Finite Group Databasebeta

← #46848965

Subgroups — #46848965

Order 34688.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774599C211yesmaximal, characteristic, complemented
2#31774594C211yesmaximal, characteristic, complemented
3#31774297C211yesmaximal, characteristic, complemented
4#31774220C211yesmaximal, characteristic, complemented
5#31774218C211yesmaximal, characteristic, complemented
6#31774209C211yesmaximal, characteristic, complemented
7#31774208C211yesmaximal, characteristic, complemented
8#31774199C211yesmaximal, characteristic, complemented
9#31774195C211yesmaximal, characteristic, complemented
10#31774147C211yesmaximal, characteristic, complemented
11D8×Dic271C211yesmaximal, characteristic, complemented
12#31773973C211yesmaximal, characteristic, complemented
13#31773939C211yesmaximal, characteristic, complemented
14#31773936C211yesmaximal, characteristic, complemented
15#469×C271C211yesmaximal, characteristic, complemented
16C2×#16864637C2211yescharacteristic
17#22576153—21no—
18#22576153—21no—
19#22576151C2211yescharacteristic
20#22576151C2211yescharacteristic
21#22576144—21no—
22#22576141—21no—
23#22576132—21no—
24#22576131—21no—
25#22576131C2211yescharacteristic
26#22576131C2211yescharacteristic
27D4×Dic271—21no—
28D4×Dic271C2211yescharacteristic
29D4×Dic271C2211yescharacteristic
30C4×C22⋊D271C2211yescharacteristic
31(C22⋊D271)⋊C4—21no—
32#22576119C2211yescharacteristic
33#22576118C2211yescharacteristic
34C2×C4⋊Dic271C2211yescharacteristic
35C4⋊C4×D271C2211yescharacteristic
36#22576094C2211yescharacteristic
37#22576094C2211yescharacteristic
38#22576093C2211yescharacteristic
39#22576093C2211yescharacteristic
40#22576093C2211yescharacteristic
41#22576092C2211yescharacteristic
42#22576092C2211yescharacteristic
43#22576091C2211yescharacteristic
44Dic271⋊C8C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×D8×C271C2211yescharacteristic
53C2×M16×C271C2211yescharacteristic
54#121×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#106×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59C2×C22⋊D271—21no—
60C2×C22⋊D271—21no—
61C2×C22⋊D271D411yescharacteristic
62C22×Dic271—21no—
63C22×Dic271—21no—
64C22×Dic271—21no—
65C2×C4×D271—21no—
66C2×C4×D271C2311yescharacteristic
67#16864637—22no—
68C2×C271⋊C8C2311yescharacteristic
69C2×C271⋊C8C2311yescharacteristic
70C4⋊Dic271—21no—
71C4⋊Dic271—21no—
72C4⋊Dic271—21no—
73C4⋊Dic271C2311yescharacteristic
74C4⋊Dic271C2311yescharacteristic
75C4⋊Dic271C2311yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77Dic271⋊C4—21no—
78Dic271⋊C4—21no—
79Dic271⋊C4—21no—
80Dic271⋊C4D411yescharacteristic
81C22⋊Dic271—21no—
82C22⋊Dic271—21no—
83C22⋊Dic271—21no—
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271D411yescharacteristic
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4D411yescharacteristic
89C4×Dic271—21no—
90C4×Dic271—21no—
91C4×Dic271C2311yescharacteristic
92C2×D4×C271—21no—
93C2×D4×C271—21no—
94C2×D4×C271C2311yescharacteristic
95C2×D4×C271C2311yescharacteristic
96C22×C4×C271C2311yescharacteristic
97D8×C271—22no—
98M16×C271—22no—
99C2×C8×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C4⋊C4×C271C2311yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C22⋊C4×C271—21no—
104C22⋊C4×C271—21no—
105C22×D271C2×D411yescharacteristic
106C22⋊D271—41no—
107C22⋊D271—42no—
108C22⋊D271—81no—
109C2×Dic271—21no—
110C2×Dic271—21no—
111C2×Dic271—21no—
112C2×Dic271—21no—
113C2×Dic271—81no—
114C2×Dic271—81no—
115C2×Dic271—81no—
116C2×Dic271C2×D411yescharacteristic
117C2×Dic271C2×D411yescharacteristic
118C2×Dic271C2×D411yescharacteristic
119C4×D271—21no—
120C4×D271—21no—
121C4×D271—81no—
122C271⋊C8—21no—
123C271⋊C8—21no—
124C23×C271—21no—
125C23×C271—21no—
126C23×C271C2×D411yescharacteristic
127D4×C271—21no—
128D4×C271—21no—
129D4×C271—21no—
130D4×C271—21no—
131D4×C271—42no—
132D4×C271—81no—
133C2×C4×C271—21no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271C2×D411yescharacteristic
138C2×C4×C271C2411yescharacteristic
139C8×C271—21no—
140C8×C271#4112yes—
141C22×C271—21no—
142C22×C271—21no—
143C22×C271—21no—
144C22×C271—41no—
145C22×C271—41no—
146C22×C271—41no—
147C22×C271—41no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271C22×D411yescharacteristic
151C2×D271—21no—
152C2×D271—21no—
153C2×D271—21no—
154C4×C271—21no—
155C4×C271—41no—
156C4×C271—41no—
157C4×C271C2×#4111yescharacteristic
158C4×C2712−1+411yescharacteristic
159Dic271—21no—
160Dic271—21no—
161Dic271—21no—
162Dic271—41no—
163Dic271—41no—
164Dic271—41no—
165Dic271—41no—
166C2×C271—41no—
167C2×C271—81no—
168C2×C271—81no—
169C2×C271#54711yescharacteristic
170C2×C271#57511yescharacteristic
171C2×C271#57611yescharacteristic
172D271—41no—
173C271#335511yescharacteristic, complemented, Sylow
174#3355—2711nomaximal, Sylow
175#553—2711no—
176#553—2711no—
177#499—2711no—
178#482—2711no—
179#482—2711no—
180#480—2711no—
181#480—2711no—
182#478—2711no—
183#478—2711no—
184#469—2711no—
185#469D27111yescharacteristic, complemented
186C4×D8—2711no—
187#436—2711no—
188#428—2711no—
189#428—2711no—
190C2×D8C2×D27111yescharacteristic
191C2×M16—2711no—
192C2×M16C2×D27111yescharacteristic
193#126—5421no—
194#126—5421no—
195#125—2711no—
196#125—2711no—
197#123—5421no—
198#123—5421no—
199#121—2711no—
200#121—2711no—
201#121—5421no—
202#121—5421no—
203#121C2×D27111yescharacteristic
204#121C2×D27111yescharacteristic
205C4×D4—2711no—
206C4×D4—2711no—
207C4×D4—2711no—
208C4×D4—5421no—
209C4×D4—5421no—
210#117—2711no—
211#117—2711no—
212C2×C4⋊C4—2711no—
213C2×C4⋊C4—2711no—
214#107—2711no—
215#107—2711no—
216#107—2711no—
217#107—2711no—
218#107—2711no—
219#106—2711no—
220#106—2711no—
221#106—2711no—
222#106C2×D27111yescharacteristic
223C4⋊C8—2711no—
224#102—2711no—
225#102—2711no—
226#102—2711no—
227#102—2711no—
228#102C2×D27111yescharacteristic
229#102C2×D27111yescharacteristic
230C22⋊C8—2711no—
231C22⋊C8—2711no—
232C4×C8—2711no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—2711no—
236C2×D4—5421no—
237C2×D4—5421no—
238C2×D4C22×D27111yescharacteristic
239C2×D4C22×D27111yescharacteristic
240C22×C4—2711no—
241C22×C4—5421no—
242C22×C4—5421no—
243C22×C4—5421no—
244C22×C4—5421no—
245C22×C4C22×D27111yescharacteristic
246D8—22no—
247M16—22no—
248M16—5422no—
249C2×C8—2711no—
250C2×C8—2711no—
251C2×C8C22×D27111yescharacteristic
252C2×C8C22×D27111yescharacteristic
253C4⋊C4—2711no—
254C4⋊C4—2711no—
255C4⋊C4—2711no—
256C4⋊C4—2711no—
257C4⋊C4—2711no—
258C4⋊C4—5421no—
259C4⋊C4—5421no—
260C4⋊C4—5421no—
261C4⋊C4—5421no—
262C4⋊C4—5421no—
263C4⋊C4—5421no—
264C4⋊C4C22×D27111yescharacteristic
265C4⋊C4C22×D27111yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—2711no—
269C22⋊C4—2711no—
270C22⋊C4—5421no—
271C22⋊C4—5421no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C42—2711no—
277C42—5421no—
278C42—5421no—
279C23—21no—
280C23—21no—
281C23—2711no—
282C23D4×D27111yescharacteristic
283D4—21no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—42no—
288D4—81no—
289D4—10841no—
290D4—10842no—
291D4—21681no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—2711no—
297C2×C4—2711no—
298C2×C4—2711no—
299C2×C4—5421no—
300C2×C4—5421no—
301C2×C4—5421no—
302C2×C4—5421no—
303C2×C4—5421no—
304C2×C4—5421no—
305C2×C4—21681no—
306C2×C4—21681no—
307C2×C4—21681no—
308C2×C4—21681no—
309C2×C4D4×D27111yescharacteristic
310C2×C4C23×D27111yescharacteristic
311C8—21no—
312C8—5421no—
313C8—5421no—
314C8#1686465312yes—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—5421no—
325C22—5421no—
326C22—5421no—
327C22C2×D4×D27111yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—5421no—
332C4—5421no—
333C4—5421no—
334C4—10841no—
335C4—10841no—
336C4—10841no—
337C4—10841no—
338C4C2×#1686465311yescharacteristic
339C4#2257620711yescharacteristic
340C2—41no—
341C2—81no—
342C2—81no—
343C2—10841no—
344C2#3177454711yescharacteristic
345C2#3177469311yescharacteristic
346C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#469×C271C2
socleC22×C271C22×D4
hypercenter#469D271
hypocenterC271#3355
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #31774599C2semidirect
2. #31774594C2semidirect
3. #31774297C2semidirect
4. #31774220C2semidirect
5. #31774218C2semidirect
6. #31774209C2semidirect
7. #31774208C2semidirect
8. #31774199C2semidirect
9. #31774195C2semidirect
10. #31774147C2semidirect
11. D8×Dic271C2semidirect
12. #31773973C2semidirect
13. #31773939C2semidirect
14. #31773936C2semidirect
15. #469×C271C2semidirect
173. C271#3355semidirect
185. #469D271semidirect

Maximal subgroups

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

subgroupindex
#317745992
#317745942
#317742972
#317742202
#317742182
#317742092
#317742082
#317741992
#317741952
#317741472
D8×Dic2712
#317739732
#317739392
#317739362
#469×C2712
#3355271