Finite Group Databasebeta

← #46847929

Subgroups — #46847929

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×#22576189C211yesmaximal, characteristic, complemented
2C2×Q16×D271C211yesmaximal, characteristic, complemented
3Q8×D1084C211yesmaximal, characteristic, complemented
4#31774555C211yesmaximal, characteristic
5#31774292C211yesmaximal, characteristic, complemented
6#31774154C211yesmaximal, characteristic, complemented
7#31774125C211yesmaximal, characteristic, complemented
8#31774108C211yesmaximal, characteristic, complemented
9#31774060C211yesmaximal, characteristic, complemented
10#31774054C211yesmaximal, characteristic, complemented
11#31774051C211yesmaximal, characteristic, complemented
12#31774034C211yesmaximal, characteristic, complemented
13#31774002C211yesmaximal, characteristic, complemented
14#31773967C211yesmaximal, characteristic, complemented
15#464×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19C2×Q8×D271C2211yescharacteristic
20#22576189—24no—
21C2×Dic1084C2211yescharacteristic
22C2×Dic1084C2211yescharacteristic
23C2×#16864644C2211yescharacteristic
24C2×#16864644C2211yescharacteristic
25C2×#16864644C2211yescharacteristic
26Q16×D271—24no—
27C2×#16864642C2211yescharacteristic
28C2×#16864642C2211yescharacteristic
29C2×#16864637C2211yescharacteristic
30C2×C8×D271C2211yescharacteristic
31#22576160—21no—
32#22576160C2211yescharacteristic
33#22576146—21no—
34#22576140—21no—
35#22576140—21no—
36#22576140C2211yescharacteristic
37#22576140C2211yescharacteristic
38#22576137—21no—
39#22576136C2211yescharacteristic
40#22576133—21no—
41#22576131C2211yescharacteristic
42C4×Dic542C2211yescharacteristic
43C4×D1084—21no—
44C4×D1084C2211yescharacteristic
45(C22⋊D271)⋊C4—21no—
46#22576091C2211yescharacteristic
47#22576085C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576084C2211yescharacteristic
50#22576083C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52(C2×D271)⋊C8C2211yescharacteristic
53#22576071C2211yescharacteristic
54C2×Q16×C271C2211yescharacteristic
55C2×SD16×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic
57C4×Q8×C271C2211yescharacteristic
58C4⋊C8×C271C2211yescharacteristic
59#103×C271C2211yescharacteristic
60#102×C271C2211yescharacteristic
61#16864653—22no—
62#16864653—41no—
63C2×Dic542—21no—
64C2×Dic542—21no—
65C2×Dic542—21no—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2311yescharacteristic
68C2×Dic542C2311yescharacteristic
69Q8×D271—22no—
70Q8×D271—22no—
71Q8×D271—22no—
72Q8×D271—41no—
73Q8×D271—41no—
74Q8×D271—41no—
75C2×D1084D411yescharacteristic
76C2×C22⋊D271—21no—
77C2×C22⋊D271—21no—
78C22×Dic271—21no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271C2311yescharacteristic
83C2×C4×D271C2311yescharacteristic
84Dic1084—22no—
85Dic1084—22no—
86#16864644—22no—
87#16864644—41no—
88#16864644D414yescomplemented
89#16864642—22no—
90#16864642—41no—
91#16864637—22no—
92C2×C271⋊C8C2311yescharacteristic
93C2×C271⋊C8C2311yescharacteristic
94C8×D271—22no—
95C4⋊Dic271—21no—
96C4⋊Dic271—21no—
97C4⋊Dic271—21no—
98C4⋊Dic271—21no—
99C4⋊Dic271D411yescharacteristic
100C4⋊Dic271C2311yescharacteristic
101Dic271⋊C4—21no—
102Dic271⋊C4—21no—
103C22⋊Dic271—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4D411yescharacteristic
108(C2×D271)⋊C4D411yescharacteristic
109C4×Dic271—21no—
110C2×Q8×C271C2311yescharacteristic
111C2×Q8×C271C2311yescharacteristic
112C2×D4×C271C2311yescharacteristic
113Q16×C271—22no—
114SD16×C271—22no—
115C2×C8×C271C2311yescharacteristic
116C2×C8×C271C2311yescharacteristic
117C4⋊C4×C271—21no—
118C4⋊C4×C271C2311yescharacteristic
119C22⋊C4×C271—21no—
120C22⋊C4×C271—21no—
121C42×C271—21no—
122C42×C271C2311yescharacteristic
123C22×D271C2×D411yescharacteristic
124C22×D271C2×D411yescharacteristic
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—21no—
129Dic542—21no—
130Dic542—42no—
131Dic542—42no—
132Dic542—42no—
133Dic542C2×D412yes—
134D1084—22no—
135C22⋊D271—42no—
136C22⋊D271—42no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—21no—
141C2×Dic271—42no—
142C2×Dic271C2×D411yescharacteristic
143C2×Dic271C2×D411yescharacteristic
144C4×D271—21no—
145C4×D271—21no—
146C4×D271—21no—
147C4×D271—21no—
148C4×D271—42no—
149C4×D271—42no—
150C4×D271—42no—
151C271⋊C8—21no—
152C271⋊C8C2×D412yes—
153C23×C271—21no—
154Q8×C271—21no—
155Q8×C271—21no—
156Q8×C271—21no—
157Q8×C271C2×D412yes—
158D4×C271—21no—
159D4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2411yescharacteristic
166C8×C271—21no—
167C8×C271—21no—
168C22×C271—41no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271C22×D411yescharacteristic
172C2×D271—21no—
173C2×D271—21no—
174C2×D271—21no—
175C2×D271—21no—
176C2×D271—21no—
177C2×D271#13712yescomplemented
178C4×C271—22no—
179C4×C271—21no—
180C4×C271—21no—
181C4×C271—41no—
182C4×C271—41no—
183C4×C271C22×D411yescharacteristic
184C4×C271H(5,𝔽2)11yescharacteristic
185Dic271—21no—
186Dic271—21no—
187Dic271—22no—
188Dic271—41no—
189Dic271—41no—
190Dic271—41no—
191C2×C271—81no—
192C2×C271D4211yescharacteristic
193C2×C271C2×#13711yescharacteristic
194C2×C271#57811yescharacteristic
195D271—22no—
196D271—41no—
197C271#327011yescharacteristic, complemented, Sylow
198#3270—2711nomaximal, Sylow
199C2×#137—2711no—
200C22×Q16—2711no—
201D4×Q8—2711no—
202#548—2711no—
203#497—2711no—
204#470—2711no—
205#465—2711no—
206#464D27111yescharacteristic, complemented
207#462—2711no—
208#452—2711no—
209#451—2711no—
210#451—2711no—
211#448—2711no—
212#441—2711no—
213#435—2711no—
214C2×#41—2711no—
215C22×Q8—2711no—
216C22×Q8—2711no—
217C22×Q8—2711no—
218#137—5424no—
219C2×Q16—2711no—
220C2×Q16—2711no—
221C2×Q16—2711no—
222C2×Q16—2711no—
223C2×Q16—2711no—
224C2×Q16—5424no—
225C2×Q16C2×D27111yescharacteristic
226C2×SD16—2711no—
227C2×SD16—2711no—
228C2×SD16C2×D27111yescharacteristic
229C2×M16—2711no—
230C22×C8—2711no—
231C4⋊Q8—2711no—
232C4⋊Q8—5421no—
233#124—5421no—
234#124C2×D27111yescharacteristic
235C22⋊Q8—2711no—
236C22⋊Q8—2711no—
237C22⋊Q8—2711no—
238C22⋊Q8—5421no—
239C22⋊Q8—5421no—
240C22⋊Q8—5421no—
241#121—2711no—
242#121—5421no—
243C4×Q8—2711no—
244C4×Q8C2×D27111yescharacteristic
245C4×D4—2711no—
246C4×D4—5421no—
247C4×D4—5421no—
248#106—2711no—
249C4⋊C8C2×D27111yescharacteristic
250#103—2711no—
251#103—2711no—
252#103—2711no—
253#103—2711no—
254#103C2×D27111yescharacteristic
255#102C2×D27111yescharacteristic
256C22⋊C8—2711no—
257C22⋊C8—2711no—
258#97—2711no—
259#41—5422no—
260#41—10841no—
261C2×Q8—2711no—
262C2×Q8—2711no—
263C2×Q8—2711no—
264C2×Q8—5421no—
265C2×Q8—5421no—
266C2×Q8—5421no—
267C2×Q8—5422no—
268C2×Q8—5422no—
269C2×Q8—5422no—
270C2×Q8—10841no—
271C2×Q8—10841no—
272C2×Q8—10841no—
273C2×Q8C22×D27111yescharacteristic
274C2×Q8C22×D27111yescharacteristic
275C2×D4—2711no—
276C2×D4—5421no—
277C2×D4—5421no—
278C2×D4C22×D27111yescharacteristic
279C22×C4—2711no—
280C22×C4—2711no—
281C22×C4—5421no—
282C22×C4—5421no—
283C22×C4—5421no—
284C22×C4—5421no—
285Q16—22no—
286Q16—2714no—
287Q16—5422no—
288Q16—5422no—
289Q16—5422no—
290Q16—10841no—
291SD16—22no—
292SD16—5422no—
293SD16—10841no—
294M16—5422no—
295C2×C8—2711no—
296C2×C8—2711no—
297C2×C8—5422no—
298C2×C8C22×D27111yescharacteristic
299C2×C8C22×D27111yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—2711no—
302C4⋊C4—2711no—
303C4⋊C4—5421no—
304C4⋊C4—5421no—
305C4⋊C4—5421no—
306C4⋊C4—5421no—
307C4⋊C4—5421no—
308C4⋊C4—5421no—
309C4⋊C4C22×D27111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—2711no—
313C22⋊C4—2711no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C42—21no—
319C42—5421no—
320C42C22×D27111yescharacteristic
321C23—21no—
322C23—2711no—
323C23—2711no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—2712no—
328Q8—5421no—
329Q8—5421no—
330Q8—5421no—
331Q8—5421no—
332Q8—5421no—
333Q8—10842no—
334Q8—10842no—
335Q8—10842no—
336Q8D4×D27112yes—
337D4—21no—
338D4—21no—
339D4—5422no—
340D4—10842no—
341D4—10842no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—2711no—
346C2×C4—2711no—
347C2×C4—5421no—
348C2×C4—5421no—
349C2×C4—5421no—
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×C4D4×D27111yescharacteristic
360C2×C4D4×D27111yescharacteristic
361C2×C4C23×D27111yescharacteristic
362C8—21no—
363C8—21no—
364C8—2712no—
365C8—5421no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—2712no—
370C22—5421no—
371C22—5421no—
372C22—5421no—
373C22—5421no—
374C22—5421no—
375C22C2×D4×D27111yescharacteristic
376C4—22no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—5421no—
382C4—5421no—
383C4—5422no—
384C4—10841no—
385C4—10841no—
386C4—10841no—
387C4C2×D4×D27111yescharacteristic
388C4#2257620611yescharacteristic
389C2—81no—
390C2—5422no—
391C2—10841no—
392C2#3177453111yescharacteristic
393C2#137×D27111yescharacteristic
394C2#3177471111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#464×C271C2
socleC22×C271C22×D4
prime radical#31774555C2
hypercenter#464D271
hypocenterC271#3270
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#22576189C2semidirect
2. C2×Q16×D271C2semidirect
3. Q8×D1084C2semidirect
5. #31774292C2semidirect
6. #31774154C2semidirect
7. #31774125C2semidirect
8. #31774108C2semidirect
9. #31774060C2semidirect
10. #31774054C2semidirect
11. #31774051C2semidirect
12. #31774034C2semidirect
13. #31774002C2semidirect
14. #31773967C2semidirect
15. #464×C271C2semidirect
88. #16864644D4semidirect
177. C2×D271#137semidirect
197. C271#3270semidirect
206. #464D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761892
C2×Q16×D2712
Q8×D10842
#317745552
#317742922
#317741542
#317741252
#317741082
#317740602
#317740542
#317740512
#317740342
#317740022
#317739672
#464×C2712
#3270271