Finite Group Databasebeta

← #46839114

Subgroups — #46839114

Order 34688.

484 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576137C211yesmaximal, characteristic
2#31773803C211yesmaximal, characteristic, complemented
3#31773801C211yesmaximal, characteristic, complemented
4#31773790C211yesmaximal, characteristic
5#31773785C211yesmaximal, characteristic
6#31773785C211yesmaximal, characteristic
7#31773784C211yesmaximal, characteristic, complemented
8#31773782C211yesmaximal, characteristic, complemented
9#31773779C211yesmaximal, characteristic, complemented
10#31773778C211yesmaximal, characteristic, complemented
11#31773758C211yesmaximal, characteristic
12C4⋊C4×Dic271C211yesmaximal, characteristic, complemented
13C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic271C211yesmaximal, characteristic
15#388×C271C211yesmaximal, characteristic
16C22×Dic542C2211yescharacteristic
17C23×Dic271C2211yescharacteristic
18#22576137C418yescomplemented
19C2×C4⋊Dic271C2211yescharacteristic
20C2×C4⋊Dic271C2211yescharacteristic
21C2×Dic271⋊C4C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×C22⋊Dic271C2211yescharacteristic
26C2×C22⋊Dic271C2211yescharacteristic
27C2×C22⋊Dic271C2211yescharacteristic
28C2×C4×Dic271C2211yescharacteristic
29C2×C4×Dic271C2211yescharacteristic
30C2×C4×Dic271C2211yescharacteristic
31C2×C4×Dic271C2211yescharacteristic
32C2×C4×Dic271C2211yescharacteristic
33C271⋊#95C2211yescharacteristic
34C271⋊#95C2211yescharacteristic
35C271⋊#95C2211yescharacteristic
36C271⋊#95C2211yescharacteristic
37C271⋊#95C2211yescharacteristic
38C271⋊#95C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C2×C4⋊C4×C271C2211yescharacteristic
46C2×C22⋊C4×C271C2211yescharacteristic
47C2×C22⋊C4×C271C2211yescharacteristic
48C2×C22⋊C4×C271C2211yescharacteristic
49C2×C42×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52C2×Dic542—41no—
53C2×Dic542—41no—
54C2×Dic542C2×C414yes—
55C22×Dic271—41no—
56C22×Dic271—41no—
57C22×Dic271C2×C414yes—
58C22×Dic271C2311yescharacteristic
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C22×Dic271C2311yescharacteristic
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66C4⋊Dic271—41no—
67C4⋊Dic271C2×C414yes—
68Dic271⋊C4—22no—
69Dic271⋊C4—41no—
70Dic271⋊C4C2×C414yes—
71Dic271⋊C4C2×C414yes—
72C22⋊Dic271—22no—
73C22⋊Dic271—41no—
74C22⋊Dic271C2×C414yes—
75C4×Dic271—22no—
76C4×Dic271—22no—
77C4×Dic271—41no—
78C4×Dic271—41no—
79C4×Dic271—41no—
80C24×C271C2311yescharacteristic
81C22×C4×C271C2311yescharacteristic
82C22×C4×C271C2311yescharacteristic
83C22×C4×C271C2311yescharacteristic
84C22×C4×C271C2311yescharacteristic
85C22×C4×C271C2311yescharacteristic
86C22×C4×C271C2311yescharacteristic
87C4⋊C4×C271—22no—
88C22⋊C4×C271—22no—
89C22⋊C4×C271—41no—
90C22⋊C4×C271C2×C414yes—
91C42×C271—41no—
92Dic542—44no—
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×C271—21no—
123C23×C271—21no—
124C23×C271C22×C412yes—
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×C271C22×C412yes—
142C2×C4×C271C22×C412yes—
143C2×C4×C271#4112yes—
144C22×C271—22no—
145C22×C271—22no—
146C22×C271—22no—
147C22×C271—41no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271—41no—
151C22×C271C23×C411yescharacteristic
152C22×C271C2×#4111yescharacteristic
153C22×C271C2×#4111yescharacteristic
154C22×C271H(5,𝔽2)11yescharacteristic
155C22×C271H(5,𝔽2)11yescharacteristic
156C22×C2712−1+411yescharacteristic
157C22×C2712−1+411yescharacteristic
158C4×C271—22no—
159C4×C271—22no—
160C4×C271—22no—
161C4×C271—41no—
162C4×C271—41no—
163C4×C271—41no—
164Dic271—22no—
165Dic271—22no—
166Dic271—22no—
167Dic271—22no—
168Dic271—22no—
169Dic271—41no—
170Dic271—41no—
171Dic271—41no—
172C2×C271—42no—
173C2×C271C4×#4111yescharacteristic
174C2×C271#51811yescharacteristic
175C2×C271#51911yescharacteristic
176C2×C271#53911yescharacteristic
177C2×C271#53911yescharacteristic
178C2×C271#54211yescharacteristic
179C2×C271#54211yescharacteristic
180C271#233211yescharacteristic, complemented, Sylow
181#2332—2711nomaximal, Sylow
182C2×C22⋊Q8—2711no—
183#391—2711no—
184#391—2711no—
185#388—2711no—
186#388—2711no—
187#388—2711no—
188#388D27111yescharacteristic
189#387—2711no—
190#387—2711no—
191#387—2711no—
192#387—2711no—
193#381—2711no—
194C4×C4⋊C4—2711no—
195C4×C4⋊C4—2711no—
196C4×C22⋊C4—2711no—
197C22×Q8—2711no—
198C23×C4—2711no—
199C22⋊Q8—2718no—
200C2×C4⋊C4—2711no—
201C2×C4⋊C4—2711no—
202C2×C4⋊C4—2711no—
203C2×C4⋊C4—2711no—
204C2×C4⋊C4—2711no—
205C2×C4⋊C4—2711no—
206C2×C4⋊C4C2×D27111yescharacteristic
207C2×C22⋊C4—2711no—
208C2×C22⋊C4—2711no—
209C2×C22⋊C4—2711no—
210C2×C22⋊C4C2×D27111yescharacteristic
211C2×C22⋊C4C2×D27111yescharacteristic
212C2×C22⋊C4C2×D27111yescharacteristic
213C2×C42—2711no—
214C2×C42—2711no—
215C2×C42—2711no—
216C2×C42—2711no—
217C2×C42—2711no—
218C2×C42C2×D27111yescharacteristic
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#95C2×D27111yescharacteristic
232#95C2×D27111yescharacteristic
233C24C22×D27111yescharacteristic
234C2×Q8—2714no—
235C2×Q8—10841no—
236C2×Q8—10841no—
237C22×C4—2711no—
238C22×C4—2711no—
239C22×C4—2711no—
240C22×C4—2711no—
241C22×C4—2711no—
242C22×C4—2714no—
243C22×C4—2711no—
244C22×C4—2711no—
245C22×C4—2711no—
246C22×C4—10841no—
247C22×C4—10841no—
248C22×C4C22×D27111yescharacteristic
249C22×C4C22×D27111yescharacteristic
250C22×C4C22×D27111yescharacteristic
251C22×C4C22×D27111yescharacteristic
252C22×C4C22×D27111yescharacteristic
253C22×C4C22×D27111yescharacteristic
254C4⋊C4—22no—
255C4⋊C4—2714no—
256C4⋊C4—2714no—
257C4⋊C4—2714no—
258C4⋊C4—5422no—
259C4⋊C4—10841no—
260C4⋊C4—10841no—
261C22⋊C4—22no—
262C22⋊C4—41no—
263C22⋊C4—2714no—
264C22⋊C4—5422no—
265C22⋊C4—10841no—
266C22⋊C4C4×D27114yes—
267C42—41no—
268C42—5422no—
269C42—5422no—
270C42—10841no—
271C42—10841no—
272C42—10841no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23C2×C4×D27112yes—
280C23C23×D27111yescharacteristic
281Q8—10844no—
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—2712no—
298C2×C4—2712no—
299C2×C4—2712no—
300C2×C4—2712no—
301C2×C4—2712no—
302C2×C4—5421no—
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—5421no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—5421no—
321C2×C4—10844no—
322C2×C4C2×C4×D27112yes—
323C2×C4C2×C4×D27112yes—
324C2×C4#1686465312yes—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C22×C4×D27111yescharacteristic
333C22#41×D27111yescharacteristic
334C22C2×#1686465311yescharacteristic
335C22#2257620511yescharacteristic
336C22#2257620611yescharacteristic
337C22#2257620711yescharacteristic
338C22#2257620711yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—5422no—
346C4—5422no—
347C4—5422no—
348C4—5422no—
349C4—5422no—
350C4—10841no—
351C4—10841no—
352C4—10841no—
353C2—42no—
354C2C4×#1686465311yescharacteristic
355C2#3177436911yescharacteristic
356C2#3177437411yescharacteristic
357C2#3177449011yescharacteristic
358C2#3177449211yescharacteristic
359C2#3177451311yescharacteristic
360C2#3177451611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
Fitting#388×C271C2
socle, cosocleC23×C271C24
prime radicalC24×C271C23
hypercenter#388D271
hypocenterC271#2332
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #31773803C2semidirect
3. #31773801C2semidirect
7. #31773784C2semidirect
8. #31773782C2semidirect
9. #31773779C2semidirect
10. #31773778C2semidirect
12. C4⋊C4×Dic271C2semidirect
13. C4×Dic271⋊C4C2semidirect
18. #22576137C4semidirect
180. C271#2332semidirect

Maximal subgroups

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

subgroupindex
C2×#225761372
#317738032
#317738012
#317737902
#317737852
#317737842
#317737822
#317737792
#317737782
#317737582
C4⋊C4×Dic2712
C4×Dic271⋊C42
C22⋊C4×Dic2712
#388×C2712
#2332271