31771267C2semidirect
8. #31771229C2semidirect
9. #31771211C2semidirect
10. #31771142C2semidirect
11. #31771140C2semidirect
12. #31771087C2semidirect
13. #31771081C2semidirect
14. C4×#16864403C2semidirect
15. #557×C269C2semidirect
177. C269#3524semidirect
180. #557D269semidirect

Maximal subgroups

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

subgroupindex
#317713402
#317713212
#317713082
#317712932
#317712772
#317712672
#317712292
#317712112
#317711422
#317711402
#317710872
#317710812
C4×#168644032
#557×C2692
#3524269