Finite Group Databasebeta

← #46839664

Subgroups — #46839664

Order 34688.

432 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
1C271⋊#401C211yesmaximal, characteristic, complemented
2#31773839C211yesmaximal, characteristic, complemented
3#31773838C211yesmaximal, characteristic, complemented
4#31773837C211yesmaximal, characteristic, complemented
5#31773825C211yesmaximal, characteristic
6#31773825C211yesmaximal, characteristic
7#31773822C211yesmaximal, characteristic
8#31773820C211yesmaximal, characteristic
9C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
10C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
11C4×C4⋊Dic271C211yesmaximal, characteristic, complemented
12C4×C22⋊Dic271C211yesmaximal, characteristic
13C4×C22⋊Dic271C211yesmaximal, characteristic
14#31773743C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C271C211yesmaximal, characteristic
16C2×C4⋊Dic271C2211yescharacteristic
17C2×C4⋊Dic271C2211yescharacteristic
18C2×Dic271⋊C4C2211yescharacteristic
19C2×Dic271⋊C4C2211yescharacteristic
20C2×Dic271⋊C4C2211yescharacteristic
21C2×Dic271⋊C4C2211yescharacteristic
22C2×C22⋊Dic271C2211yescharacteristic
23C2×C22⋊Dic271C2211yescharacteristic
24C2×C22⋊Dic271C2211yescharacteristic
25C2×C22⋊Dic271C2211yescharacteristic
26C2×C4×Dic271C2211yescharacteristic
27C2×C4×Dic271C2211yescharacteristic
28C2×C4×Dic271C2211yescharacteristic
29C2×C4×Dic271C2211yescharacteristic
30C271⋊#95C2211yescharacteristic
31C271⋊#95C2211yescharacteristic
32C271⋊#95C2211yescharacteristic
33C271⋊#95C2211yescharacteristic
34C271⋊#95C2211yescharacteristic
35C271⋊#95C2211yescharacteristic
36C271⋊#95C2211yescharacteristic
37C271⋊#95C2211yescharacteristic
38C271⋊#95C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C23×C4×C271C2211yescharacteristic
45C2×C22⋊C4×C271C2211yescharacteristic
46C2×C22⋊C4×C271C2211yescharacteristic
47C2×C42×C271C2211yescharacteristic
48C2×C42×C271C2211yescharacteristic
49#95×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51C22×Dic271C2311yescharacteristic
52C22×Dic271C2311yescharacteristic
53C22×Dic271C2311yescharacteristic
54C22×Dic271C2311yescharacteristic
55C22×Dic271C2311yescharacteristic
56C22×Dic271C2311yescharacteristic
57C22×Dic271C2311yescharacteristic
58C22×Dic271C2311yescharacteristic
59C4⋊Dic271—22no—
60C4⋊Dic271—22no—
61Dic271⋊C4—22no—
62Dic271⋊C4—22no—
63Dic271⋊C4—22no—
64Dic271⋊C4—22no—
65C22⋊Dic271—22no—
66C22⋊Dic271—22no—
67C22⋊Dic271—22no—
68C22⋊Dic271—22no—
69C4×Dic271—22no—
70C4×Dic271—22no—
71C4×Dic271—22no—
72C4×Dic271—22no—
73C24×C271C2311yescharacteristic
74C22×C4×C271—22no—
75C22×C4×C271—22no—
76C22×C4×C271—22no—
77C22×C4×C271C2311yescharacteristic
78C22×C4×C271C2311yescharacteristic
79C22×C4×C271C2311yescharacteristic
80C22×C4×C271C2311yescharacteristic
81C22×C4×C271C2311yescharacteristic
82C22×C4×C271C2311yescharacteristic
83C22⋊C4×C271—22no—
84C22⋊C4×C271—22no—
85C42×C271—22no—
86C42×C271—22no—
87C2×Dic271—21no—
88C2×Dic271—21no—
89C2×Dic271—21no—
90C2×Dic271—21no—
91C2×Dic271—21no—
92C2×Dic271—21no—
93C2×Dic271—21no—
94C2×Dic271—21no—
95C2×Dic271—21no—
96C2×Dic271—21no—
97C2×Dic271—21no—
98C2×Dic271—21no—
99C2×Dic271—21no—
100C2×Dic271—21no—
101C2×Dic271—21no—
102C2×Dic271—21no—
103C2×Dic271—21no—
104C2×Dic271—21no—
105C2×Dic271—21no—
106C2×Dic271—21no—
107C2×Dic271—21no—
108C2×Dic271—21no—
109C2×Dic271#4112yes—
110C2×Dic271#4112yes—
111C23×C271—21no—
112C23×C271—21no—
113C23×C271—21no—
114C23×C271—21no—
115C23×C271—21no—
116C23×C271—21no—
117C23×C271#4112yes—
118C23×C271C2411yescharacteristic
119C2×C4×C271—21no—
120C2×C4×C271—21no—
121C2×C4×C271—21no—
122C2×C4×C271—21no—
123C2×C4×C271—21no—
124C2×C4×C271—21no—
125C2×C4×C271—21no—
126C2×C4×C271—21no—
127C2×C4×C271—21no—
128C2×C4×C271—21no—
129C2×C4×C271—21no—
130C2×C4×C271—21no—
131C2×C4×C271—21no—
132C2×C4×C271—21no—
133C2×C4×C271—44no—
134C2×C4×C271#4112yes—
135C2×C4×C271#4112yes—
136C2×C4×C271#4112yes—
137C2×C4×C271#4112yes—
138C22×C271—22no—
139C22×C271—22no—
140C22×C271—22no—
141C22×C271—41no—
142C22×C271—41no—
143C22×C271—41no—
144C22×C271—41no—
145C22×C271C2×#4111yescharacteristic
146C22×C271C2×#4111yescharacteristic
147C22×C271C2×#4111yescharacteristic
148C22×C271C2×#4111yescharacteristic
149C22×C271C2×#4111yescharacteristic
150C22×C271C2×#4111yescharacteristic
151C22×C271C2×#4111yescharacteristic
152C4×C271—22no—
153C4×C271—41no—
154C4×C271—41no—
155C4×C271—41no—
156C4×C271—41no—
157C4×C271#12614yes—
158Dic271—22no—
159Dic271—22no—
160Dic271—41no—
161Dic271—41no—
162Dic271—41no—
163Dic271—41no—
164Dic271—41no—
165Dic271—41no—
166C2×C271—42no—
167C2×C271C2×#12611yescharacteristic
168C2×C271#52911yescharacteristic
169C2×C271#52911yescharacteristic
170C2×C271#52911yescharacteristic
171C2×C271#52911yescharacteristic
172C2×C271#52911yescharacteristic
173C2×C271#52911yescharacteristic
174C271#240111yescharacteristic, complemented, Sylow
175#2401—2711nomaximal, Sylow
176#401—2711no—
177#400—2711no—
178#400—2711no—
179#400—2711no—
180#397—2711no—
181#397—2711no—
182#397—2711no—
183#396—2711no—
184C4×C4⋊C4—2711no—
185C4×C4⋊C4—2711no—
186C4×C4⋊C4—2711no—
187C4×C22⋊C4—2711no—
188C4×C22⋊C4—2711no—
189C4×C22⋊C4D27111yescharacteristic
190#376—2711no—
191C23×C4C2×D27111yescharacteristic
192C2×C4⋊C4—2711no—
193C2×C4⋊C4—2711no—
194C2×C4⋊C4—2711no—
195C2×C4⋊C4—2711no—
196C2×C4⋊C4—2711no—
197C2×C4⋊C4—2711no—
198C2×C22⋊C4—2711no—
199C2×C22⋊C4—2711no—
200C2×C22⋊C4—2711no—
201C2×C22⋊C4—2711no—
202C2×C22⋊C4C2×D27111yescharacteristic
203C2×C22⋊C4C2×D27111yescharacteristic
204C2×C42—2711no—
205C2×C42—2711no—
206C2×C42—2711no—
207C2×C42—2711no—
208C2×C42C2×D27111yescharacteristic
209C2×C42C2×D27111yescharacteristic
210#95—2711no—
211#95—2711no—
212#95—2711no—
213#95—2711no—
214#95—2711no—
215#95—2711no—
216#95—2711no—
217#95—2711no—
218#95—2711no—
219#95—2711no—
220#95—2711no—
221#95—2711no—
222#95—2711no—
223#95—2711no—
224#95C2×D27111yescharacteristic
225#95C2×D27111yescharacteristic
226C24C22×D27111yescharacteristic
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—2711no—
231C22×C4—2711no—
232C22×C4—2711no—
233C22×C4—2711no—
234C22×C4—2711no—
235C22×C4—2711no—
236C22×C4—2711no—
237C22×C4—2711no—
238C22×C4C22×D27111yescharacteristic
239C22×C4C22×D27111yescharacteristic
240C22×C4C22×D27111yescharacteristic
241C22×C4C22×D27111yescharacteristic
242C22×C4C22×D27111yescharacteristic
243C22×C4C22×D27111yescharacteristic
244C4⋊C4—5422no—
245C4⋊C4—5422no—
246C4⋊C4—5422no—
247C4⋊C4—5422no—
248C4⋊C4—5422no—
249C4⋊C4—5422no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—5422no—
253C22⋊C4—5422no—
254C22⋊C4—5422no—
255C22⋊C4—5422no—
256C42—22no—
257C42—22no—
258C42—5422no—
259C42—5422no—
260C42—5422no—
261C42—5422no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#1686465412yes—
269C23C23×D27111yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—44no—
285C2×C4—2712no—
286C2×C4—2712no—
287C2×C4—5421no—
288C2×C4—5421no—
289C2×C4—5421no—
290C2×C4—5421no—
291C2×C4—5421no—
292C2×C4—5421no—
293C2×C4—5421no—
294C2×C4—5421no—
295C2×C4—5421no—
296C2×C4—5421no—
297C2×C4—5421no—
298C2×C4—5421no—
299C2×C4—5421no—
300C2×C4—5421no—
301C2×C4—5421no—
302C2×C4—5421no—
303C2×C4—5421no—
304C2×C4—5421no—
305C2×C4—5421no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4#1686465312yes—
310C2×C4#1686465312yes—
311C2×C4#1686465412yes—
312C2×C4#1686465412yes—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22#41×D27111yescharacteristic
321C22#41×D27111yescharacteristic
322C22C2×#1686465311yescharacteristic
323C22C2×#1686465311yescharacteristic
324C22C2×#1686465411yescharacteristic
325C22C2×#1686465411yescharacteristic
326C22C2×#1686465411yescharacteristic
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—5422no—
333C4—5422no—
334C4—10841no—
335C4—10841no—
336C4—10841no—
337C4—10841no—
338C4—10841no—
339C4—10841no—
340C4#2257615314yes—
341C2—42no—
342C2C2×#2257615311yescharacteristic
343C2#3177442511yescharacteristic
344C2#3177442711yescharacteristic
345C2#3177442711yescharacteristic
346C2#3177442711yescharacteristic
347C2#3177442911yescharacteristic
348C2#3177443111yescharacteristic
subgroupG/H
centerC22×C4C22×D271
commutator, socle, cosocleC23×C271C24
FrattiniC23C23×D271
FittingC4×C22⋊C4×C271C2
prime radicalC24×C271C23
hypercenterC4×C22⋊C4D271
hypocenterC271#2401
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C271⋊#401C2semidirect
2. #31773839C2semidirect
3. #31773838C2semidirect
4. #31773837C2semidirect
9. C4×Dic271⋊C4C2semidirect
10. C4×Dic271⋊C4C2semidirect
11. C4×C4⋊Dic271C2semidirect
14. #31773743C2semidirect
174. C271#2401semidirect

Maximal subgroups

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

subgroupindex
C271⋊#4012
#317738392
#317738382
#317738372
#317738252
#317738222
#317738202
C4×Dic271⋊C42
C4×C4⋊Dic2712
C4×C22⋊Dic2712
#317737432
C4×C22⋊C4×C2712
#2401271