Finite Group Databasebeta

← #46846918

Subgroups — #46846918

Order 34688.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774449C211yesmaximal, characteristic, complemented
2#31774442C211yesmaximal, characteristic
3#31774294C211yesmaximal, characteristic, complemented
4#31774268C211yesmaximal, characteristic, complemented
5#31774246C211yesmaximal, characteristic, complemented
6#31774242C211yesmaximal, characteristic, complemented
7#31774128C211yesmaximal, characteristic, complemented
8#31774126C211yesmaximal, characteristic, complemented
9#31774124C211yesmaximal, characteristic, complemented
10#31773988C211yesmaximal, characteristic, complemented
11SD16×Dic271C211yesmaximal, characteristic, complemented
12#31773984C211yesmaximal, characteristic, complemented
13#31773982C211yesmaximal, characteristic, complemented
14C2×C8×Dic271C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×#16864643C2211yescharacteristic
18C2×#16864642C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C22×C271⋊C8C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576159C2211yescharacteristic
23#22576158C2211yescharacteristic
24#22576156C2211yescharacteristic
25#22576150—21no—
26#22576146—21no—
27#22576139C2211yescharacteristic
28#22576137—21no—
29#22576134C2211yescharacteristic
30#22576132—21no—
31Q8×Dic271C2211yescharacteristic
32#22576125—21no—
33#22576125C2211yescharacteristic
34D4×Dic271C2211yescharacteristic
35#22576122C2211yescharacteristic
36(C22⋊D271)⋊C4—21no—
37#22576116—21no—
38C2×C4×Dic271C2211yescharacteristic
39#22576092C2211yescharacteristic
40#22576091C2211yescharacteristic
41#22576091C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576083C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576080C2211yescharacteristic
48C8×Dic271C2211yescharacteristic
49C8×Dic271C2211yescharacteristic
50C8×Dic271C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×SD16×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54C22⋊Q8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#106×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864653—44no—
60C2×Dic542—21no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63C2×D1084C2311yescharacteristic
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C22×Dic271—21no—
68C22×Dic271D411yescharacteristic
69C22×Dic271D411yescharacteristic
70C2×C4×D271—21no—
71#16864643—22no—
72#16864642—22no—
73#16864641—22no—
74C2×C271⋊C8—22no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C4⋊Dic271—21no—
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90C4×Dic271—21no—
91C4×Dic271—21no—
92C4×Dic271D411yescharacteristic
93C4×Dic271D411yescharacteristic
94C4×Dic271C2311yescharacteristic
95C4×Dic271C2311yescharacteristic
96C2×Q8×C271C2311yescharacteristic
97C2×D4×C271—21no—
98C2×D4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100SD16×C271—22no—
101C2×C8×C271—22no—
102C2×C8×C271C2311yescharacteristic
103C2×C8×C271C2311yescharacteristic
104C4⋊C4×C271—21no—
105C4⋊C4×C271C2311yescharacteristic
106C4⋊C4×C271C2311yescharacteristic
107C22⋊C4×C271—21no—
108C22⋊C4×C271—21no—
109C22×D271—21no—
110Dic542—21no—
111Dic542—21no—
112Dic542—42no—
113Dic542—42no—
114D1084—21no—
115D1084—21no—
116C22⋊D271—42no—
117C22⋊D271—42no—
118C22⋊D271—42no—
119C2×Dic271—22no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—42no—
124C2×Dic271SD1614yescomplemented
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C4×D271—42no—
130C271⋊C8—21no—
131C271⋊C8#4112yescomplemented
132C23×C271—21no—
133C23×C271C2×D411yescharacteristic
134Q8×C271—21no—
135Q8×C271—21no—
136D4×C271—21no—
137D4×C271—21no—
138D4×C271—42no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2411yescharacteristic
146C8×C271—21no—
147C8×C271#4112yescomplemented
148C22×C271—21no—
149C22×C271—21no—
150C22×C271—41no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271C2×SD1612yes—
154C22×C271C22×D411yescharacteristic
155C2×D271—41no—
156C2×D271—41no—
157C2×D271—41no—
158C4×C271—21no—
159C4×C271—41no—
160C4×C271—41no—
161C4×C271—41no—
162C4×C271C2×#4111yescharacteristic
163C4×C271C2×#4111yescharacteristic
164Dic271—21no—
165Dic271—41no—
166Dic271—41no—
167Dic271—41no—
168Dic271C2×SD1612yes—
169Dic271C2×SD1612yes—
170Dic271#13512yes—
171C2×C271—22no—
172C2×C271—81no—
173C2×C271#53211yescharacteristic
174C2×C271C22×SD1611yescharacteristic
175C2×C271C2×#13511yescharacteristic
176D271—81no—
177C271#316711yescharacteristic, complemented, Sylow
178#3167—2711nomaximal, Sylow
179#533—2711no—
180#532—2711no—
181#498—2711no—
182#492—2711no—
183#487—2711no—
184#486—2711no—
185#465—2711no—
186#465—2711no—
187#465—2711no—
188#465D27111yescharacteristic, complemented
189C4×SD16—2711no—
190C4×SD16—2711no—
191C4×SD16—2711no—
192C4×SD16—2711no—
193C2×C4×C8—2711no—
194C2×#41—5421no—
195C2×SD16—2711no—
196C2×SD16—2711no—
197C2×SD16—2711no—
198C2×SD16C2×D27111yescharacteristic
199C22×C8—2711no—
200C22×C8C2×D27111yescharacteristic
201C4⋊Q8—2711no—
202C4⋊Q8—2711no—
203C4⋊Q8—2711no—
204#127—2711no—
205#125—5421no—
206#124—5421no—
207C22⋊Q8—2711no—
208C22⋊Q8—5421no—
209C22⋊Q8C2×D27111yescharacteristic
210#121—2711no—
211#121—5421no—
212#121C2×D27111yescharacteristic
213C4×Q8—2711no—
214C4×Q8—2711no—
215C4×Q8—5421no—
216C4×D4—2711no—
217C4×D4—2711no—
218C4×D4—5421no—
219#117—5421no—
220C2×C42—2711no—
221#106—2711no—
222#106—2711no—
223#106—2711no—
224#106C2×D27111yescharacteristic
225#103—2711no—
226#103—2711no—
227#103—2711no—
228#103C2×D27111yescharacteristic
229#102—2711no—
230#102—2711no—
231#102—2711no—
232#102C2×D27111yescharacteristic
233C4×C8—2711no—
234C4×C8—2711no—
235C4×C8—2711no—
236C4×C8—2711no—
237#41—10844no—
238C2×Q8—2711no—
239C2×Q8—5421no—
240C2×Q8—5421no—
241C2×Q8C22×D27111yescharacteristic
242C2×D4—21no—
243C2×D4—2711no—
244C2×D4—5421no—
245C2×D4—5421no—
246C2×D4—5421no—
247C2×D4C22×D27111yescharacteristic
248C22×C4—2711no—
249C22×C4—2711no—
250C22×C4—5421no—
251C22×C4—5421no—
252C22×C4C22×D27111yescharacteristic
253SD16—22no—
254SD16—5422no—
255SD16—5422no—
256SD16—5422no—
257C2×C8—22no—
258C2×C8—2711no—
259C2×C8—2711no—
260C2×C8—5422no—
261C2×C8C22×D27111yescharacteristic
262C2×C8C22×D27111yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—2711no—
265C4⋊C4—2711no—
266C4⋊C4—5421no—
267C4⋊C4—5421no—
268C4⋊C4—5421no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4C22×D27111yescharacteristic
274C4⋊C4C22×D27111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C42—2711no—
282C42—2711no—
283C42—2711no—
284C42—2711no—
285C42—5421no—
286C42—5421no—
287C23—21no—
288C23—5421no—
289C23D4×D27111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—5421no—
293Q8—5421no—
294Q8—10842no—
295Q8—10842no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—5421no—
300D4—5421no—
301D4—10842no—
302D4—10842no—
303D4—10842no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—2714no—
310C2×C4—2711no—
311C2×C4—2711no—
312C2×C4—2711no—
313C2×C4—2711no—
314C2×C4—5422no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—10842no—
319C2×C4—10842no—
320C2×C4D4×D27111yescharacteristic
321C2×C4C23×D27111yescharacteristic
322C8—21no—
323C8—2712no—
324C8—5421no—
325C8#1686465312yescomplemented
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—10841no—
332C22—10841no—
333C22—10841no—
334C22SD16×D27112yes—
335C22C2×D4×D27111yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—2712no—
341C4—2712no—
342C4—2712no—
343C4—5421no—
344C4—10841no—
345C4—10841no—
346C4—10841no—
347C4#41×D27111yescharacteristic
348C4C2×#1686465311yescharacteristic
349C2—22no—
350C2—81no—
351C2—21681no—
352C2#3177444211yescharacteristic
353C2C2×SD16×D27111yescharacteristic
354C2#135×D27111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
prime radical#31774442C2
hypercenter#465D271
hypocenterC271#3167
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #31774449C2semidirect
3. #31774294C2semidirect
4. #31774268C2semidirect
5. #31774246C2semidirect
6. #31774242C2semidirect
7. #31774128C2semidirect
8. #31774126C2semidirect
9. #31774124C2semidirect
10. #31773988C2semidirect
11. SD16×Dic271C2semidirect
12. #31773984C2semidirect
13. #31773982C2semidirect
14. C2×C8×Dic271C2semidirect
15. #465×C271C2semidirect
124. C2×Dic271SD16semidirect
131. C271⋊C8#41semidirect
147. C8×C271#41semidirect
177. C271#3167semidirect
188. #465D271semidirect
325. C8#16864653semidirect

Maximal subgroups

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

subgroupindex
#317744492
#317744422
#317742942
#317742682
#317742462
#317742422
#317741282
#317741262
#317741242
#317739882
SD16×Dic2712
#317739842
#317739822
C2×C8×Dic2712
#465×C2712
#3167271