Finite Group Databasebeta

← #46846468

Subgroups — #46846468

Order 34688.

392 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#31774428C211yesmaximal, characteristic, complemented
2#31774365C211yesmaximal, characteristic, complemented
3#31774238C211yesmaximal, characteristic, complemented
4#31774203C211yesmaximal, characteristic, complemented
5#31774196C211yesmaximal, characteristic, complemented
6#31774118C211yesmaximal, characteristic, complemented
7#31774111C211yesmaximal, characteristic, complemented
8#31774061C211yesmaximal, characteristic, complemented
9#31774040C211yesmaximal, characteristic
10#31774006C211yesmaximal, characteristic, complemented
11#31774003C211yesmaximal, characteristic, complemented
12#31773996C211yesmaximal, characteristic, complemented
13#31773994C211yesmaximal, characteristic, complemented
14#31773951C211yesmaximal, characteristic, complemented
15#482×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×Dic1084C2211yescharacteristic
18C2×#16864643C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21#22576153—21no—
22#22576151C2211yescharacteristic
23#22576147C2211yescharacteristic
24#22576141—21no—
25#22576139C2211yescharacteristic
26#22576125C2211yescharacteristic
27#22576125C2211yescharacteristic
28#22576125C2211yescharacteristic
29D4×Dic271—21no—
30D4×Dic271C2211yescharacteristic
31#22576122C2211yescharacteristic
32#22576122C2211yescharacteristic
33(C22⋊D271)⋊C4—21no—
34(C22⋊D271)⋊C4—21no—
35#22576117—21no—
36#22576116—21no—
37C2×C4×Dic271C2211yescharacteristic
38C42×D271—21no—
39#22576093C2211yescharacteristic
40#22576091C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576083C2211yescharacteristic
45#22576083C2211yescharacteristic
46#22576083C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50#22576070C2211yescharacteristic
51#22576070C2211yescharacteristic
52#121×C271C2211yescharacteristic
53#117×C271C2211yescharacteristic
54#107×C271C2211yescharacteristic
55#106×C271C2211yescharacteristic
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59#16864654—22no—
60#16864654—22no—
61C2×Dic542C2311yescharacteristic
62C2×Dic542C2311yescharacteristic
63C2×D1084C2311yescharacteristic
64C2×C22⋊D271—21no—
65C22×Dic271—21no—
66C22×Dic271D411yescharacteristic
67C22×Dic271D411yescharacteristic
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71Dic1084—22no—
72#16864643—22no—
73#16864642—22no—
74#16864639—22no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C4⋊Dic271—21no—
78C4⋊Dic271C2311yescharacteristic
79Dic271⋊C4—21no—
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90C4×Dic271—21no—
91C4×Dic271—21no—
92C4×Dic271—21no—
93C4×Dic271D411yescharacteristic
94C4×Dic271D411yescharacteristic
95C4×Dic271C2311yescharacteristic
96C4×Dic271C2311yescharacteristic
97C2×D4×C271—21no—
98C2×D4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C2×C8×C271C2311yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C42×C271—21no—
108C22×D271—21no—
109Dic542—21no—
110Dic542—21no—
111Dic542—21no—
112Dic542#4112yes—
113D1084—21no—
114D1084#4112yes—
115C22⋊D271—24no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—41no—
120C2×Dic271—41no—
121C2×Dic271—42no—
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C4×D271—24no—
127C4×D271—42no—
128C4×D271—42no—
129C271⋊C8—21no—
130C271⋊C8—21no—
131C23×C271—21no—
132C23×C271C2×D411yescharacteristic
133D4×C271—21no—
134D4×C271—21no—
135D4×C271—42no—
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—21no—
147C22×C271—21no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271—41no—
151C22×C271C22×D411yescharacteristic
152C2×D271—22no—
153C2×D271—41no—
154C2×D271—41no—
155C4×C271—21no—
156C4×C271—41no—
157C4×C271—41no—
158C4×C271—41no—
159C4×C271C2×#4111yescharacteristic
160C4×C271C2×#4111yescharacteristic
161Dic271—22no—
162Dic271—21no—
163Dic271—21no—
164Dic271—21no—
165Dic271—21no—
166Dic271—41no—
167Dic271—41no—
168C2×C271—41no—
169C2×C271—81no—
170C2×C271#52511yescharacteristic
171C2×C271#57511yescharacteristic
172C2×C271#57511yescharacteristic
173D271—42no—
174C271#311111yescharacteristic, complemented, Sylow
175#3111—2711nomaximal, Sylow
176#529—2711no—
177C4×#41—2711no—
178#485—2711no—
179#482D27111yescharacteristic, complemented
180#479—2711no—
181#478—2711no—
182#464—2711no—
183#463—2711no—
184#452—2711no—
185#449—2711no—
186#442—2711no—
187#441—2711no—
188#440—2711no—
189#440—2711no—
190#432—2711no—
191C2×#41—2711no—
192C2×Q16—2711no—
193C2×SD16—2711no—
194C2×SD16—2711no—
195C2×D8—2711no—
196#126—5421no—
197#125—2711no—
198#124—2711no—
199#123—5421no—
200C22⋊Q8—2711no—
201#121C2×D27111yescharacteristic
202C4×Q8—2711no—
203C4×Q8—2711no—
204C4×Q8—2711no—
205C4×D4—2711no—
206C4×D4—2711no—
207C4×D4—2711no—
208C4×D4—5421no—
209C4×D4—5421no—
210C4×D4—5421no—
211#117—5421no—
212#117—5421no—
213#117C2×D27111yescharacteristic
214C2×C42—2711no—
215C2×C42—5421no—
216#107—2711no—
217#107C2×D27111yescharacteristic
218#106—2711no—
219#106C2×D27111yescharacteristic
220C4⋊C8—2711no—
221C4⋊C8—2711no—
222#103—2711no—
223#103—2711no—
224#103—2711no—
225#103—2711no—
226#102—2711no—
227#102—2711no—
228#102C2×D27111yescharacteristic
229#102C2×D27111yescharacteristic
230C22⋊C8—2711no—
231C22⋊C8C2×D27111yescharacteristic
232#97—2711no—
233#97—2711no—
234#41—5422no—
235#41—5422no—
236C2×Q8—2711no—
237C2×Q8—2711no—
238C2×D4—21no—
239C2×D4—2711no—
240C2×D4—5421no—
241C2×D4C22×D27111yescharacteristic
242C22×C4—2711no—
243C22×C4—2711no—
244C22×C4—5421no—
245C22×C4—5421no—
246C22×C4—5421no—
247C22×C4—5421no—
248C22×C4C22×D27111yescharacteristic
249Q16—5422no—
250SD16—5422no—
251SD16—5422no—
252D8—5422no—
253C2×C8—2711no—
254C2×C8—2711no—
255C2×C8C22×D27111yescharacteristic
256C2×C8C22×D27111yescharacteristic
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
266C4⋊C4C22×D27111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—5421no—
270C22⋊C4—5421no—
271C22⋊C4—5421no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C42—21no—
276C42—2711no—
277C42—2711no—
278C42—2711no—
279C42—2711no—
280C42—5421no—
281C42—5421no—
282C42—5421no—
283C23—21no—
284C23—5421no—
285C23D4×D27111yescharacteristic
286Q8—2712no—
287Q8—5421no—
288Q8—5421no—
289Q8—5421no—
290D4—21no—
291D4—21no—
292D4—42no—
293D4—2712no—
294D4—5424no—
295D4—5421no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—2711no—
302C2×C4—2711no—
303C2×C4—2711no—
304C2×C4—2711no—
305C2×C4—5424no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4—10842no—
310C2×C4—10842no—
311C2×C4—10841no—
312C2×C4—10841no—
313C2×C4—10842no—
314C2×C4D4×D27111yescharacteristic
315C2×C4C23×D27111yescharacteristic
316C8—21no—
317C8—21no—
318C8—5421no—
319C8—5421no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—5422no—
327C22—10841no—
328C22—10841no—
329C22C2×D4×D27111yescharacteristic
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—5422no—
335C4—5421no—
336C4—5421no—
337C4—5421no—
338C4—5421no—
339C4—10841no—
340C4—10841no—
341C4#41×D27111yescharacteristic
342C4#41×D27111yescharacteristic
343C2—41no—
344C2—81no—
345C2—10842no—
346C2#3177440511yescharacteristic
347C2#3177469311yescharacteristic
348C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#482×C271C2
socleC22×C271C22×D4
prime radical#31774040C2
hypercenter#482D271
hypocenterC271#3111
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #31774428C2semidirect
2. #31774365C2semidirect
3. #31774238C2semidirect
4. #31774203C2semidirect
5. #31774196C2semidirect
6. #31774118C2semidirect
7. #31774111C2semidirect
8. #31774061C2semidirect
10. #31774006C2semidirect
11. #31774003C2semidirect
12. #31773996C2semidirect
13. #31773994C2semidirect
14. #31773951C2semidirect
15. #482×C271C2semidirect
174. C271#3111semidirect
179. #482D271semidirect

Maximal subgroups

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

subgroupindex
#317744282
#317743652
#317742382
#317742032
#317741962
#317741182
#317741112
#317740612
#317740402
#317740062
#317740032
#317739962
#317739942
#317739512
#482×C2712
#3111271