Finite Group Databasebeta

← #46824483

Subgroups — #46824483

Order 34432.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771338C213yesmaximal, complemented
2#31771331C211yesmaximal, characteristic, complemented
3#31771323C213yesmaximal, complemented
4#31771305C213yesmaximal, complemented
5#31771299C213yesmaximal, complemented
6#31771274C211yesmaximal, characteristic, complemented
7#31771267C211yesmaximal, characteristic, complemented
8#31771262C211yesmaximal, characteristic, complemented
9#31771152C213yesmaximal, complemented
10#31771139C213yesmaximal, complemented
11#31771137C213yesmaximal, complemented
12C4×#16864403C211yesmaximal, characteristic, complemented
13C4×#16864402C213yesmaximal, complemented
14C4×Q8×D269C211yesmaximal, characteristic, complemented
15C4×#41×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864402C2213yes—
18C2×Q8×D269C2211yescharacteristic
19#22575452C2213yes—
20#22575449C2213yes—
21#22575447C2216yes—
22#22575446C2216yes—
23#22575445C2213yes—
24#22575444C2213yes—
25#22575441C2213yes—
26#22575440C2213yes—
27#22575439C2216yes—
28#22575436C2216yes—
29#22575435C2216yes—
30#22575433C2213yes—
31#22575432C2213yes—
32#22575428C2213yes—
33#22575427C2213yes—
34#22575426C2216yes—
35#22575424C2213yes—
36#22575424C2213yes—
37Q8×Dic269C2211yescharacteristic
38Q8×Dic269C2211yescharacteristic
39C4×Dic538C2213yes—
40C4×Dic538C2213yes—
41C4×Dic538C2213yes—
42#22575418C2213yes—
43C4×D1076C2213yes—
44C4×D1076C2213yes—
45C4×D1076C2213yes—
46#22575415C2213yes—
47C4×C22⋊D269C2216yes—
48C4×C22⋊D269C2213yes—
49C4×C22⋊D269C2213yes—
50C4×C22⋊D269C2213yes—
51C4×C22⋊D269C2213yes—
52#22575412C2213yes—
53#22575411C2213yes—
54#22575410C2213yes—
55#22575410C2211yescharacteristic
56C4⋊C4×D269C2213yes—
57C42×D269C2213yes—
58C42×D269C2213yes—
59C2×#41×C269C2211yescharacteristic
60C4×Q8×C269C2211yescharacteristic
61C4×Q8×C269C2211yescharacteristic
62C4×D4×C269C2213yes—
63C4×D4×C269C2213yes—
64#117×C269C2213yes—
65C2×C42×C269C2213yes—
66#16864403—24no—
67#16864402—212no—
68C2×Dic538C2313yes—
69C2×Dic538C2313yes—
70Q8×D269—24no—
71C2×D1076C2313yes—
72C2×D1076C2313yes—
73C2×C22⋊D269C2313yes—
74C2×C22⋊D269C2313yes—
75C2×C22⋊D269C2316yes—
76C2×C4×D269C2313yes—
77C2×C4×D269C2313yes—
78C2×C4×D269C2313yes—
79C2×C4×D269C2313yes—
80C2×C4×D269C2313yes—
81C2×C4×D269C2311yescharacteristic
82C4⋊Dic269C2313yes—
83C4⋊Dic269C2313yes—
84C4⋊Dic269C2313yes—
85C4⋊Dic269C2313yes—
86#16864381C2316yes—
87#16864381C2311yescharacteristic
88#16864381C2311yescharacteristic
89#16864381C2313yes—
90#16864381C2313yes—
91#16864381C2316yes—
92C22⋊Dic269C2313yes—
93C22⋊Dic269C2313yes—
94C22⋊Dic269C2316yes—
95#16864379C2311yescharacteristic
96#16864379C2311yescharacteristic
97#16864379C2313yes—
98#16864379C2313yes—
99#16864379C2316yes—
100#16864379C2316yes—
101C4×Dic269C2311yescharacteristic
102C4×Dic269C2313yes—
103C4×Dic269C2313yes—
104C4×Dic269C2313yes—
105C4×Dic269C2313yes—
106C4×Dic269C2313yes—
107#41×C269—24no—
108C2×Q8×C269C2311yescharacteristic
109C2×D4×C269C2313yes—
110C22×C4×C269C2313yes—
111C22×C4×C269C2313yes—
112C22×C4×C269C2313yes—
113C4⋊C4×C269C2313yes—
114C4⋊C4×C269C2313yes—
115C22⋊C4×C269C2316yes—
116C42×C269C2313yes—
117C42×C269C2311yescharacteristic
118C42×C269C2313yes—
119C42×C269C2313yes—
120C22×D269C2413yes—
121C22×D269C2411yescharacteristic
122Dic538—26no—
123Dic538—26no—
124D1076—26no—
125D1076—26no—
126C22⋊D269—26no—
127C22⋊D269—26no—
128C22⋊D269—46no—
129C2×Dic269C2411yescharacteristic
130C2×Dic269C2411yescharacteristic
131C2×Dic269C2413yes—
132C2×Dic269C2413yes—
133C2×Dic269C2413yes—
134C2×Dic269C2411yescharacteristic
135C4×D269—26no—
136C4×D269—26no—
137C4×D269—26no—
138C4×D269—43no—
139C4×D269—43no—
140C4×D269#4114yescomplemented
141C23×C269C2413yes—
142Q8×C269#4114yescomplemented
143D4×C269—26no—
144C2×C4×C269—26no—
145C2×C4×C269—26no—
146C2×C4×C269—43no—
147C2×C4×C269C2411yescharacteristic
148C2×C4×C269C2411yescharacteristic
149C2×C4×C269C2413yes—
150C2×C4×C269C2413yes—
151C2×C4×C269C2413yes—
152C2×C4×C269C2411yescharacteristic
153C22×C269—23no—
154C22×C269—23no—
155C22×C269—23no—
156C22×C269C2511yescharacteristic
157C2×D269—21no—
158C2×D269—21no—
159C2×D269—23no—
160C2×D269—23no—
161C2×D269—23no—
162C2×D269C2×#4112yes—
163C4×C269—21no—
164C4×C269—23no—
165C4×C269—21no—
166C4×C269—23no—
167C4×C269C2×#4112yes—
168C4×C269C2×#4116yes—
169Dic269—21no—
170Dic269—23no—
171Dic269—23no—
172Dic269—21no—
173Dic269—23no—
174Dic269C2×#4112yes—
175C2×C269—43no—
176C2×C269C22×#4111yescharacteristic
177C2×C269C22×#4111yescharacteristic
178C2×C269#58511yescharacteristic
179D269—22no—
180D269—43no—
181C269#348011yescharacteristic, complemented, Sylow
182#3480—2691nomaximal, Sylow
183#559—2693no—
184#557—2691no—
185#556—2693no—
186#553—2693no—
187#553—2693no—
188#550—2691no—
189#548—2691no—
190#548—2691no—
191#532—2693no—
192#529—2693no—
193#529—2693no—
194C4×#41—2691no—
195C4×#41—2693no—
196C4×#41D26911yescharacteristic, complemented
197C2×C4×Q8—2691no—
198C2×#41—2691no—
199C2×#41—2693no—
200C2×#41C2×D26911yescharacteristic
201C22×Q8—2691no—
202C4⋊Q8—2693no—
203#127—2693no—
204#126—2696no—
205#126—2696no—
206#125—2693no—
207#125—2693no—
208#124—2693no—
209#124—2696no—
210#124—2693no—
211#123—2696no—
212#123—2696no—
213C22⋊Q8—2693no—
214C22⋊Q8—2693no—
215#121—2693no—
216#121—2693no—
217#121—2693no—
218#121—2693no—
219#121—2696no—
220C4×Q8—2691no—
221C4×Q8—2691no—
222C4×Q8—2693no—
223C4×Q8—2693no—
224C4×Q8—2693no—
225C4×Q8—2693no—
226C4×Q8C2×D26911yescharacteristic
227C4×Q8C2×D26911yescharacteristic
228C4×D4—2693no—
229C4×D4—2693no—
230C4×D4—2693no—
231C4×D4—2693no—
232C4×D4—2696no—
233C4×D4—2693no—
234C4×D4—2693no—
235C4×D4—2693no—
236C4×D4—2693no—
237C4×D4C2×D26913yes—
238C4×D4C2×D26913yes—
239#117—2693no—
240#117—2693no—
241#117—2691no—
242#117—2693no—
243#117C2×D26913yes—
244C2×C4⋊C4—2693no—
245C2×C42—2693no—
246C2×C42—2693no—
247C2×C42C2×D26913yes—
248#41—24no—
249#41—53812no—
250#41—5384no—
251C2×Q8—2693no—
252C2×Q8—2693no—
253C2×Q8—5384no—
254C2×Q8C22×D26911yescharacteristic
255C2×D4—2693no—
256C2×D4—2693no—
257C2×D4—2696no—
258C2×D4—2693no—
259C2×D4—2693no—
260C2×D4C22×D26913yes—
261C22×C4—2691no—
262C22×C4—2693no—
263C22×C4—2693no—
264C22×C4—2693no—
265C22×C4—2693no—
266C22×C4—2693no—
267C22×C4C22×D26913yes—
268C22×C4C22×D26913yes—
269C22×C4C22×D26913yes—
270C4⋊C4—2693no—
271C4⋊C4—2696no—
272C4⋊C4—2691no—
273C4⋊C4—2691no—
274C4⋊C4—2693no—
275C4⋊C4—2693no—
276C4⋊C4—2693no—
277C4⋊C4—2693no—
278C4⋊C4—2693no—
279C4⋊C4—2696no—
280C4⋊C4C22×D26913yes—
281C4⋊C4C22×D26913yes—
282C22⋊C4—2693no—
283C22⋊C4—2693no—
284C22⋊C4—2696no—
285C22⋊C4—2691no—
286C22⋊C4—2691no—
287C22⋊C4—2693no—
288C22⋊C4—2696no—
289C22⋊C4—2693no—
290C22⋊C4—2696no—
291C22⋊C4C22×D26916yes—
292C42—2693no—
293C42—2693no—
294C42—2693no—
295C42—2693no—
296C42—2691no—
297C42—2693no—
298C42C22×D26913yes—
299C42C22×D26913yes—
300C42C22×D26913yes—
301C42C22×D26911yescharacteristic
302C23—2691no—
303C23—2693no—
304C23C23×D26913yes—
305Q8—5386no—
306Q8—5386no—
307Q8#1686440214yescomplemented
308D4—26no—
309D4—5386no—
310D4—5386no—
311D4—5386no—
312D4—5386no—
313D4—10766no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—43no—
317C2×C4—2691no—
318C2×C4—2694no—
319C2×C4—2691no—
320C2×C4—2691no—
321C2×C4—2693no—
322C2×C4—2693no—
323C2×C4—2693no—
324C2×C4—5386no—
325C2×C4—5386no—
326C2×C4—5386no—
327C2×C4—10763no—
328C2×C4—10763no—
329C2×C4C23×D26913yes—
330C2×C4C23×D26911yescharacteristic
331C2×C4C23×D26911yescharacteristic
332C2×C4C23×D26913yes—
333C2×C4C23×D26913yes—
334C2×C4C23×D26911yescharacteristic
335C22—23no—
336C22—23no—
337C22—23no—
338C22—2692no—
339C22—5381no—
340C22—5381no—
341C22—5383no—
342C22—5383no—
343C22—5383no—
344C22C24×D26911yescharacteristic
345C4—23no—
346C4—21no—
347C4—21no—
348C4—23no—
349C4—2692no—
350C4—5383no—
351C4—5381no—
352C4—5383no—
353C4—5381no—
354C4—5383no—
355C4#41×D26912yes—
356C4C2×#1686440216yes—
357C2—43no—
358C2—5382no—
359C2—10763no—
360C2C2×#41×D26911yescharacteristic
361C2C22×#1686440211yescharacteristic
362C2#3177144111yescharacteristic
subgroupG/H
centerC2×C4C23×D269
commutator, socle, cosocleC22×C269C25
FrattiniC22C24×D269
FittingC4×#41×C269C2
hypercenterC4×#41D269
hypocenterC269#3480
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C2×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #31771338C2semidirect
2. #31771331C2semidirect
3. #31771323C2semidirect
4. #31771305C2semidirect
5. #31771299C2semidirect
6. #31771274C2semidirect
7. #31771267C2semidirect
8. #31771262C2semidirect
9. #31771152C2semidirect
10. #31771139C2semidirect
11. #31771137C2semidirect
12. C4×#16864403C2semidirect
13. C4×#16864402C2semidirect
14. C4×Q8×D269C2semidirect
15. C4×#41×C269C2semidirect
140. C4×D269#41semidirect
142. Q8×C269#41semidirect
181. C269#3480semidirect
196. C4×#41D269semidirect
307. Q8#16864402semidirect

Maximal subgroups

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

subgroupindex
#317713382
#317713312
#317713232
#317713052
#317712992
#317712742
#317712672
#317712622
#317711522
#317711392
#317711372
C4×#168644032
C4×#168644022
C4×Q8×D2692
C4×#41×C2692
#3480269