Finite Group Databasebeta

← #46839956

Subgroups — #46839956

Order 34688.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576159C211yesmaximal, characteristic, complemented
2C2×#22576137C211yesmaximal, characteristic
3C2×#22576137C211yesmaximal, characteristic
4C2×#22576125C211yesmaximal, characteristic, complemented
5#31773831C211yesmaximal, characteristic, complemented
6#31773827C211yesmaximal, characteristic, complemented
7#31773820C211yesmaximal, characteristic
8#31773801C211yesmaximal, characteristic, complemented
9#31773796C211yesmaximal, characteristic, complemented
10#31773792C211yesmaximal, characteristic, complemented
11#31773792C211yesmaximal, characteristic, complemented
12#31773791C211yesmaximal, characteristic
13#31773785C211yesmaximal, characteristic
14C22⋊C4×Dic271C211yesmaximal, characteristic
15#396×C271C211yesmaximal, characteristic
16C22×Dic542C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18C23×Dic271C2211yescharacteristic
19#22576159—24no—
20#22576137—24no—
21#22576137—24no—
22#22576125—24no—
23C2×C4⋊Dic271C2211yescharacteristic
24C2×C4⋊Dic271C2211yescharacteristic
25C2×C4⋊Dic271C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C2×Dic271⋊C4C2211yescharacteristic
30C2×Dic271⋊C4C2211yescharacteristic
31C2×Dic271⋊C4C2211yescharacteristic
32C2×Dic271⋊C4C2211yescharacteristic
33C2×Dic271⋊C4C2211yescharacteristic
34C2×C22⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×C4×Dic271C2211yescharacteristic
38C2×C4×Dic271C2211yescharacteristic
39C2×C4×Dic271C2211yescharacteristic
40C2×C4×Dic271C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C2×C4⋊C4×C271C2211yescharacteristic
49C2×C4⋊C4×C271C2211yescharacteristic
50C2×C4⋊C4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54#95×C271C2211yescharacteristic
55C2×Dic542—22no—
56C2×Dic542—22no—
57C2×Dic542—22no—
58C2×Dic542—22no—
59C2×Dic542—22no—
60C2×Dic542D414yes—
61C22×Dic271—22no—
62C22×Dic271—22no—
63C22×Dic271—22no—
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C4⋊Dic271—22no—
73C4⋊Dic271—22no—
74C4⋊Dic271—22no—
75Dic271⋊C4—22no—
76Dic271⋊C4—22no—
77Dic271⋊C4—22no—
78Dic271⋊C4—22no—
79Dic271⋊C4—22no—
80Dic271⋊C4—22no—
81Dic271⋊C4—22no—
82Dic271⋊C4D414yes—
83C22⋊Dic271—22no—
84C22⋊Dic271—22no—
85C22⋊Dic271Q814yescomplemented
86C4×Dic271—22no—
87C4×Dic271—22no—
88C4×Dic271—22no—
89C4×Dic271—22no—
90C24×C271C2311yescharacteristic
91C22×C4×C271C2311yescharacteristic
92C22×C4×C271C2311yescharacteristic
93C22×C4×C271C2311yescharacteristic
94C22×C4×C271C2311yescharacteristic
95C22×C4×C271C2311yescharacteristic
96C22×C4×C271C2311yescharacteristic
97C4⋊C4×C271—22no—
98C4⋊C4×C271—22no—
99C4⋊C4×C271—22no—
100C22⋊C4×C271—22no—
101C22⋊C4×C271—22no—
102C22⋊C4×C271—22no—
103Dic542—28no—
104Dic542—44no—
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×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—44no—
123C2×Dic271C2×D412yes—
124C2×Dic271C2×D412yes—
125C2×Dic271C2×D412yes—
126C2×Dic271C2×D412yes—
127C2×Dic271C2×Q812yes—
128C2×Dic271C2×Q812yes—
129C2×Dic271#4112yes—
130C23×C271—21no—
131C23×C271—21no—
132C23×C271—21no—
133C23×C271—21no—
134C23×C271—21no—
135C23×C271—21no—
136C23×C271C2×Q812yes—
137C23×C271C2411yescharacteristic
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×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271C2×D412yes—
154C2×C4×C271C2×D412yes—
155C2×C4×C271#4112yes—
156C22×C271—22no—
157C22×C271—22no—
158C22×C271—22no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271C22×D411yescharacteristic
164C22×C271C22×D411yescharacteristic
165C22×C271C22×Q811yescharacteristic
166C22×C271C2×#4111yescharacteristic
167C22×C271C2×#4111yescharacteristic
168C22×C2712−1+411yescharacteristic
169C22×C2712−1+411yescharacteristic
170C4×C271—22no—
171C4×C271—22no—
172C4×C271—22no—
173C4×C271—41no—
174C4×C271—41no—
175C4×C271—41no—
176Dic271—22no—
177Dic271—22no—
178Dic271—22no—
179Dic271—22no—
180Dic271—41no—
181Dic271—41no—
182Dic271—41no—
183Dic271#12114yes—
184C2×C271—42no—
185C2×C271C2×#12111yescharacteristic
186C2×C271#53311yescharacteristic
187C2×C271#54111yescharacteristic
188C2×C271#54711yescharacteristic
189C2×C271#54811yescharacteristic
190C2×C271D4×Q811yescharacteristic
191C2×C271D4×Q811yescharacteristic
192C271#242911yescharacteristic, complemented, Sylow
193#2429—2711nomaximal, Sylow
194C2×C4⋊Q8—2711no—
195C2×C22⋊Q8—2711no—
196C2×C22⋊Q8—2711no—
197C2×C4×Q8—2711no—
198#398—2711no—
199#398—2711no—
200#396—2711no—
201#396D27111yescharacteristic
202#391—2711no—
203#389—2711no—
204#389—2711no—
205#389—2711no—
206#388—2711no—
207#388—2711no—
208C4×C22⋊C4—2711no—
209C22×Q8—2711no—
210C22×Q8—2711no—
211C23×C4—2711no—
212C4⋊Q8—5424no—
213C22⋊Q8—5424no—
214C22⋊Q8—5424no—
215C4×Q8—5424no—
216C2×C4⋊C4—2711no—
217C2×C4⋊C4—2711no—
218C2×C4⋊C4—2711no—
219C2×C4⋊C4—2711no—
220C2×C4⋊C4—2711no—
221C2×C4⋊C4—2711no—
222C2×C4⋊C4—2711no—
223C2×C4⋊C4—2711no—
224C2×C4⋊C4—2711no—
225C2×C4⋊C4—2711no—
226C2×C4⋊C4—2711no—
227C2×C4⋊C4C2×D27111yescharacteristic
228C2×C4⋊C4C2×D27111yescharacteristic
229C2×C4⋊C4C2×D27111yescharacteristic
230C2×C22⋊C4—2711no—
231C2×C22⋊C4—2711no—
232C2×C22⋊C4—2711no—
233C2×C22⋊C4C2×D27111yescharacteristic
234C2×C22⋊C4C2×D27111yescharacteristic
235C2×C22⋊C4C2×D27111yescharacteristic
236C2×C42—2711no—
237C2×C42—2711no—
238C2×C42—2711no—
239C2×C42—2711no—
240#95—2711no—
241#95—2711no—
242#95—2711no—
243#95—2711no—
244#95—2711no—
245#95—2711no—
246#95—2711no—
247#95C2×D27111yescharacteristic
248C24C22×D27111yescharacteristic
249C2×Q8—2714no—
250C2×Q8—5422no—
251C2×Q8—5422no—
252C2×Q8—5422no—
253C2×Q8—5422no—
254C2×Q8—5422no—
255C22×C4—2711no—
256C22×C4—2711no—
257C22×C4—2711no—
258C22×C4—2711no—
259C22×C4—2711no—
260C22×C4—2711no—
261C22×C4—2711no—
262C22×C4—2711no—
263C22×C4—5422no—
264C22×C4—5422no—
265C22×C4—5422no—
266C22×C4C22×D27111yescharacteristic
267C22×C4C22×D27111yescharacteristic
268C22×C4C22×D27111yescharacteristic
269C22×C4C22×D27111yescharacteristic
270C22×C4C22×D27111yescharacteristic
271C22×C4C22×D27111yescharacteristic
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—2714no—
276C4⋊C4—5422no—
277C4⋊C4—5422no—
278C4⋊C4—5422no—
279C4⋊C4—5422no—
280C4⋊C4—5422no—
281C4⋊C4—5422no—
282C4⋊C4—5422no—
283C4⋊C4—5422no—
284C4⋊C4—5422no—
285C4⋊C4—5422no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—2714no—
290C22⋊C4—5422no—
291C22⋊C4—5422no—
292C42—5422no—
293C42—5422no—
294C42—5422no—
295C42—5422no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23Q8×D27112yes—
303C23C23×D27111yescharacteristic
304Q8—5428no—
305Q8—10844no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—2712no—
322C2×C4—2712no—
323C2×C4—2712no—
324C2×C4—2712no—
325C2×C4—2712no—
326C2×C4—2712no—
327C2×C4—2712no—
328C2×C4—5421no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—5421no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—5421no—
338C2×C4—5421no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—5421no—
344C2×C4—5421no—
345C2×C4—10844no—
346C2×C4D4×D27112yes—
347C2×C4D4×D27112yes—
348C2×C4#1686465312yes—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22C2×D4×D27111yescharacteristic
357C22C2×D4×D27111yescharacteristic
358C22C2×Q8×D27111yescharacteristic
359C22#41×D27111yescharacteristic
360C22C2×#1686465311yescharacteristic
361C22#2257620711yescharacteristic
362C22#2257620711yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—2714no—
370C4—5422no—
371C4—5422no—
372C4—5422no—
373C4—5422no—
374C4—10841no—
375C4—10841no—
376C4—10841no—
377C2—42no—
378C2#121×D27111yescharacteristic
379C2#3177444911yescharacteristic
380C2#3177450411yescharacteristic
381C2#3177454711yescharacteristic
382C2#3177455311yescharacteristic
383C2#3177456011yescharacteristic
384C2#3177456011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#396×C271C2
prime radicalC24×C271C23
hypercenter#396D271
hypocenterC271#2429
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#22576159C2semidirect
4. C2×#22576125C2semidirect
5. #31773831C2semidirect
6. #31773827C2semidirect
8. #31773801C2semidirect
9. #31773796C2semidirect
10. #31773792C2semidirect
11. #31773792C2semidirect
85. C22⋊Dic271Q8semidirect
192. C271#2429semidirect

Maximal subgroups

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

subgroupindex
C2×#225761592
C2×#225761372
C2×#225761252
#317738312
#317738272
#317738202
#317738012
#317737962
#317737922
#317737912
#317737852
C22⋊C4×Dic2712
#396×C2712
#2429271