Finite Group Databasebeta

← #46846952

Subgroups — #46846952

Order 34688.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774442C211yesmaximal, characteristic, complemented
2#31774433C211yesmaximal, characteristic, complemented
3#31774305C211yesmaximal, characteristic, complemented
4#31774283C211yesmaximal, characteristic, complemented
5#31774260C211yesmaximal, characteristic, complemented
6#31774258C211yesmaximal, characteristic, complemented
7#31774150C211yesmaximal, characteristic, complemented
8#31774148C211yesmaximal, characteristic, complemented
9#31774145C211yesmaximal, characteristic, complemented
10SD16×Dic271C211yesmaximal, characteristic, complemented
11#31773982C211yesmaximal, characteristic, complemented
12#31773979C211yesmaximal, characteristic, complemented
13#31773977C211yesmaximal, characteristic, complemented
14M16×Dic271C211yesmaximal, characteristic, complemented
15#468×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×#16864641C2211yescharacteristic
18C2×D2168C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20C2×#16864638C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576156C2211yescharacteristic
23#22576155—21no—
24#22576151C2211yescharacteristic
25#22576149C2211yescharacteristic
26#22576146—21no—
27#22576144—21no—
28#22576134C2211yescharacteristic
29#22576134C2211yescharacteristic
30#22576132—21no—
31Q8×Dic271C2211yescharacteristic
32D4×Dic271C2211yescharacteristic
33#22576122C2211yescharacteristic
34#22576122C2211yescharacteristic
35(C22⋊D271)⋊C4—21no—
36(C22⋊D271)⋊C4—21no—
37#22576118—21no—
38C2×C4×Dic271C2211yescharacteristic
39#22576093C2211yescharacteristic
40#22576092C2211yescharacteristic
41#22576091C2211yescharacteristic
42#22576084C2211yescharacteristic
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576080C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576070C2211yescharacteristic
49#22576070C2211yescharacteristic
50C8×Dic271C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×SD16×C271C2211yescharacteristic
53C2×M16×C271C2211yescharacteristic
54C22⋊Q8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#107×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864653—44no—
60C2×Dic542—21no—
61C2×D1084C2311yescharacteristic
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C22×Dic271—21no—
68C22×Dic271D411yescharacteristic
69C22×Dic271D411yescharacteristic
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72#16864641—22no—
73D2168—22no—
74#16864639—22no—
75#16864638—22no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C4⋊Dic271—21no—
79C4⋊Dic271—21no—
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84C22⋊Dic271—21no—
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—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—
101M16×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—
110C22×D271—21no—
111Dic542—42no—
112D1084—21no—
113D1084—21no—
114D1084—21no—
115D1084—21no—
116C22⋊D271—42no—
117C22⋊D271—42no—
118C22⋊D271—42no—
119C22⋊D271—42no—
120C2×Dic271—22no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—41no—
124C2×Dic271—42no—
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C4×D271—42no—
130C4×D271—42no—
131C271⋊C8—21no—
132C271⋊C8#4112yescomplemented
133C23×C271—21no—
134C23×C271C2×D411yescharacteristic
135Q8×C271—21no—
136Q8×C271—21no—
137D4×C271—21no—
138D4×C271—21no—
139D4×C271—42no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2411yescharacteristic
147C8×C271—21no—
148C8×C271#4112yescomplemented
149C22×C271—21no—
150C22×C271—21no—
151C22×C271—21no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271C22×D411yescharacteristic
156C2×D271—41no—
157C2×D271—41no—
158C2×D271—41no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—41no—
162C4×C271—21no—
163C4×C271—41no—
164C4×C271—41no—
165C4×C271—41no—
166C4×C271C2×#4111yescharacteristic
167C4×C271C2×#4111yescharacteristic
168Dic271—21no—
169Dic271—21no—
170Dic271—21no—
171Dic271—41no—
172Dic271—41no—
173Dic271AGL(1,ℤ/8)12yes—
174C2×C271—41no—
175C2×C271—81no—
176C2×C271#53211yescharacteristic
177C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
178C2×C271#57511yescharacteristic
179D271—81no—
180D271—81no—
181C271#317111yescharacteristic, complemented, Sylow
182#3171—2711nomaximal, Sylow
183#532—2711no—
184#529—2711no—
185#501—2711no—
186#496—2711no—
187#490—2711no—
188#489—2711no—
189#469—2711no—
190#469—2711no—
191#468—2711no—
192#468D27111yescharacteristic, complemented
193C4×SD16—2711no—
194C4×SD16—2711no—
195C4×D8—2711no—
196C4×D8—2711no—
197C4×M16—2711no—
198C2×#41—5421no—
199C2×SD16—2711no—
200C2×SD16C2×D27111yescharacteristic
201C2×D8—2711no—
202C2×D8—2711no—
203C2×M16—2711no—
204C2×M16C2×D27111yescharacteristic
205C4⋊Q8—2711no—
206#127—2711no—
207#126—5421no—
208#125—2711no—
209#124—2711no—
210#124—5421no—
211#123—5421no—
212C22⋊Q8C2×D27111yescharacteristic
213#121—2711no—
214#121—2711no—
215#121—5421no—
216#121C2×D27111yescharacteristic
217C4×Q8—2711no—
218C4×D4—2711no—
219C4×D4—2711no—
220C4×D4—2711no—
221C4×D4—5421no—
222C4×D4—5421no—
223#117—5421no—
224C2×C42—2711no—
225#107—2711no—
226#107C2×D27111yescharacteristic
227#106—2711no—
228#106—2711no—
229#103—2711no—
230#103C2×D27111yescharacteristic
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235#102—2711no—
236#102C2×D27111yescharacteristic
237#97—2711no—
238#97—2711no—
239C4×C8—2711no—
240C4×C8—2711no—
241#41—10844no—
242C2×Q8—5421no—
243C2×Q8C22×D27111yescharacteristic
244C2×D4—21no—
245C2×D4—2711no—
246C2×D4—2711no—
247C2×D4—5421no—
248C2×D4—5421no—
249C2×D4—5421no—
250C2×D4—5421no—
251C2×D4C22×D27111yescharacteristic
252C22×C4—2711no—
253C22×C4—2711no—
254C22×C4—5421no—
255C22×C4—5421no—
256C22×C4—5421no—
257C22×C4C22×D27111yescharacteristic
258SD16—22no—
259SD16—5422no—
260D8—5422no—
261D8—5422no—
262M16—22no—
263M16—5422no—
264C2×C8—2711no—
265C2×C8—2711no—
266C2×C8C22×D27111yescharacteristic
267C2×C8C22×D27111yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—2711no—
270C4⋊C4—2711no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4C22×D27111yescharacteristic
276C4⋊C4C22×D27111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—5421no—
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—
291C23—21no—
292C23—5421no—
293C23—5421no—
294C23D4×D27111yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—10842no—
298D4—21no—
299D4—21no—
300D4—42no—
301D4—5421no—
302D4—5421no—
303D4—5421no—
304D4—5421no—
305D4—10842no—
306D4—10842no—
307D4—10842no—
308D4—10842no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—2711no—
315C2×C4—2711no—
316C2×C4—2711no—
317C2×C4—2711no—
318C2×C4—5422no—
319C2×C4—5421no—
320C2×C4—5421no—
321C2×C4—10842no—
322C2×C4—10842no—
323C2×C4—10842no—
324C2×C4—10841no—
325C2×C4D4×D27111yescharacteristic
326C2×C4C23×D27111yescharacteristic
327C8—21no—
328C8—2712no—
329C8—5421no—
330C8#1686465312yescomplemented
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—10841no—
338C22—10841no—
339C22—10841no—
340C22—10841no—
341C22—10841no—
342C22—10841no—
343C22C2×D4×D27111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—2712no—
349C4—5421no—
350C4—5421no—
351C4—5421no—
352C4—10841no—
353C4—10841no—
354C4#41×D27111yescharacteristic
355C4C2×#1686465311yescharacteristic
356C2—41no—
357C2—81no—
358C2—21681no—
359C2—21681no—
360C2#3177444211yescharacteristic
361C2AGL(1,ℤ/8)×D27111yescharacteristic
362C2#3177469611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#468×C271C2
socleC22×C271C22×D4
hypercenter#468D271
hypocenterC271#3171
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #31774442C2semidirect
2. #31774433C2semidirect
3. #31774305C2semidirect
4. #31774283C2semidirect
5. #31774260C2semidirect
6. #31774258C2semidirect
7. #31774150C2semidirect
8. #31774148C2semidirect
9. #31774145C2semidirect
10. SD16×Dic271C2semidirect
11. #31773982C2semidirect
12. #31773979C2semidirect
13. #31773977C2semidirect
14. M16×Dic271C2semidirect
15. #468×C271C2semidirect
132. C271⋊C8#41semidirect
148. C8×C271#41semidirect
181. C271#3171semidirect
192. #468D271semidirect
330. C8#16864653semidirect

Maximal subgroups

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

subgroupindex
#317744422
#317744332
#317743052
#317742832
#317742602
#317742582
#317741502
#317741482
#317741452
SD16×Dic2712
#317739822
#317739792
#317739772
M16×Dic2712
#468×C2712
#3171271