Finite Group Databasebeta

← #46846508

Subgroups — #46846508

Order 34688.

430 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
1#31774442C211yesmaximal, characteristic, complemented
2#41×Dic271C211yesmaximal, characteristic, complemented
3#31774222C211yesmaximal, characteristic, complemented
4#31774218C211yesmaximal, characteristic, complemented
5#31774180C211yesmaximal, characteristic, complemented
6#31774175C211yesmaximal, characteristic, complemented
7#31774095C211yesmaximal, characteristic, complemented
8#31774090C211yesmaximal, characteristic, complemented
9#31774042C211yesmaximal, characteristic, complemented
10#31774009C211yesmaximal, characteristic, complemented
11#31774006C211yesmaximal, characteristic, complemented
12#31774000C211yesmaximal, characteristic, complemented
13#31773998C211yesmaximal, characteristic, complemented
14#31773951C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×#16864641C2211yescharacteristic
18C2×#16864639C2211yescharacteristic
19#22576159C2211yescharacteristic
20#22576156C2211yescharacteristic
21#22576146—21no—
22#22576134C2211yescharacteristic
23#22576132—21no—
24Q8×Dic271C2211yescharacteristic
25Q8×Dic271C2211yescharacteristic
26D4×Dic271—21no—
27D4×Dic271—21no—
28D4×Dic271C2211yescharacteristic
29D4×Dic271C2211yescharacteristic
30D4×Dic271C2211yescharacteristic
31#22576122C2211yescharacteristic
32(C22⋊D271)⋊C4—21no—
33#22576119—21no—
34#22576119C2211yescharacteristic
35C2×C4×Dic271—21no—
36C2×C4×Dic271C2211yescharacteristic
37#22576094C2211yescharacteristic
38#22576093C2211yescharacteristic
39#22576092C2211yescharacteristic
40#22576091C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50#22576070C2211yescharacteristic
51#22576070C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×D8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59#16864653—44no—
60C2×Dic542—21no—
61C2×D1084C2311yescharacteristic
62C2×C22⋊D271—21no—
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C22×Dic271—21no—
66C22×Dic271—21no—
67C22×Dic271—21no—
68C22×Dic271D411yescharacteristic
69C22×Dic271D411yescharacteristic
70C2×C4×D271—21no—
71#16864641—22no—
72#16864639—22no—
73C2×C271⋊C8C2311yescharacteristic
74C2×C271⋊C8C2311yescharacteristic
75C4⋊Dic271—21no—
76C4⋊Dic271—21no—
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81C22⋊Dic271—21no—
82C22⋊Dic271—21no—
83C22⋊Dic271—21no—
84C22⋊Dic271—21no—
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87C4×Dic271—21no—
88C4×Dic271—21no—
89C4×Dic271—21no—
90C4×Dic271D411yescharacteristic
91C4×Dic271D411yescharacteristic
92C4×Dic271C2311yescharacteristic
93C4×Dic271C2311yescharacteristic
94#41×C271—22no—
95#41×C271—22no—
96C2×Q8×C271C2311yescharacteristic
97C2×D4×C271—21no—
98C2×D4×C271—21no—
99C2×D4×C271C2311yescharacteristic
100C2×D4×C271C2311yescharacteristic
101C22×C4×C271—21no—
102C22×C4×C271C2311yescharacteristic
103SD16×C271—22no—
104D8×C271—22no—
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C22⋊C4×C271—21no—
109C22×D271—21no—
110Dic542—42no—
111D1084—21no—
112D1084—21no—
113C22⋊D271—42no—
114C22⋊D271—42no—
115C22⋊D271—42no—
116C2×Dic271—22no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—41no—
121C2×Dic271—42no—
122C2×Dic271—42no—
123C2×Dic271—42no—
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C4×D271—42no—
129C271⋊C8—21no—
130C271⋊C8—21no—
131C23×C271—21no—
132C23×C271—21no—
133C23×C271C2×D411yescharacteristic
134Q8×C271—21no—
135Q8×C271#4112yescomplemented
136D4×C271—24no—
137D4×C271—21no—
138D4×C271—21no—
139D4×C271—21no—
140D4×C271—42no—
141D4×C271#4112yescomplemented
142C2×C4×C271—24no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2411yescharacteristic
149C8×C271—21no—
150C8×C271—21no—
151C22×C271—21no—
152C22×C271—22no—
153C22×C271—21no—
154C22×C271—21no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271C22×D411yescharacteristic
161C2×D271—41no—
162C2×D271—41no—
163C2×D271—41no—
164C4×C271—21no—
165C4×C271—22no—
166C4×C271—41no—
167C4×C271C2×#4111yescharacteristic
168C4×C271C2×#4111yescharacteristic
169Dic271—21no—
170Dic271—21no—
171Dic271—21no—
172Dic271—41no—
173Dic271—41no—
174Dic271—41no—
175Dic271AGL(1,ℤ/8)12yes—
176C2×C271—41no—
177C2×C271—42no—
178C2×C271—81no—
179C2×C271#52511yescharacteristic
180C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C271#57511yescharacteristic
182D271—81no—
183C271#311511yescharacteristic, complemented, Sylow
184#3115—2711nomaximal, Sylow
185#532—2711no—
186C4×#41—2711no—
187#482—2711no—
188#482—2711no—
189#475—2711no—
190#474—2711no—
191#460—2711no—
192#459—2711no—
193#449—2711no—
194#449D27111yescharacteristic, complemented
195#442—2711no—
196#442—2711no—
197#440—2711no—
198#440—2711no—
199#432—2711no—
200C2×#41—5421no—
201C2×#41C2×D27111yescharacteristic
202C2×SD16—2711no—
203C2×SD16C2×D27111yescharacteristic
204C2×D8—2711no—
205C2×D8C2×D27111yescharacteristic
206C4⋊Q8—2711no—
207#127—2711no—
208#124—5421no—
209#121—2711no—
210#121—5421no—
211#121C2×D27111yescharacteristic
212C4×Q8—2711no—
213C4×Q8—2711no—
214C4×D4—2711no—
215C4×D4—2711no—
216C4×D4—2711no—
217C4×D4—2711no—
218C4×D4—5421no—
219C4×D4—5421no—
220C4×D4—5421no—
221#117—2711no—
222#117—5421no—
223C2×C42—2711no—
224C2×C42—5421no—
225#107—2711no—
226#107—2711no—
227#106—2711no—
228#106—2711no—
229C4⋊C8—2711no—
230C4⋊C8—2711no—
231#103—2711no—
232#103C2×D27111yescharacteristic
233#102—2711no—
234#102—2711no—
235#102—2711no—
236#102—2711no—
237#102—2711no—
238#102C2×D27111yescharacteristic
239C22⋊C8—2711no—
240C22⋊C8C2×D27111yescharacteristic
241#97—2711no—
242#97—2711no—
243#41—22no—
244#41—22no—
245#41—10844no—
246C2×Q8—5421no—
247C2×Q8C22×D27111yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—2711no—
251C2×D4—5421no—
252C2×D4—5421no—
253C2×D4—5421no—
254C2×D4C22×D27111yescharacteristic
255C2×D4C22×D27111yescharacteristic
256C22×C4—21no—
257C22×C4—2711no—
258C22×C4—2711no—
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4C22×D27111yescharacteristic
264SD16—22no—
265SD16—5422no—
266D8—22no—
267D8—5422no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—2711no—
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—5421no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4C22×D27111yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C42—2711no—
287C42—2711no—
288C42—2711no—
289C42—2711no—
290C42—5421no—
291C42—5421no—
292C42—5421no—
293C23—21no—
294C23—21no—
295C23—5421no—
296C23D4×D27111yescharacteristic
297Q8—21no—
298Q8—10842no—
299Q8#1686465312yescomplemented
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—5421no—
306D4—5421no—
307D4—10842no—
308D4—10842no—
309D4—10842no—
310D4#1686465312yescomplemented
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—2711no—
317C2×C4—2711no—
318C2×C4—2711no—
319C2×C4—2711no—
320C2×C4—5422no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—10842no—
325C2×C4—10842no—
326C2×C4—10841no—
327C2×C4—10842no—
328C2×C4—10842no—
329C2×C4D4×D27111yescharacteristic
330C2×C4C23×D27111yescharacteristic
331C8—21no—
332C8—21no—
333C8—5421no—
334C8—5421no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—10841no—
345C22—10841no—
346C22—10841no—
347C22C2×D4×D27111yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—2712no—
352C4—5421no—
353C4—5421no—
354C4—5421no—
355C4—10841no—
356C4—10841no—
357C4—10841no—
358C4C2×#1686465311yescharacteristic
359C4C2×#1686465311yescharacteristic
360C2—41no—
361C2—42no—
362C2—81no—
363C2—21681no—
364C2#3177440311yescharacteristic
365C2AGL(1,ℤ/8)×D27111yescharacteristic
366C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3115
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31774442C2semidirect
2. #41×Dic271C2semidirect
3. #31774222C2semidirect
4. #31774218C2semidirect
5. #31774180C2semidirect
6. #31774175C2semidirect
7. #31774095C2semidirect
8. #31774090C2semidirect
9. #31774042C2semidirect
10. #31774009C2semidirect
11. #31774006C2semidirect
12. #31774000C2semidirect
13. #31773998C2semidirect
14. #31773951C2semidirect
15. #449×C271C2semidirect
135. Q8×C271#41semidirect
141. D4×C271#41semidirect
183. C271#3115semidirect
194. #449D271semidirect
299. Q8#16864653semidirect
310. D4#16864653semidirect

Maximal subgroups

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

subgroupindex
#317744422
#41×Dic2712
#317742222
#317742182
#317741802
#317741752
#317740952
#317740902
#317740422
#317740092
#317740062
#317740002
#317739982
#317739512
#449×C2712
#3115271