Finite Group Databasebeta

← #46848351

Subgroups — #46848351

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#31774558C211yesmaximal, characteristic, complemented
2#31774548C211yesmaximal, characteristic, complemented
3#31774258C211yesmaximal, characteristic, complemented
4#31774231C211yesmaximal, characteristic, complemented
5#31774210C211yesmaximal, characteristic, complemented
6#31774190C211yesmaximal, characteristic, complemented
7#31774146C211yesmaximal, characteristic, complemented
8#31774133C211yesmaximal, characteristic, complemented
9#31774115C211yesmaximal, characteristic, complemented
10#31774045C211yesmaximal, characteristic, complemented
11#31774036C211yesmaximal, characteristic, complemented
12#31774000C211yesmaximal, characteristic, complemented
13#31773968C211yesmaximal, characteristic, complemented
14#31773897C211yesmaximal, characteristic, complemented
15C2×#103×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654—21no—
18C2×#16864641C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C2×D2168C2211yescharacteristic
21C2×#16864637C2211yescharacteristic
22#22576159C2211yescharacteristic
23#22576149—21no—
24#22576149C2211yescharacteristic
25#22576145—21no—
26#22576143—21no—
27#22576138—21no—
28#22576136—21no—
29#22576136C2211yescharacteristic
30#22576135C2211yescharacteristic
31#22576134—21no—
32#22576134C2211yescharacteristic
33#22576134C2211yescharacteristic
34Q8×Dic271C2211yescharacteristic
35#22576122C2211yescharacteristic
36C4×C22⋊D271—21no—
37C4×C22⋊D271C2211yescharacteristic
38C4⋊C4×D271C2211yescharacteristic
39#22576094C2211yescharacteristic
40#22576093C2211yescharacteristic
41#22576093C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576070C2211yescharacteristic
52C22×Q8×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54C2×C4⋊C4×C271C2211yescharacteristic
55#103×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic
59#16864655—22no—
60#16864655—41no—
61#16864654—44no—
62C2×Dic542—21no—
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271D411yescharacteristic
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271C2311yescharacteristic
74#16864641—22no—
75#16864641—41no—
76D2168—41no—
77#16864637—41no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C4⋊Dic271—21no—
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87C22⋊Dic271D411yescharacteristic
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4D411yescharacteristic
96C4×Dic271—21no—
97C4×Dic271C2311yescharacteristic
98C2×Q8×C271—21no—
99C2×Q8×C271—22no—
100C2×Q8×C271—41no—
101C2×Q8×C271C2311yescharacteristic
102C2×Q8×C271C2311yescharacteristic
103C22×C4×C271—21no—
104C22×C4×C271—21no—
105C22×C4×C271C2311yescharacteristic
106C2×C8×C271—41no—
107C2×C8×C271C2311yescharacteristic
108C2×C8×C271C2311yescharacteristic
109C4⋊C4×C271—21no—
110C4⋊C4×C271C2311yescharacteristic
111C4⋊C4×C271C2311yescharacteristic
112C22×D271—21no—
113C22×D271—21no—
114C22×D271C2×D411yescharacteristic
115Dic542—42no—
116D1084—21no—
117D1084—21no—
118D1084—21no—
119D1084—21no—
120D1084—42no—
121D1084—42no—
122C22⋊D271—22no—
123C22⋊D271—42no—
124C22⋊D271—42no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—42no—
133C4×D271—42no—
134C4×D271—42no—
135C271⋊C8—21no—
136C271⋊C8—21no—
137C23×C271C2×D411yescharacteristic
138Q8×C271—21no—
139Q8×C271—21no—
140Q8×C271—21no—
141Q8×C271—42no—
142Q8×C271#4112yescomplemented
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—42no—
150C2×C4×C271—42no—
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271AGL(1,ℤ/8)12yes—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—21no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—41no—
168C4×C271—21no—
169C4×C271—22no—
170C4×C271—41no—
171C4×C271—41no—
172C4×C271—41no—
173C4×C271C2×#4111yescharacteristic
174C4×C271H(5,𝔽2)11yescharacteristic
175Dic271—21no—
176Dic271—21no—
177Dic271—21no—
178Dic271—41no—
179Dic271—41no—
180C2×C271—22no—
181C2×C271#54611yescharacteristic
182C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
183C2×C271#57811yescharacteristic
184D271—41no—
185D271—81no—
186D271—81no—
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#463—2711no—
198#449—2711no—
199#448—2711no—
200#440—2711no—
201#435—2711no—
202#417—2711no—
203C2×#103D27111yescharacteristic, complemented
204C2×#41—2711no—
205C2×#41—5421no—
206C22×Q8C2×D27111yescharacteristic
207C2×SD16—2711no—
208C2×SD16—2711no—
209C2×D8—2711no—
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—2711no—
222#121—2711no—
223#121—5421no—
224C4×Q8—2711no—
225C4×D4—2711no—
226C4×D4—2711no—
227C4×D4—5421no—
228C2×C4⋊C4—2711no—
229C2×C4⋊C4C2×D27111yescharacteristic
230#107—2711no—
231#107—2711no—
232#107—2711no—
233C4⋊C8—2711no—
234#103—2711no—
235#103—2711no—
236#103C2×D27111yescharacteristic
237#103C2×D27111yescharacteristic
238#103C2×D27111yescharacteristic
239#103C2×D27111yescharacteristic
240#102—2711no—
241#102—2711no—
242#102—2711no—
243#102—2711no—
244C22⋊C8—2711no—
245C22⋊C8—2711no—
246#97—2711no—
247#41—5422no—
248#41—10844no—
249#41—10841no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—5421no—
254C2×Q8C22×D27111yescharacteristic
255C2×Q8C22×D27111yescharacteristic
256C2×D4—2711no—
257C2×D4—2711no—
258C2×D4—2711no—
259C2×D4—5421no—
260C2×D4—5421no—
261C2×D4—5421no—
262C2×D4—5421no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—2711no—
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4—5421no—
269C22×C4C22×D27111yescharacteristic
270SD16—5422no—
271SD16—10841no—
272D8—10841no—
273M16—10841no—
274C2×C8—41no—
275C2×C8—2711no—
276C2×C8—2711no—
277C2×C8C22×D27111yescharacteristic
278C2×C8C22×D27111yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—2711no—
281C4⋊C4—2711no—
282C4⋊C4—2711no—
283C4⋊C4—5421no—
284C4⋊C4—5421no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4C22×D27111yescharacteristic
288C4⋊C4C22×D27111yescharacteristic
289C22⋊C4—2711no—
290C22⋊C4—2711no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C42—2711no—
299C42—5421no—
300C23—2711no—
301C23—5421no—
302C23—5421no—
303C23D4×D27111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—10842no—
309Q8#1686465412yescomplemented
310D4—5422no—
311D4—5421no—
312D4—5421no—
313D4—5421no—
314D4—5421no—
315D4—10842no—
316D4—10842no—
317D4—10842no—
318D4—10842no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—2711no—
328C2×C4—2711no—
329C2×C4—2711no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—5421no—
333C2×C4—5421no—
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—5421no—
346C22—5421no—
347C22—5421no—
348C22—10841no—
349C22—10841no—
350C22—10841no—
351C22—10841no—
352C22—10841no—
353C22—10841no—
354C22#2257618712yes—
355C22C2×D4×D27111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—5421no—
362C4—5421no—
363C4—5421no—
364C4—10841no—
365C4—10841no—
366C4C2×#1686465411yescharacteristic
367C4#2257620511yescharacteristic
368C2—22no—
369C2—10841no—
370C2—21681no—
371C2—21681no—
372C2#3177453711yescharacteristic
373C2C2×#2257618711yescharacteristic
374C2#3177471011yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#103×C271C2
socleC22×C271C22×D4
hypercenterC2×#103D271
hypocenterC271#3308
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. #31774558C2semidirect
2. #31774548C2semidirect
3. #31774258C2semidirect
4. #31774231C2semidirect
5. #31774210C2semidirect
6. #31774190C2semidirect
7. #31774146C2semidirect
8. #31774133C2semidirect
9. #31774115C2semidirect
10. #31774045C2semidirect
11. #31774036C2semidirect
12. #31774000C2semidirect
13. #31773968C2semidirect
14. #31773897C2semidirect
15. C2×#103×C271C2semidirect
142. Q8×C271#41semidirect
187. C271#3308semidirect
203. C2×#103D271semidirect
309. Q8#16864654semidirect

Maximal subgroups

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

subgroupindex
#317745582
#317745482
#317742582
#317742312
#317742102
#317741902
#317741462
#317741332
#317741152
#317740452
#317740362
#317740002
#317739682
#317738972
C2×#103×C2712
#3308271