Finite Group Databasebeta

← #46824501

Subgroups — #46824501

Order 34432.

1094 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#31771331C211yesmaximal, characteristic, complemented
2#31771324C213yesmaximal, complemented
3#31771295C213yesmaximal, complemented
4#31771289C213yesmaximal, complemented
5Q8×C22⋊D269C211yesmaximal, characteristic, complemented
6#31771262C211yesmaximal, characteristic, complemented
7#31771255C213yesmaximal, complemented
8#31771157C213yesmaximal, complemented
9#31771136C213yesmaximal, complemented
10#31771134C213yesmaximal, complemented
11#41×Dic269C211yesmaximal, characteristic, complemented
12C4×#16864401C213yesmaximal, complemented
13C2×Q8×Dic269C211yesmaximal, characteristic, complemented
14C4×Q8×D269C211yesmaximal, characteristic, complemented
15#548×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2213yes—
17C2×Q8×D269C2211yescharacteristic
18#22575452C2213yes—
19#22575451C2213yes—
20#22575446C2216yes—
21#22575446C2216yes—
22#22575445C2213yes—
23#22575444C2213yes—
24#22575443C2216yes—
25#22575440C2213yes—
26#22575435C2216yes—
27#22575434C2216yes—
28#22575432C2213yes—
29#22575432C2213yes—
30#22575430C2216yes—
31#22575429C2213yes—
32#22575424C2213yes—
33Q8×Dic269C2213yes—
34Q8×Dic269C2211yescharacteristic
35Q8×Dic269C2211yescharacteristic
36Q8×Dic269C2213yes—
37Q8×Dic269C2211yescharacteristic
38C4×Dic538C2213yes—
39C4×Dic538C2213yes—
40#22575418C2213yes—
41#22575418C2213yes—
42D4×Dic269C2213yes—
43D4×Dic269C2213yes—
44D4×Dic269C2213yes—
45C4×C22⋊D269C2213yes—
46C4×C22⋊D269C2213yes—
47#22575413C2211yescharacteristic
48#22575413C2213yes—
49#22575412C2213yes—
50#22575411C2213yes—
51#22575409C2216yes—
52#22575409C2213yes—
53#22575409C2213yes—
54C2×C4⋊Dic269C2213yes—
55C4⋊C4×D269C2213yes—
56C2×C4×Dic269C2213yes—
57C2×C4×Dic269C2213yes—
58C42×D269C2213yes—
59C2×#41×C269C2211yescharacteristic
60C22×Q8×C269C2211yescharacteristic
61#124×C269C2213yes—
62C22⋊Q8×C269C2213yes—
63#121×C269C2213yes—
64C4×Q8×C269C2211yescharacteristic
65C4×D4×C269C2213yes—
66#16864401—212no—
67C2×Dic538C2313yes—
68C2×Dic538C2313yes—
69Q8×D269—24no—
70C2×C22⋊D269C2313yes—
71C2×C22⋊D269C2311yescharacteristic
72C22×Dic269C2311yescharacteristic
73C22×Dic269C2313yes—
74C22×Dic269C2313yes—
75C22×Dic269C2313yes—
76C2×C4×D269C2311yescharacteristic
77C2×C4×D269C2313yes—
78C2×C4×D269C2313yes—
79C4⋊Dic269C2313yes—
80C4⋊Dic269C2313yes—
81C4⋊Dic269C2316yes—
82C4⋊Dic269C2313yes—
83C4⋊Dic269C2313yes—
84#16864381C2316yes—
85#16864381C2311yescharacteristic
86#16864381C2313yes—
87#16864381C2313yes—
88#16864381C2313yes—
89#16864381C2316yes—
90#16864381C2316yes—
91C22⋊Dic269C2313yes—
92C22⋊Dic269C2313yes—
93C22⋊Dic269C2316yes—
94C22⋊Dic269C2316yes—
95#16864379C2311yescharacteristic
96#16864379C2313yes—
97C4×Dic269C2313yes—
98C4×Dic269C2313yes—
99C4×Dic269C2313yes—
100C4×Dic269C2311yescharacteristic
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×C269C2311yescharacteristic
117C22⋊C4×C269C2313yes—
118C22⋊C4×C269C2316yes—
119C42×C269C2313yes—
120C22×D269C2411yescharacteristic
121Dic538—26no—
122Dic538—26no—
123C22⋊D269—26no—
124C22⋊D269#4114yescomplemented
125C2×Dic269—26no—
126C2×Dic269—26no—
127C2×Dic269—22no—
128C2×Dic269—26no—
129C2×Dic269C2411yescharacteristic
130C2×Dic269C2413yes—
131C2×Dic269C2413yes—
132C2×Dic269C2413yes—
133C2×Dic269C2413yes—
134C2×Dic269C2411yescharacteristic
135C2×Dic269C2411yescharacteristic
136C4×D269—26no—
137C4×D269—26no—
138C4×D269—22no—
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×C269C2×#4112yes—
158C22×C269C2511yescharacteristic
159C2×D269—21no—
160C2×D269—21no—
161C2×D269C2×#4112yes—
162C4×C269—21no—
163C4×C269—23no—
164C4×C269—23no—
165C4×C269—21no—
166C4×C269C2×#4116yes—
167Dic269—23no—
168Dic269—23no—
169Dic269—23no—
170Dic269—21no—
171Dic269—23no—
172Dic269C2×#4112yes—
173Dic2692−1+412yes—
174C2×C269—22no—
175C2×C269—43no—
176C2×C269C22×#4111yescharacteristic
177C2×C269C22×#4111yescharacteristic
178C2×C269C2×2−1+411yescharacteristic
179D269—22no—
180C269#348211yescharacteristic, complemented, Sylow
181#3482—2691nomaximal, Sylow
182#557—2691no—
183#556—2693no—
184#552—2693no—
185#552—2693no—
186D4×Q8—2691no—
187#548—2691no—
188#548D26911yescharacteristic, complemented
189#547—2693no—
190#533—2693no—
191#529—2693no—
192#529—2693no—
193C4×#41—2691no—
194C4×#41—2693no—
195C2×C4×Q8—2691no—
196C2×C4×Q8—2691no—
197C2×#41—2693no—
198C2×#41C2×D26911yescharacteristic
199C22×Q8—2691no—
200C22×Q8C2×D26911yescharacteristic
201C4⋊Q8—2693no—
202C4⋊Q8—2693no—
203#126—2696no—
204#126—2696no—
205#125—2693no—
206#125—2696no—
207#125—2693no—
208#124—2693no—
209#124C2×D26913yes—
210#123—2696no—
211#123—2696no—
212C22⋊Q8—2693no—
213C22⋊Q8—2693no—
214C22⋊Q8—2693no—
215C22⋊Q8—2696no—
216C22⋊Q8C2×D26913yes—
217#121—2693no—
218#121C2×D26913yes—
219C4×Q8—2691no—
220C4×Q8—2693no—
221C4×Q8—2691no—
222C4×Q8—2693no—
223C4×Q8—2691no—
224C4×Q8—2693no—
225C4×Q8—2693no—
226C4×Q8—2693no—
227C4×Q8—2693no—
228C4×Q8C2×D26911yescharacteristic
229C4×D4—2693no—
230C4×D4—2693no—
231C4×D4—2693no—
232C4×D4—2693no—
233C4×D4—2693no—
234C4×D4—2691no—
235C4×D4—2693no—
236C4×D4C2×D26913yes—
237#117—2693no—
238#117—2693no—
239#117—2693no—
240#117—2693no—
241#117—2696no—
242C2×C4⋊C4—2693no—
243C2×C4⋊C4—2693no—
244C2×C42—2693no—
245C2×C42—2693no—
246C2×C42—2693no—
247#41—24no—
248#41—53812no—
249C2×Q8—24no—
250C2×Q8—2693no—
251C2×Q8—2693no—
252C2×Q8—5384no—
253C2×Q8C22×D26913yes—
254C2×Q8C22×D26911yescharacteristic
255C2×D4—2691no—
256C2×D4—2693no—
257C2×D4C22×D26913yes—
258C2×D4C22×D26913yes—
259C22×C4—2691no—
260C22×C4—2693no—
261C22×C4—2693no—
262C22×C4—2693no—
263C22×C4—2691no—
264C22×C4—2693no—
265C22×C4—2693no—
266C22×C4C22×D26913yes—
267C22×C4C22×D26913yes—
268C4⋊C4—2693no—
269C4⋊C4—2693no—
270C4⋊C4—2696no—
271C4⋊C4—2696no—
272C4⋊C4—2691no—
273C4⋊C4—2693no—
274C4⋊C4—2693no—
275C4⋊C4—2693no—
276C4⋊C4—2693no—
277C4⋊C4—2693no—
278C4⋊C4—2696no—
279C4⋊C4—2696no—
280C4⋊C4C22×D26913yes—
281C4⋊C4C22×D26913yes—
282C22⋊C4—2693no—
283C22⋊C4—2696no—
284C22⋊C4—2693no—
285C22⋊C4—2696no—
286C22⋊C4—2691no—
287C22⋊C4—2693no—
288C22⋊C4C22×D26911yescharacteristic
289C22⋊C4C22×D26913yes—
290C22⋊C4C22×D26916yes—
291C42—2691no—
292C42—2693no—
293C42—2693no—
294C42—2693no—
295C42—2693no—
296C42—2693no—
297C42—2693no—
298C42—2693no—
299C42—2693no—
300C42C22×D26913yes—
301C23—2691no—
302C23C23×D26913yes—
303C23C23×D26911yescharacteristic
304Q8—26no—
305Q8—5386no—
306Q8—5386no—
307Q8#1686440114yescomplemented
308D4—26no—
309D4—26no—
310D4—2694no—
311D4—5386no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—2691no—
315C2×C4—2691no—
316C2×C4—2691no—
317C2×C4—2693no—
318C2×C4—2693no—
319C2×C4—2693no—
320C2×C4—2693no—
321C2×C4—5382no—
322C2×C4—5386no—
323C2×C4—5386no—
324C2×C4—5382no—
325C2×C4—5386no—
326C2×C4—5386no—
327C2×C4—5386no—
328C2×C4C23×D26911yescharacteristic
329C2×C4C23×D26911yescharacteristic
330C2×C4C23×D26913yes—
331C2×C4C23×D26913yes—
332C2×C4C23×D26913yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—2692no—
339C22—5381no—
340C22—5381no—
341C22#41×D26912yes—
342C22C24×D26911yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—2692no—
348C4—2692no—
349C4—5383no—
350C4—5383no—
351C4—5383no—
352C4—5381no—
353C4—5383no—
354C4C2×#1686440116yes—
355C2—22no—
356C2—43no—
357C2—5382no—
358C2C2×#41×D26911yescharacteristic
359C2C22×#1686440111yescharacteristic
360C22−1+4×D26911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#548×C269C2
hypercenter#548D269
hypocenterC269#3482
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31771331C2semidirect
2. #31771324C2semidirect
3. #31771295C2semidirect
4. #31771289C2semidirect
5. Q8×C22⋊D269C2semidirect
6. #31771262C2semidirect
7. #31771255C2semidirect
8. #31771157C2semidirect
9. #31771136C2semidirect
10. #31771134C2semidirect
11. #41×Dic269C2semidirect
12. C4×#16864401C2semidirect
13. C2×Q8×Dic269C2semidirect
14. C4×Q8×D269C2semidirect
15. #548×C269C2semidirect
124. C22⋊D269#41semidirect
142. Q8×C269#41semidirect
180. C269#3482semidirect
188. #548D269semidirect
307. Q8#16864401semidirect

Maximal subgroups

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

subgroupindex
#317713312
#317713242
#317712952
#317712892
Q8×C22⋊D2692
#317712622
#317712552
#317711572
#317711362
#317711342
#41×Dic2692
C4×#168644012
C2×Q8×Dic2692
C4×Q8×D2692
#548×C2692
#3482269