Finite Group Databasebeta

← #46824972

Subgroups — #46824972

Order 34432.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771331C211yesmaximal, characteristic, complemented
2#31771330C213yesmaximal, complemented
3#31771326C213yesmaximal, complemented
4#31771297C213yesmaximal, complemented
5#31771295C213yesmaximal, complemented
6Q8×C22⋊D269C211yesmaximal, characteristic, complemented
7Q8×C22⋊D269C211yesmaximal, characteristic, complemented
8#31771216C213yesmaximal, complemented
9#31771215C213yesmaximal, complemented
10#31771157C213yesmaximal, complemented
11#31771135C213yesmaximal
12#31771083C211yesmaximal, characteristic, complemented
13#31771081C211yesmaximal, characteristic, complemented
14C2×Q8×Dic269C211yesmaximal, characteristic, complemented
15D4×Q8×C269C211yesmaximal, characteristic, complemented
16C2×Q8×D269C2211yescharacteristic
17#22575452C2213yes—
18#22575451C2213yes—
19#22575451C2213yes—
20#22575448C2213yes—
21#22575448C2216yes—
22#22575446C2213yes—
23#22575446C2216yes—
24#22575445C2213yes—
25#22575444C2216yes—
26#22575444C2213yes—
27#22575443C2216yes—
28#22575443C2216yes—
29#22575443C2213yes—
30#22575439C2213yes—
31#22575437C2213yes—
32#22575434C2213yes—
33#22575432C2213yes—
34#22575432C2213yes—
35#22575430C2213yes—
36#22575429C2213yes—
37#22575429C2213yes—
38#22575425C2211yescharacteristic
39Q8×Dic269C2213yes—
40Q8×Dic269C2213yes—
41Q8×Dic269C2211yescharacteristic
42Q8×Dic269C2211yescharacteristic
43Q8×Dic269C2213yes—
44Q8×Dic269C2213yes—
45Q8×Dic269C2211yescharacteristic
46C4×Dic538C2213yes—
47C4×Dic538C2213yes—
48#22575418C2213yes—
49#22575418C2216yes—
50C4×C22⋊D269C2213yes—
51C4×C22⋊D269C2213yes—
52#22575412C2213yes—
53#22575412C2213yes—
54#22575411C2213yes—
55#22575410C2213yes—
56#22575409C2216yes—
57C2×C4⋊Dic269C2213yes—
58C2×C4×Dic269C2213yes—
59C22×Q8×C269C2211yescharacteristic
60C22×Q8×C269C2211yescharacteristic
61C4⋊Q8×C269C2213yes—
62C22⋊Q8×C269C2213yes—
63C22⋊Q8×C269C2213yes—
64C4×Q8×C269C2211yescharacteristic
65C4×D4×C269C2213yes—
66C2×Dic538C2313yes—
67C2×Dic538C2313yes—
68Q8×D269—24no—
69C2×C22⋊D269C2311yescharacteristic
70C2×C22⋊D269C2311yescharacteristic
71C22×Dic269C2311yescharacteristic
72C22×Dic269C2313yes—
73C2×C4×D269C2313yes—
74C4⋊Dic269C2313yes—
75C4⋊Dic269C2313yes—
76C4⋊Dic269C2316yes—
77C4⋊Dic269C2313yes—
78C4⋊Dic269C2313yes—
79C4⋊Dic269C2313yes—
80C4⋊Dic269C2313yes—
81#16864381C2316yes—
82#16864381C2316yes—
83#16864381C2313yes—
84#16864381C2313yes—
85#16864381C2313yes—
86#16864381C2313yes—
87#16864381C2316yes—
88#16864381C2311yescharacteristic
89#16864381C2313yes—
90C22⋊Dic269C2313yes—
91C22⋊Dic269C2311yescharacteristic
92C22⋊Dic269C2313yes—
93C22⋊Dic269C2313yes—
94#16864379C2311yescharacteristic
95#16864379C2313yes—
96#16864379C2313yes—
97#16864379C2313yes—
98C4×Dic269C2313yes—
99C4×Dic269C2313yes—
100C4×Dic269C2313yes—
101C4×Dic269C2313yes—
102C4×Dic269C2316yes—
103C4×Dic269C2316yes—
104C4×Dic269C2313yes—
105C2×Q8×C269—24no—
106C2×Q8×C269—24no—
107C2×Q8×C269C2313yes—
108C2×Q8×C269C2313yes—
109C2×Q8×C269C2311yescharacteristic
110C2×D4×C269C2311yescharacteristic
111C22×C4×C269C2313yes—
112C22×C4×C269C2313yes—
113C4⋊C4×C269C2313yes—
114C4⋊C4×C269C2316yes—
115C4⋊C4×C269C2313yes—
116C22⋊C4×C269C2313yes—
117C22⋊C4×C269C2313yes—
118C42×C269C2313yes—
119C22×D269C2411yescharacteristic
120Dic538—26no—
121Dic538—26no—
122C22⋊D269—22no—
123C22⋊D269—22no—
124C2×Dic269—26no—
125C2×Dic269—22no—
126C2×Dic269C2411yescharacteristic
127C2×Dic269C2413yes—
128C2×Dic269C2413yes—
129C2×Dic269C2411yescharacteristic
130C2×Dic269C2413yes—
131C2×Dic269C2413yes—
132C2×Dic269C2411yescharacteristic
133C4×D269—26no—
134C23×C269C2411yescharacteristic
135C23×C269C2411yescharacteristic
136Q8×C269—26no—
137Q8×C269—26no—
138Q8×C269#4114yes—
139D4×C269—22no—
140C2×C4×C269—26no—
141C2×C4×C269—26no—
142C2×C4×C269C2411yescharacteristic
143C2×C4×C269C2413yes—
144C2×C4×C269C2413yes—
145C2×C4×C269C2413yes—
146C2×C4×C269C2413yes—
147C22×C269—21no—
148C22×C269—21no—
149C22×C269—21no—
150C22×C269—21no—
151C22×C269—21no—
152C22×C2692−1+412yes—
153C22×C269C2511yescharacteristic
154C2×D269—21no—
155C2×D269—21no—
156C2×D269—21no—
157C4×C269—23no—
158C4×C269—23no—
159C4×C269—21no—
160C4×C269—23no—
161C4×C269C2×#4116yes—
162Dic269—23no—
163Dic269—21no—
164Dic269—23no—
165Dic269—23no—
166Dic269—21no—
167Dic269—23no—
168Dic2692−1+412yes—
169C2×C269—22no—
170C2×C269—41no—
171C2×C269C22×#4111yescharacteristic
172C2×C269C2×2−1+411yescharacteristic
173C2×C269C2×2−1+411yescharacteristic
174D269—41no—
175C269#352311yescharacteristic, complemented, Sylow
176#3523—2691nomaximal, Sylow
177#557—2693no—
178#557—2691no—
179#556—2693no—
180#552—2693no—
181#552—2693no—
182D4×Q8—2691no—
183D4×Q8—2691no—
184D4×Q8D26911yescharacteristic, complemented
185#541—2693no—
186#541—2693no—
187#533—2693no—
188#529—2693no—
189#519—2691no—
190#519—2691no—
191C2×C4×Q8—2691no—
192C22×Q8—2691no—
193C22×Q8C2×D26911yescharacteristic
194C22×Q8C2×D26911yescharacteristic
195C4⋊Q8—2693no—
196C4⋊Q8—2693no—
197C4⋊Q8—2693no—
198C4⋊Q8C2×D26913yes—
199#126—2693no—
200#126—2696no—
201#126—2693no—
202#126—2696no—
203#125—2693no—
204#125—2696no—
205#125—2696no—
206#125—2696no—
207#125—2693no—
208#125—2693no—
209#124—2693no—
210#123—2693no—
211#123—2693no—
212C22⋊Q8—2693no—
213C22⋊Q8—2693no—
214C22⋊Q8—2693no—
215C22⋊Q8—2693no—
216C22⋊Q8—2693no—
217C22⋊Q8C2×D26913yes—
218C22⋊Q8C2×D26913yes—
219#121—2691no—
220C4×Q8—2691no—
221C4×Q8—2693no—
222C4×Q8—2691no—
223C4×Q8—2693no—
224C4×Q8—2693no—
225C4×Q8—2691no—
226C4×Q8—2693no—
227C4×Q8—2693no—
228C4×Q8—2696no—
229C4×Q8—2693no—
230C4×Q8—2693no—
231C4×Q8C2×D26911yescharacteristic
232C4×D4—2693no—
233C4×D4—2693no—
234C4×D4C2×D26913yes—
235#117—2693no—
236#117—2693no—
237#117—2693no—
238#117—2696no—
239#117—2693no—
240C2×C4⋊C4—2693no—
241C2×C42—2693no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—2693no—
245C2×Q8—2693no—
246C2×Q8—5384no—
247C2×Q8C22×D26913yes—
248C2×Q8C22×D26913yes—
249C2×Q8C22×D26911yescharacteristic
250C2×D4—2691no—
251C2×D4—2691no—
252C2×D4C22×D26911yescharacteristic
253C22×C4—2691no—
254C22×C4—2693no—
255C22×C4—2693no—
256C22×C4C22×D26913yes—
257C22×C4C22×D26913yes—
258C4⋊C4—2693no—
259C4⋊C4—2693no—
260C4⋊C4—2696no—
261C4⋊C4—2693no—
262C4⋊C4—2693no—
263C4⋊C4—2693no—
264C4⋊C4—2693no—
265C4⋊C4—2693no—
266C4⋊C4—2696no—
267C4⋊C4—2696no—
268C4⋊C4—2693no—
269C4⋊C4—2693no—
270C4⋊C4—2691no—
271C4⋊C4—2696no—
272C4⋊C4—2693no—
273C4⋊C4—2693no—
274C4⋊C4C22×D26913yes—
275C4⋊C4C22×D26913yes—
276C4⋊C4C22×D26916yes—
277C22⋊C4—2691no—
278C22⋊C4—2693no—
279C22⋊C4—2693no—
280C22⋊C4—2693no—
281C22⋊C4—2691no—
282C22⋊C4—2693no—
283C22⋊C4—2693no—
284C22⋊C4—2693no—
285C22⋊C4C22×D26913yes—
286C22⋊C4C22×D26913yes—
287C42—2693no—
288C42—2693no—
289C42—2693no—
290C42—2693no—
291C42—2696no—
292C42—2696no—
293C42—2693no—
294C42C22×D26913yes—
295C23—2691no—
296C23C23×D26911yescharacteristic
297C23C23×D26911yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—5386no—
301Q8—5386no—
302Q8#1686440114yes—
303D4—22no—
304D4—5382no—
305D4—5382no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—2691no—
309C2×C4—2691no—
310C2×C4—2693no—
311C2×C4—2693no—
312C2×C4—2691no—
313C2×C4—2693no—
314C2×C4—2693no—
315C2×C4—5382no—
316C2×C4—5386no—
317C2×C4—5386no—
318C2×C4C23×D26913yes—
319C2×C4C23×D26913yes—
320C2×C4C23×D26913yes—
321C2×C4C23×D26911yescharacteristic
322C2×C4C23×D26913yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—5381no—
329C22—5381no—
330C22—5381no—
331C22#2257550112yes—
332C22C24×D26911yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—2692no—
338C4—5383no—
339C4—5381no—
340C4—5383no—
341C4—5383no—
342C4—5383no—
343C4—5381no—
344C4C2×#1686440116yes—
345C2—22no—
346C2—41no—
347C2—10761no—
348C2C22×#1686440111yescharacteristic
349C22−1+4×D26911yescharacteristic
350C2C2×#2257550111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
FittingD4×Q8×C269C2
prime radical#22575425C22
hypercenterD4×Q8D269
hypocenterC269#3523
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #31771331C2semidirect
2. #31771330C2semidirect
3. #31771326C2semidirect
4. #31771297C2semidirect
5. #31771295C2semidirect
6. Q8×C22⋊D269C2semidirect
7. Q8×C22⋊D269C2semidirect
8. #31771216C2semidirect
9. #31771215C2semidirect
10. #31771157C2semidirect
12. #31771083C2semidirect
13. #31771081C2semidirect
14. C2×Q8×Dic269C2semidirect
15. D4×Q8×C269C2semidirect
175. C269#3523semidirect
184. D4×Q8D269semidirect

Maximal subgroups

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

subgroupindex
#317713312
#317713302
#317713262
#317712972
#317712952
Q8×C22⋊D2692
#317712162
#317712152
#317711572
#317711352
#317710832
#317710812
C2×Q8×Dic2692
D4×Q8×C2692
#3523269