Finite Group Databasebeta

← #46846627

Subgroups — #46846627

Order 34688.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774520C211yesmaximal, characteristic, complemented
2#31774380C211yesmaximal, characteristic, complemented
3#31774230C211yesmaximal, characteristic, complemented
4#31774211C211yesmaximal, characteristic, complemented
5#31774190C211yesmaximal, characteristic, complemented
6#31774175C211yesmaximal, characteristic, complemented
7#31774121C211yesmaximal, characteristic, complemented
8#31774114C211yesmaximal, characteristic, complemented
9#31774042C211yesmaximal, characteristic
10#31774000C211yesmaximal, characteristic, complemented
11#31773998C211yesmaximal, characteristic, complemented
12Q16×Dic271C211yesmaximal, characteristic, complemented
13#31773977C211yesmaximal, characteristic, complemented
14#31773955C211yesmaximal, characteristic, complemented
15#452×C271C211yesmaximal, characteristic, complemented
16C2×#16864641C2211yescharacteristic
17C2×#16864639C2211yescharacteristic
18#22576159C2211yescharacteristic
19#22576155—21no—
20#22576149C2211yescharacteristic
21#22576146—21no—
22#22576142—21no—
23#22576140—21no—
24#22576134C2211yescharacteristic
25Q8×Dic271C2211yescharacteristic
26Q8×Dic271C2211yescharacteristic
27Q8×Dic271C2211yescharacteristic
28D4×Dic271—21no—
29D4×Dic271—21no—
30D4×Dic271C2211yescharacteristic
31D4×Dic271C2211yescharacteristic
32#22576122C2211yescharacteristic
33#22576119—21no—
34#22576119C2211yescharacteristic
35C2×C4⋊Dic271—21no—
36C2×C4⋊Dic271C2211yescharacteristic
37#22576094C2211yescharacteristic
38#22576094C2211yescharacteristic
39#22576093C2211yescharacteristic
40#22576093C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50#22576070C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C2×Q16×C271C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59C2×Dic542—21no—
60C2×D1084C2311yescharacteristic
61C2×C22⋊D271—21no—
62C22×Dic271—21no—
63C22×Dic271—21no—
64C22×Dic271—21no—
65C2×C4×D271—21no—
66#16864641—22no—
67#16864639—22no—
68C2×C271⋊C8C2311yescharacteristic
69C2×C271⋊C8C2311yescharacteristic
70C4⋊Dic271—21no—
71C4⋊Dic271—21no—
72C4⋊Dic271—21no—
73C4⋊Dic271—21no—
74C4⋊Dic271D411yescharacteristic
75C4⋊Dic271D411yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79Dic271⋊C4—21no—
80C22⋊Dic271—21no—
81C22⋊Dic271—21no—
82C22⋊Dic271D411yescharacteristic
83C22⋊Dic271D411yescharacteristic
84(C2×D271)⋊C4—21no—
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87(C2×D271)⋊C4—21no—
88C4×Dic271—21no—
89C4×Dic271—21no—
90C4×Dic271—21no—
91C4×Dic271C2311yescharacteristic
92C4×Dic271C2311yescharacteristic
93#41×C271—22no—
94#41×C271—22no—
95C2×Q8×C271C2311yescharacteristic
96C2×Q8×C271C2311yescharacteristic
97C2×D4×C271—21no—
98C2×D4×C271C2311yescharacteristic
99C22×C4×C271—21no—
100C22×C4×C271C2311yescharacteristic
101Q16×C271—22no—
102SD16×C271—22no—
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C4⋊C4×C271—21no—
106C4⋊C4×C271C2311yescharacteristic
107C22⋊C4×C271—21no—
108C22×D271—21no—
109Dic542—81no—
110D1084—21no—
111D1084—21no—
112C22⋊D271—81no—
113C2×Dic271—21no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—81no—
117C2×Dic271—81no—
118C2×Dic271—81no—
119C2×Dic271C2×D411yescharacteristic
120C2×Dic271C2×D411yescharacteristic
121C2×Dic271C2×D411yescharacteristic
122C2×Dic271C2×D411yescharacteristic
123C4×D271—81no—
124C271⋊C8—21no—
125C271⋊C8—21no—
126C23×C271—21no—
127C23×C271C2×D411yescharacteristic
128Q8×C271—21no—
129Q8×C271—21no—
130Q8×C271—21no—
131Q8×C271#4112yes—
132D4×C271—24no—
133D4×C271—21no—
134D4×C271#4112yes—
135C2×C4×C271—24no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271C2×D411yescharacteristic
142C2×C4×C271C2411yescharacteristic
143C8×C271—21no—
144C8×C271—21no—
145C22×C271—21no—
146C22×C271—22no—
147C22×C271—21no—
148C22×C271—21no—
149C22×C271—41no—
150C22×C271—41no—
151C22×C271C22×D411yescharacteristic
152C2×D271—41no—
153C2×D271—41no—
154C2×D271—41no—
155C4×C271—21no—
156C4×C271—22no—
157C4×C271—41no—
158C4×C271—41no—
159C4×C271C2×#4111yescharacteristic
160C4×C271C2×#4111yescharacteristic
161Dic271—21no—
162Dic271—21no—
163Dic271—21no—
164Dic271—21no—
165Dic271—41no—
166Dic271—41no—
167Dic271—41no—
168C2×C271—41no—
169C2×C271—42no—
170C2×C271#52511yescharacteristic
171C2×C271#57611yescharacteristic
172C2×C271#57811yescharacteristic
173D271—81no—
174C271#312411yescharacteristic, complemented, Sylow
175#3124—2711nomaximal, Sylow
176#542—2711no—
177#520—2711no—
178#484—2711no—
179#480—2711no—
180#477—2711no—
181#474—2711no—
182#464—2711no—
183#463—2711no—
184#452D27111yescharacteristic, complemented
185#449—2711no—
186#440—2711no—
187#440—2711no—
188C4×Q16—2711no—
189C4×D8—2711no—
190#433—2711no—
191C2×#41C2×D27111yescharacteristic
192C2×Q16C2×D27111yescharacteristic
193C2×SD16—2711no—
194C2×SD16C2×D27111yescharacteristic
195C2×D8—2711no—
196C4⋊Q8—2711no—
197#126—5421no—
198#124—2711no—
199#124—5421no—
200#123—5421no—
201C22⋊Q8—5421no—
202C22⋊Q8C2×D27111yescharacteristic
203#121—2711no—
204C4×Q8—2711no—
205C4×Q8—2711no—
206C4×Q8—2711no—
207C4×D4—2711no—
208C4×D4—2711no—
209C4×D4—2711no—
210C4×D4—5421no—
211C4×D4—5421no—
212#117—2711no—
213#117—5421no—
214C2×C4⋊C4—2711no—
215C2×C4⋊C4—5421no—
216#107—2711no—
217#107—2711no—
218#107—2711no—
219#107—2711no—
220C4⋊C8—2711no—
221C4⋊C8—2711no—
222#103—2711no—
223#103—2711no—
224#103—2711no—
225#103C2×D27111yescharacteristic
226#102—2711no—
227#102—2711no—
228#102—2711no—
229#102C2×D27111yescharacteristic
230C22⋊C8—2711no—
231C22⋊C8C2×D27111yescharacteristic
232#97—2711no—
233C4×C8—2711no—
234#41—22no—
235#41—22no—
236C2×Q8—5421no—
237C2×Q8C22×D27111yescharacteristic
238C2×Q8C22×D27111yescharacteristic
239C2×D4—21no—
240C2×D4—2711no—
241C2×D4—5421no—
242C2×D4C22×D27111yescharacteristic
243C22×C4—21no—
244C22×C4—5421no—
245C22×C4—5421no—
246C22×C4—5421no—
247C22×C4—5421no—
248C22×C4C22×D27111yescharacteristic
249Q16—22no—
250SD16—22no—
251SD16—5422no—
252D8—5422no—
253C2×C8—2711no—
254C2×C8—2711no—
255C2×C8C22×D27111yescharacteristic
256C2×C8C22×D27111yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—2711no—
259C4⋊C4—2711no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—2711no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4—5421no—
267C4⋊C4—5421no—
268C4⋊C4C22×D27111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—2711no—
271C22⋊C4—2711no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C22⋊C4—5421no—
277C22⋊C4—5421no—
278C42—2711no—
279C42—2711no—
280C42—5421no—
281C42—5421no—
282C42—5421no—
283C23—21no—
284C23—5421no—
285C23D4×D27111yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—21681no—
290Q8#1686465312yes—
291D4—24no—
292D4—21no—
293D4—5421no—
294D4—5421no—
295D4—21681no—
296D4#1686465312yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—2711no—
304C2×C4—2711no—
305C2×C4—2711no—
306C2×C4—2711no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4—5421no—
310C2×C4—21681no—
311C2×C4—21681no—
312C2×C4—21681no—
313C2×C4—21681no—
314C2×C4D4×D27111yescharacteristic
315C2×C4C23×D27111yescharacteristic
316C8—21no—
317C8—21no—
318C8—5421no—
319C8—5421no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—10841no—
327C22—10841no—
328C22—10841no—
329C22C2×D4×D27111yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—5421no—
335C4—5421no—
336C4—5421no—
337C4—5421no—
338C4—10841no—
339C4—10841no—
340C4—10841no—
341C4C2×#1686465311yescharacteristic
342C4C2×#1686465311yescharacteristic
343C2—41no—
344C2—42no—
345C2—21681no—
346C2#3177440311yescharacteristic
347C2#3177469911yescharacteristic
348C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#452×C271C2
socleC22×C271C22×D4
prime radical#31774042C2
hypercenter#452D271
hypocenterC271#3124
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #31774520C2semidirect
2. #31774380C2semidirect
3. #31774230C2semidirect
4. #31774211C2semidirect
5. #31774190C2semidirect
6. #31774175C2semidirect
7. #31774121C2semidirect
8. #31774114C2semidirect
10. #31774000C2semidirect
11. #31773998C2semidirect
12. Q16×Dic271C2semidirect
13. #31773977C2semidirect
14. #31773955C2semidirect
15. #452×C271C2semidirect
174. C271#3124semidirect
184. #452D271semidirect

Maximal subgroups

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

subgroupindex
#317745202
#317743802
#317742302
#317742112
#317741902
#317741752
#317741212
#317741142
#317740422
#317740002
#317739982
Q16×Dic2712
#317739772
#317739552
#452×C2712
#3124271