Finite Group Databasebeta

← #46848259

Subgroups — #46848259

Order 34688.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774594C211yesmaximal, characteristic, complemented
2#31774554C211yesmaximal, characteristic, complemented
3#31774250C211yesmaximal, characteristic, complemented
4#31774239C211yesmaximal, characteristic, complemented
5#31774214C211yesmaximal, characteristic, complemented
6#31774206C211yesmaximal, characteristic, complemented
7#31774130C211yesmaximal, characteristic, complemented
8#31774124C211yesmaximal, characteristic, complemented
9#31774110C211yesmaximal, characteristic, complemented
10#31774034C211yesmaximal, characteristic, complemented
11SD16×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773893C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×Q8×D271C2211yescharacteristic
17C2×#16864642C2211yescharacteristic
18C22×C271⋊C8C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576153—21no—
21#22576151C2211yescharacteristic
22#22576147—21no—
23#22576147C2211yescharacteristic
24#22576141—21no—
25#22576139—21no—
26#22576139C2211yescharacteristic
27#22576131—21no—
28#22576131—21no—
29#22576131C2211yescharacteristic
30#22576131C2211yescharacteristic
31Q8×Dic271C2211yescharacteristic
32D4×Dic271C2211yescharacteristic
33C4×C22⋊D271—21no—
34C4×C22⋊D271C2211yescharacteristic
35(C22⋊D271)⋊C4—21no—
36#22576119C2211yescharacteristic
37C4⋊C4×D271C2211yescharacteristic
38#22576094C2211yescharacteristic
39#22576092C2211yescharacteristic
40#22576091C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576083C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576081C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×D8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59C2×Dic542—21no—
60C2×Dic542C2311yescharacteristic
61Q8×D271—22no—
62Q8×D271—41no—
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C2×C22⋊D271D411yescharacteristic
66C22×Dic271—21no—
67C2×C4×D271—21no—
68C2×C4×D271—21no—
69C2×C4×D271C2311yescharacteristic
70#16864642—41no—
71C2×C271⋊C8—41no—
72C2×C271⋊C8C2311yescharacteristic
73C2×C271⋊C8C2311yescharacteristic
74C8×D271—41no—
75C4⋊Dic271—21no—
76C4⋊Dic271—21no—
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4D411yescharacteristic
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91C22⋊Dic271D411yescharacteristic
92(C2×D271)⋊C4D411yescharacteristic
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96#41×C271—22no—
97#41×C271—41no—
98C2×Q8×C271C2311yescharacteristic
99C2×D4×C271—21no—
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271—21no—
104C22×C4×C271C2311yescharacteristic
105SD16×C271—22no—
106D8×C271—41no—
107C2×C8×C271C2311yescharacteristic
108C2×C8×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271C2×D411yescharacteristic
112Dic542—21no—
113Dic542—21no—
114Dic542—42no—
115C22⋊D271—22no—
116C22⋊D271—42no—
117C22⋊D271—42no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—81no—
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C4×D271—21no—
127C4×D271—21no—
128C4×D271—42no—
129C4×D271—81no—
130C271⋊C8—21no—
131C271⋊C8—21no—
132C23×C271—21no—
133C23×C271—21no—
134C23×C271C2×D411yescharacteristic
135Q8×C271—21no—
136Q8×C271#4112yes—
137D4×C271—21no—
138D4×C271—21no—
139D4×C271—21no—
140D4×C271—21no—
141D4×C271—42no—
142D4×C271—81no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—42no—
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—21no—
153C22×C271—21no—
154C22×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#13512yes—
165C4×C271—21no—
166C4×C271—22no—
167C4×C271—41no—
168C4×C271C2×#4111yescharacteristic
169C4×C271H(5,𝔽2)11yescharacteristic
170Dic271—21no—
171Dic271—21no—
172Dic271—21no—
173Dic271—41no—
174Dic271—41no—
175Dic271—41no—
176Dic271—41no—
177C2×C271—41no—
178C2×C271—81no—
179C2×C271—81no—
180C2×C271#54611yescharacteristic
181C2×C271C2×#13511yescharacteristic
182C2×C271#57711yescharacteristic
183D271—22no—
184C271#330111yescharacteristic, complemented, Sylow
185#3301—2711nomaximal, Sylow
186#553—2711no—
187#548—2711no—
188#488—2711no—
189#485—2711no—
190#481—2711no—
191#479—2711no—
192#466—2711no—
193#465—2711no—
194#463—2711no—
195#449D27111yescharacteristic, complemented
196#448—2711no—
197C4×SD16—2711no—
198C8×D4—2711no—
199#416—2711no—
200C2×#103—2711no—
201C2×#41C2×D27111yescharacteristic
202C22×Q8—2711no—
203C2×SD16—2711no—
204C2×SD16C2×D27111yescharacteristic
205C2×D8C2×D27111yescharacteristic
206C22×C8—2711no—
207C22×C8—2711no—
208#126—5421no—
209#125—2711no—
210#124—2711no—
211#124—5421no—
212#123—5421no—
213C22⋊Q8—2711no—
214C22⋊Q8—5421no—
215#121—2711no—
216#121—2711no—
217#121—5421no—
218#121—5421no—
219#121C2×D27111yescharacteristic
220C4×Q8—2711no—
221C4×D4—2711no—
222C4×D4—2711no—
223C4×D4—5421no—
224C4×D4—5421no—
225#117—2711no—
226C2×C4⋊C4—2711no—
227#107—2711no—
228#106—2711no—
229#106—2711no—
230C4⋊C8—2711no—
231#103—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103—2711no—
236#103C2×D27111yescharacteristic
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102C2×D27111yescharacteristic
241C22⋊C8—2711no—
242C22⋊C8C2×D27111yescharacteristic
243C4×C8—2711no—
244#41—22no—
245#41—41no—
246C2×Q8—2711no—
247C2×Q8—5421no—
248C2×Q8—5422no—
249C2×Q8—10841no—
250C2×Q8C22×D27111yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—2711no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4C22×D27111yescharacteristic
257C2×D4C22×D27111yescharacteristic
258C22×C4—21no—
259C22×C4—2711no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4C22×D27111yescharacteristic
264SD16—22no—
265SD16—10841no—
266D8—41no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8—10841no—
270C2×C8—10841no—
271C2×C8C22×D27111yescharacteristic
272C2×C8C22×D27111yescharacteristic
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—2711no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4—5421no—
282C4⋊C4C22×D27111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—2711no—
285C22⋊C4—2711no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C42—2711no—
294C42—5421no—
295C42—5421no—
296C23—21no—
297C23—21no—
298C23—2711no—
299C23D4×D27111yescharacteristic
300Q8—21no—
301Q8—5421no—
302Q8—5421no—
303Q8—10842no—
304Q8#1686465312yes—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—42no—
310D4—81no—
311D4—5422no—
312D4—10842no—
313D4—10842no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—2711no—
320C2×C4—2711no—
321C2×C4—2711no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—10842no—
329C2×C4—21681no—
330C2×C4—21681no—
331C2×C4D4×D27111yescharacteristic
332C2×C4C23×D27111yescharacteristic
333C8—21no—
334C8—21no—
335C8—5421no—
336C8—5421no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—2712no—
347C22—5421no—
348C22—5421no—
349C22C2×D4×D27111yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—5421no—
354C4—5421no—
355C4—5421no—
356C4—10841no—
357C4—10841no—
358C4—10841no—
359C4—10841no—
360C4C2×#1686465311yescharacteristic
361C4#2257620511yescharacteristic
362C2—41no—
363C2—81no—
364C2—81no—
365C2—5422no—
366C2#3177453411yescharacteristic
367C2#135×D27111yescharacteristic
368C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3301
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. #31774594C2semidirect
2. #31774554C2semidirect
3. #31774250C2semidirect
4. #31774239C2semidirect
5. #31774214C2semidirect
6. #31774206C2semidirect
7. #31774130C2semidirect
8. #31774124C2semidirect
9. #31774110C2semidirect
10. #31774034C2semidirect
11. SD16×Dic271C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773893C2semidirect
14. #103×D271C2semidirect
15. #449×C271C2semidirect
184. C271#3301semidirect
195. #449D271semidirect

Maximal subgroups

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

subgroupindex
#317745942
#317745542
#317742502
#317742392
#317742142
#317742062
#317741302
#317741242
#317741102
#317740342
SD16×Dic2712
(C22⋊D271)⋊C82
#317738932
#103×D2712
#449×C2712
#3301271