Finite Group Databasebeta

← #46848362

Subgroups — #46848362

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#31774555C211yesmaximal, characteristic, complemented
2#31774552C211yesmaximal, characteristic, complemented
3#31774256C211yesmaximal, characteristic, complemented
4#31774231C211yesmaximal, characteristic, complemented
5#31774209C211yesmaximal, characteristic, complemented
6#31774193C211yesmaximal, characteristic, complemented
7#31774140C211yesmaximal, characteristic, complemented
8#31774130C211yesmaximal, characteristic, complemented
9#31774039C211yesmaximal, characteristic, complemented
10#31774034C211yesmaximal, characteristic, complemented
11#31773997C211yesmaximal, characteristic, complemented
12#31773967C211yesmaximal, characteristic, complemented
13#31773895C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C2×#16864654—21no—
17C2×#16864653C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×#16864637C2211yescharacteristic
21C2×C8×D271C2211yescharacteristic
22#22576160C2211yescharacteristic
23#22576146—21no—
24#22576142—21no—
25#22576142—21no—
26#22576139—21no—
27#22576137—21no—
28#22576136C2211yescharacteristic
29#22576133—21no—
30#22576131C2211yescharacteristic
31#22576131C2211yescharacteristic
32#22576131C2211yescharacteristic
33C4×Dic542C2211yescharacteristic
34C4×D1084C2211yescharacteristic
35(C22⋊D271)⋊C4—21no—
36C4⋊C4×D271C2211yescharacteristic
37C4⋊C4×D271C2211yescharacteristic
38#22576094C2211yescharacteristic
39#22576094C2211yescharacteristic
40#22576093C2211yescharacteristic
41#22576085C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576083C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576071C2211yescharacteristic
52C2×SD16×C271C2211yescharacteristic
53C2×D8×C271C2211yescharacteristic
54#124×C271C2211yescharacteristic
55C4×D4×C271C2211yescharacteristic
56C4⋊C8×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864654—44no—
60#16864653—22no—
61#16864653—41no—
62C2×Dic542—21no—
63C2×Dic542—21no—
64C2×Dic542C2311yescharacteristic
65Q8×D271—22no—
66Q8×D271—41no—
67C2×D1084D411yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C22×Dic271—21no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271C2311yescharacteristic
77C2×C4×D271C2311yescharacteristic
78#16864642—22no—
79#16864637—41no—
80C2×C271⋊C8C2311yescharacteristic
81C2×C271⋊C8C2311yescharacteristic
82C8×D271—41no—
83C4⋊Dic271—21no—
84C4⋊Dic271D411yescharacteristic
85C4⋊Dic271C2311yescharacteristic
86C4⋊Dic271C2311yescharacteristic
87C4⋊Dic271C2311yescharacteristic
88Dic271⋊C4—21no—
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91Dic271⋊C4—21no—
92C22⋊Dic271—21no—
93C22⋊Dic271—21no—
94C22⋊Dic271—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4D411yescharacteristic
97(C2×D271)⋊C4D411yescharacteristic
98C4×Dic271—21no—
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271C2311yescharacteristic
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271—21no—
103SD16×C271—41no—
104D8×C271—41no—
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C22⋊C4×C271—21no—
109C22⋊C4×C271—21no—
110C22⋊C4×C271—21no—
111C42×C271C2311yescharacteristic
112C22×D271C2×D411yescharacteristic
113C22×D271C2×D411yescharacteristic
114Dic542—21no—
115Dic542—42no—
116Dic542—42no—
117Dic542#4112yescomplemented
118D1084—22no—
119C22⋊D271—42no—
120C22⋊D271—42no—
121C22⋊D271—42no—
122C22⋊D271—42no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—42no—
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—21no—
133C4×D271—21no—
134C4×D271—42no—
135C4×D271—42no—
136C4×D271—42no—
137C271⋊C8—21no—
138C271⋊C8—21no—
139C23×C271—21no—
140C23×C271—21no—
141Q8×C271—21no—
142Q8×C271—21no—
143D4×C271—21no—
144D4×C271—21no—
145D4×C271—21no—
146D4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—42no—
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271C22×D411yescharacteristic
162C2×D271—21no—
163C2×D271—21no—
164C2×D271—21no—
165C2×D271—21no—
166C2×D271—21no—
167C2×D271AGL(1,ℤ/8)12yes—
168C4×C271—21no—
169C4×C271—21no—
170C4×C271—41no—
171C4×C271—41no—
172C4×C271C2×#4111yescharacteristic
173C4×C271H(5,𝔽2)11yescharacteristic
174Dic271—21no—
175Dic271—21no—
176Dic271—22no—
177Dic271—41no—
178Dic271—41no—
179Dic271—41no—
180C2×C271—81no—
181C2×C271—81no—
182C2×C271#54611yescharacteristic
183C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C271#57811yescharacteristic
185D271—22no—
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#466—2711no—
197#463D27111yescharacteristic, complemented
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×SD16C2×D27111yescharacteristic
209C2×D8C2×D27111yescharacteristic
210C2×M16—2711no—
211C22×C8—2711no—
212C4⋊Q8—2711no—
213#124—5421no—
214#124C2×D27111yescharacteristic
215#123—5421no—
216#123—5421no—
217C22⋊Q8—2711no—
218C22⋊Q8—5421no—
219C22⋊Q8—5421no—
220#121—2711no—
221#121—2711no—
222#121—2711no—
223#121—5421no—
224C4×Q8—2711no—
225C4×D4—2711no—
226C4×D4—5421no—
227C4×D4C2×D27111yescharacteristic
228C2×C4⋊C4—2711no—
229C2×C4⋊C4—2711no—
230#107—2711no—
231#107—2711no—
232#107—2711no—
233C4⋊C8C2×D27111yescharacteristic
234#103—2711no—
235#103—2711no—
236#103—2711no—
237#103—2711no—
238#103—2711no—
239#103C2×D27111yescharacteristic
240#102—2711no—
241#102—2711no—
242#102—2711no—
243#102C2×D27111yescharacteristic
244C22⋊C8—2711no—
245C22⋊C8—2711no—
246#97—2711no—
247#41—5422no—
248#41—10844no—
249#41—10841no—
250C2×Q8—2711no—
251C2×Q8—5421no—
252C2×Q8—5421no—
253C2×Q8—5422no—
254C2×Q8—10841no—
255C2×Q8C22×D27111yescharacteristic
256C2×D4—2711no—
257C2×D4—5421no—
258C2×D4—5421no—
259C2×D4—5421no—
260C2×D4—5421no—
261C2×D4C22×D27111yescharacteristic
262C2×D4C22×D27111yescharacteristic
263C22×C4—21no—
264C22×C4—2711no—
265C22×C4—2711no—
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4—5421no—
269C22×C4—5421no—
270SD16—41no—
271SD16—5422no—
272D8—41no—
273M16—10841no—
274C2×C8—2711no—
275C2×C8—2711no—
276C2×C8—10841no—
277C2×C8C22×D27111yescharacteristic
278C2×C8C22×D27111yescharacteristic
279C4⋊C4—2711no—
280C4⋊C4—2711no—
281C4⋊C4—2711no—
282C4⋊C4—2711no—
283C4⋊C4—5421no—
284C4⋊C4—5421no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4—5421no—
288C4⋊C4C22×D27111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—2711no—
293C22⋊C4—2711no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C42—5421no—
299C42C22×D27111yescharacteristic
300C23—21no—
301C23—21no—
302C23—2711no—
303C23—2711no—
304Q8—21no—
305Q8—21no—
306Q8—2712no—
307Q8—5421no—
308Q8—10842no—
309Q8—10842no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—5422no—
315D4—10842no—
316D4—10842no—
317D4—10842no—
318D4—10842no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—2711no—
323C2×C4—2711no—
324C2×C4—5421no—
325C2×C4—5421no—
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×C4D4×D27111yescharacteristic
337C2×C4D4×D27111yescharacteristic
338C2×C4C23×D27111yescharacteristic
339C8—21no—
340C8—21no—
341C8—5421no—
342C8—5421no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—2712no—
350C22—5421no—
351C22—5421no—
352C22—5421no—
353C22—5421no—
354C22—5421no—
355C22C2×D4×D27111yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—5421no—
361C4—5421no—
362C4—5422no—
363C4—10841no—
364C4—10841no—
365C4—10841no—
366C4#41×D27111yescharacteristic
367C4#2257620511yescharacteristic
368C2—81no—
369C2—81no—
370C2—5422no—
371C2—10841no—
372C2#3177454411yescharacteristic
373C2AGL(1,ℤ/8)×D27111yescharacteristic
374C2#3177471111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
hypercenter#463D271
hypocenterC271#3308
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #31774555C2semidirect
2. #31774552C2semidirect
3. #31774256C2semidirect
4. #31774231C2semidirect
5. #31774209C2semidirect
6. #31774193C2semidirect
7. #31774140C2semidirect
8. #31774130C2semidirect
9. #31774039C2semidirect
10. #31774034C2semidirect
11. #31773997C2semidirect
12. #31773967C2semidirect
13. #31773895C2semidirect
14. #103×D271C2semidirect
15. #463×C271C2semidirect
117. Dic542#41semidirect
187. C271#3308semidirect
197. #463D271semidirect

Maximal subgroups

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

subgroupindex
#317745552
#317745522
#317742562
#317742312
#317742092
#317741932
#317741402
#317741302
#317740392
#317740342
#317739972
#317739672
#317738952
#103×D2712
#463×C2712
#3308271