Finite Group Databasebeta

← #46824964

Subgroups — #46824964

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#31771338C213yesmaximal, complemented
2Q8×Dic538C211yesmaximal, characteristic, complemented
3#31771310C213yesmaximal, complemented
4#31771274C211yesmaximal, characteristic, complemented
5#31771267C211yesmaximal, characteristic, complemented
6#31771262C211yesmaximal, characteristic, complemented
7#31771260C213yesmaximal, complemented
8#31771256C213yesmaximal, complemented
9#31771226C213yesmaximal, complemented
10#31771220C213yesmaximal, complemented
11#31771152C213yesmaximal, complemented
12#31771133C213yesmaximal, complemented
13#31771087C211yesmaximal, characteristic, complemented
14C4×Q8×D269C211yesmaximal, characteristic, complemented
15#520×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864402C2216yes—
18C2×Q8×D269C2211yescharacteristic
19#22575453C2213yes—
20#22575452C2213yes—
21#22575452C2213yes—
22#22575449C2213yes—
23#22575448C2213yes—
24#22575446C2213yes—
25#22575443C2213yes—
26#22575441C2213yes—
27#22575440C2213yes—
28#22575439C2216yes—
29#22575439C2213yes—
30#22575438C2213yes—
31#22575436C2216yes—
32#22575435C2216yes—
33#22575435C2213yes—
34#22575433C2213yes—
35#22575432C2216yes—
36#22575432C2213yes—
37#22575431C2216yes—
38#22575431C2211yescharacteristic
39#22575428C2213yes—
40#22575427C2213yes—
41#22575426C2213yes—
42#22575424C2213yes—
43#22575424C2213yes—
44Q8×Dic269C2211yescharacteristic
45Q8×Dic269C2211yescharacteristic
46C4×Dic538C2213yes—
47C4×Dic538C2213yes—
48#22575418C2213yes—
49C4×D1076C2213yes—
50C4×D1076C2213yes—
51#22575415C2213yes—
52C4×C22⋊D269C2216yes—
53C4×C22⋊D269C2213yes—
54C4×C22⋊D269C2213yes—
55#22575410C2213yes—
56C4⋊C4×D269C2213yes—
57C4⋊C4×D269C2213yes—
58C42×D269C2213yes—
59C2×#41×C269C2211yescharacteristic
60C4×Q8×C269C2211yescharacteristic
61C4×Q8×C269C2211yescharacteristic
62C4×D4×C269C2213yes—
63C4×D4×C269C2213yes—
64#117×C269C2213yes—
65C2×C4⋊C4×C269C2213yes—
66#16864403—24no—
67#16864402—412no—
68C2×Dic538C2313yes—
69C2×Dic538C2311yescharacteristic
70C2×Dic538C2316yes—
71Q8×D269—24no—
72C2×D1076C2313yes—
73C2×D1076C2313yes—
74C2×C22⋊D269C2313yes—
75C2×C22⋊D269C2313yes—
76C2×C22⋊D269C2316yes—
77C2×C4×D269C2311yescharacteristic
78C2×C4×D269C2313yes—
79C2×C4×D269C2313yes—
80C2×C4×D269C2313yes—
81C2×C4×D269C2316yes—
82C4⋊Dic269C2313yes—
83C4⋊Dic269C2313yes—
84C4⋊Dic269C2313yes—
85C4⋊Dic269C2313yes—
86#16864381C2313yes—
87#16864381C2316yes—
88#16864381C2311yescharacteristic
89#16864381C2311yescharacteristic
90#16864381C2313yes—
91#16864381C2313yes—
92#16864381C2313yes—
93C22⋊Dic269C2313yes—
94C22⋊Dic269C2316yes—
95C22⋊Dic269C2313yes—
96#16864379C2313yes—
97#16864379C2313yes—
98#16864379C2311yescharacteristic
99#16864379C2313yes—
100#16864379C2316yes—
101#16864379C2311yescharacteristic
102#16864379C2313yes—
103C4×Dic269C2313yes—
104C4×Dic269C2313yes—
105C4×Dic269C2313yes—
106C4×Dic269C2313yes—
107#41×C269—24no—
108C2×Q8×C269C2311yescharacteristic
109C2×D4×C269C2313yes—
110C22×C4×C269C2313yes—
111C22×C4×C269C2316yes—
112C4⋊C4×C269C2313yes—
113C4⋊C4×C269C2313yes—
114C4⋊C4×C269C2313yes—
115C4⋊C4×C269C2311yescharacteristic
116C22⋊C4×C269C2313yes—
117C22⋊C4×C269C2313yes—
118C42×C269C2313yes—
119C42×C269C2313yes—
120C22×D269C2413yes—
121C22×D269C2411yescharacteristic
122Dic538—26no—
123Dic538—22no—
124Dic538—46no—
125D1076—26no—
126D1076—26no—
127C22⋊D269—26no—
128C22⋊D269—26no—
129C22⋊D269—46no—
130C2×Dic269C2411yescharacteristic
131C2×Dic269C2413yes—
132C2×Dic269C2413yes—
133C2×Dic269C2413yes—
134C2×Dic269C2411yescharacteristic
135C2×Dic269C2411yescharacteristic
136C4×D269—26no—
137C4×D269—26no—
138C4×D269—26no—
139C4×D269—22no—
140C4×D269—46no—
141C23×C269C2413yes—
142Q8×C269#4114yescomplemented
143D4×C269—26no—
144C2×C4×C269—26no—
145C2×C4×C269—46no—
146C2×C4×C269C2411yescharacteristic
147C2×C4×C269C2411yescharacteristic
148C2×C4×C269C2413yes—
149C2×C4×C269C2413yes—
150C2×C4×C269C2411yescharacteristic
151C2×C4×C269C2413yes—
152C22×C269—23no—
153C22×C269—23no—
154C22×C269—23no—
155C22×C269C2511yescharacteristic
156C2×D269—21no—
157C2×D269—21no—
158C2×D269—23no—
159C2×D269—23no—
160C2×D269—23no—
161C2×D269H(5,𝔽2)12yes—
162C4×C269—21no—
163C4×C269—23no—
164C4×C269—23no—
165C4×C269—21no—
166C4×C269—21no—
167C4×C269C2×#4116yes—
168Dic269—21no—
169Dic269—23no—
170Dic269—23no—
171Dic269—23no—
172Dic269—21no—
173Dic2692−1+412yes—
174C2×C269—43no—
175C2×C269C22×#4111yescharacteristic
176C2×C269C2×H(5,𝔽2)11yescharacteristic
177C2×C269C2×2−1+411yescharacteristic
178D269—22no—
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#548—2691no—
188#547—2693no—
189#547—2693no—
190#542—2693no—
191#542—2693no—
192#532—2693no—
193#529—2693no—
194#520—2691no—
195#520D26911yescharacteristic, complemented
196C2×C4×Q8—2691no—
197C2×#41—2691no—
198C2×#41—2696no—
199C2×#41C2×D26911yescharacteristic
200C22×Q8—2691no—
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#124—2693no—
212#123—2696no—
213#123—2693no—
214#123—2693no—
215#123—2696no—
216C22⋊Q8—2696no—
217C22⋊Q8—2693no—
218C22⋊Q8—2691no—
219C22⋊Q8—2696no—
220C22⋊Q8—2693no—
221#121—2693no—
222#121—2693no—
223#121—2693no—
224#121—2693no—
225#121—2693no—
226C4×Q8—2691no—
227C4×Q8—2691no—
228C4×Q8—2693no—
229C4×Q8—2693no—
230C4×Q8—2693no—
231C4×Q8C2×D26911yescharacteristic
232C4×Q8C2×D26911yescharacteristic
233C4×D4—2693no—
234C4×D4—2693no—
235C4×D4—2696no—
236C4×D4—2693no—
237C4×D4—2693no—
238C4×D4—2693no—
239C4×D4C2×D26913yes—
240C4×D4C2×D26913yes—
241#117—2693no—
242#117C2×D26913yes—
243C2×C4⋊C4—2693no—
244C2×C4⋊C4—2693no—
245C2×C4⋊C4C2×D26913yes—
246C2×C42—2693no—
247#41—24no—
248#41—5384no—
249#41—107612no—
250C2×Q8—2691no—
251C2×Q8—2693no—
252C2×Q8—2696no—
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—2696no—
266C22×C4C22×D26913yes—
267C22×C4C22×D26916yes—
268C4⋊C4—2693no—
269C4⋊C4—2693no—
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⋊C4C22×D26913yes—
280C4⋊C4C22×D26913yes—
281C4⋊C4C22×D26913yes—
282C4⋊C4C22×D26911yescharacteristic
283C22⋊C4—2693no—
284C22⋊C4—2696no—
285C22⋊C4—2693no—
286C22⋊C4—2693no—
287C22⋊C4—2693no—
288C22⋊C4—2691no—
289C22⋊C4—2691no—
290C22⋊C4—2693no—
291C22⋊C4—2696no—
292C22⋊C4—2693no—
293C22⋊C4C22×D26913yes—
294C22⋊C4C22×D26913yes—
295C42—2693no—
296C42—2693no—
297C42—2693no—
298C42—2693no—
299C42C22×D26913yes—
300C42C22×D26913yes—
301C23—2691no—
302C23—2693no—
303C23C23×D26913yes—
304Q8—5382no—
305Q8—5386no—
306Q8—10766no—
307Q8#1686440214yescomplemented
308D4—26no—
309D4—5386no—
310D4—5386no—
311D4—5386no—
312D4—5386no—
313D4—10766no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—2691no—
317C2×C4—2691no—
318C2×C4—2693no—
319C2×C4—2693no—
320C2×C4—2693no—
321C2×C4—2691no—
322C2×C4—5382no—
323C2×C4—5386no—
324C2×C4—5386no—
325C2×C4—5386no—
326C2×C4—10766no—
327C2×C4C23×D26911yescharacteristic
328C2×C4C23×D26913yes—
329C2×C4C23×D26911yescharacteristic
330C2×C4C23×D26913yes—
331C2×C4C23×D26911yescharacteristic
332C2×C4C23×D26913yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—2692no—
337C22—5381no—
338C22—5381no—
339C22—5383no—
340C22—5383no—
341C22—5383no—
342C22C24×D26911yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—2692no—
349C4—5381no—
350C4—5383no—
351C4—5383no—
352C4—5383no—
353C4—5381no—
354C4C2×#1686440216yes—
355C2—43no—
356C2—5382no—
357C2—10763no—
358C2C22×#1686440211yescharacteristic
359C2H(5,𝔽2)×D26911yescharacteristic
360C22−1+4×D26911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#520×C269C2
hypercenter#520D269
hypocenterC269#3522
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #31771338C2semidirect
2. Q8×Dic538C2semidirect
3. #31771310C2semidirect
4. #31771274C2semidirect
5. #31771267C2semidirect
6. #31771262C2semidirect
7. #31771260C2semidirect
8. #31771256C2semidirect
9. #31771226C2semidirect
10. #31771220C2semidirect
11. #31771152C2semidirect
12. #31771133C2semidirect
13. #31771087C2semidirect
14. C4×Q8×D269C2semidirect
15. #520×C269C2semidirect
142. Q8×C269#41semidirect
180. C269#3522semidirect
195. #520D269semidirect
307. Q8#16864402semidirect

Maximal subgroups

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

subgroupindex
#317713382
Q8×Dic5382
#317713102
#317712742
#317712672
#317712622
#317712602
#317712562
#317712262
#317712202
#317711522
#317711332
#317710872
C4×Q8×D2692
#520×C2692
#3522269