Finite Group Databasebeta

← #46824961

Subgroups — #46824961

Order 34432.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771337C213yesmaximal, complemented
2Q8×Dic538C211yesmaximal, characteristic, complemented
3#31771313C213yesmaximal, complemented
4#31771274C211yesmaximal, characteristic, complemented
5#31771266C211yesmaximal, characteristic, complemented
6#31771259C213yesmaximal, complemented
7#31771255C213yesmaximal, complemented
8#31771228C213yesmaximal, complemented
9#31771224C213yesmaximal, complemented
10#31771150C213yesmaximal, complemented
11#31771135C213yesmaximal, complemented
12#31771088C211yesmaximal, characteristic, complemented
13#31771087C211yesmaximal, characteristic, complemented
14C2×Q8×Dic269C211yesmaximal, characteristic, complemented
15#548×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864401C2216yes—
18#22575453C2213yes—
19#22575452C2213yes—
20#22575452C2213yes—
21#22575449C2213yes—
22#22575448C2213yes—
23#22575446C2213yes—
24#22575443C2213yes—
25#22575442C2213yes—
26#22575439C2216yes—
27#22575439C2213yes—
28#22575437C2216yes—
29#22575437C2213yes—
30#22575435C2216yes—
31#22575434C2213yes—
32#22575433C2216yes—
33#22575430C2216yes—
34#22575430C2211yescharacteristic
35#22575429C2213yes—
36#22575427C2213yes—
37#22575427C2213yes—
38#22575425C2213yes—
39#22575424C2213yes—
40Q8×Dic269C2213yes—
41Q8×Dic269C2211yescharacteristic
42Q8×Dic269C2211yescharacteristic
43Q8×Dic269C2213yes—
44Q8×Dic269C2211yescharacteristic
45C4×Dic538C2213yes—
46C4×D1076C2213yes—
47D4×Dic269C2213yes—
48D4×Dic269C2213yes—
49#22575415C2213yes—
50C4×C22⋊D269C2213yes—
51C4×C22⋊D269C2213yes—
52#22575413C2216yes—
53#22575412C2213yes—
54#22575410C2213yes—
55C2×C4⋊Dic269C2213yes—
56C2×C4⋊Dic269C2213yes—
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—
67#16864401—412no—
68C2×Dic538C2311yescharacteristic
69C2×Dic538C2316yes—
70C2×D1076C2313yes—
71C2×C22⋊D269C2313yes—
72C2×C22⋊D269C2313yes—
73C2×C22⋊D269C2316yes—
74C22×Dic269C2311yescharacteristic
75C22×Dic269C2313yes—
76C22×Dic269C2316yes—
77C2×C4×D269C2313yes—
78C2×C4×D269C2316yes—
79C4⋊Dic269C2313yes—
80C4⋊Dic269C2313yes—
81C4⋊Dic269C2313yes—
82C4⋊Dic269C2313yes—
83C4⋊Dic269C2316yes—
84C4⋊Dic269C2311yescharacteristic
85C4⋊Dic269C2313yes—
86#16864381C2313yes—
87#16864381C2311yescharacteristic
88#16864381C2313yes—
89#16864381C2311yescharacteristic
90#16864381C2313yes—
91#16864381C2313yes—
92C22⋊Dic269C2313yes—
93C22⋊Dic269C2313yes—
94C22⋊Dic269C2311yescharacteristic
95C22⋊Dic269C2313yes—
96#16864379C2313yes—
97#16864379C2313yes—
98#16864379C2316yes—
99#16864379C2313yes—
100#16864379C2313yes—
101C4×Dic269C2313yes—
102C4×Dic269C2313yes—
103C4×Dic269C2313yes—
104C4×Dic269C2313yes—
105C4×Dic269C2313yes—
106#41×C269—24no—
107C2×Q8×C269—24no—
108C2×Q8×C269C2313yes—
109C2×Q8×C269C2311yescharacteristic
110C2×D4×C269C2313yes—
111C2×D4×C269C2313yes—
112C22×C4×C269C2313yes—
113C22×C4×C269C2313yes—
114C4⋊C4×C269C2313yes—
115C4⋊C4×C269C2313yes—
116C22⋊C4×C269C2316yes—
117C22⋊C4×C269C2311yescharacteristic
118C22⋊C4×C269C2313yes—
119C42×C269C2313yes—
120C22×D269C2413yes—
121Dic538—22no—
122Dic538—46no—
123D1076—26no—
124C22⋊D269—26no—
125C22⋊D269—26no—
126C22⋊D269—46no—
127C2×Dic269—26no—
128C2×Dic269—22no—
129C2×Dic269—46no—
130C2×Dic269C2411yescharacteristic
131C2×Dic269C2411yescharacteristic
132C2×Dic269C2413yes—
133C2×Dic269C2413yes—
134C2×Dic269C2413yes—
135C2×Dic269C2411yescharacteristic
136C2×Dic269C2411yescharacteristic
137C4×D269—26no—
138C4×D269—46no—
139C23×C269C2413yes—
140C23×C269C2411yescharacteristic
141Q8×C269—26no—
142Q8×C269#4114yescomplemented
143D4×C269—26no—
144D4×C269—26no—
145C2×C4×C269—26no—
146C2×C4×C269—26no—
147C2×C4×C269C2411yescharacteristic
148C2×C4×C269C2413yes—
149C2×C4×C269C2413yes—
150C2×C4×C269C2411yescharacteristic
151C2×C4×C269C2413yes—
152C22×C269—21no—
153C22×C269—21no—
154C22×C269—23no—
155C22×C269—23no—
156C22×C269—23no—
157C22×C269H(5,𝔽2)12yes—
158C22×C269C2511yescharacteristic
159C2×D269—23no—
160C2×D269—23no—
161C2×D269—23no—
162C4×C269—21no—
163C4×C269—23no—
164C4×C269—23no—
165C4×C269—21no—
166C4×C269C2×#4116yes—
167Dic269—21no—
168Dic269—23no—
169Dic269—23no—
170Dic269—23no—
171Dic269—21no—
172Dic269—21no—
173Dic2692−1+412yes—
174C2×C269—22no—
175C2×C269—43no—
176C2×C269C22×#4111yescharacteristic
177C2×C269C2×H(5,𝔽2)11yescharacteristic
178C2×C269C2×2−1+411yescharacteristic
179D269—43no—
180C269#352211yescharacteristic, complemented, Sylow
181#3522—2691nomaximal, Sylow
182#559—2693no—
183Q82—2691no—
184#554—2693no—
185#550—2691no—
186#548—2691no—
187#548D26911yescharacteristic, complemented
188#547—2693no—
189#547—2693no—
190#542—2693no—
191#542—2693no—
192#532—2693no—
193#529—2693no—
194#520—2691no—
195#520—2691no—
196C2×C4×Q8—2691no—
197C2×#41—2691no—
198C2×#41—2696no—
199C2×#41C2×D26911yescharacteristic
200C22×Q8C2×D26911yescharacteristic
201C4⋊Q8—2693no—
202C4⋊Q8—2693no—
203C4⋊Q8—2693no—
204#127—2693no—
205#126—2693no—
206#126—2693no—
207#125—2693no—
208#124—2693no—
209#124—2693no—
210#124—2696no—
211#124C2×D26913yes—
212#123—2693no—
213#123—2693no—
214#123—2696no—
215#123—2696no—
216C22⋊Q8—2693no—
217C22⋊Q8—2696no—
218C22⋊Q8—2691no—
219C22⋊Q8—2696no—
220C22⋊Q8C2×D26913yes—
221#121—2693no—
222#121—2693no—
223#121—2693no—
224#121—2693no—
225#121C2×D26913yes—
226C4×Q8—2691no—
227C4×Q8—2693no—
228C4×Q8—2691no—
229C4×Q8—2693no—
230C4×Q8—2691no—
231C4×Q8—2693no—
232C4×Q8C2×D26911yescharacteristic
233C4×D4—2693no—
234C4×D4—2693no—
235C4×D4—2693no—
236C4×D4—2693no—
237C4×D4—2693no—
238C4×D4—2696no—
239C4×D4—2693no—
240C4×D4C2×D26913yes—
241#117—2693no—
242#117—2693no—
243C2×C4⋊C4—2693no—
244C2×C4⋊C4—2693no—
245C2×C4⋊C4—2693no—
246C2×C42—2693no—
247#41—24no—
248#41—5384no—
249#41—107612no—
250C2×Q8—24no—
251C2×Q8—2691no—
252C2×Q8—2696no—
253C2×Q8C22×D26913yes—
254C2×Q8C22×D26911yescharacteristic
255C2×D4—2693no—
256C2×D4—2693no—
257C2×D4—2696no—
258C2×D4—2693no—
259C2×D4C22×D26913yes—
260C2×D4C22×D26913yes—
261C22×C4—2691no—
262C22×C4—2693no—
263C22×C4—2696no—
264C22×C4—2693no—
265C22×C4—2696no—
266C22×C4C22×D26913yes—
267C22×C4C22×D26913yes—
268C4⋊C4—2693no—
269C4⋊C4—2693no—
270C4⋊C4—2693no—
271C4⋊C4—2693no—
272C4⋊C4—2693no—
273C4⋊C4—2696no—
274C4⋊C4—2691no—
275C4⋊C4—2691no—
276C4⋊C4—2691no—
277C4⋊C4—2693no—
278C4⋊C4—2693no—
279C4⋊C4—2693no—
280C4⋊C4—2693no—
281C4⋊C4C22×D26913yes—
282C4⋊C4C22×D26913yes—
283C22⋊C4—2693no—
284C22⋊C4—2693no—
285C22⋊C4—2691no—
286C22⋊C4—2693no—
287C22⋊C4—2693no—
288C22⋊C4—2693no—
289C22⋊C4—2693no—
290C22⋊C4—2696no—
291C22⋊C4—2693no—
292C22⋊C4C22×D26916yes—
293C22⋊C4C22×D26911yescharacteristic
294C22⋊C4C22×D26913yes—
295C42—2693no—
296C42—2693no—
297C42—2693no—
298C42—2693no—
299C42—2693no—
300C42C22×D26913yes—
301C23—2693no—
302C23C23×D26913yes—
303C23C23×D26911yescharacteristic
304Q8—26no—
305Q8—5382no—
306Q8—10766no—
307Q8#1686440114yescomplemented
308D4—26no—
309D4—26no—
310D4—5386no—
311D4—5386no—
312D4—5386no—
313D4—10766no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—2691no—
317C2×C4—2691no—
318C2×C4—2691no—
319C2×C4—2693no—
320C2×C4—2693no—
321C2×C4—2693no—
322C2×C4—2691no—
323C2×C4—5382no—
324C2×C4—5386no—
325C2×C4—5386no—
326C2×C4—10766no—
327C2×C4—10766no—
328C2×C4C23×D26913yes—
329C2×C4C23×D26911yescharacteristic
330C2×C4C23×D26913yes—
331C2×C4C23×D26911yescharacteristic
332C2×C4C23×D26913yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—5383no—
339C22—5383no—
340C22—5383no—
341C22#2257549912yes—
342C22C24×D26911yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—2692no—
348C4—5381no—
349C4—5383no—
350C4—5383no—
351C4—5383no—
352C4—5381no—
353C4—5381no—
354C4C2×#1686440116yes—
355C2—22no—
356C2—43no—
357C2—10763no—
358C2C22×#1686440111yescharacteristic
359C2C2×#2257549911yescharacteristic
360C22−1+4×D26911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#548×C269C2
hypercenter#548D269
hypocenterC269#3522
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31771337C2semidirect
2. Q8×Dic538C2semidirect
3. #31771313C2semidirect
4. #31771274C2semidirect
5. #31771266C2semidirect
6. #31771259C2semidirect
7. #31771255C2semidirect
8. #31771228C2semidirect
9. #31771224C2semidirect
10. #31771150C2semidirect
11. #31771135C2semidirect
12. #31771088C2semidirect
13. #31771087C2semidirect
14. C2×Q8×Dic269C2semidirect
15. #548×C269C2semidirect
142. Q8×C269#41semidirect
180. C269#3522semidirect
187. #548D269semidirect
307. Q8#16864401semidirect

Maximal subgroups

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

subgroupindex
#317713372
Q8×Dic5382
#317713132
#317712742
#317712662
#317712592
#317712552
#317712282
#317712242
#317711502
#317711352
#317710882
#317710872
C2×Q8×Dic2692
#548×C2692
#3522269