Finite Group Databasebeta

← #46847930

Subgroups — #46847930

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×#22576190C211yesmaximal, characteristic, complemented
2C22×#16864644C211yesmaximal, characteristic, complemented
3D4×Dic542C211yesmaximal, characteristic, complemented
4#31774292C211yesmaximal, characteristic, complemented
5#31774153C211yesmaximal, characteristic, complemented
6#31774124C211yesmaximal, characteristic, complemented
7#31774116C211yesmaximal, characteristic, complemented
8#31774107C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774053C211yesmaximal, characteristic, complemented
11#31774052C211yesmaximal, characteristic, complemented
12#31774035C211yesmaximal, characteristic, complemented
13#31774002C211yesmaximal, characteristic, complemented
14#31773969C211yesmaximal, characteristic, complemented
15#548×C271C211yesmaximal, characteristic
16C22×Dic542C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18#22576190—24no—
19C2×#16864644—24no—
20C2×#16864644C2211yescharacteristic
21C2×#16864644C2211yescharacteristic
22C2×#16864644C2211yescharacteristic
23C2×#16864644C2211yescharacteristic
24C2×#16864644C2211yescharacteristic
25C2×#16864644C2211yescharacteristic
26C2×#16864642C2211yescharacteristic
27C2×#16864642C2211yescharacteristic
28C2×#16864642C2211yescharacteristic
29C2×#16864638C2211yescharacteristic
30C22×C271⋊C8C2211yescharacteristic
31#22576160C2211yescharacteristic
32#22576159—21no—
33#22576139C2211yescharacteristic
34#22576139C2211yescharacteristic
35#22576137—21no—
36#22576137—21no—
37C4×Dic542C2211yescharacteristic
38D4×Dic271—21no—
39#22576091C2211yescharacteristic
40C4⋊(C271⋊C8)C2211yescharacteristic
41#22576084C2211yescharacteristic
42#22576084C2211yescharacteristic
43#22576083C2211yescharacteristic
44#22576083C2211yescharacteristic
45#22576083C2211yescharacteristic
46#22576081C2211yescharacteristic
47C22⋊(C271⋊C8)C2211yescharacteristic
48C22⋊(C271⋊C8)C2211yescharacteristic
49#22576071C2211yescharacteristic
50C2×#41×C271C2211yescharacteristic
51C22×Q8×C271C2211yescharacteristic
52#124×C271—21no—
53#124×C271C2211yescharacteristic
54C22⋊Q8×C271—21no—
55C22⋊Q8×C271C2211yescharacteristic
56#121×C271—21no—
57#121×C271C2211yescharacteristic
58C4×Q8×C271C2211yescharacteristic
59C4×D4×C271—21no—
60C4×D4×C271C2211yescharacteristic
61C2×Dic542—22no—
62C2×Dic542—22no—
63C2×Dic542—21no—
64C2×Dic542—21no—
65C2×Dic542—41no—
66C2×Dic542—41no—
67C2×Dic542C2311yescharacteristic
68C2×Dic542C2311yescharacteristic
69C2×Dic542C2311yescharacteristic
70C22×Dic271—21no—
71C22×Dic271—21no—
72#16864644—22no—
73#16864644—22no—
74#16864644—22no—
75#16864644—22no—
76#16864644—41no—
77#16864644D414yescomplemented
78#16864642—22no—
79#16864642—22no—
80#16864642—41no—
81#16864638—22no—
82C2×C271⋊C8—22no—
83C2×C271⋊C8C2311yescharacteristic
84C2×C271⋊C8C2311yescharacteristic
85C2×C271⋊C8C2311yescharacteristic
86C2×C271⋊C8C2311yescharacteristic
87C4⋊Dic271—21no—
88C4⋊Dic271—21no—
89C4⋊Dic271C2311yescharacteristic
90Dic271⋊C4—21no—
91Dic271⋊C4—21no—
92Dic271⋊C4—21no—
93C22⋊Dic271—21no—
94C22⋊Dic271—21no—
95C4×Dic271—21no—
96#41×C271—22no—
97#41×C271—41no—
98C2×Q8×C271—21no—
99C2×Q8×C271—22no—
100C2×Q8×C271—41no—
101C2×Q8×C271C2311yescharacteristic
102C2×Q8×C271C2311yescharacteristic
103C2×D4×C271—21no—
104C2×D4×C271—21no—
105C2×D4×C271D411yescharacteristic
106C2×D4×C271C2311yescharacteristic
107C22×C4×C271—21no—
108C22×C4×C271—21no—
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111C4⋊C4×C271—21no—
112C4⋊C4×C271—21no—
113C4⋊C4×C271D411yescharacteristic
114C4⋊C4×C271C2311yescharacteristic
115C22⋊C4×C271—21no—
116C22⋊C4×C271—21no—
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C22⋊C4×C271D411yescharacteristic
120C22⋊C4×C271D411yescharacteristic
121C42×C271—21no—
122C42×C271C2311yescharacteristic
123Dic542—21no—
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—42no—
129Dic542—42no—
130Dic542C2×D412yes—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—42no—
136C2×Dic271—42no—
137C271⋊C8—21no—
138C271⋊C8—21no—
139C271⋊C8—21no—
140C271⋊C8C2×D412yes—
141C23×C271—21no—
142C23×C271C2×D411yescharacteristic
143C23×C271C2×D411yescharacteristic
144Q8×C271—21no—
145Q8×C271—21no—
146Q8×C271—21no—
147Q8×C271—42no—
148Q8×C271C2×D412yes—
149D4×C271—22no—
150D4×C271—21no—
151D4×C271—21no—
152D4×C271—42no—
153D4×C271—42no—
154C2×C4×C271—21no—
155C2×C4×C271—21no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—42no—
162C2×C4×C271—42no—
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2×D411yescharacteristic
166C2×C4×C271C2×D411yescharacteristic
167C2×C4×C271C2411yescharacteristic
168C22×C271—21no—
169C22×C271—21no—
170C22×C271—21no—
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—41no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271#13712yescomplemented
177C22×C271C22×D411yescharacteristic
178C4×C271—21no—
179C4×C271—21no—
180C4×C271—22no—
181C4×C271—21no—
182C4×C271—21no—
183C4×C271—41no—
184C4×C271—41no—
185C4×C271C22×D411yescharacteristic
186C4×C271H(5,𝔽2)11yescharacteristic
187Dic271—22no—
188Dic271—41no—
189Dic271—41no—
190Dic271—41no—
191C2×C271—22no—
192C2×C271—41no—
193C2×C271—81no—
194C2×C271D4211yescharacteristic
195C2×C271C2×#13711yescharacteristic
196C2×C271#57811yescharacteristic
197C271#327011yescharacteristic, complemented, Sylow
198#3270—2711nomaximal, Sylow
199C2×#137—2711no—
200C22×Q16—2711no—
201D4×Q8—2711no—
202#548D27111yescharacteristic
203#497—2711no—
204#470—2711no—
205#465—2711no—
206#464—2711no—
207#462—2711no—
208#452—2711no—
209#451—2711no—
210#451—2711no—
211#448—2711no—
212#441—2711no—
213#435—2711no—
214C2×#41C2×D27111yescharacteristic
215C22×Q8—2711no—
216C22×Q8—2711no—
217C22×Q8C2×D27111yescharacteristic
218#137—5424no—
219C2×Q16—2711no—
220C2×Q16—2711no—
221C2×Q16—2711no—
222C2×Q16—2711no—
223C2×Q16—2711no—
224C2×Q16—2711no—
225C2×Q16—5424no—
226C2×SD16—2711no—
227C2×SD16—2711no—
228C2×SD16—2711no—
229C2×M16—2711no—
230C22×C8—2711no—
231C4⋊Q8—2711no—
232C4⋊Q8—5421no—
233#124—21no—
234#124C2×D27111yescharacteristic
235C22⋊Q8—21no—
236C22⋊Q8—2711no—
237C22⋊Q8—2711no—
238C22⋊Q8—5421no—
239C22⋊Q8—5421no—
240C22⋊Q8C2×D27111yescharacteristic
241#121—21no—
242#121C2×D27111yescharacteristic
243C4×Q8—2711no—
244C4×Q8C2×D27111yescharacteristic
245C4×D4—21no—
246C4×D4—5421no—
247C4×D4C2×D27111yescharacteristic
248#106—2711no—
249C4⋊C8—2711no—
250#103—2711no—
251#103—2711no—
252#103—2711no—
253#103—2711no—
254#103—2711no—
255#102—2711no—
256C22⋊C8—2711no—
257C22⋊C8—2711no—
258#97—2711no—
259#41—22no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—2711no—
265C2×Q8—2711no—
266C2×Q8—2711no—
267C2×Q8—5422no—
268C2×Q8—5422no—
269C2×Q8—5421no—
270C2×Q8—5421no—
271C2×Q8—10841no—
272C2×Q8—10841no—
273C2×Q8C22×D27111yescharacteristic
274C2×Q8C22×D27111yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C22⋊D27111yescharacteristic
278C2×D4C22×D27111yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—5421no—
282C22×C4—5421no—
283C22×C4C22×D27111yescharacteristic
284C22×C4C22×D27111yescharacteristic
285Q16—2714no—
286Q16—5422no—
287Q16—5422no—
288Q16—5422no—
289Q16—5422no—
290Q16—10841no—
291SD16—5422no—
292SD16—5422no—
293SD16—10841no—
294M16—5422no—
295C2×C8—2711no—
296C2×C8—2711no—
297C2×C8—2711no—
298C2×C8—2711no—
299C2×C8—5422no—
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⋊C4C22⋊D27111yescharacteristic
309C4⋊C4C22×D27111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C22⋊C4C22⋊D27111yescharacteristic
317C22⋊C4C22⋊D27111yescharacteristic
318C42—21no—
319C42—5421no—
320C42C22×D27111yescharacteristic
321C23—21no—
322C23C2×C22⋊D27111yescharacteristic
323C23C2×C22⋊D27111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—2712no—
329Q8—5421no—
330Q8—5421no—
331Q8—5421no—
332Q8—5421no—
333Q8—5421no—
334Q8—10842no—
335Q8—10842no—
336Q8D4×D27112yes—
337D4—22no—
338D4—21no—
339D4—21no—
340D4—42no—
341D4—42no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—10842no—
356C2×C4—10842no—
357C2×C4C2×C22⋊D27111yescharacteristic
358C2×C4C2×C22⋊D27111yescharacteristic
359C2×C4C2×C22⋊D27111yescharacteristic
360C2×C4C2×C22⋊D27111yescharacteristic
361C2×C4C23×D27111yescharacteristic
362C8—2712no—
363C8—5421no—
364C8—5421no—
365C8—5421no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22#2257619012yescomplemented
375C22C22×C22⋊D27111yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—5422no—
384C4—10841no—
385C4—10841no—
386C4—10841no—
387C4C2×D4×D27111yescharacteristic
388C4#2257620511yescharacteristic
389C2—22no—
390C2—41no—
391C2—81no—
392C2D4×C22⋊D27111yescharacteristic
393C2C2×#2257619011yescharacteristic
394C2#3177471211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting, prime radical#548×C271C2
socleC22×C271C22×D4
hypercenter#548D271
hypocenterC271#3270
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#22576190C2semidirect
2. C22×#16864644C2semidirect
3. D4×Dic542C2semidirect
4. #31774292C2semidirect
5. #31774153C2semidirect
6. #31774124C2semidirect
7. #31774116C2semidirect
8. #31774107C2semidirect
9. #31774059C2semidirect
10. #31774053C2semidirect
11. #31774052C2semidirect
12. #31774035C2semidirect
13. #31774002C2semidirect
14. #31773969C2semidirect
77. #16864644D4semidirect
176. C22×C271#137semidirect
197. C271#3270semidirect
374. C22#22576190semidirect

Maximal subgroups

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

subgroupindex
C2×#225761902
C22×#168646442
D4×Dic5422
#317742922
#317741532
#317741242
#317741162
#317741072
#317740592
#317740532
#317740522
#317740352
#317740022
#317739692
#548×C2712
#3270271