Finite Group Databasebeta

← #46848007

Subgroups — #46848007

Order 34688.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576193C211yesmaximal, characteristic, complemented
2C2×#22576192C211yesmaximal, characteristic, complemented
3#31774560C211yesmaximal, characteristic, complemented
4#31774549C211yesmaximal, characteristic
5#31774275C211yesmaximal, characteristic, complemented
6#31774156C211yesmaximal, characteristic, complemented
7#31774106C211yesmaximal, characteristic, complemented
8#31774101C211yesmaximal, characteristic, complemented
9#31774062C211yesmaximal, characteristic, complemented
10#31774057C211yesmaximal, characteristic, complemented
11#31774054C211yesmaximal, characteristic, complemented
12#31774052C211yesmaximal, characteristic, complemented
13#31773984C211yesmaximal, characteristic, complemented
14#31773973C211yesmaximal, characteristic, complemented
15#470×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654C2211yescharacteristic
18C22×Dic542C2211yescharacteristic
19C2×Q8×D271C2211yescharacteristic
20#22576193—24no—
21#22576192—24no—
22C2×Dic1084C2211yescharacteristic
23C2×Dic1084C2211yescharacteristic
24C2×#16864644C2211yescharacteristic
25C2×#16864644C2211yescharacteristic
26C2×#16864644C2211yescharacteristic
27C2×#16864643C2211yescharacteristic
28C2×#16864643C2211yescharacteristic
29C2×#16864641C2211yescharacteristic
30C2×#16864637C2211yescharacteristic
31#22576159—21no—
32#22576158C2211yescharacteristic
33#22576158C2211yescharacteristic
34#22576145—21no—
35#22576142—21no—
36#22576140C2211yescharacteristic
37#22576140C2211yescharacteristic
38#22576138—21no—
39#22576137—21no—
40#22576133—21no—
41#22576125C2211yescharacteristic
42#22576122C2211yescharacteristic
43C4×C22⋊D271C2211yescharacteristic
44(C22⋊D271)⋊C4—21no—
45C4⋊C4×D271—21no—
46Dic271⋊C8C2211yescharacteristic
47#22576083C2211yescharacteristic
48#22576083C2211yescharacteristic
49#22576083C2211yescharacteristic
50#22576080C2211yescharacteristic
51C22⋊(C271⋊C8)C2211yescharacteristic
52(C2×D271)⋊C8C2211yescharacteristic
53C8×Dic271C2211yescharacteristic
54C2×Q16×C271C2211yescharacteristic
55C2×M16×C271C2211yescharacteristic
56C22⋊Q8×C271C2211yescharacteristic
57C22⋊Q8×C271C2211yescharacteristic
58#106×C271C2211yescharacteristic
59#103×C271C2211yescharacteristic
60#103×C271C2211yescharacteristic
61#16864655—22no—
62#16864655—41no—
63#16864654—22no—
64#16864654—41no—
65C2×Dic542—21no—
66C2×Dic542—22no—
67C2×Dic542—21no—
68C2×Dic542—41no—
69C2×Dic542C2311yescharacteristic
70C2×Dic542C2311yescharacteristic
71C2×Dic542C2311yescharacteristic
72Q8×D271—22no—
73Q8×D271—41no—
74C2×D1084—21no—
75C2×D1084C2311yescharacteristic
76C2×C22⋊D271—21no—
77C2×C22⋊D271D411yescharacteristic
78C22×Dic271—21no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271—21no—
83C2×C4×D271C2311yescharacteristic
84Dic1084—22no—
85Dic1084—22no—
86#16864644—22no—
87#16864644—41no—
88#16864644—41no—
89#16864643—22no—
90#16864643D414yescomplemented
91#16864641—22no—
92#16864637—22no—
93C2×C271⋊C8C2311yescharacteristic
94C2×C271⋊C8C2311yescharacteristic
95C4⋊Dic271—21no—
96C4⋊Dic271—21no—
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99Dic271⋊C4—21no—
100Dic271⋊C4—21no—
101Dic271⋊C4—21no—
102Dic271⋊C4D411yescharacteristic
103C22⋊Dic271D411yescharacteristic
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4D411yescharacteristic
108C4×Dic271—21no—
109C4×Dic271C2311yescharacteristic
110C2×Q8×C271C2311yescharacteristic
111C2×Q8×C271C2311yescharacteristic
112C22×C4×C271C2311yescharacteristic
113Q16×C271—22no—
114M16×C271—22no—
115C2×C8×C271C2311yescharacteristic
116C2×C8×C271C2311yescharacteristic
117C4⋊C4×C271—21no—
118C4⋊C4×C271—21no—
119C4⋊C4×C271C2311yescharacteristic
120C4⋊C4×C271C2311yescharacteristic
121C22⋊C4×C271—21no—
122C22⋊C4×C271—21no—
123C22×D271—21no—
124C22×D271C2×D411yescharacteristic
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—21no—
129Dic542—21no—
130Dic542—42no—
131Dic542—42no—
132Dic542C2×D412yes—
133D1084—21no—
134D1084—42no—
135D1084C2×D412yes—
136C22⋊D271—22no—
137C22⋊D271—42no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—21no—
141C2×Dic271—42no—
142C2×Dic271C2×D411yescharacteristic
143C2×Dic271C2×D411yescharacteristic
144C2×Dic271C2×D411yescharacteristic
145C4×D271—21no—
146C4×D271—21no—
147C4×D271—42no—
148C4×D271—42no—
149C4×D271—42no—
150C4×D271—42no—
151C271⋊C8—21no—
152C271⋊C8—21no—
153C23×C271C2×D411yescharacteristic
154Q8×C271—21no—
155Q8×C271—21no—
156Q8×C271—21no—
157Q8×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2411yescharacteristic
166C8×C271—21no—
167C8×C271C2×D412yes—
168C22×C271—21no—
169C22×C271—21no—
170C22×C271—21no—
171C22×C271C22×D411yescharacteristic
172C2×D271—21no—
173C2×D271—22no—
174C2×D271—21no—
175C2×D271—21no—
176C2×D271—41no—
177C2×D271—41no—
178C4×C271—21no—
179C4×C271—41no—
180C4×C271—41no—
181C4×C271—41no—
182C4×C271—41no—
183C4×C271C22×D411yescharacteristic
184C4×C271H(5,𝔽2)11yescharacteristic
185Dic271—21no—
186Dic271—22no—
187Dic271—21no—
188Dic271—41no—
189Dic271—41no—
190Dic271#13712yescomplemented
191C2×C271—41no—
192C2×C271D4211yescharacteristic
193C2×C271C2×#13711yescharacteristic
194C2×C271#57811yescharacteristic
195D271—41no—
196D271—42no—
197C271#327611yescharacteristic, complemented, Sylow
198#3276—2711nomaximal, Sylow
199C2×#137—2711no—
200C2×#137—2711no—
201D4×Q8—2711no—
202#547—2711no—
203#494—2711no—
204#470—2711no—
205#470D27111yescharacteristic, complemented
206#462—2711no—
207#461—2711no—
208#452—2711no—
209#452—2711no—
210#451—2711no—
211#451—2711no—
212C4×SD16—2711no—
213#436—2711no—
214C2×#41—2711no—
215C2×#41—2711no—
216C22×Q8—2711no—
217C22×Q8—2711no—
218#137—5424no—
219#137—5424no—
220C2×Q16—2711no—
221C2×Q16—2711no—
222C2×Q16—2711no—
223C2×Q16—2711no—
224C2×Q16—2711no—
225C2×Q16C2×D27111yescharacteristic
226C2×SD16—2711no—
227C2×SD16—2711no—
228C2×SD16—2711no—
229C2×M16—2711no—
230C2×M16C2×D27111yescharacteristic
231C4⋊Q8—2711no—
232C4⋊Q8—2711no—
233C4⋊Q8—5421no—
234#123—5421no—
235#123—5421no—
236C22⋊Q8—2711no—
237C22⋊Q8—2711no—
238C22⋊Q8—5421no—
239C22⋊Q8—5421no—
240C22⋊Q8C2×D27111yescharacteristic
241C22⋊Q8C2×D27111yescharacteristic
242#121—5421no—
243C4×Q8—2711no—
244C4×D4—2711no—
245C4×D4—2711no—
246C4×D4—5421no—
247C2×C4⋊C4—5421no—
248#106C2×D27111yescharacteristic
249C4⋊C8—2711no—
250#103—2711no—
251#103—2711no—
252#103—2711no—
253#103C2×D27111yescharacteristic
254#103C2×D27111yescharacteristic
255#102—2711no—
256C22⋊C8—2711no—
257C22⋊C8—2711no—
258C4×C8—2711no—
259#41—5422no—
260#41—5422no—
261#41—10841no—
262#41—10841no—
263C2×Q8—2711no—
264C2×Q8—2711no—
265C2×Q8—2711no—
266C2×Q8—5421no—
267C2×Q8—5422no—
268C2×Q8—5421no—
269C2×Q8—5422no—
270C2×Q8—10841no—
271C2×Q8—10841no—
272C2×Q8C22×D27111yescharacteristic
273C2×Q8C22×D27111yescharacteristic
274C2×D4—2711no—
275C2×D4—2711no—
276C2×D4—5421no—
277C2×D4—5421no—
278C22×C4—2711no—
279C22×C4—5421no—
280C22×C4—5421no—
281C22×C4—5421no—
282C22×C4—5421no—
283C22×C4—5421no—
284C22×C4C22×D27111yescharacteristic
285Q16—22no—
286Q16—5422no—
287Q16—5422no—
288Q16—5422no—
289Q16—10841no—
290Q16—10841no—
291SD16—2714no—
292SD16—5422no—
293SD16—5422no—
294M16—22no—
295M16—5422no—
296C2×C8—2711no—
297C2×C8—2711no—
298C2×C8C22×D27111yescharacteristic
299C2×C8C22×D27111yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—2711no—
303C4⋊C4—5421no—
304C4⋊C4—5421no—
305C4⋊C4—5421no—
306C4⋊C4—5421no—
307C4⋊C4—5421no—
308C4⋊C4—5421no—
309C4⋊C4—5421no—
310C4⋊C4C22×D27111yescharacteristic
311C4⋊C4C22×D27111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—2711no—
315C22⋊C4—2711no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C42—2711no—
320C42—5421no—
321C23—2711no—
322C23—5421no—
323C23D4×D27111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—2712no—
329Q8—5421no—
330Q8—5421no—
331Q8—5421no—
332Q8—5421no—
333Q8—5421no—
334Q8—10842no—
335Q8—10842no—
336D4—2712no—
337D4—5422no—
338D4—5421no—
339D4—10842no—
340D4—10842no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—2711no—
348C2×C4—2711no—
349C2×C4—2711no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—10842no—
356C2×C4—10842no—
357C2×C4—10842no—
358C2×C4—10842no—
359C2×C4—10842no—
360C2×C4D4×D27111yescharacteristic
361C2×C4C23×D27111yescharacteristic
362C8—21no—
363C8—5421no—
364C8—5421no—
365C8D4×D27112yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—5421no—
370C22—5422no—
371C22—5421no—
372C22—5421no—
373C22—10841no—
374C22—10841no—
375C22C2×D4×D27111yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—2712no—
382C4—5421no—
383C4—5422no—
384C4—5421no—
385C4—10841no—
386C4—10841no—
387C4C2×D4×D27111yescharacteristic
388C4#2257620611yescharacteristic
389C2—41no—
390C2—10841no—
391C2—10842no—
392C2#3177453111yescharacteristic
393C2#137×D27111yescharacteristic
394C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#470×C271C2
socleC22×C271C22×D4
prime radical#31774549C2
hypercenter#470D271
hypocenterC271#3276
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#22576193C2semidirect
2. C2×#22576192C2semidirect
3. #31774560C2semidirect
5. #31774275C2semidirect
6. #31774156C2semidirect
7. #31774106C2semidirect
8. #31774101C2semidirect
9. #31774062C2semidirect
10. #31774057C2semidirect
11. #31774054C2semidirect
12. #31774052C2semidirect
13. #31773984C2semidirect
14. #31773973C2semidirect
15. #470×C271C2semidirect
90. #16864643D4semidirect
190. Dic271#137semidirect
197. C271#3276semidirect
205. #470D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761932
C2×#225761922
#317745602
#317745492
#317742752
#317741562
#317741062
#317741012
#317740622
#317740572
#317740542
#317740522
#317739842
#317739732
#470×C2712
#3276271