Finite Group Databasebeta

← #46848348

Subgroups — #46848348

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#31774554C211yesmaximal, characteristic, complemented
2#31774547C211yesmaximal, characteristic, complemented
3#31774255C211yesmaximal, characteristic, complemented
4#31774233C211yesmaximal, characteristic, complemented
5#31774209C211yesmaximal, characteristic, complemented
6#31774190C211yesmaximal, characteristic, complemented
7#31774141C211yesmaximal, characteristic, complemented
8#31774130C211yesmaximal, characteristic, complemented
9#31774110C211yesmaximal, characteristic, complemented
10#31774034C211yesmaximal, characteristic, complemented
11#31773998C211yesmaximal, characteristic, complemented
12#31773971C211yesmaximal, characteristic, complemented
13#31773898C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×Q8×D271C2211yescharacteristic
18C2×#16864642C2211yescharacteristic
19C2×#16864638C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576147—21no—
23#22576147C2211yescharacteristic
24#22576142—21no—
25#22576141—21no—
26#22576139—21no—
27#22576139C2211yescharacteristic
28#22576137—21no—
29#22576131—21no—
30#22576131C2211yescharacteristic
31#22576131C2211yescharacteristic
32Q8×Dic271C2211yescharacteristic
33D4×Dic271C2211yescharacteristic
34C4×C22⋊D271—21no—
35C4×C22⋊D271C2211yescharacteristic
36C2×C4⋊Dic271C2211yescharacteristic
37C4⋊C4×D271C2211yescharacteristic
38#22576094C2211yescharacteristic
39#22576094C2211yescharacteristic
40#22576093C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576083C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576081C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576070C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×D8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59#16864653—44no—
60C2×Dic542—21no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63Q8×D271—22no—
64Q8×D271—41no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271D411yescharacteristic
68C22×Dic271—21no—
69C22×Dic271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271C2311yescharacteristic
73#16864642—41no—
74#16864638—41no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C8×D271—41no—
78C4⋊Dic271—21no—
79C4⋊Dic271—21no—
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271—21no—
93C22⋊Dic271D411yescharacteristic
94(C2×D271)⋊C4D411yescharacteristic
95C4×Dic271—21no—
96C4×Dic271C2311yescharacteristic
97#41×C271—22no—
98#41×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271—21no—
102C2×D4×C271C2311yescharacteristic
103C2×D4×C271C2311yescharacteristic
104C22×C4×C271—21no—
105C22×C4×C271C2311yescharacteristic
106SD16×C271—22no—
107D8×C271—41no—
108C2×C8×C271C2311yescharacteristic
109C2×C8×C271C2311yescharacteristic
110C4⋊C4×C271C2311yescharacteristic
111C22⋊C4×C271—21no—
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—21no—
115Dic542—42no—
116Dic542—42no—
117C22⋊D271—22no—
118C22⋊D271—42no—
119C22⋊D271—42no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—42no—
125C2×Dic271—42no—
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C4×D271—21no—
130C4×D271—21no—
131C4×D271—42no—
132C4×D271—42no—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271—21no—
136C23×C271—21no—
137C23×C271C2×D411yescharacteristic
138Q8×C271—21no—
139Q8×C271#4112yescomplemented
140D4×C271—21no—
141D4×C271—21no—
142D4×C271—21no—
143D4×C271—21no—
144D4×C271—42no—
145D4×C271—42no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—42no—
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—21no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271—41no—
164C22×C271C22×D411yescharacteristic
165C2×D271—21no—
166C2×D271—21no—
167C2×D271AGL(1,ℤ/8)12yes—
168C4×C271—21no—
169C4×C271—22no—
170C4×C271—41no—
171C4×C271C2×#4111yescharacteristic
172C4×C271H(5,𝔽2)11yescharacteristic
173Dic271—21no—
174Dic271—21no—
175Dic271—21no—
176Dic271—41no—
177Dic271—41no—
178Dic271—41no—
179Dic271—41no—
180C2×C271—41no—
181C2×C271—81no—
182C2×C271—81no—
183C2×C271#54611yescharacteristic
184C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C271#57811yescharacteristic
186D271—22no—
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#449D27111yescharacteristic, complemented
199#448—2711no—
200#440—2711no—
201#435—2711no—
202#417—2711no—
203C2×#103—2711no—
204C2×#41—5421no—
205C2×#41C2×D27111yescharacteristic
206C22×Q8—2711no—
207C2×SD16—2711no—
208C2×SD16C2×D27111yescharacteristic
209C2×D8C2×D27111yescharacteristic
210C2×M16—2711no—
211C22×C8—2711no—
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—5421no—
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#107—2711no—
233C4⋊C8—2711no—
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⋊C8C2×D27111yescharacteristic
246#97—2711no—
247#41—22no—
248#41—41no—
249#41—10844no—
250C2×Q8—2711no—
251C2×Q8—5421no—
252C2×Q8—5421no—
253C2×Q8—5422no—
254C2×Q8—10841no—
255C2×Q8C22×D27111yescharacteristic
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—2711no—
265C22×C4—5421no—
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4—5421no—
269C22×C4C22×D27111yescharacteristic
270SD16—22no—
271SD16—10841no—
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—2711no—
291C22⋊C4—2711no—
292C22⋊C4—5421no—
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—21no—
305Q8—5421no—
306Q8—5421no—
307Q8—10842no—
308Q8—10842no—
309Q8#1686465312yescomplemented
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—42no—
324C2×C4—2711no—
325C2×C4—2711no—
326C2×C4—2711no—
327C2×C4—5421no—
328C2×C4—5421no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—5421no—
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—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—2712no—
353C22—5421no—
354C22—5421no—
355C22C2×D4×D27111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—5421no—
360C4—5421no—
361C4—5421no—
362C4—10841no—
363C4—10841no—
364C4—10841no—
365C4—10841no—
366C4C2×#1686465311yescharacteristic
367C4#2257620511yescharacteristic
368C2—41no—
369C2—81no—
370C2—81no—
371C2—5422no—
372C2#3177453411yescharacteristic
373C2AGL(1,ℤ/8)×D27111yescharacteristic
374C2#3177471111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3308
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31774554C2semidirect
2. #31774547C2semidirect
3. #31774255C2semidirect
4. #31774233C2semidirect
5. #31774209C2semidirect
6. #31774190C2semidirect
7. #31774141C2semidirect
8. #31774130C2semidirect
9. #31774110C2semidirect
10. #31774034C2semidirect
11. #31773998C2semidirect
12. #31773971C2semidirect
13. #31773898C2semidirect
14. #103×D271C2semidirect
15. #449×C271C2semidirect
139. Q8×C271#41semidirect
187. C271#3308semidirect
198. #449D271semidirect
309. Q8#16864653semidirect

Maximal subgroups

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

subgroupindex
#317745542
#317745472
#317742552
#317742332
#317742092
#317741902
#317741412
#317741302
#317741102
#317740342
#317739982
#317739712
#317738982
#103×D2712
#449×C2712
#3308271