Finite Group Databasebeta

← #46822743

Subgroups — #46822743

Order 34432.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D269C211yesmaximal, characteristic, complemented
2#31771255C211yesmaximal, characteristic
3#31770954C211yesmaximal, characteristic, complemented
4#31770936C211yesmaximal, characteristic, complemented
5#31770934C211yesmaximal, characteristic, complemented
6#31770888C211yesmaximal, characteristic, complemented
7#31770832C211yesmaximal, characteristic, complemented
8#31770831C211yesmaximal, characteristic, complemented
9#31770808C211yesmaximal, characteristic, complemented
10#31770742C211yesmaximal, characteristic, complemented
11SD16×Dic269C211yesmaximal, characteristic, complemented
12#31770668C211yesmaximal, characteristic, complemented
13C2×#22575377C211yesmaximal, characteristic, complemented
14#103×D269C211yesmaximal, characteristic, complemented
15#448×C269C211yesmaximal, characteristic, complemented
16C2×#16864401—21no—
17C2×Q8×D269C2211yescharacteristic
18C2×#16864390C2211yescharacteristic
19C22×#14524130C2211yescharacteristic
20C2×C8×D269C2211yescharacteristic
21#22575452C2211yescharacteristic
22#22575451—21no—
23#22575451C2211yescharacteristic
24#22575435—21no—
25#22575434—21no—
26#22575432—21no—
27#22575432C2211yescharacteristic
28#22575430—21no—
29#22575429—21no—
30#22575429C2211yescharacteristic
31#22575424C2211yescharacteristic
32Q8×Dic269C2211yescharacteristic
33D4×Dic269C2211yescharacteristic
34C4×C22⋊D269—21no—
35C4×C22⋊D269C2211yescharacteristic
36C2×C4⋊Dic269C2211yescharacteristic
37C4⋊C4×D269C2211yescharacteristic
38#22575385C2211yescharacteristic
39#22575385C2211yescharacteristic
40#22575384C2211yescharacteristic
41#22575382C2211yescharacteristic
42#22575378C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575377C2211yescharacteristic
47#22575377C2211yescharacteristic
48#22575376C2211yescharacteristic
49#22575374C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C22×Q8×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54C2×SD16×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic
59#16864401—44no—
60C2×Dic538—21no—
61C2×Dic538—21no—
62C2×Dic538C2311yescharacteristic
63Q8×D269—22no—
64Q8×D269—41no—
65C2×C22⋊D269—21no—
66C2×C22⋊D269D411yescharacteristic
67C22×Dic269—21no—
68C22×Dic269—21no—
69C2×C4×D269—21no—
70C2×C4×D269—21no—
71C2×C4×D269C2311yescharacteristic
72#16864390—41no—
73C2×#14524130—41no—
74C2×#14524130C2311yescharacteristic
75C2×#14524130C2311yescharacteristic
76C8×D269—41no—
77C4⋊Dic269—21no—
78C4⋊Dic269—21no—
79C4⋊Dic269C2311yescharacteristic
80C4⋊Dic269C2311yescharacteristic
81C4⋊Dic269C2311yescharacteristic
82#16864381—21no—
83#16864381—21no—
84#16864381—21no—
85#16864381—21no—
86#16864381—21no—
87#16864381—21no—
88#16864381D411yescharacteristic
89C22⋊Dic269—21no—
90C22⋊Dic269—21no—
91C22⋊Dic269—21no—
92C22⋊Dic269D411yescharacteristic
93#16864379—21no—
94#16864379D411yescharacteristic
95C4×Dic269—21no—
96C4×Dic269C2311yescharacteristic
97C2×Q8×C269—21no—
98C2×Q8×C269—22no—
99C2×Q8×C269—41no—
100C2×Q8×C269C2311yescharacteristic
101C2×Q8×C269C2311yescharacteristic
102C2×D4×C269—21no—
103C2×D4×C269C2311yescharacteristic
104C22×C4×C269—21no—
105C22×C4×C269C2311yescharacteristic
106SD16×C269—22no—
107SD16×C269—41no—
108C2×C8×C269C2311yescharacteristic
109C2×C8×C269C2311yescharacteristic
110C4⋊C4×C269C2311yescharacteristic
111C22⋊C4×C269—21no—
112C22×D269C2×D411yescharacteristic
113Dic538—21no—
114Dic538—21no—
115Dic538—42no—
116Dic538—42no—
117C22⋊D269—42no—
118C22⋊D269SD1614yescomplemented
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—21no—
122C2×Dic269—21no—
123C2×Dic269—42no—
124C2×Dic269—42no—
125C2×Dic269C2×D411yescharacteristic
126C2×Dic269C2×D411yescharacteristic
127C2×Dic269C2×D411yescharacteristic
128C4×D269—21no—
129C4×D269—21no—
130C4×D269—42no—
131C4×D269—42no—
132#14524130—21no—
133#14524130—21no—
134C23×C269—21no—
135C23×C269C2×D411yescharacteristic
136Q8×C269—21no—
137Q8×C269—21no—
138Q8×C269—21no—
139Q8×C269—42no—
140Q8×C269#4112yescomplemented
141D4×C269—21no—
142D4×C269—21no—
143D4×C269—42no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269—21no—
149C2×C4×C269—42no—
150C2×C4×C269C2×D411yescharacteristic
151C2×C4×C269C2411yescharacteristic
152C8×C269—21no—
153C8×C269—21no—
154C22×C269—21no—
155C22×C269—21no—
156C22×C269—41no—
157C22×C269—41no—
158C22×C269—41no—
159C22×C269C2×SD1612yes—
160C22×C269C22×D411yescharacteristic
161C2×D269—21no—
162C2×D269—21no—
163C2×D269C2×SD1612yes—
164C4×C269—21no—
165C4×C269—22no—
166C4×C269—41no—
167C4×C269—41no—
168C4×C269C2×#4111yescharacteristic
169C4×C269H(5,𝔽2)11yescharacteristic
170Dic269—21no—
171Dic269—21no—
172Dic269—41no—
173Dic269—41no—
174Dic269—41no—
175Dic269—41no—
176Dic269C2×SD1612yes—
177C2×C269—22no—
178C2×C269—81no—
179C2×C269#54611yescharacteristic
180C2×C269C22×SD1611yescharacteristic
181C2×C269#57811yescharacteristic
182D269—22no—
183C269#329811yescharacteristic, complemented, Sylow
184#3298—2691nomaximal, Sylow
185D4×Q8—2691no—
186#547—2691no—
187#487—2691no—
188#483—2691no—
189#483—2691no—
190#475—2691no—
191#465—2691no—
192#465—2691no—
193#461—2691no—
194#448—2691no—
195#448D26911yescharacteristic, complemented
196C4×SD16—2691no—
197C8×D4—2691no—
198C2×#103—2691no—
199C2×#103—2691no—
200C2×#41—5381no—
201C22×Q8—2691no—
202C22×Q8C2×D26911yescharacteristic
203C2×SD16—2691no—
204C2×SD16C2×D26911yescharacteristic
205C2×SD16C2×D26911yescharacteristic
206C22×C8—2691no—
207C22×C8—2691no—
208C4⋊Q8—2691no—
209C4⋊Q8—2691no—
210C4⋊Q8—5381no—
211#123—5381no—
212#123—5381no—
213C22⋊Q8—2691no—
214C22⋊Q8—2691no—
215C22⋊Q8—5381no—
216C22⋊Q8—5381no—
217C22⋊Q8—5381no—
218#121—2691no—
219#121C2×D26911yescharacteristic
220C4×Q8—2691no—
221C4×D4—2691no—
222C4×D4—2691no—
223C4×D4—5381no—
224C2×C4⋊C4—2691no—
225C2×C4⋊C4—2691no—
226#106—2691no—
227#106—2691no—
228#106—2691no—
229C4⋊C8—2691no—
230#103—2691no—
231#103—2691no—
232#103—2691no—
233#103—2691no—
234#103—2691no—
235#103—2691no—
236#103—2691no—
237#103C2×D26911yescharacteristic
238#103C2×D26911yescharacteristic
239#102—2691no—
240C22⋊C8—2691no—
241C22⋊C8C2×D26911yescharacteristic
242C4×C8—2691no—
243#41—10764no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—2691no—
248C2×Q8—5381no—
249C2×Q8—5381no—
250C2×Q8—5382no—
251C2×Q8—10761no—
252C2×Q8C22×D26911yescharacteristic
253C2×Q8C22×D26911yescharacteristic
254C2×D4—21no—
255C2×D4—2691no—
256C2×D4—5381no—
257C2×D4C22×D26911yescharacteristic
258C22×C4—21no—
259C22×C4—2691no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4—5381no—
263C22×C4—5381no—
264C22×C4C22×D26911yescharacteristic
265SD16—22no—
266SD16—41no—
267SD16—10761no—
268C2×C8—2691no—
269C2×C8—2691no—
270C2×C8—10761no—
271C2×C8—10761no—
272C2×C8C22×D26911yescharacteristic
273C2×C8C22×D26911yescharacteristic
274C4⋊C4—2691no—
275C4⋊C4—2691no—
276C4⋊C4—2691no—
277C4⋊C4—2691no—
278C4⋊C4—5381no—
279C4⋊C4—5381no—
280C4⋊C4—5381no—
281C4⋊C4—5381no—
282C4⋊C4—5381no—
283C4⋊C4—5381no—
284C4⋊C4—5381no—
285C4⋊C4—5381no—
286C4⋊C4C22×D26911yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—2691no—
289C22⋊C4—2691no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4—5381no—
293C22⋊C4—5381no—
294C42—2691no—
295C42—5381no—
296C23—21no—
297C23—2691no—
298C23D4×D26911yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—5381no—
304Q8—5381no—
305Q8—10762no—
306Q8—10762no—
307Q8#1686440112yescomplemented
308D4—21no—
309D4—21no—
310D4—42no—
311D4—2694no—
312D4—10762no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—2691no—
320C2×C4—2691no—
321C2×C4—2691no—
322C2×C4—5381no—
323C2×C4—5381no—
324C2×C4—5381no—
325C2×C4—5381no—
326C2×C4—5381no—
327C2×C4—5381no—
328C2×C4—10762no—
329C2×C4—10762no—
330C2×C4—10762no—
331C2×C4—10762no—
332C2×C4D4×D26911yescharacteristic
333C2×C4C23×D26911yescharacteristic
334C8—21no—
335C8—21no—
336C8—5381no—
337C8—5381no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—2692no—
344C22—5381no—
345C22—5381no—
346C22SD16×D26912yes—
347C22C2×D4×D26911yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—2692no—
353C4—5381no—
354C4—5381no—
355C4—10761no—
356C4—10761no—
357C4—10761no—
358C4—10761no—
359C4C2×#1686440111yescharacteristic
360C4#2257549811yescharacteristic
361C2—22no—
362C2—81no—
363C2—5382no—
364C2#3177124211yescharacteristic
365C2C2×SD16×D26911yescharacteristic
366C2#3177142111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#448×C269C2
socleC22×C269C22×D4
prime radical#31771255C2
hypercenter#448D269
hypocenterC269#3298
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D269C2semidirect
3. #31770954C2semidirect
4. #31770936C2semidirect
5. #31770934C2semidirect
6. #31770888C2semidirect
7. #31770832C2semidirect
8. #31770831C2semidirect
9. #31770808C2semidirect
10. #31770742C2semidirect
11. SD16×Dic269C2semidirect
12. #31770668C2semidirect
13. C2×#22575377C2semidirect
14. #103×D269C2semidirect
15. #448×C269C2semidirect
118. C22⋊D269SD16semidirect
140. Q8×C269#41semidirect
183. C269#3298semidirect
195. #448D269semidirect
307. Q8#16864401semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D2692
#317712552
#317709542
#317709362
#317709342
#317708882
#317708322
#317708312
#317708082
#317707422
SD16×Dic2692
#317706682
C2×#225753772
#103×D2692
#448×C2692
#3298269