Finite Group Databasebeta

← #46839218

Subgroups — #46839218

Order 34688.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576116C211yesmaximal, characteristic
2#31773781C211yesmaximal, characteristic, complemented
3#31773781C211yesmaximal, characteristic, complemented
4#31773779C211yesmaximal, characteristic, complemented
5#31773779C211yesmaximal, characteristic, complemented
6#31773770C211yesmaximal, characteristic
7#31773769C211yesmaximal, characteristic
8#31773764C211yesmaximal, characteristic, complemented
9#31773761C211yesmaximal, characteristic, complemented
10#31773758C211yesmaximal, characteristic
11C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic271C211yesmaximal, characteristic
13#31773743C211yesmaximal, characteristic, complemented
14C2×C271⋊#95C211yesmaximal, characteristic
15C4×C22⋊C4×C271C211yesmaximal, characteristic
16C23×Dic271C2211yescharacteristic
17C23×Dic271C2211yescharacteristic
18#22576116C418yescomplemented
19C2×Dic271⋊C4C2211yescharacteristic
20C2×Dic271⋊C4C2211yescharacteristic
21C2×Dic271⋊C4C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×C22⋊Dic271C2211yescharacteristic
26C2×C22⋊Dic271C2211yescharacteristic
27C2×C4×Dic271C2211yescharacteristic
28C2×C4×Dic271C2211yescharacteristic
29C2×C4×Dic271C2211yescharacteristic
30C2×C4×Dic271C2211yescharacteristic
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
44C271⋊#95C2211yescharacteristic
45C23×C4×C271C2211yescharacteristic
46C2×C22⋊C4×C271C2211yescharacteristic
47C2×C22⋊C4×C271C2211yescharacteristic
48C2×C42×C271C2211yescharacteristic
49C2×C42×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52C22×Dic271—22no—
53C22×Dic271—22no—
54C22×Dic271—22no—
55C22×Dic271—22no—
56C22×Dic271—41no—
57C22×Dic271C2×C414yes—
58C22×Dic271C2311yescharacteristic
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C22×Dic271C2311yescharacteristic
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66Dic271⋊C4—22no—
67Dic271⋊C4—22no—
68Dic271⋊C4—41no—
69Dic271⋊C4—41no—
70Dic271⋊C4C2×C414yescomplemented
71Dic271⋊C4C2×C414yescomplemented
72C22⋊Dic271—22no—
73C22⋊Dic271C2×C414yes—
74C4×Dic271—22no—
75C4×Dic271—41no—
76C4×Dic271C2×C414yescomplemented
77C4×Dic271C2×C414yescomplemented
78C24×C271C2311yescharacteristic
79C22×C4×C271—22no—
80C22×C4×C271—22no—
81C22×C4×C271—41no—
82C22×C4×C271C2311yescharacteristic
83C22×C4×C271C2311yescharacteristic
84C22×C4×C271C2311yescharacteristic
85C22×C4×C271C2311yescharacteristic
86C22×C4×C271C2311yescharacteristic
87C22×C4×C271C2311yescharacteristic
88C22⋊C4×C271—22no—
89C22⋊C4×C271C2×C414yes—
90C42×C271—22no—
91C42×C271—41no—
92C2×Dic271—28no—
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—21no—
110C2×Dic271—21no—
111C2×Dic271—21no—
112C2×Dic271—44no—
113C2×Dic271C22×C412yes—
114C2×Dic271C22×C412yes—
115C2×Dic271C22×C412yes—
116C2×Dic271C22×C412yes—
117C2×Dic271#4112yes—
118C23×C271—21no—
119C23×C271—21no—
120C23×C271—21no—
121C23×C271—21no—
122C23×C271C22×C412yes—
123C23×C271#4112yes—
124C23×C271#4112yes—
125C23×C271C2411yescharacteristic
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—21no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—44no—
142C2×C4×C271C22×C412yes—
143C2×C4×C271C22×C412yes—
144C2×C4×C271#4112yes—
145C22×C271—22no—
146C22×C271—22no—
147C22×C271—22no—
148C22×C271—22no—
149C22×C271—22no—
150C22×C271—22no—
151C22×C271#11714yes—
152C22×C271C23×C411yescharacteristic
153C22×C271C2×#4111yescharacteristic
154C22×C271C2×#4111yescharacteristic
155C22×C271C2×#4111yescharacteristic
156C22×C271C2×#4111yescharacteristic
157C22×C271H(5,𝔽2)11yescharacteristic
158C22×C2712−1+411yescharacteristic
159C4×C271—22no—
160C4×C271—22no—
161C4×C271—22no—
162C4×C271—41no—
163C4×C271—41no—
164C4×C271—41no—
165Dic271—22no—
166Dic271—22no—
167Dic271—22no—
168Dic271—22no—
169Dic271—22no—
170Dic271—41no—
171Dic271—41no—
172Dic271—41no—
173C2×C271—24no—
174C2×C271C2×#11711yescharacteristic
175C2×C271C4×#4111yescharacteristic
176C2×C271#52011yescharacteristic
177C2×C271#55111yescharacteristic
178C2×C271#55111yescharacteristic
179C2×C271#55211yescharacteristic
180C2×C271#55211yescharacteristic
181C271#234311yescharacteristic, complemented, Sylow
182#2343—2711nomaximal, Sylow
183C2×#117—2711no—
184#387—2711no—
185#387—2711no—
186#387—2711no—
187#387—2711no—
188#385—2711no—
189#385—2711no—
190#383—2711no—
191#382—2711no—
192#381—2711no—
193C4×C4⋊C4—2711no—
194C4×C22⋊C4—2711no—
195C4×C22⋊C4D27111yescharacteristic
196#376—2711no—
197C2×#95—2711no—
198C23×C4—2711no—
199C23×C4—2711no—
200C23×C4C2×D27111yescharacteristic
201#117—2718no—
202C2×C4⋊C4—2711no—
203C2×C4⋊C4—2711no—
204C2×C4⋊C4—2711no—
205C2×C4⋊C4—2711no—
206C2×C4⋊C4—2711no—
207C2×C4⋊C4—2711no—
208C2×C22⋊C4—2711no—
209C2×C22⋊C4—2711no—
210C2×C22⋊C4C2×D27111yescharacteristic
211C2×C22⋊C4C2×D27111yescharacteristic
212C2×C42—2711no—
213C2×C42—2711no—
214C2×C42—2711no—
215C2×C42—2711no—
216C2×C42C2×D27111yescharacteristic
217C2×C42C2×D27111yescharacteristic
218#95—2711no—
219#95—2711no—
220#95—2711no—
221#95—2711no—
222#95—2711no—
223#95—2711no—
224#95—2711no—
225#95—2711no—
226#95—2711no—
227#95—2711no—
228#95—2711no—
229#95—2711no—
230#95—2711no—
231#95—2711no—
232#95C2×D27111yescharacteristic
233#95C2×D27111yescharacteristic
234C24C22×D27111yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—2714no—
239C22×C4—2711no—
240C22×C4—2711no—
241C22×C4—2711no—
242C22×C4—2711no—
243C22×C4—2711no—
244C22×C4—2711no—
245C22×C4—2711no—
246C22×C4—2711no—
247C22×C4—5422no—
248C22×C4—5422no—
249C22×C4—5422no—
250C22×C4—5422no—
251C22×C4—10841no—
252C22×C4C22×D27111yescharacteristic
253C22×C4C22×D27111yescharacteristic
254C22×C4C22×D27111yescharacteristic
255C22×C4C22×D27111yescharacteristic
256C22×C4C22×D27111yescharacteristic
257C22×C4C22×D27111yescharacteristic
258C4⋊C4—2714no—
259C4⋊C4—2714no—
260C4⋊C4—5422no—
261C4⋊C4—5422no—
262C4⋊C4—10841no—
263C4⋊C4—10841no—
264C22⋊C4—22no—
265C22⋊C4—2714no—
266C22⋊C4—5422no—
267C22⋊C4C4×D27114yes—
268C42—22no—
269C42—41no—
270C42—2714no—
271C42—2714no—
272C42—5422no—
273C42—10841no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D27112yes—
279C23#1686465412yes—
280C23#1686465412yes—
281C23C23×D27111yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—2712no—
299C2×C4—2712no—
300C2×C4—2712no—
301C2×C4—2712no—
302C2×C4—2712no—
303C2×C4—5421no—
304C2×C4—5421no—
305C2×C4—5421no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4—5421no—
310C2×C4—5421no—
311C2×C4—5421no—
312C2×C4—5421no—
313C2×C4—5428no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—5421no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—10844no—
324C2×C4C2×C4×D27112yes—
325C2×C4C2×C4×D27112yes—
326C2×C4#1686465312yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22#2257611714yes—
334C22C22×C4×D27111yescharacteristic
335C22#41×D27111yescharacteristic
336C22C2×#1686465311yescharacteristic
337C22C2×#1686465411yescharacteristic
338C22C2×#1686465411yescharacteristic
339C22#2257620511yescharacteristic
340C22#2257620711yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—5422no—
348C4—5422no—
349C4—5422no—
350C4—5422no—
351C4—5422no—
352C4—10841no—
353C4—10841no—
354C4—10841no—
355C2—24no—
356C2C2×#2257611711yescharacteristic
357C2C4×#1686465311yescharacteristic
358C2#3177437611yescharacteristic
359C2#3177457311yescharacteristic
360C2#3177457611yescharacteristic
361C2#3177458011yescharacteristic
362C2#3177458711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
FittingC4×C22⋊C4×C271C2
socle, cosocleC23×C271C24
prime radicalC24×C271C23
hypercenterC4×C22⋊C4D271
hypocenterC271#2343
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #31773781C2semidirect
3. #31773781C2semidirect
4. #31773779C2semidirect
5. #31773779C2semidirect
8. #31773764C2semidirect
9. #31773761C2semidirect
11. C4×Dic271⋊C4C2semidirect
13. #31773743C2semidirect
18. #22576116C4semidirect
70. Dic271⋊C4C2×C4semidirect
71. Dic271⋊C4C2×C4semidirect
76. C4×Dic271C2×C4semidirect
77. C4×Dic271C2×C4semidirect
181. C271#2343semidirect

Maximal subgroups

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

subgroupindex
C2×#225761162
#317737812
#317737792
#317737702
#317737692
#317737642
#317737612
#317737582
C4×Dic271⋊C42
C22⋊C4×Dic2712
#317737432
C2×C271⋊#952
C4×C22⋊C4×C2712
#2343271