Finite Group Databasebeta

← #46849117

Subgroups — #46849117

Order 34688.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774632C211yesmaximal, characteristic, complemented
2#31774263C211yesmaximal, characteristic, complemented
3#31774259C211yesmaximal, characteristic, complemented
4#31774254C211yesmaximal, characteristic, complemented
5#31774201C211yesmaximal, characteristic, complemented
6#31774194C211yesmaximal, characteristic, complemented
7#31774174C211yesmaximal, characteristic, complemented
8#31774112C211yesmaximal, characteristic, complemented
9#31774112C211yesmaximal, characteristic, complemented
10#31774092C211yesmaximal, characteristic
11#31774028C211yesmaximal, characteristic, complemented
12#31774007C211yesmaximal, characteristic, complemented
13#31773997C211yesmaximal, characteristic, complemented
14C4×#16864639C211yesmaximal, characteristic, complemented
15#554×C271C211yesmaximal, characteristic, complemented
16C2×#16864642C2211yescharacteristic
17C2×#16864639C2211yescharacteristic
18C2×#16864639C2211yescharacteristic
19#22576157C2211yescharacteristic
20#22576149—21no—
21#22576148C2211yescharacteristic
22#22576148C2211yescharacteristic
23#22576145—21no—
24#22576145—21no—
25C4×Dic542C2211yescharacteristic
26C4×D1084C2211yescharacteristic
27C4×D1084C2211yescharacteristic
28#22576122—21no—
29#22576093C2211yescharacteristic
30#22576093C2211yescharacteristic
31#22576091C2211yescharacteristic
32C4⋊(C271⋊C8)C2211yescharacteristic
33C4⋊(C271⋊C8)C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35#22576083C2211yescharacteristic
36#22576083C2211yescharacteristic
37#22576083C2211yescharacteristic
38#22576081C2211yescharacteristic
39#22576080C2211yescharacteristic
40#22576080C2211yescharacteristic
41#22576080C2211yescharacteristic
42#22576080C2211yescharacteristic
43#22576080C2211yescharacteristic
44#22576080C2211yescharacteristic
45#22576071C2211yescharacteristic
46#22576071C2211yescharacteristic
47C4×C271⋊C8C2211yescharacteristic
48C4⋊Q8×C271C2211yescharacteristic
49#125×C271C2211yescharacteristic
50#125×C271C2211yescharacteristic
51C22⋊Q8×C271—21no—
52C22⋊Q8×C271—21no—
53C22⋊Q8×C271—21no—
54C4×Q8×C271C2211yescharacteristic
55C4×D4×C271C2211yescharacteristic
56C4×D4×C271C2211yescharacteristic
57C4×D4×C271C2211yescharacteristic
58C2×C4⋊C4×C271—21no—
59C2×Dic542C2311yescharacteristic
60C2×D1084—21no—
61C2×D1084—21no—
62C2×D1084C2311yescharacteristic
63C2×D1084C2311yescharacteristic
64C2×C4×D271—21no—
65C2×C4×D271—21no—
66#16864642—22no—
67#16864639—22no—
68#16864639—22no—
69C2×C271⋊C8C2311yescharacteristic
70C2×C271⋊C8C2311yescharacteristic
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C4⋊Dic271C2311yescharacteristic
74Dic271⋊C4—21no—
75(C2×D271)⋊C4—21no—
76(C2×D271)⋊C4—21no—
77(C2×D271)⋊C4—21no—
78(C2×D271)⋊C4—21no—
79(C2×D271)⋊C4—21no—
80(C2×D271)⋊C4—21no—
81C4×Dic271—21no—
82C2×Q8×C271—21no—
83C2×Q8×C271D411yescharacteristic
84C2×D4×C271C2311yescharacteristic
85C22×C4×C271—21no—
86C22×C4×C271—21no—
87C22×C4×C271—21no—
88C4⋊C4×C271—21no—
89C4⋊C4×C271—21no—
90C4⋊C4×C271—21no—
91C4⋊C4×C271—21no—
92C4⋊C4×C271—21no—
93C4⋊C4×C271D411yescharacteristic
94C4⋊C4×C271D411yescharacteristic
95C4⋊C4×C271D411yescharacteristic
96C4⋊C4×C271C2311yescharacteristic
97C4⋊C4×C271C2311yescharacteristic
98C4⋊C4×C271C2311yescharacteristic
99C22⋊C4×C271—21no—
100C22⋊C4×C271—21no—
101C22⋊C4×C271—21no—
102C42×C271C2311yescharacteristic
103C42×C271C2311yescharacteristic
104C42×C271C2311yescharacteristic
105C22×D271—21no—
106C22×D271—21no—
107Dic542—21no—
108Dic542—21no—
109D1084—21no—
110D1084—21no—
111D1084—21no—
112D1084—21no—
113D1084—42no—
114D1084—81no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C4×D271—81no—
118C4×D271—81no—
119C271⋊C8—21no—
120C271⋊C8—21no—
121C271⋊C8—21no—
122C271⋊C8—21no—
123C23×C271—21no—
124Q8×C271—41no—
125Q8×C271—81no—
126D4×C271—21no—
127D4×C271#4112yes—
128C2×C4×C271—21no—
129C2×C4×C271—21no—
130C2×C4×C271—21no—
131C2×C4×C271—42no—
132C2×C4×C271—81no—
133C2×C4×C271—81no—
134C2×C4×C271C2×D411yescharacteristic
135C2×C4×C271C2×D411yescharacteristic
136C2×C4×C271C2×D411yescharacteristic
137C2×C4×C271C2×D411yescharacteristic
138C2×C4×C271C2×D411yescharacteristic
139C2×C4×C271C2×D411yescharacteristic
140C2×C4×C271C2411yescharacteristic
141C22×C271—22no—
142C22×C271—41no—
143C22×C271—41no—
144C22×C271C22×D411yescharacteristic
145C2×D271—41no—
146C2×D271—41no—
147C2×D271—41no—
148C2×D271—41no—
149C2×D271—41no—
150C2×D271—41no—
151C4×C271—21no—
152C4×C271—21no—
153C4×C271—21no—
154C4×C271—21no—
155C4×C271—21no—
156C4×C271—21no—
157C4×C271—41no—
158C4×C271—41no—
159C4×C271—41no—
160C4×C271C2×#4111yescharacteristic
161C4×C2712−1+411yescharacteristic
162Dic271—41no—
163Dic271—41no—
164C2×C271—42no—
165C2×C271#54811yescharacteristic
166C2×C271#57511yescharacteristic
167C2×C271#57611yescharacteristic
168D271—81no—
169D271—81no—
170C271#337011yescharacteristic, complemented, Sylow
171#3370—2711nomaximal, Sylow
172#559—2711no—
173#554D27111yescharacteristic, complemented
174#490—2711no—
175#489—2711no—
176#488—2711no—
177#479—2711no—
178#478—2711no—
179#474—2711no—
180#463—2711no—
181#463—2711no—
182#459—2711no—
183#446—2711no—
184#442—2711no—
185#440—2711no—
186C4×D8—2711no—
187C2×SD16—2711no—
188C2×D8—2711no—
189C2×D8—2711no—
190C4⋊Q8C2×D27111yescharacteristic
191#127—2711no—
192#125C2×D27111yescharacteristic
193#125C2×D27111yescharacteristic
194#124—2711no—
195#124—2711no—
196#124—5421no—
197#123—5421no—
198#123—5421no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C22⋊Q8—21no—
202C4×Q8—2711no—
203C4×Q8C2×D27111yescharacteristic
204C4×D4—2711no—
205C4×D4—2711no—
206C4×D4—5421no—
207C4×D4C2×D27111yescharacteristic
208C4×D4C2×D27111yescharacteristic
209C4×D4C2×D27111yescharacteristic
210C2×C4⋊C4—21no—
211#107—2711no—
212#107—2711no—
213#106—2711no—
214C4⋊C8—2711no—
215C4⋊C8—2711no—
216C4⋊C8—2711no—
217#103—2711no—
218#103—2711no—
219#103—2711no—
220#102—2711no—
221#102—2711no—
222#102—2711no—
223#102—2711no—
224#102—2711no—
225#102—2711no—
226#102—2711no—
227#97—2711no—
228#97—2711no—
229C4×C8—2711no—
230C2×Q8—21no—
231C2×Q8—2711no—
232C2×Q8C22⋊D27111yescharacteristic
233C2×D4—2711no—
234C2×D4—2711no—
235C2×D4—5421no—
236C2×D4—5421no—
237C2×D4C22×D27111yescharacteristic
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—5421no—
242C22×C4—5421no—
243SD16—5422no—
244D8—5422no—
245D8—5422no—
246C2×C8—2711no—
247C2×C8—2711no—
248C2×C8—2711no—
249C2×C8—2711no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—2711no—
256C4⋊C4—5421no—
257C4⋊C4C22⋊D27111yescharacteristic
258C4⋊C4C22⋊D27111yescharacteristic
259C4⋊C4C22⋊D27111yescharacteristic
260C4⋊C4C22×D27111yescharacteristic
261C4⋊C4C22×D27111yescharacteristic
262C4⋊C4C22×D27111yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—5421no—
267C22⋊C4—5421no—
268C22⋊C4—5421no—
269C22⋊C4—5421no—
270C22⋊C4—5421no—
271C22⋊C4—5421no—
272C42—5421no—
273C42C22×D27111yescharacteristic
274C42C22×D27111yescharacteristic
275C42C22×D27111yescharacteristic
276C23—21no—
277C23—5421no—
278C23—5421no—
279Q8—41no—
280Q8—81no—
281Q8—5421no—
282Q8—5421no—
283D4—21no—
284D4—5421no—
285D4—5421no—
286D4—5421no—
287D4—5421no—
288D4—10842no—
289D4—21681no—
290D4#1686465512yes—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—5421no—
298C2×C4—5421no—
299C2×C4—21681no—
300C2×C4—21681no—
301C2×C4C2×C22⋊D27111yescharacteristic
302C2×C4C2×C22⋊D27111yescharacteristic
303C2×C4C2×C22⋊D27111yescharacteristic
304C2×C4C2×C22⋊D27111yescharacteristic
305C2×C4C2×C22⋊D27111yescharacteristic
306C2×C4C2×C22⋊D27111yescharacteristic
307C2×C4C23×D27111yescharacteristic
308C8—5421no—
309C8—5421no—
310C8—5421no—
311C8—5421no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—10841no—
316C22—10841no—
317C22—10841no—
318C22—10841no—
319C22—10841no—
320C22—10841no—
321C22C22×C22⋊D27111yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—10841no—
332C4—10841no—
333C4C2×#1686465511yescharacteristic
334C4#2257620811yescharacteristic
335C2—42no—
336C2—21681no—
337C2—21681no—
338C2#3177455811yescharacteristic
339C2#3177469511yescharacteristic
340C2#3177470011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#554×C271C2
socleC22×C271C22×D4
prime radical#31774092C2
hypercenter#554D271
hypocenterC271#3370
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #31774632C2semidirect
2. #31774263C2semidirect
3. #31774259C2semidirect
4. #31774254C2semidirect
5. #31774201C2semidirect
6. #31774194C2semidirect
7. #31774174C2semidirect
8. #31774112C2semidirect
9. #31774112C2semidirect
11. #31774028C2semidirect
12. #31774007C2semidirect
13. #31773997C2semidirect
14. C4×#16864639C2semidirect
15. #554×C271C2semidirect
170. C271#3370semidirect
173. #554D271semidirect

Maximal subgroups

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

subgroupindex
#317746322
#317742632
#317742592
#317742542
#317742012
#317741942
#317741742
#317741122
#317740922
#317740282
#317740072
#317739972
C4×#168646392
#554×C2712
#3370271