Finite Group Databasebeta

← #46845987

Subgroups — #46845987

Order 34688.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576189C211yesmaximal, characteristic, complemented
2C22×#16864637C211yesmaximal, characteristic, complemented
3#31774407C211yesmaximal, characteristic
4C22⋊Q8×D271C211yesmaximal, characteristic, complemented
5#31774156C211yesmaximal, characteristic, complemented
6#31774154C211yesmaximal, characteristic, complemented
7#31774142C211yesmaximal, characteristic, complemented
8#31774140C211yesmaximal, characteristic, complemented
9#31774138C211yesmaximal, characteristic, complemented
10#31774136C211yesmaximal, characteristic, complemented
11#31774124C211yesmaximal, characteristic, complemented
12#31773936C211yesmaximal, characteristic, complemented
13#31773904C211yesmaximal, characteristic, complemented
14#31773895C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C22×C4×D271C2211yescharacteristic
19#22576189—24no—
20C2×Dic1084C2211yescharacteristic, complemented
21C2×#16864644C2211yescharacteristic, complemented
22C2×#16864642C2211yescharacteristic, complemented
23C2×#16864637—24no—
24C2×#16864637C2211yescharacteristic
25C2×#16864637C2211yescharacteristic
26C2×#16864637C2211yescharacteristic
27C2×#16864637C2211yescharacteristic
28C22×C271⋊C8C2211yescharacteristic
29#22576142—21no—
30#22576140C2211yescharacteristic
31#22576140C2211yescharacteristic
32#22576139C2211yescharacteristic
33#22576139C2211yescharacteristic
34#22576138—21no—
35#22576137—21no—
36#22576136C2211yescharacteristic
37#22576131C2211yescharacteristic
38#22576128—21no—
39D4×Dic271—21no—
40(C22⋊D271)⋊C4—21no—
41#22576118C2211yescharacteristic
42C4⋊C4×D271—21no—
43C4⋊C4×D271C2211yescharacteristic
44C22⋊C4×D271—21no—
45#22576094C2211yescharacteristic, complemented
46#22576093C2211yescharacteristic, complemented
47#22576091C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576085C2211yescharacteristic, complemented
50#22576084C2211yescharacteristic, complemented
51#22576083C2211yescharacteristic, complemented
52#22576083C2211yescharacteristic, complemented
53#22576081C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic, complemented
55C22×C8×C271C2211yescharacteristic
56C22⋊Q8×C271C2211yescharacteristic
57#121×C271C2211yescharacteristic
58#106×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271D411yescharacteristic
62#16864653—22no—
63#16864653—22no—
64C2×Dic542—21no—
65C2×Dic542C2311yescharacteristic
66C2×Dic542C2311yescharacteristic
67Q8×D271—22no—
68Q8×D271—22no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C22×Dic271—21no—
72C22×Dic271D411yescharacteristic
73C2×C4×D271—22no—
74C2×C4×D271—22no—
75C2×C4×D271—21no—
76C2×C4×D271—21no—
77C2×C4×D271D411yescharacteristic
78C2×C4×D271D411yescharacteristic
79C2×C4×D271C2311yescharacteristic
80C2×C4×D271C2311yescharacteristic
81Dic1084—22no—
82#16864644—22no—
83#16864642—22no—
84#16864637—22no—
85#16864637—22no—
86#16864637—22no—
87#16864637D414yescomplemented
88C2×C271⋊C8—22no—
89C2×C271⋊C8C2311yescharacteristic
90C2×C271⋊C8C2311yescharacteristic
91C4⋊Dic271—21no—
92C4⋊Dic271—21no—
93C4⋊Dic271C2311yescharacteristic
94C4⋊Dic271C2311yescharacteristic
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99C22⋊Dic271—21no—
100C22⋊Dic271—21no—
101C22⋊Dic271—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106C4×Dic271—21no—
107C2×Q8×C271C2311yescharacteristic
108C2×D4×C271—21no—
109C2×D4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111SD16×C271—22no—
112C2×C8×C271—22no—
113C2×C8×C271C2311yescharacteristic
114C2×C8×C271C2311yescharacteristic
115C4⋊C4×C271—21no—
116C4⋊C4×C271C2311yescharacteristic
117C4⋊C4×C271C2311yescharacteristic
118C22⋊C4×C271—21no—
119C22⋊C4×C271—21no—
120C22×D271—22no—
121C22×D271—22no—
122C22×D271—41no—
123C22×D271C2×D411yescharacteristic
124C22×D271C2×D411yescharacteristic
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—21no—
129Dic542—42no—
130C22⋊D271—42no—
131C22⋊D271—42no—
132C2×Dic271—22no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—42no—
138C2×Dic271C2×D411yescharacteristic
139C2×Dic271C2×D411yescharacteristic
140C4×D271—22no—
141C4×D271—21no—
142C4×D271—21no—
143C4×D271—22no—
144C4×D271—42no—
145C4×D271—42no—
146C4×D271C2×D412yes—
147C4×D271#4112yes—
148C271⋊C8—21no—
149C271⋊C8C2×D412yes—
150C23×C271—21no—
151C23×C271C2×D411yescharacteristic
152Q8×C271—21no—
153Q8×C271—21no—
154D4×C271—21no—
155D4×C271—21no—
156D4×C271—42no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271C2×D411yescharacteristic
163C2×C4×C271C2411yescharacteristic
164C8×C271—21no—
165C8×C271C2×D412yes—
166C22×C271—21no—
167C22×C271—21no—
168C22×C271—41no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271#13712yescomplemented
172C22×C271C22×D411yescharacteristic
173C2×D271—21no—
174C2×D271—21no—
175C2×D271—21no—
176C2×D271—21no—
177C2×D271—21no—
178C2×D271—44no—
179C2×D271#12112yes—
180C4×C271—21no—
181C4×C271—41no—
182C4×C271—41no—
183C4×C271—41no—
184C4×C271C22×D411yescharacteristic
185C4×C271C2×#4111yescharacteristic
186Dic271—21no—
187Dic271—41no—
188Dic271—41no—
189Dic271—41no—
190Dic271—41no—
191Dic271#12112yes—
192C2×C271—22no—
193C2×C271—81no—
194C2×C271C2×#12111yescharacteristic
195C2×C271C2×#13711yescharacteristic
196C2×C271#57511yescharacteristic
197D271—22no—
198D271—41no—
199C271#305611yescharacteristic, complemented, Sylow
200#3056—2711nomaximal, Sylow
201C2×#137—2711no—
202C22×M16—2711no—
203#525—2711no—
204C2×C22⋊Q8—2711no—
205#470—2711no—
206#470—2711no—
207#468—2711no—
208#468—2711no—
209#467—2711no—
210#467—2711no—
211#465—2711no—
212#465D27111yescharacteristic, complemented
213#428—2711no—
214#419—2711no—
215#417—2711no—
216C2×#41—2711no—
217C22×Q8—2711no—
218C23×C4—2711no—
219#137—5424no—
220C2×Q16—2711no—
221C2×Q16—2711no—
222C2×SD16—2711no—
223C2×SD16C2×D27111yescharacteristic, complemented
224C2×M16—2711no—
225C2×M16—2711no—
226C2×M16—2711no—
227C2×M16—2711no—
228C2×M16—5424no—
229C22×C8—2711no—
230C22×C8C2×D27111yescharacteristic
231#123—5421no—
232C22⋊Q8—2711no—
233C22⋊Q8—2711no—
234C22⋊Q8—2711no—
235C22⋊Q8—2711no—
236C22⋊Q8—2711no—
237C22⋊Q8—5421no—
238C22⋊Q8—5421no—
239C22⋊Q8C2×D27111yescharacteristic
240#121—2711no—
241#121C2×D27111yescharacteristic
242#120—5421no—
243C4×D4—5421no—
244C4×D4—5421no—
245#117—2711no—
246C2×C4⋊C4—2711no—
247C2×C4⋊C4—5421no—
248C2×C22⋊C4—5421no—
249#107—2711no—
250#107—2711no—
251#106—2711no—
252#106C2×D27111yescharacteristic, complemented
253#103—2711no—
254#103—2711no—
255#103—2711no—
256#103—2711no—
257#103—2711no—
258#103C2×D27111yescharacteristic, complemented
259#102—2711no—
260#102C2×D27111yescharacteristic, complemented
261C24—2711no—
262#41—5422no—
263#41—5422no—
264C2×Q8—2711no—
265C2×Q8—2711no—
266C2×Q8—5421no—
267C2×Q8—5422no—
268C2×Q8—5422no—
269C2×Q8C22×D27111yescharacteristic
270C2×D4—21no—
271C2×D4—5421no—
272C2×D4—5421no—
273C2×D4C22×D27111yescharacteristic
274C22×C4—2711no—
275C22×C4—2711no—
276C22×C4—2711no—
277C22×C4—2711no—
278C22×C4—2711no—
279C22×C4—5422no—
280C22×C4—5422no—
281C22×C4—5421no—
282C22×C4—5421no—
283C22×C4—5421no—
284C22×C4C22×D27111yescharacteristic
285Q16—5422no—
286Q16—5422no—
287SD16—22no—
288SD16—5422no—
289M16—2714no—
290M16—5422no—
291M16—5422no—
292M16—5422no—
293C2×C8—22no—
294C2×C8—2711no—
295C2×C8—2711no—
296C2×C8—5422no—
297C2×C8C22×D27111yescharacteristic
298C2×C8C22×D27111yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—2711no—
301C4⋊C4—2711no—
302C4⋊C4—5421no—
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—5421no—
313C22⋊C4—5421no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C42—5421no—
320C23—21no—
321C23—2711no—
322C23—2711no—
323C23—5422no—
324C23—5422no—
325C23—10841no—
326C23D4×D27111yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—5421no—
330Q8—5421no—
331Q8—5421no—
332Q8—5421no—
333Q8—10842no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—10842no—
338D4—10842no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—2711no—
345C2×C4—2712no—
346C2×C4—2712no—
347C2×C4—2711no—
348C2×C4—5422no—
349C2×C4—5422no—
350C2×C4—5422no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—5421no—
356C2×C4—5421no—
357C2×C4—10842no—
358C2×C4—10842no—
359C2×C4—10842no—
360C2×C4D4×D27111yescharacteristic
361C2×C4C23×D27111yescharacteristic
362C8—21no—
363C8—2712no—
364C8—5421no—
365C8D4×D27112yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—2712no—
372C22—5421no—
373C22—5421no—
374C22—5421no—
375C22—5421no—
376C22—5421no—
377C22—10844no—
378C22#2257618912yescomplemented
379C22C2×D4×D27111yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—2712no—
385C4—5421no—
386C4—10841no—
387C4—10841no—
388C4—10841no—
389C4—10841no—
390C4C2×D4×D27111yescharacteristic
391C4#41×D27111yescharacteristic
392C2—22no—
393C2—81no—
394C2—5422no—
395C2—10841no—
396C2#121×D27111yescharacteristic
397C2C2×#2257618911yescharacteristic
398C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
prime radical#31774407C2
hypercenter#465D271
hypocenterC271#3056
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#22576189C2semidirect
2. C22×#16864637C2semidirect
4. C22⋊Q8×D271C2semidirect
5. #31774156C2semidirect
6. #31774154C2semidirect
7. #31774142C2semidirect
8. #31774140C2semidirect
9. #31774138C2semidirect
10. #31774136C2semidirect
11. #31774124C2semidirect
12. #31773936C2semidirect
13. #31773904C2semidirect
14. #31773895C2semidirect
15. #465×C271C2semidirect
20. C2×Dic1084C22semidirect
21. C2×#16864644C22semidirect
22. C2×#16864642C22semidirect
45. #22576094C22semidirect
46. #22576093C22semidirect
47. #22576091C22semidirect
48. #22576085C22semidirect
49. #22576085C22semidirect
50. #22576084C22semidirect
51. #22576083C22semidirect
52. #22576083C22semidirect
53. #22576081C22semidirect
54. C2×SD16×C271C22semidirect
58. #106×C271C22semidirect
59. #103×C271C22semidirect
60. #102×C271C22semidirect
87. #16864637D4semidirect
171. C22×C271#137semidirect
199. C271#3056semidirect
212. #465D271semidirect
223. C2×SD16C2×D271semidirect
252. #106C2×D271semidirect
258. #103C2×D271semidirect
260. #102C2×D271semidirect
378. C22#22576189semidirect

Maximal subgroups

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

subgroupindex
C2×#225761892
C22×#168646372
#317744072
C22⋊Q8×D2712
#317741562
#317741542
#317741422
#317741402
#317741382
#317741362
#317741242
#317739362
#317739042
#317738952
#465×C2712
#3056271