Finite Group Databasebeta

← #46848349

Subgroups — #46848349

Order 34688.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774556C211yesmaximal, characteristic, complemented
2#31774259C211yesmaximal, characteristic, complemented
3#31774232C211yesmaximal, characteristic, complemented
4#31774210C211yesmaximal, characteristic, complemented
5#31774189C211yesmaximal, characteristic, complemented
6#31774141C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774112C211yesmaximal, characteristic, complemented
9#31774040C211yesmaximal, characteristic, complemented
10#31774035C211yesmaximal, characteristic, complemented
11#31773997C211yesmaximal, characteristic, complemented
12#31773969C211yesmaximal, characteristic, complemented
13#31773896C211yesmaximal, characteristic, complemented
14C2×#22576083C211yesmaximal, characteristic, complemented
15#547×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18C2×#16864642C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21C2×#16864638C2211yescharacteristic
22C22×C271⋊C8C2211yescharacteristic
23#22576148C2211yescharacteristic
24#22576146—21no—
25#22576139C2211yescharacteristic
26#22576138—21no—
27#22576134C2211yescharacteristic
28#22576132—21no—
29C4×Dic542C2211yescharacteristic
30(C22⋊D271)⋊C4—21no—
31#22576093C2211yescharacteristic
32#22576093C2211yescharacteristic
33#22576093C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35#22576083C2211yescharacteristic
36#22576083C2211yescharacteristic
37#22576083C2211yescharacteristic
38#22576083C2211yescharacteristic
39#22576083C2211yescharacteristic
40#22576083C2211yescharacteristic
41#22576081C2211yescharacteristic
42#22576080C2211yescharacteristic
43#22576080C2211yescharacteristic
44#22576080C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46C22⋊(C271⋊C8)C2211yescharacteristic
47#22576071C2211yescharacteristic
48C2×#41×C271—21no—
49C4⋊Q8×C271C2211yescharacteristic
50#123×C271—21no—
51#123×C271—21no—
52C22⋊Q8×C271—21no—
53#121×C271C2211yescharacteristic
54#121×C271C2211yescharacteristic
55C4×D4×C271C2211yescharacteristic
56C4×D4×C271C2211yescharacteristic
57C2×C4⋊C4×C271C2211yescharacteristic
58C2×C4⋊C4×C271C2211yescharacteristic
59#16864654—22no—
60#16864654—41no—
61C2×Dic542—22no—
62C2×Dic542—21no—
63C2×Dic542—41no—
64C2×Dic542C2311yescharacteristic
65C2×Dic542C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C22×Dic271—21no—
70C2×C4×D271—21no—
71#16864642—22no—
72#16864642—41no—
73#16864639—41no—
74#16864638—41no—
75C2×C271⋊C8—41no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83C22⋊Dic271—21no—
84(C2×D271)⋊C4—21no—
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87C4×Dic271—21no—
88#41×C271—44no—
89C2×Q8×C271—21no—
90C2×D4×C271—21no—
91C2×D4×C271—21no—
92C2×D4×C271D411yescharacteristic
93C2×D4×C271C2311yescharacteristic
94C22×C4×C271—21no—
95C22×C4×C271—21no—
96C22×C4×C271—21no—
97C22×C4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C4⋊C4×C271—21no—
100C4⋊C4×C271—21no—
101C4⋊C4×C271—21no—
102C4⋊C4×C271D411yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C22⋊C4×C271—21no—
107C22⋊C4×C271—21no—
108C22⋊C4×C271—21no—
109C22⋊C4×C271D411yescharacteristic
110C22⋊C4×C271D411yescharacteristic
111C42×C271C2311yescharacteristic
112C22×D271—21no—
113Dic542—21no—
114Dic542—21no—
115Dic542—21no—
116Dic542—42no—
117Dic542#4112yescomplemented
118D1084—21no—
119D1084—21no—
120C22⋊D271—42no—
121C22⋊D271—42no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—42no—
126C4×D271—42no—
127C271⋊C8—21no—
128C271⋊C8—21no—
129C271⋊C8—21no—
130C271⋊C8—21no—
131C23×C271—21no—
132C23×C271C2×D411yescharacteristic
133C23×C271C2×D411yescharacteristic
134Q8×C271—42no—
135D4×C271—22no—
136D4×C271—21no—
137D4×C271—21no—
138D4×C271—42no—
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×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—42no—
148C2×C4×C271—42no—
149C2×C4×C271—42no—
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2×D411yescharacteristic
153C2×C4×C271C2×D411yescharacteristic
154C2×C4×C271C2411yescharacteristic
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—21no—
158C22×C271—21no—
159C22×C271—21no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271AGL(1,ℤ/8)12yes—
164C22×C271C22×D411yescharacteristic
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—41no—
168C4×C271—21no—
169C4×C271—21no—
170C4×C271—21no—
171C4×C271—21no—
172C4×C271—41no—
173C4×C271—41no—
174C4×C271—41no—
175C4×C271C2×#4111yescharacteristic
176C4×C271H(5,𝔽2)11yescharacteristic
177Dic271—22no—
178Dic271—41no—
179Dic271—41no—
180C2×C271—22no—
181C2×C271—41no—
182C2×C271—81no—
183C2×C271#54611yescharacteristic
184C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C271#57811yescharacteristic
186D271—81no—
187C271#330811yescharacteristic, complemented, Sylow
188#3308—2711nomaximal, Sylow
189#548—2711no—
190#547D27111yescharacteristic, complemented
191#489—2711no—
192#484—2711no—
193#480—2711no—
194#477—2711no—
195#468—2711no—
196#466—2711no—
197#463—2711no—
198#449—2711no—
199#448—2711no—
200#440—2711no—
201#435—2711no—
202#417—2711no—
203C2×#103—2711no—
204C2×#41—21no—
205C2×#41—2711no—
206C22×Q8—2711no—
207C2×SD16—2711no—
208C2×SD16—2711no—
209C2×D8—2711no—
210C2×M16—2711no—
211C22×C8—2711no—
212C4⋊Q8C2×D27111yescharacteristic
213#124—2711no—
214#124—5421no—
215#123—21no—
216#123—21no—
217C22⋊Q8—21no—
218C22⋊Q8—2711no—
219C22⋊Q8—5421no—
220#121—2711no—
221#121—5421no—
222#121C2×D27111yescharacteristic
223#121C2×D27111yescharacteristic
224C4×Q8—2711no—
225C4×D4—5421no—
226C4×D4C2×D27111yescharacteristic
227C4×D4C2×D27111yescharacteristic
228C2×C4⋊C4C2×D27111yescharacteristic
229C2×C4⋊C4C2×D27111yescharacteristic
230#107—2711no—
231#107—2711no—
232#107—2711no—
233C4⋊C8—2711no—
234#103—2711no—
235#103—2711no—
236#103—2711no—
237#103—2711no—
238#103—2711no—
239#103—2711no—
240#102—2711no—
241#102—2711no—
242#102—2711no—
243#102—2711no—
244C22⋊C8—2711no—
245C22⋊C8—2711no—
246#97—2711no—
247#41—44no—
248#41—5422no—
249#41—10841no—
250C2×Q8—21no—
251C2×Q8—2711no—
252C2×Q8—2711no—
253C2×Q8—5422no—
254C2×Q8—5421no—
255C2×Q8—10841no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—2711no—
259C2×D4—5421no—
260C2×D4—5421no—
261C2×D4C22⋊D27111yescharacteristic
262C2×D4C22×D27111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4C22×D27111yescharacteristic
269C22×C4C22×D27111yescharacteristic
270SD16—5422no—
271SD16—10841no—
272D8—10841no—
273M16—10841no—
274C2×C8—2711no—
275C2×C8—2711no—
276C2×C8—2711no—
277C2×C8—2711no—
278C2×C8—10841no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—2711no—
283C4⋊C4—5421no—
284C4⋊C4—5421no—
285C4⋊C4C22⋊D27111yescharacteristic
286C4⋊C4C22×D27111yescharacteristic
287C4⋊C4C22×D27111yescharacteristic
288C4⋊C4C22×D27111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4C22⋊D27111yescharacteristic
297C22⋊C4C22⋊D27111yescharacteristic
298C42—5421no—
299C42C22×D27111yescharacteristic
300C23—21no—
301C23—5421no—
302C23C2×C22⋊D27111yescharacteristic
303C23C2×C22⋊D27111yescharacteristic
304Q8—42no—
305Q8—2712no—
306Q8—5421no—
307Q8—5421no—
308Q8—5421no—
309Q8—10842no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—5421no—
316D4—5421no—
317D4—10842no—
318D4—10842no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—10842no—
333C2×C4—10842no—
334C2×C4C2×C22⋊D27111yescharacteristic
335C2×C4C2×C22⋊D27111yescharacteristic
336C2×C4C2×C22⋊D27111yescharacteristic
337C2×C4C2×C22⋊D27111yescharacteristic
338C2×C4C23×D27111yescharacteristic
339C8—5421no—
340C8—5421no—
341C8—5421no—
342C8—5421no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—10841no—
352C22—10841no—
353C22—10841no—
354C22#2257618612yes—
355C22C22×C22⋊D27111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—5422no—
364C4—10841no—
365C4—10841no—
366C4#41×D27111yescharacteristic
367C4#2257620611yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—21681no—
372C2#3177453511yescharacteristic
373C2C2×#2257618611yescharacteristic
374C2#3177471211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#547×C271C2
socleC22×C271C22×D4
hypercenter#547D271
hypocenterC271#3308
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #31774556C2semidirect
2. #31774259C2semidirect
3. #31774232C2semidirect
4. #31774210C2semidirect
5. #31774189C2semidirect
6. #31774141C2semidirect
7. #31774131C2semidirect
8. #31774112C2semidirect
9. #31774040C2semidirect
10. #31774035C2semidirect
11. #31773997C2semidirect
12. #31773969C2semidirect
13. #31773896C2semidirect
14. C2×#22576083C2semidirect
15. #547×C271C2semidirect
117. Dic542#41semidirect
187. C271#3308semidirect
190. #547D271semidirect

Maximal subgroups

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

subgroupindex
#317745562
#317742592
#317742322
#317742102
#317741892
#317741412
#317741312
#317741122
#317740402
#317740352
#317739972
#317739692
#317738962
C2×#225760832
#547×C2712
#3308271