Finite Group Databasebeta

← #46839443

Subgroups — #46839443

Order 34688.

548 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576136C211yesmaximal, characteristic
2#31773804C211yesmaximal, characteristic, complemented
3#31773784C211yesmaximal, characteristic, complemented
4#31773782C211yesmaximal, characteristic, complemented
5#31773780C211yesmaximal, characteristic, complemented
6#31773778C211yesmaximal, characteristic, complemented
7#31773771C211yesmaximal, characteristic
8#31773768C211yesmaximal, characteristic
9C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
10#31773748C211yesmaximal, characteristic, complemented
11C4×(C2×D271)⋊C4C211yesmaximal, characteristic
12#31773744C211yesmaximal, characteristic
13#31773744C211yesmaximal, characteristic
14#95×D271C211yesmaximal, characteristic
15#391×C271C211yesmaximal, characteristic, complemented
16C22×C4×D271C2211yescharacteristic
17C22×C4×D271C2211yescharacteristic
18C22×C4×D271C2211yescharacteristic
19#22576136C418yescomplemented
20C2×C4⋊Dic271C2211yescharacteristic
21C2×C4⋊Dic271C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊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
33C2×C4×Dic271C2211yescharacteristic
34C2×C4×Dic271C2211yescharacteristic
35C271⋊#95C2211yescharacteristic
36C271⋊#95C2211yescharacteristic
37C271⋊#95C2211yescharacteristic
38C271⋊#95C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C22×Q8×C271C2211yescharacteristic
46C2×C4⋊C4×C271C2211yescharacteristic
47C2×C42×C271C2211yescharacteristic
48#95×C271C2211yescharacteristic
49#95×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52C23×D271C2311yescharacteristic
53C22×Dic271C2311yescharacteristic
54C22×Dic271C2311yescharacteristic
55C22×Dic271C2311yescharacteristic
56C22×Dic271C2311yescharacteristic
57C22×Dic271C2311yescharacteristic
58C22×Dic271C2311yescharacteristic
59C22×Dic271C2311yescharacteristic
60C2×C4×D271—22no—
61C2×C4×D271—22no—
62C2×C4×D271—22no—
63C2×C4×D271—22no—
64C2×C4×D271—41no—
65C2×C4×D271—41no—
66C2×C4×D271—41no—
67C2×C4×D271—41no—
68C2×C4×D271C2×C414yes—
69C4⋊Dic271—41no—
70C4⋊Dic271C2×C414yescomplemented
71Dic271⋊C4—22no—
72Dic271⋊C4—22no—
73Dic271⋊C4C2×C414yescomplemented
74Dic271⋊C4C2×C414yescomplemented
75(C2×D271)⋊C4—22no—
76(C2×D271)⋊C4C2×C414yes—
77(C2×D271)⋊C4C2×C414yes—
78(C2×D271)⋊C4Q814yescomplemented
79C4×Dic271—22no—
80C4×Dic271—22no—
81C4×Dic271—41no—
82C4×Dic271—41no—
83C4×Dic271—41no—
84C2×Q8×C271—41no—
85C2×Q8×C271—41no—
86C2×Q8×C271C2×C414yescomplemented
87C22×C4×C271C2311yescharacteristic
88C22×C4×C271C2311yescharacteristic
89C22×C4×C271C2311yescharacteristic
90C22×C4×C271C2311yescharacteristic
91C22×C4×C271C2311yescharacteristic
92C22×C4×C271C2311yescharacteristic
93C22×C4×C271C2311yescharacteristic
94C4⋊C4×C271—41no—
95C42×C271—41no—
96C22×D271—21no—
97C22×D271—21no—
98C22×D271—21no—
99C22×D271—21no—
100C22×D271C22×C412yes—
101C22×D271C2×Q812yes—
102C22×D271#4112yes—
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×Dic271—21no—
119C2×Dic271C22×C412yes—
120C2×Dic271C22×C412yes—
121C2×Dic271C22×C412yes—
122C2×Dic271C2×Q812yes—
123C2×Dic271#4112yes—
124C4×D271—44no—
125C4×D271—44no—
126C4×D271—44no—
127C23×C271C2411yescharacteristic
128Q8×C271—44no—
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—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271C22×C412yes—
146C2×C4×C271C22×C412yes—
147C2×C4×C271C22×C412yes—
148C2×C4×C271C2×Q812yes—
149C2×C4×C271#4112yes—
150C22×C271C23×C411yescharacteristic
151C22×C271C22×Q811yescharacteristic
152C22×C271C2×#4111yescharacteristic
153C22×C271C2×#4111yescharacteristic
154C22×C271C2×#4111yescharacteristic
155C22×C271H(5,𝔽2)11yescharacteristic
156C22×C271H(5,𝔽2)11yescharacteristic
157C2×D271—22no—
158C2×D271—22no—
159C2×D271—22no—
160C2×D271—22no—
161C2×D271—22no—
162C2×D271—22no—
163C2×D271C4×Q814yes—
164C4×C271—22no—
165C4×C271—22no—
166C4×C271—22no—
167C4×C271—22no—
168C4×C271—22no—
169C4×C271—41no—
170C4×C271—41no—
171Dic271—22no—
172Dic271—22no—
173Dic271—22no—
174Dic271—22no—
175Dic271—22no—
176Dic271—41no—
177Dic271—41no—
178C2×C271C2×C4×Q811yescharacteristic
179C2×C271C4×#4111yescharacteristic
180C2×C271#51811yescharacteristic
181C2×C271#55111yescharacteristic
182C2×C271#55111yescharacteristic
183C2×C271#55411yescharacteristic
184C2×C271#55411yescharacteristic
185D271—24no—
186C271#236811yescharacteristic, complemented, Sylow
187#2368—2711nomaximal, Sylow
188C2×C22⋊Q8—2711no—
189#391—2711no—
190#391D27111yescharacteristic, complemented
191#387—2711no—
192#387—2711no—
193#387—2711no—
194#387—2711no—
195#385—2711no—
196#385—2711no—
197C4×C4⋊C4—2711no—
198C4×C4⋊C4—2711no—
199C4×C22⋊C4—2711no—
200C4×C22⋊C4—2711no—
201C4×C22⋊C4—2711no—
202C2×#95—2711no—
203C22×Q8C2×D27111yescharacteristic
204C23×C4—2711no—
205C23×C4—2711no—
206C23×C4—2711no—
207C22⋊Q8—2718no—
208C2×C4⋊C4—2711no—
209C2×C4⋊C4—2711no—
210C2×C4⋊C4—2711no—
211C2×C4⋊C4—2711no—
212C2×C4⋊C4—2711no—
213C2×C4⋊C4—2711no—
214C2×C4⋊C4C2×D27111yescharacteristic
215C2×C22⋊C4—2711no—
216C2×C22⋊C4—2711no—
217C2×C22⋊C4—2711no—
218C2×C22⋊C4—2711no—
219C2×C42—2711no—
220C2×C42—2711no—
221C2×C42—2711no—
222C2×C42—2711no—
223C2×C42—2711no—
224C2×C42C2×D27111yescharacteristic
225#95—2711no—
226#95—2711no—
227#95—2711no—
228#95—2711no—
229#95—2711no—
230#95—2711no—
231#95—2711no—
232#95—2711no—
233#95—2711no—
234#95—2711no—
235#95C2×D27111yescharacteristic
236#95C2×D27111yescharacteristic
237#95C2×D27111yescharacteristic
238#95C2×D27111yescharacteristic
239C24—2711no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8C4×D27114yescomplemented
243C22×C4—2711no—
244C22×C4—2711no—
245C22×C4—2711no—
246C22×C4—2711no—
247C22×C4—2711no—
248C22×C4—2714no—
249C22×C4—2711no—
250C22×C4—2711no—
251C22×C4—5422no—
252C22×C4—5422no—
253C22×C4—5422no—
254C22×C4—5422no—
255C22×C4—10841no—
256C22×C4—10841no—
257C22×C4—10841no—
258C22×C4—10841no—
259C22×C4C22×D27111yescharacteristic
260C22×C4C22×D27111yescharacteristic
261C22×C4C22×D27111yescharacteristic
262C22×C4C22×D27111yescharacteristic
263C22×C4C22×D27111yescharacteristic
264C22×C4C22×D27111yescharacteristic
265C22×C4C22×D27111yescharacteristic
266C4⋊C4—41no—
267C4⋊C4—2714no—
268C4⋊C4—2714no—
269C4⋊C4—2714no—
270C4⋊C4—5422no—
271C4⋊C4—5422no—
272C4⋊C4—10841no—
273C22⋊C4—2714no—
274C22⋊C4—2714no—
275C22⋊C4—2714no—
276C22⋊C4—5422no—
277C42—41no—
278C42—5422no—
279C42—5422no—
280C42—10841no—
281C42—10841no—
282C42—10841no—
283C23—2712no—
284C23—2712no—
285C23—2712no—
286C23—5421no—
287C23—5421no—
288C23—5421no—
289C23—5421no—
290C23C23×D27111yescharacteristic
291Q8—44no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—2712no—
309C2×C4—2712no—
310C2×C4—2712no—
311C2×C4—2712no—
312C2×C4—2712no—
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—5421no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—5421no—
329C2×C4—10844no—
330C2×C4—10844no—
331C2×C4—10844no—
332C2×C4C2×C4×D27112yes—
333C2×C4C2×C4×D27112yes—
334C2×C4C2×C4×D27112yes—
335C2×C4Q8×D27112yes—
336C2×C4#1686465512yes—
337C22—2714no—
338C22—5422no—
339C22—5422no—
340C22—5422no—
341C22—5422no—
342C22—5422no—
343C22—5422no—
344C22C22×C4×D27111yescharacteristic
345C22C2×Q8×D27111yescharacteristic
346C22#41×D27111yescharacteristic
347C22#41×D27111yescharacteristic
348C22C2×#1686465511yescharacteristic
349C22#2257620511yescharacteristic
350C22#2257620511yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—5422no—
359C4—5422no—
360C4—5422no—
361C4—5422no—
362C4—5422no—
363C4—10841no—
364C4—10841no—
365C2—5424no—
366C2C4×Q8×D27111yescharacteristic
367C2C4×#1686465511yescharacteristic
368C2#3177437011yescharacteristic
369C2#3177457411yescharacteristic
370C2#3177457711yescharacteristic
371C2#3177460111yescharacteristic
372C2#3177460811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
Fitting#391×C271C2
socle, cosocleC23×C271C24
prime radicalC23×D271C23
hypercenter#391D271
hypocenterC271#2368
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #31773804C2semidirect
3. #31773784C2semidirect
4. #31773782C2semidirect
5. #31773780C2semidirect
6. #31773778C2semidirect
9. C4×Dic271⋊C4C2semidirect
10. #31773748C2semidirect
15. #391×C271C2semidirect
19. #22576136C4semidirect
70. C4⋊Dic271C2×C4semidirect
73. Dic271⋊C4C2×C4semidirect
74. Dic271⋊C4C2×C4semidirect
78. (C2×D271)⋊C4Q8semidirect
86. C2×Q8×C271C2×C4semidirect
186. C271#2368semidirect
190. #391D271semidirect
242. C2×Q8C4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761362
#317738042
#317737842
#317737822
#317737802
#317737782
#317737712
#317737682
C4×Dic271⋊C42
#317737482
C4×(C2×D271)⋊C42
#317737442
#95×D2712
#391×C2712
#2368271