Finite Group Databasebeta

← #46839115

Subgroups — #46839115

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×#22576138C211yesmaximal, characteristic
2#31773803C211yesmaximal, characteristic, complemented
3#31773801C211yesmaximal, characteristic, complemented
4#31773789C211yesmaximal, characteristic
5#31773788C211yesmaximal, characteristic
6#31773788C211yesmaximal, characteristic
7#31773787C211yesmaximal, characteristic
8#31773783C211yesmaximal, characteristic, complemented
9#31773781C211yesmaximal, characteristic, complemented
10#31773779C211yesmaximal, characteristic, complemented
11#31773778C211yesmaximal, characteristic, complemented
12#31773759C211yesmaximal, characteristic
13C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
14C4×(C2×D271)⋊C4C211yesmaximal, characteristic
15C4×C4⋊C4×C271C211yesmaximal, characteristic, complemented
16C22×Dic542C2211yescharacteristic
17C22×C4×D271C2211yescharacteristic
18#22576138C418yescomplemented
19C2×C4⋊Dic271C2211yescharacteristic
20C2×Dic271⋊C4C2211yescharacteristic
21C2×Dic271⋊C4C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×Dic271⋊C4C2211yescharacteristic
24C2×(C2×D271)⋊C4C2211yescharacteristic
25C2×(C2×D271)⋊C4C2211yescharacteristic
26C2×(C2×D271)⋊C4C2211yescharacteristic
27C2×(C2×D271)⋊C4C2211yescharacteristic
28C2×(C2×D271)⋊C4C2211yescharacteristic
29C2×(C2×D271)⋊C4C2211yescharacteristic
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×C4⋊C4×C271C2211yescharacteristic
47C2×C42×C271C2211yescharacteristic
48C2×C42×C271C2211yescharacteristic
49C2×C42×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52C23×D271C2311yescharacteristic
53C2×Dic542—41no—
54C2×Dic542—41no—
55C2×Dic542C2×C414yes—
56C22×Dic271C2311yescharacteristic
57C22×Dic271C2311yescharacteristic
58C22×Dic271C2311yescharacteristic
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C2×C4×D271—41no—
64C2×C4×D271—41no—
65C2×C4×D271C2×C414yes—
66C4⋊Dic271—41no—
67Dic271⋊C4—22no—
68Dic271⋊C4—41no—
69Dic271⋊C4C2×C414yes—
70Dic271⋊C4C2×C414yes—
71(C2×D271)⋊C4—22no—
72(C2×D271)⋊C4—22no—
73(C2×D271)⋊C4—41no—
74(C2×D271)⋊C4—41no—
75(C2×D271)⋊C4C2×C414yes—
76(C2×D271)⋊C4C2×C414yes—
77C4×Dic271—22no—
78C4×Dic271—41no—
79C4×Dic271—41no—
80C22×C4×C271C2311yescharacteristic
81C22×C4×C271C2311yescharacteristic
82C22×C4×C271C2311yescharacteristic
83C22×C4×C271C2311yescharacteristic
84C22×C4×C271C2311yescharacteristic
85C22×C4×C271C2311yescharacteristic
86C22×C4×C271C2311yescharacteristic
87C4⋊C4×C271—22no—
88C4⋊C4×C271C2×C414yes—
89C42×C271—22no—
90C42×C271—41no—
91C42×C271—41no—
92C22×D271—21no—
93C22×D271—21no—
94C22×D271—21no—
95C22×D271—21no—
96C22×D271—21no—
97C22×D271—21no—
98C22×D271C22×C412yes—
99Dic542—44no—
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—21no—
113C2×Dic271—21no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271C22×C412yes—
119C2×Dic271C22×C412yes—
120C2×Dic271C22×C412yes—
121C4×D271—44no—
122C23×C271C2411yescharacteristic
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—21no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271C22×C412yes—
140C2×C4×C271C22×C412yes—
141C2×C4×C271C22×C412yes—
142C2×C4×C271#4112yes—
143C2×C4×C271#4112yes—
144C22×C271C23×C411yescharacteristic
145C22×C271C2×#4111yescharacteristic
146C22×C271C2×#4111yescharacteristic
147C22×C271H(5,𝔽2)11yescharacteristic
148C22×C271H(5,𝔽2)11yescharacteristic
149C22×C2712−1+411yescharacteristic
150C22×C2712−1+411yescharacteristic
151C2×D271—22no—
152C2×D271—22no—
153C2×D271—22no—
154C2×D271—41no—
155C2×D271—41no—
156C2×D271—41no—
157C2×D271—41no—
158C4×C271—22no—
159C4×C271—22no—
160C4×C271—22no—
161C4×C271—22no—
162C4×C271—22no—
163C4×C271—41no—
164C4×C271—41no—
165Dic271—22no—
166Dic271—22no—
167Dic271—22no—
168Dic271—41no—
169Dic271—41no—
170Dic271—41no—
171Dic271—41no—
172C2×C271C4×#4111yescharacteristic
173C2×C271#51811yescharacteristic
174C2×C271#51911yescharacteristic
175C2×C271#53911yescharacteristic
176C2×C271#53911yescharacteristic
177C2×C271#54211yescharacteristic
178C2×C271#54211yescharacteristic
179D271—42no—
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#388—2711no—
189#387—2711no—
190#387—2711no—
191#387—2711no—
192#387—2711no—
193#381—2711no—
194C4×C4⋊C4—2711no—
195C4×C4⋊C4D27111yescharacteristic, complemented
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⋊C4C2×D27111yescharacteristic
206C2×C4⋊C4C2×D27111yescharacteristic
207C2×C22⋊C4—2711no—
208C2×C22⋊C4—2711no—
209C2×C22⋊C4—2711no—
210C2×C22⋊C4—2711no—
211C2×C22⋊C4—2711no—
212C2×C22⋊C4—2711no—
213C2×C42—2711no—
214C2×C42—2711no—
215C2×C42—2711no—
216C2×C42C2×D27111yescharacteristic
217C2×C42C2×D27111yescharacteristic
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
233C24—2711no—
234C2×Q8—2714no—
235C2×Q8—10841no—
236C2×Q8—10841no—
237C22×C4—2711no—
238C22×C4—2711no—
239C22×C4—2711no—
240C22×C4—2714no—
241C22×C4—2711no—
242C22×C4—2711no—
243C22×C4—2711no—
244C22×C4—2711no—
245C22×C4—10841no—
246C22×C4—10841no—
247C22×C4C22×D27111yescharacteristic
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—5422no—
258C4⋊C4—10841no—
259C4⋊C4—10841no—
260C4⋊C4C4×D27114yes—
261C22⋊C4—2714no—
262C22⋊C4—2714no—
263C22⋊C4—5422no—
264C22⋊C4—5422no—
265C22⋊C4—10841no—
266C22⋊C4—10841no—
267C42—22no—
268C42—41no—
269C42—41no—
270C42—5422no—
271C42—10841no—
272C42—10841no—
273C23—2712no—
274C23—5421no—
275C23—5421no—
276C23—5421no—
277C23—5421no—
278C23—5421no—
279C23—5421no—
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—21no—
298C2×C4—2712no—
299C2×C4—2712no—
300C2×C4—2712no—
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—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—10844no—
320C2×C4C2×C4×D27112yes—
321C2×C4C2×C4×D27112yes—
322C2×C4C2×C4×D27112yes—
323C2×C4#1686465412yes—
324C2×C4#1686465412yes—
325C22—5422no—
326C22—5422no—
327C22—5422no—
328C22—10841no—
329C22—10841no—
330C22—10841no—
331C22—10841no—
332C22C22×C4×D27111yescharacteristic
333C22C2×#1686465411yescharacteristic
334C22C2×#1686465411yescharacteristic
335C22#2257620511yescharacteristic
336C22#2257620611yescharacteristic
337C22#2257620711yescharacteristic
338C22#2257620811yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—5422no—
347C4—5422no—
348C4—5422no—
349C4—10841no—
350C4—10841no—
351C4—10841no—
352C4—10841no—
353C2—10842no—
354C2C4×#1686465411yescharacteristic
355C2#3177436911yescharacteristic
356C2#3177437311yescharacteristic
357C2#3177448911yescharacteristic
358C2#3177449411yescharacteristic
359C2#3177451811yescharacteristic
360C2#3177451911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
FittingC4×C4⋊C4×C271C2
socle, cosocleC23×C271C24
prime radicalC23×D271C23
hypercenterC4×C4⋊C4D271
hypocenterC271#2332
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #31773803C2semidirect
3. #31773801C2semidirect
8. #31773783C2semidirect
9. #31773781C2semidirect
10. #31773779C2semidirect
11. #31773778C2semidirect
13. C4×Dic271⋊C4C2semidirect
15. C4×C4⋊C4×C271C2semidirect
18. #22576138C4semidirect
180. C271#2332semidirect
195. C4×C4⋊C4D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761382
#317738032
#317738012
#317737892
#317737882
#317737872
#317737832
#317737812
#317737792
#317737782
#317737592
C4×Dic271⋊C42
C4×(C2×D271)⋊C42
C4×C4⋊C4×C2712
#2332271