Finite Group Databasebeta

← #46839430

Subgroups — #46839430

Order 34688.

632 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576146C211yesmaximal, characteristic
2C2×#22576125C211yesmaximal, characteristic, complemented
3C2×(C22⋊D271)⋊C4C211yesmaximal, characteristic
4#31773801C211yesmaximal, characteristic, complemented
5#31773797C211yesmaximal, characteristic
6#31773786C211yesmaximal, characteristic, complemented
7#31773786C211yesmaximal, characteristic, complemented
8#31773785C211yesmaximal, characteristic, complemented
9#31773769C211yesmaximal, characteristic, complemented
10#31773768C211yesmaximal, characteristic, complemented
11C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
12#31773748C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic271C211yesmaximal, characteristic, complemented
14C4×(C2×D271)⋊C4C211yesmaximal, characteristic, complemented
15#388×C271C211yesmaximal, characteristic, complemented
16C22×Dic542C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C23×Dic271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20#22576146C418yescomplemented
21#22576125—24no—
22(C22⋊D271)⋊C4—24no—
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×C22⋊Dic271C2211yescharacteristic
29C2×C22⋊Dic271C2211yescharacteristic
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×C4×Dic271C2211yescharacteristic
36C2×C4×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
47C2×C4⋊C4×C271C2211yescharacteristic
48C2×C22⋊C4×C271C2211yescharacteristic
49C2×C22⋊C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271C2211yescharacteristic
51C2×C42×C271C2211yescharacteristic
52#95×C271C2211yescharacteristic
53#95×C271C2211yescharacteristic
54C23×D271C2311yescharacteristic
55C2×Dic542—22no—
56C2×Dic542—22no—
57C2×Dic542C2×C414yescomplemented
58C2×C22⋊D271—22no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271C2×C414yes—
61C22×Dic271—22no—
62C22×Dic271—22no—
63C22×Dic271—41no—
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C2×C4×D271—22no—
72C2×C4×D271—22no—
73C2×C4×D271—41no—
74Dic271⋊C4—22no—
75Dic271⋊C4—22no—
76Dic271⋊C4—22no—
77Dic271⋊C4—22no—
78Dic271⋊C4D414yes—
79C22⋊Dic271—22no—
80C22⋊Dic271C2×C414yescomplemented
81(C2×D271)⋊C4—22no—
82(C2×D271)⋊C4—22no—
83(C2×D271)⋊C4—41no—
84(C2×D271)⋊C4C2×C414yescomplemented
85(C2×D271)⋊C4C2×C414yescomplemented
86C4×Dic271—22no—
87C4×Dic271—22no—
88C4×Dic271—22no—
89C4×Dic271—22no—
90C4×Dic271—41no—
91C4×Dic271C2×C414yescomplemented
92C24×C271C2311yescharacteristic
93C22×C4×C271C2311yescharacteristic
94C22×C4×C271C2311yescharacteristic
95C22×C4×C271C2311yescharacteristic
96C22×C4×C271C2311yescharacteristic
97C22×C4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C4⋊C4×C271—22no—
100C22⋊C4×C271—22no—
101C22⋊C4×C271—41no—
102C22⋊C4×C271C2×C414yescomplemented
103C42×C271—41no—
104C22×D271—21no—
105C22×D271—21no—
106C22×D271—21no—
107C22×D271—21no—
108C22×D271—21no—
109C22×D271—21no—
110C22×D271C22×C412yes—
111Dic542—28no—
112C22⋊D271—28no—
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—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—44no—
127C2×Dic271C22×C412yes—
128C2×Dic271C22×C412yes—
129C2×Dic271C22×C412yes—
130C2×Dic271C2×D412yes—
131C2×Dic271C2×D412yes—
132C2×Dic271#4112yes—
133C2×Dic271#4112yes—
134C2×Dic271#4112yes—
135C4×D271—44no—
136C23×C271—21no—
137C23×C271—21no—
138C23×C271—21no—
139C23×C271—21no—
140C23×C271—21no—
141C23×C271—21no—
142C23×C271C22×C412yes—
143C23×C271C2411yescharacteristic
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×C271—21no—
154C2×C4×C271—21no—
155C2×C4×C271—21no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271C22×C412yes—
160C2×C4×C271C22×C412yes—
161C2×C4×C271C2×D412yes—
162C22×C271—22no—
163C22×C271—22no—
164C22×C271—22no—
165C22×C271—41no—
166C22×C271—41no—
167C22×C271—41no—
168C22×C271—41no—
169C22×C271C23×C411yescharacteristic
170C22×C271C22×D411yescharacteristic
171C22×C271C2×#4111yescharacteristic
172C22×C271C2×#4111yescharacteristic
173C22×C271C2×#4111yescharacteristic
174C22×C271H(5,𝔽2)11yescharacteristic
175C22×C2712−1+411yescharacteristic
176C2×D271—22no—
177C2×D271—22no—
178C2×D271—22no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271—41no—
183C4×C271—22no—
184C4×C271—22no—
185C4×C271—22no—
186C4×C271—41no—
187C4×C271—41no—
188C4×C271—41no—
189Dic271—22no—
190Dic271—22no—
191Dic271—22no—
192Dic271—22no—
193Dic271—22no—
194Dic271—41no—
195Dic271C4×D414yes—
196C2×C271—42no—
197C2×C271C2×C4×D411yescharacteristic
198C2×C271C4×#4111yescharacteristic
199C2×C271#52011yescharacteristic
200C2×C271#54611yescharacteristic
201C2×C271#54811yescharacteristic
202C2×C271#55611yescharacteristic
203C2×C271#55911yescharacteristic
204D271—42no—
205C271#236711yescharacteristic, complemented, Sylow
206#2367—2711nomaximal, Sylow
207C2×#124—2711no—
208C2×C4×Q8—2711no—
209C2×C4×D4—2711no—
210#391—2711no—
211#390—2711no—
212#388—2711no—
213#388—2711no—
214#388—2711no—
215#388D27111yescharacteristic, complemented
216#385—2711no—
217#385—2711no—
218C4×C4⋊C4—2711no—
219C4×C4⋊C4—2711no—
220C4×C22⋊C4—2711no—
221C4×C22⋊C4—2711no—
222C22×Q8—2711no—
223C22×D4—2711no—
224C23×C4—2711no—
225C23×C4—2711no—
226#124—2718no—
227C4×Q8—5424no—
228C4×D4—5424no—
229C2×C4⋊C4—2711no—
230C2×C4⋊C4—2711no—
231C2×C4⋊C4—2711no—
232C2×C4⋊C4—2711no—
233C2×C4⋊C4—2711no—
234C2×C4⋊C4C2×D27111yescharacteristic
235C2×C22⋊C4—2711no—
236C2×C22⋊C4—2711no—
237C2×C22⋊C4—2711no—
238C2×C22⋊C4—2711no—
239C2×C22⋊C4—2711no—
240C2×C22⋊C4—2711no—
241C2×C22⋊C4—2711no—
242C2×C22⋊C4C2×D27111yescharacteristic
243C2×C22⋊C4C2×D27111yescharacteristic
244C2×C22⋊C4C2×D27111yescharacteristic
245C2×C42—2711no—
246C2×C42—2711no—
247C2×C42—2711no—
248C2×C42—2711no—
249C2×C42—2711no—
250C2×C42—2711no—
251C2×C42C2×D27111yescharacteristic
252#95—2711no—
253#95—2711no—
254#95—2711no—
255#95—2711no—
256#95—2711no—
257#95—2711no—
258#95C2×D27111yescharacteristic
259#95C2×D27111yescharacteristic
260C24—2711no—
261C24C22×D27111yescharacteristic
262C2×Q8—2714no—
263C2×Q8—5422no—
264C2×Q8—5422no—
265C2×D4—2714no—
266C2×D4—5422no—
267C2×D4—5422no—
268C22×C4—2711no—
269C22×C4—2711no—
270C22×C4—2711no—
271C22×C4—2711no—
272C22×C4—2711no—
273C22×C4—2711no—
274C22×C4—2711no—
275C22×C4—5422no—
276C22×C4—5422no—
277C22×C4—5422no—
278C22×C4—5422no—
279C22×C4—10841no—
280C22×C4—10841no—
281C22×C4C22×D27111yescharacteristic
282C22×C4C22×D27111yescharacteristic
283C22×C4C22×D27111yescharacteristic
284C22×C4C22×D27111yescharacteristic
285C22×C4C22×D27111yescharacteristic
286C22×C4C22×D27111yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—2714no—
289C4⋊C4—5422no—
290C4⋊C4—5422no—
291C4⋊C4—5422no—
292C4⋊C4—5422no—
293C22⋊C4—22no—
294C22⋊C4—41no—
295C22⋊C4—2714no—
296C22⋊C4—2714no—
297C22⋊C4—2714no—
298C22⋊C4—5422no—
299C22⋊C4—5422no—
300C22⋊C4—5422no—
301C22⋊C4—10841no—
302C22⋊C4C4×D27114yescomplemented
303C42—41no—
304C42—2714no—
305C42—5422no—
306C42—5422no—
307C42—5422no—
308C42—5422no—
309C42—10841no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—2712no—
317C23—5421no—
318C23—5421no—
319C23—5421no—
320C23—5421no—
321C23—5421no—
322C23—5421no—
323C23C2×C4×D27112yes—
324C23C23×D27111yescharacteristic
325Q8—5428no—
326D4—5428no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2712no—
343C2×C4—2712no—
344C2×C4—2712no—
345C2×C4—2712no—
346C2×C4—2712no—
347C2×C4—2712no—
348C2×C4—2712no—
349C2×C4—2712no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—5421no—
356C2×C4—5421no—
357C2×C4—5421no—
358C2×C4—5421no—
359C2×C4—5421no—
360C2×C4—5421no—
361C2×C4—5421no—
362C2×C4—5421no—
363C2×C4—10844no—
364C2×C4—10844no—
365C2×C4C2×C4×D27112yes—
366C2×C4C2×C4×D27112yes—
367C2×C4D4×D27112yes—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—5422no—
376C22—5422no—
377C22—5422no—
378C22—10841no—
379C22—10841no—
380C22—10841no—
381C22—10841no—
382C22C22×C4×D27111yescharacteristic
383C22C2×D4×D27111yescharacteristic
384C22#41×D27111yescharacteristic
385C22#41×D27111yescharacteristic
386C22#41×D27111yescharacteristic
387C22#2257620511yescharacteristic
388C22#2257620711yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—2714no—
396C4—5422no—
397C4—5422no—
398C4—5422no—
399C4—5422no—
400C4—5422no—
401C4—10841no—
402C2—42no—
403C2—10842no—
404C2C4×D4×D27111yescharacteristic
405C2#3177436511yescharacteristic
406C2#3177437611yescharacteristic
407C2#3177453611yescharacteristic
408C2#3177455511yescharacteristic
409C2#3177461411yescharacteristic
410C2#3177463111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
Fitting#388×C271C2
socle, cosocleC23×C271C24
prime radicalC22×C22⋊D271C22
hypercenter#388D271
hypocenterC271#2367
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#22576125C2semidirect
4. #31773801C2semidirect
6. #31773786C2semidirect
7. #31773786C2semidirect
8. #31773785C2semidirect
9. #31773769C2semidirect
10. #31773768C2semidirect
11. C4×Dic271⋊C4C2semidirect
12. #31773748C2semidirect
13. C22⋊C4×Dic271C2semidirect
14. C4×(C2×D271)⋊C4C2semidirect
15. #388×C271C2semidirect
20. #22576146C4semidirect
57. C2×Dic542C2×C4semidirect
80. C22⋊Dic271C2×C4semidirect
84. (C2×D271)⋊C4C2×C4semidirect
85. (C2×D271)⋊C4C2×C4semidirect
91. C4×Dic271C2×C4semidirect
102. C22⋊C4×C271C2×C4semidirect
205. C271#2367semidirect
215. #388D271semidirect
302. C22⋊C4C4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761462
C2×#225761252
C2×(C22⋊D271)⋊C42
#317738012
#317737972
#317737862
#317737852
#317737692
#317737682
C4×Dic271⋊C42
#317737482
C22⋊C4×Dic2712
C4×(C2×D271)⋊C42
#388×C2712
#2367271