Finite Group Databasebeta

← #46845436

Subgroups — #46845436

Order 34688.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774444C211yesmaximal, characteristic
2#31773973C211yesmaximal, characteristic, complemented
3#31773972C211yesmaximal, characteristic, complemented
4#31773970C211yesmaximal, characteristic, complemented
5#31773969C211yesmaximal, characteristic, complemented
6#31773967C211yesmaximal, characteristic, complemented
7C8×C22⋊D271C211yesmaximal, characteristic, complemented
8(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
9#31773952C211yesmaximal, characteristic, complemented
10#31773949C211yesmaximal, characteristic, complemented
11#31773865C211yesmaximal, characteristic, complemented
12#31773863C211yesmaximal, characteristic, complemented
13#31773862C211yesmaximal, characteristic, complemented
14#31773862C211yesmaximal, characteristic, complemented
15#435×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×#16864654C2211yescharacteristic
18C2×#16864638C2211yescharacteristic
19C2×#16864637C2211yescharacteristic
20C2×#16864637C2211yescharacteristic
21C22×C271⋊C8C2211yescharacteristic
22C2×C8×D271C2211yescharacteristic
23C2×C8×D271C2211yescharacteristic
24#22576159C411yescharacteristic
25#22576156C411yescharacteristic
26#22576146C411yescharacteristic
27#22576146C411yescharacteristic
28#22576135C411yescharacteristic
29#22576133C411yescharacteristic
30#22576133C411yescharacteristic
31#22576131C411yescharacteristic
32C4×D1084C2211yescharacteristic
33C4×C22⋊D271C2211yescharacteristic
34C4×C22⋊D271C2211yescharacteristic
35C42×D271C2211yescharacteristic
36C4⋊(C271⋊C8)C2211yescharacteristic
37Dic271⋊C8C2211yescharacteristic
38Dic271⋊C8C2211yescharacteristic
39C22⋊(C271⋊C8)C2211yescharacteristic
40C22⋊(C271⋊C8)C2211yescharacteristic
41(C2×D271)⋊C8C2211yescharacteristic
42(C2×D271)⋊C8C2211yescharacteristic
43(C2×D271)⋊C8C2211yescharacteristic
44(C2×D271)⋊C8C2211yescharacteristic
45(C2×D271)⋊C8C2211yescharacteristic
46(C2×D271)⋊C8C2211yescharacteristic
47(C2×D271)⋊C8C2211yescharacteristic
48(C2×D271)⋊C8C2211yescharacteristic
49#22576071C2211yescharacteristic
50C8×Dic271C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×M16×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54C4×D4×C271C2211yescharacteristic
55C4⋊C8×C271C2211yescharacteristic
56C22⋊C8×C271C2211yescharacteristic
57C22⋊C8×C271C2211yescharacteristic
58#97×C271C2211yescharacteristic
59#16864654—41no—
60#16864654—41no—
61#16864654—41no—
62#16864654—41no—
63C2×Dic542C2×C411yescharacteristic
64C2×Dic542C2×C411yescharacteristic
65C2×D1084C2×C411yescharacteristic
66C2×D1084C2×C411yescharacteristic
67C2×D1084C2×C411yescharacteristic
68C2×C22⋊D271C2×C411yescharacteristic
69C2×C22⋊D271C2×C411yescharacteristic
70C2×C22⋊D271C2×C411yescharacteristic
71C2×C22⋊D271C2×C411yescharacteristic
72C2×C22⋊D271C2×C411yescharacteristic
73C2×C22⋊D271C2×C411yescharacteristic
74C2×C4×D271C2×C411yescharacteristic
75C2×C4×D271C2×C411yescharacteristic
76C2×C4×D271C2311yescharacteristic
77C2×C4×D271C2311yescharacteristic
78C2×C4×D271C2311yescharacteristic
79#16864638—41no—
80#16864637—41no—
81#16864637—41no—
82C2×C271⋊C8—41no—
83C2×C271⋊C8C2311yescharacteristic
84C2×C271⋊C8C2311yescharacteristic
85C2×C271⋊C8C2311yescharacteristic
86C2×C271⋊C8C2311yescharacteristic
87C8×D271—41no—
88C8×D271—41no—
89C4⋊Dic271C2×C411yescharacteristic
90Dic271⋊C4C2×C411yescharacteristic
91Dic271⋊C4C2×C411yescharacteristic
92C22⋊Dic271C2×C411yescharacteristic
93C22⋊Dic271C2×C411yescharacteristic
94(C2×D271)⋊C4C2×C411yescharacteristic
95(C2×D271)⋊C4C2×C411yescharacteristic
96(C2×D271)⋊C4C2×C411yescharacteristic
97(C2×D271)⋊C4C2×C411yescharacteristic
98C4×Dic271C2×C411yescharacteristic
99C4×Dic271C2×C411yescharacteristic
100C4×Dic271C2311yescharacteristic
101C2×D4×C271C2×C411yescharacteristic
102C22×C4×C271C2311yescharacteristic
103C22×C4×C271C2311yescharacteristic
104M16×C271—41no—
105C2×C8×C271—41no—
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C2×C8×C271C2311yescharacteristic
109C2×C8×C271C2311yescharacteristic
110C4⋊C4×C271C2×C411yescharacteristic
111C22⋊C4×C271C2×C411yescharacteristic
112C22⋊C4×C271C2×C411yescharacteristic
113C42×C271C2311yescharacteristic
114C22×D271C22×C411yescharacteristic
115C22×D271C22×C411yescharacteristic
116C22×D271C22×C411yescharacteristic
117Dic542—41no—
118Dic542—41no—
119D1084—41no—
120D1084—41no—
121D1084—41no—
122C22⋊D271—22no—
123C22⋊D271—22no—
124C22⋊D271—41no—
125C22⋊D271—41no—
126C22⋊D271—41no—
127C22⋊D271—41no—
128C2×Dic271C22×C411yescharacteristic
129C2×Dic271C22×C411yescharacteristic
130C2×Dic271C22×C411yescharacteristic
131C2×Dic271C22×C411yescharacteristic
132C2×Dic271C22×C411yescharacteristic
133C4×D271—21no—
134C4×D271—21no—
135C4×D271—21no—
136C4×D271—21no—
137C4×D271—21no—
138C4×D271—21no—
139C4×D271—22no—
140C4×D271—41no—
141C271⋊C8—21no—
142C271⋊C8—21no—
143C271⋊C8—21no—
144C271⋊C8—21no—
145C23×C271C22×C411yescharacteristic
146C23×C271C22×C411yescharacteristic
147D4×C271—41no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271—21no—
152C2×C4×C271C22×C411yescharacteristic
153C2×C4×C271C22×C411yescharacteristic
154C2×C4×C271C22×C411yescharacteristic
155C2×C4×C271C22×C411yescharacteristic
156C2×C4×C271C2411yescharacteristic
157C8×C271—21no—
158C8×C271—21no—
159C8×C271—21no—
160C8×C271—21no—
161C22×C271—21no—
162C22×C271—21no—
163C22×C271—21no—
164C22×C271—21no—
165C22×C271—21no—
166C22×C271—21no—
167C22×C271C23×C411yescharacteristic
168C2×D271—21no—
169C2×D271—21no—
170C2×D271—21no—
171C2×D271—21no—
172C2×D271—21no—
173C2×D271—21no—
174C2×D271—21no—
175C2×D271—21no—
176C2×D271—21no—
177C4×C271—21no—
178C4×C271—21no—
179C4×C271—21no—
180C4×C271—21no—
181C4×C271H(5,𝔽2)11yescharacteristic
182C4×C271H(5,𝔽2)11yescharacteristic
183Dic271—21no—
184Dic271—21no—
185Dic271—21no—
186Dic271—21no—
187Dic271#13112yes—
188C2×C271—41no—
189C2×C271—41no—
190C2×C271#51811yescharacteristic
191C2×C271C2×#13111yescharacteristic
192C2×C271#56811yescharacteristic
193D271—41no—
194D271—41no—
195D271—41no—
196C271#298111yescharacteristic, complemented, Sylow
197#2981—2711nomaximal, Sylow
198#532—2711no—
199#436—2711no—
200#436—2711no—
201#435—2711no—
202#435—2711no—
203#435—2711no—
204#435D27111yescharacteristic, complemented
205C8×D4—2711no—
206C8×D4—2711no—
207#432—2711no—
208#431—2711no—
209#408—2711no—
210#408—2711no—
211#408—2711no—
212#408—2711no—
213C2×#41—2711no—
214C2×#41—2711no—
215C2×M16—2711no—
216C2×M16—2711no—
217C2×M16—2711no—
218C2×M16C2×D27111yescharacteristic
219C22×C8—2711no—
220C22×C8—2711no—
221C22×C8—2711no—
222C22×C8C2×D27111yescharacteristic
223C4⋊Q8—2711no—
224#127—2711no—
225#124—2711no—
226#124—2711no—
227#121—2711no—
228#121—2711no—
229#121—2711no—
230#121—2711no—
231C4×D4—2711no—
232C4×D4—2711no—
233C4×D4—2711no—
234C4×D4C2×D27111yescharacteristic
235C2×C42—2711no—
236C4⋊C8—2711no—
237C4⋊C8—2711no—
238C4⋊C8—2711no—
239C4⋊C8C2×D27111yescharacteristic
240C22⋊C8—2711no—
241C22⋊C8—2711no—
242C22⋊C8—2711no—
243C22⋊C8—2711no—
244C22⋊C8—2711no—
245C22⋊C8—2711no—
246C22⋊C8—2711no—
247C22⋊C8—2711no—
248C22⋊C8—2711no—
249C22⋊C8—2711no—
250C22⋊C8C2×D27111yescharacteristic
251C22⋊C8C2×D27111yescharacteristic
252#97—2711no—
253#97C2×D27111yescharacteristic
254C4×C8—2711no—
255C4×C8—2711no—
256#41—10841no—
257#41—10841no—
258#41—10841no—
259#41—10841no—
260C2×Q8—2711no—
261C2×Q8—2711no—
262C2×D4—2711no—
263C2×D4—2711no—
264C2×D4—2711no—
265C2×D4—2711no—
266C2×D4—2711no—
267C2×D4—2711no—
268C2×D4—2711no—
269C2×D4—2711no—
270C2×D4—2711no—
271C2×D4C4×D27111yescharacteristic
272C22×C4—2711no—
273C22×C4—2711no—
274C22×C4—2711no—
275C22×C4—2711no—
276C22×C4—2711no—
277C22×C4C22×D27111yescharacteristic
278C22×C4C22×D27111yescharacteristic
279M16—41no—
280M16—10841no—
281M16—10841no—
282M16—10841no—
283C2×C8—41no—
284C2×C8—2711no—
285C2×C8—2711no—
286C2×C8—2711no—
287C2×C8—2711no—
288C2×C8—10841no—
289C2×C8—10841no—
290C2×C8—10841no—
291C2×C8C22×D27111yescharacteristic
292C2×C8C22×D27111yescharacteristic
293C2×C8C22×D27111yescharacteristic
294C2×C8C22×D27111yescharacteristic
295C4⋊C4—2711no—
296C4⋊C4—2711no—
297C4⋊C4—2711no—
298C4⋊C4C4×D27111yescharacteristic
299C22⋊C4—2711no—
300C22⋊C4—2711no—
301C22⋊C4—2711no—
302C22⋊C4—2711no—
303C22⋊C4—2711no—
304C22⋊C4—2711no—
305C22⋊C4C4×D27111yescharacteristic
306C22⋊C4C4×D27111yescharacteristic
307C42—2711no—
308C42—2711no—
309C42—2711no—
310C42C22×D27111yescharacteristic
311C23—2711no—
312C23—2711no—
313C23—2711no—
314C23C2×C4×D27111yescharacteristic
315C23C2×C4×D27111yescharacteristic
316Q8—10841no—
317Q8—10841no—
318D4—41no—
319D4—5422no—
320D4—5422no—
321D4—10841no—
322D4—10841no—
323D4—10841no—
324D4—10841no—
325D4—10841no—
326D4—10841no—
327D4—10841no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—2711no—
333C2×C4—2711no—
334C2×C4—2711no—
335C2×C4—2711no—
336C2×C4—2711no—
337C2×C4—5422no—
338C2×C4—5421no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—5421no—
344C2×C4—10841no—
345C2×C4C2×C4×D27111yescharacteristic
346C2×C4C2×C4×D27111yescharacteristic
347C2×C4C2×C4×D27111yescharacteristic
348C2×C4C2×C4×D27111yescharacteristic
349C2×C4C23×D27111yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—5421no—
355C8—5421no—
356C8—5421no—
357C8—5421no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—5421no—
365C22—5421no—
366C22—5421no—
367C22—5421no—
368C22—5421no—
369C22—5421no—
370C22—5421no—
371C22—5421no—
372C22—5421no—
373C22C22×C4×D27111yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—2712no—
379C4—5421no—
380C4—5421no—
381C4—5421no—
382C4—5421no—
383C4#2257620511yescharacteristic
384C4#2257620611yescharacteristic
385C2—41no—
386C2—41no—
387C2—10841no—
388C2—10841no—
389C2—10841no—
390C2#3177436911yescharacteristic
391C2#131×D27111yescharacteristic
392C2#3177466311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D271
commutator, socleC22×C271C23×C4
Fitting#435×C271C2
prime radical#31774444C2
cosocleC2×C4×C271C24
hypercenter#435D271
hypocenterC271#2981
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #31773973C2semidirect
3. #31773972C2semidirect
4. #31773970C2semidirect
5. #31773969C2semidirect
6. #31773967C2semidirect
7. C8×C22⋊D271C2semidirect
8. (C22⋊D271)⋊C8C2semidirect
9. #31773952C2semidirect
10. #31773949C2semidirect
11. #31773865C2semidirect
12. #31773863C2semidirect
13. #31773862C2semidirect
14. #31773862C2semidirect
15. #435×C271C2semidirect
196. C271#2981semidirect
204. #435D271semidirect

Maximal subgroups

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

subgroupindex
#317744442
#317739732
#317739722
#317739702
#317739692
#317739672
C8×C22⋊D2712
(C22⋊D271)⋊C82
#317739522
#317739492
#317738652
#317738632
#317738622
#435×C2712
#2981271