Finite Group Databasebeta

← #46839230

Subgroups — #46839230

Order 34688.

604 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576118C211yesmaximal, characteristic
2C2×#22576117C211yesmaximal, characteristic
3C2×C4⋊C4×D271C211yesmaximal, characteristic
4#31773794C211yesmaximal, characteristic, complemented
5#31773792C211yesmaximal, characteristic, complemented
6#31773783C211yesmaximal, characteristic, complemented
7#31773782C211yesmaximal, characteristic, complemented
8#31773780C211yesmaximal, characteristic, complemented
9#31773759C211yesmaximal, characteristic
10#31773759C211yesmaximal, characteristic
11#31773756C211yesmaximal, characteristic
12C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
13#31773748C211yesmaximal, characteristic, complemented
14#31773744C211yesmaximal, characteristic
15#387×C271C211yesmaximal, characteristic, complemented
16C22×C4×D271C2211yescharacteristic
17C22×C4×D271C2211yescharacteristic
18C22×C4×D271C2211yescharacteristic
19#22576118—24no—
20#22576117—24no—
21C2×C4⋊Dic271C2211yescharacteristic
22C2×C4⋊Dic271C2211yescharacteristic
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C4⋊C4×D271C418yescomplemented
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×C4×Dic271C2211yescharacteristic
35C2×C4×Dic271C2211yescharacteristic
36C2×C4×Dic271C2211yescharacteristic
37C2×C4×Dic271C2211yescharacteristic
38C2×C4×Dic271C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C2×C4⋊C4×C271C2211yescharacteristic
48C2×C4⋊C4×C271C2211yescharacteristic
49C2×C42×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52#95×C271C2211yescharacteristic
53#95×C271C2211yescharacteristic
54C23×D271C2311yescharacteristic
55C22×Dic271C2311yescharacteristic
56C22×Dic271C2311yescharacteristic
57C22×Dic271C2311yescharacteristic
58C22×Dic271C2311yescharacteristic
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C2×C4×D271—22no—
63C2×C4×D271—22no—
64C2×C4×D271—22no—
65C2×C4×D271—22no—
66C2×C4×D271—22no—
67C2×C4×D271—22no—
68C2×C4×D271C2×C414yes—
69C2×C4×D271C2×C414yes—
70C2×C4×D271C2×C414yes—
71C4⋊Dic271—22no—
72C4⋊Dic271C2×C414yes—
73Dic271⋊C4—22no—
74Dic271⋊C4—22no—
75Dic271⋊C4—22no—
76Dic271⋊C4—22no—
77Dic271⋊C4C2×C414yes—
78Dic271⋊C4C2×C414yes—
79(C2×D271)⋊C4—22no—
80(C2×D271)⋊C4—22no—
81(C2×D271)⋊C4—22no—
82(C2×D271)⋊C4D414yes—
83C4×Dic271—22no—
84C4×Dic271—22no—
85C4×Dic271—22no—
86C4×Dic271—22no—
87C4×Dic271—22no—
88C22×C4×C271C2311yescharacteristic
89C22×C4×C271C2311yescharacteristic
90C22×C4×C271C2311yescharacteristic
91C22×C4×C271C2311yescharacteristic
92C22×C4×C271C2311yescharacteristic
93C22×C4×C271C2311yescharacteristic
94C22×C4×C271C2311yescharacteristic
95C4⋊C4×C271—22no—
96C4⋊C4×C271C2×C414yes—
97C42×C271—22no—
98C22×D271—21no—
99C22×D271—21no—
100C22×D271—21no—
101C22×D271—21no—
102C22×D271C22×C412yes—
103C22×D271C2×D412yes—
104C22×D271#4112yes—
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×Dic271C22×C412yes—
121C2×Dic271C22×C412yes—
122C2×Dic271C22×C412yes—
123C2×Dic271C2×D412yes—
124C2×Dic271#4112yes—
125C2×Dic271#4112yes—
126C4×D271—28no—
127C4×D271—28no—
128C4×D271—28no—
129C23×C271C2411yescharacteristic
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×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271C22×C412yes—
148C2×C4×C271C22×C412yes—
149C2×C4×C271C22×C412yes—
150C2×C4×C271C2×D412yes—
151C22×C271C23×C411yescharacteristic
152C22×C271C22×D411yescharacteristic
153C22×C271C2×#4111yescharacteristic
154C22×C271C2×#4111yescharacteristic
155C22×C271C2×#4111yescharacteristic
156C22×C2712−1+411yescharacteristic
157C22×C2712−1+411yescharacteristic
158C2×D271—22no—
159C2×D271—22no—
160C2×D271—22no—
161C2×D271—22no—
162C2×D271—22no—
163C2×D271—22no—
164C2×D271C4×D414yes—
165C4×C271—22no—
166C4×C271—22no—
167C4×C271—22no—
168C4×C271—22no—
169C4×C271—41no—
170C4×C271—41no—
171C4×C271—41no—
172Dic271—22no—
173Dic271—22no—
174Dic271—22no—
175Dic271—41no—
176Dic271—41no—
177Dic271—41no—
178Dic271#11714yescomplemented
179C2×C271C2×#11711yescharacteristic
180C2×C271C2×C4×D411yescharacteristic
181C2×C271#51911yescharacteristic
182C2×C271#54711yescharacteristic
183C2×C271#54711yescharacteristic
184C2×C271#55211yescharacteristic
185C2×C271#55211yescharacteristic
186D271—24no—
187C271#234411yescharacteristic, complemented, Sylow
188#2344—2711nomaximal, Sylow
189C2×#117—2711no—
190C2×#117—2711no—
191C22×C4⋊C4—2711no—
192#389—2711no—
193#389—2711no—
194#387—2711no—
195#387—2711no—
196#387—2711no—
197#387D27111yescharacteristic, complemented
198#381—2711no—
199#381—2711no—
200#380—2711no—
201C4×C4⋊C4—2711no—
202C4×C4⋊C4—2711no—
203C4×C22⋊C4—2711no—
204C23×C4—2711no—
205C23×C4—2711no—
206C23×C4—2711no—
207#117—5424no—
208#117—5424no—
209C2×C4⋊C4—2711no—
210C2×C4⋊C4—2711no—
211C2×C4⋊C4—2711no—
212C2×C4⋊C4—2718no—
213C2×C4⋊C4—2711no—
214C2×C4⋊C4—2711no—
215C2×C4⋊C4—2711no—
216C2×C4⋊C4—2711no—
217C2×C4⋊C4—2711no—
218C2×C4⋊C4C2×D27111yescharacteristic
219C2×C4⋊C4C2×D27111yescharacteristic
220C2×C22⋊C4—2711no—
221C2×C22⋊C4—2711no—
222C2×C22⋊C4—2711no—
223C2×C22⋊C4—2711no—
224C2×C42—2711no—
225C2×C42—2711no—
226C2×C42—2711no—
227C2×C42—2711no—
228C2×C42—2711no—
229C2×C42C2×D27111yescharacteristic
230#95—2711no—
231#95—2711no—
232#95—2711no—
233#95—2711no—
234#95—2711no—
235#95—2711no—
236#95—2711no—
237#95—2711no—
238#95C2×D27111yescharacteristic
239#95C2×D27111yescharacteristic
240#95C2×D27111yescharacteristic
241#95C2×D27111yescharacteristic
242C24—2711no—
243C22×C4—2711no—
244C22×C4—2711no—
245C22×C4—2711no—
246C22×C4—2711no—
247C22×C4—2714no—
248C22×C4—2714no—
249C22×C4—2714no—
250C22×C4—2711no—
251C22×C4—2711no—
252C22×C4—2711no—
253C22×C4—5422no—
254C22×C4—5422no—
255C22×C4—5422no—
256C22×C4—5422no—
257C22×C4—5422no—
258C22×C4—5422no—
259C22×C4C22×D27111yescharacteristic
260C22×C4C22×D27111yescharacteristic
261C22×C4C22×D27111yescharacteristic
262C22×C4C22×D27111yescharacteristic
263C22×C4C22×D27111yescharacteristic
264C22×C4C22×D27111yescharacteristic
265C22×C4C22×D27111yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—2714no—
268C4⋊C4—2714no—
269C4⋊C4—2714no—
270C4⋊C4—5422no—
271C4⋊C4—5422no—
272C4⋊C4—5422no—
273C4⋊C4—5422no—
274C4⋊C4—5422no—
275C4⋊C4C4×D27114yes—
276C22⋊C4—2714no—
277C22⋊C4—5422no—
278C22⋊C4—5422no—
279C22⋊C4—5422no—
280C42—22no—
281C42—5422no—
282C42—5422no—
283C42—5422no—
284C42—5422no—
285C42—5422no—
286C23—2712no—
287C23—2712no—
288C23—2712no—
289C23—5421no—
290C23—5421no—
291C23—5421no—
292C23—5421no—
293C23C23×D27111yescharacteristic
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—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—2712no—
312C2×C4—2712no—
313C2×C4—2712no—
314C2×C4—2712no—
315C2×C4—2712no—
316C2×C4—2712no—
317C2×C4—5428no—
318C2×C4—5428no—
319C2×C4—5428no—
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—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—5421no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4C2×C4×D27112yes—
336C2×C4C2×C4×D27112yes—
337C2×C4C2×C4×D27112yes—
338C2×C4D4×D27112yes—
339C22—2714no—
340C22—5422no—
341C22—5422no—
342C22—5422no—
343C22—5422no—
344C22—5422no—
345C22—5422no—
346C22C22×C4×D27111yescharacteristic
347C22C2×D4×D27111yescharacteristic
348C22#41×D27111yescharacteristic
349C22#41×D27111yescharacteristic
350C22#41×D27111yescharacteristic
351C22#2257620711yescharacteristic
352C22#2257620811yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—2714no—
361C4—5422no—
362C4—5422no—
363C4—5422no—
364C4—10841no—
365C4—10841no—
366C4—10841no—
367C2—5424no—
368C2#117×D27111yescharacteristic
369C2C4×D4×D27111yescharacteristic
370C2#3177437311yescharacteristic
371C2#3177454511yescharacteristic
372C2#3177454911yescharacteristic
373C2#3177457911yescharacteristic
374C2#3177458311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
Fitting#387×C271C2
socle, cosocleC23×C271C24
prime radicalC23×D271C23
hypercenter#387D271
hypocenterC271#2344
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
4. #31773794C2semidirect
5. #31773792C2semidirect
6. #31773783C2semidirect
7. #31773782C2semidirect
8. #31773780C2semidirect
12. C4×Dic271⋊C4C2semidirect
13. #31773748C2semidirect
15. #387×C271C2semidirect
29. C4⋊C4×D271C4semidirect
178. Dic271#117semidirect
187. C271#2344semidirect
197. #387D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761182
C2×#225761172
C2×C4⋊C4×D2712
#317737942
#317737922
#317737832
#317737822
#317737802
#317737592
#317737562
C4×Dic271⋊C42
#317737482
#317737442
#387×C2712
#2344271