Finite Group Databasebeta

← #46848361

Subgroups — #46848361

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#31774553C211yesmaximal, characteristic, complemented
2#31774547C211yesmaximal, characteristic, complemented
3#31774255C211yesmaximal, characteristic, complemented
4#31774234C211yesmaximal, characteristic, complemented
5#31774208C211yesmaximal, characteristic, complemented
6#31774191C211yesmaximal, characteristic, complemented
7#31774142C211yesmaximal, characteristic, complemented
8#31774110C211yesmaximal, characteristic, complemented
9#31774039C211yesmaximal, characteristic, complemented
10#31774035C211yesmaximal, characteristic, complemented
11#31773994C211yesmaximal, characteristic, complemented
12#31773968C211yesmaximal, characteristic, complemented
13#31773895C211yesmaximal, characteristic, complemented
14C2×#22576085C211yesmaximal, characteristic, complemented
15#466×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×#16864653C2211yescharacteristic
18C22×Dic542C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×#16864642C2211yescharacteristic
21C2×#16864637C2211yescharacteristic
22#22576159C2211yescharacteristic
23#22576147C2211yescharacteristic
24#22576146—21no—
25#22576142—21no—
26#22576141—21no—
27#22576140C2211yescharacteristic
28#22576137—21no—
29#22576137—21no—
30#22576132—21no—
31#22576131C2211yescharacteristic
32#22576125C2211yescharacteristic
33D4×Dic271—21no—
34D4×Dic271C2211yescharacteristic
35C4×C22⋊D271C2211yescharacteristic
36C2×C4⋊Dic271C2211yescharacteristic
37C4⋊C4×D271C2211yescharacteristic
38#22576093C2211yescharacteristic
39#22576093C2211yescharacteristic
40Dic271⋊C8C2211yescharacteristic
41#22576085C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576085C2211yescharacteristic
45#22576083C2211yescharacteristic
46#22576083C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576070C2211yescharacteristic
52C2×D8×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54#121×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#107×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864653—22no—
60#16864653—41no—
61#16864653—44no—
62C2×Dic542—21no—
63C2×Dic542—22no—
64C2×Dic542—21no—
65C2×Dic542—41no—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271D411yescharacteristic
71C22×Dic271—21no—
72C22×Dic271—21no—
73C22×Dic271—21no—
74C22×Dic271—21no—
75C2×C4×D271—21no—
76C2×C4×D271C2311yescharacteristic
77#16864642—22no—
78#16864642—41no—
79#16864637—41no—
80C2×C271⋊C8C2311yescharacteristic
81C2×C271⋊C8C2311yescharacteristic
82C4⋊Dic271—21no—
83C4⋊Dic271—21no—
84C4⋊Dic271C2311yescharacteristic
85C4⋊Dic271C2311yescharacteristic
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4D411yescharacteristic
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271—21no—
93C22⋊Dic271—21no—
94C22⋊Dic271D411yescharacteristic
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4D411yescharacteristic
97C4×Dic271—21no—
98C4×Dic271C2311yescharacteristic
99C2×D4×C271—21no—
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271C2311yescharacteristic
104D8×C271—41no—
105C2×C8×C271—41no—
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22⋊C4×C271—21no—
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—21no—
115Dic542—21no—
116Dic542—42no—
117Dic542—42no—
118Dic542#4112yescomplemented
119C22⋊D271—22no—
120C22⋊D271—42no—
121C22⋊D271—42no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—42no—
127C2×Dic271—42no—
128C2×Dic271—42no—
129C2×Dic271—42no—
130C2×Dic271C2×D411yescharacteristic
131C2×Dic271C2×D411yescharacteristic
132C2×Dic271C2×D411yescharacteristic
133C4×D271—21no—
134C4×D271—21no—
135C4×D271—42no—
136C271⋊C8—21no—
137C271⋊C8—21no—
138C23×C271—21no—
139C23×C271—21no—
140C23×C271C2×D411yescharacteristic
141D4×C271—21no—
142D4×C271—21no—
143D4×C271—21no—
144D4×C271—21no—
145D4×C271—42no—
146D4×C271—42no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271AGL(1,ℤ/8)12yes—
164C22×C271C22×D411yescharacteristic
165C2×D271—21no—
166C2×D271—21no—
167C2×D271—21no—
168C4×C271—21no—
169C4×C271—41no—
170C4×C271—41no—
171C4×C271C2×#4111yescharacteristic
172C4×C271H(5,𝔽2)11yescharacteristic
173Dic271—21no—
174Dic271—22no—
175Dic271—21no—
176Dic271—21no—
177Dic271—41no—
178Dic271—41no—
179Dic271—41no—
180C2×C271—22no—
181C2×C271—81no—
182C2×C271—81no—
183C2×C271#54611yescharacteristic
184C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C271#57811yescharacteristic
186D271—41no—
187C271#330811yescharacteristic, complemented, Sylow
188#3308—2711nomaximal, Sylow
189#548—2711no—
190#547—2711no—
191#489—2711no—
192#484—2711no—
193#480—2711no—
194#477—2711no—
195#468—2711no—
196#466D27111yescharacteristic, complemented
197#463—2711no—
198#449—2711no—
199#448—2711no—
200#440—2711no—
201#435—2711no—
202#417—2711no—
203C2×#103—2711no—
204C2×#41—2711no—
205C2×#41—5421no—
206C22×Q8—2711no—
207C2×SD16—2711no—
208C2×SD16—2711no—
209C2×D8C2×D27111yescharacteristic
210C2×M16—2711no—
211C22×C8C2×D27111yescharacteristic
212C4⋊Q8—2711no—
213#124—2711no—
214#124—5421no—
215#123—5421no—
216#123—5421no—
217C22⋊Q8—2711no—
218C22⋊Q8—5421no—
219C22⋊Q8—5421no—
220#121—2711no—
221#121—5421no—
222#121C2×D27111yescharacteristic
223#121C2×D27111yescharacteristic
224C4×Q8—2711no—
225C4×D4—2711no—
226C4×D4—2711no—
227C4×D4—5421no—
228C2×C4⋊C4—2711no—
229C2×C4⋊C4—2711no—
230#107—2711no—
231#107—2711no—
232#107C2×D27111yescharacteristic
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#102C2×D27111yescharacteristic
243#102C2×D27111yescharacteristic
244C22⋊C8—2711no—
245C22⋊C8—2711no—
246#97—2711no—
247#41—5422no—
248#41—10841no—
249#41—10844no—
250C2×Q8—2711no—
251C2×Q8—2711no—
252C2×Q8—5421no—
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—2711no—
264C22×C4—5421no—
265C22×C4—5421no—
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4—5421no—
269C22×C4C22×D27111yescharacteristic
270SD16—5422no—
271SD16—10841no—
272D8—41no—
273M16—10841no—
274C2×C8—41no—
275C2×C8—2711no—
276C2×C8—2711no—
277C2×C8C22×D27111yescharacteristic
278C2×C8C22×D27111yescharacteristic
279C4⋊C4—2711no—
280C4⋊C4—2711no—
281C4⋊C4—2711no—
282C4⋊C4—5421no—
283C4⋊C4—5421no—
284C4⋊C4—5421no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4C22×D27111yescharacteristic
288C4⋊C4C22×D27111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—2711no—
292C22⋊C4—2711no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C42—2711no—
299C42—5421no—
300C23—21no—
301C23—21no—
302C23—2711no—
303C23D4×D27111yescharacteristic
304Q8—2712no—
305Q8—5421no—
306Q8—5421no—
307Q8—5421no—
308Q8—10842no—
309Q8—10842no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—42no—
316D4—5422no—
317D4—10842no—
318D4—10842no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—2711no—
324C2×C4—2711no—
325C2×C4—2711no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—5421no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—10842no—
333C2×C4—10842no—
334C2×C4—10842no—
335C2×C4—10842no—
336C2×C4—10842no—
337C2×C4D4×D27111yescharacteristic
338C2×C4C23×D27111yescharacteristic
339C8—21no—
340C8—21no—
341C8—5421no—
342C8—5421no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—5421no—
352C22—5421no—
353C22—5421no—
354C22#2257618412yes—
355C22C2×D4×D27111yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—5421no—
360C4—5422no—
361C4—5421no—
362C4—5421no—
363C4—10841no—
364C4—10841no—
365C4—10841no—
366C4#41×D27111yescharacteristic
367C4#2257620511yescharacteristic
368C2—22no—
369C2—81no—
370C2—81no—
371C2—10841no—
372C2#3177453611yescharacteristic
373C2C2×#2257618411yescharacteristic
374C2#3177471111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#466×C271C2
socleC22×C271C22×D4
hypercenter#466D271
hypocenterC271#3308
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #31774553C2semidirect
2. #31774547C2semidirect
3. #31774255C2semidirect
4. #31774234C2semidirect
5. #31774208C2semidirect
6. #31774191C2semidirect
7. #31774142C2semidirect
8. #31774110C2semidirect
9. #31774039C2semidirect
10. #31774035C2semidirect
11. #31773994C2semidirect
12. #31773968C2semidirect
13. #31773895C2semidirect
14. C2×#22576085C2semidirect
15. #466×C271C2semidirect
118. Dic542#41semidirect
187. C271#3308semidirect
196. #466D271semidirect

Maximal subgroups

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

subgroupindex
#317745532
#317745472
#317742552
#317742342
#317742082
#317741912
#317741422
#317741102
#317740392
#317740352
#317739942
#317739682
#317738952
C2×#225760852
#466×C2712
#3308271