Finite Group Databasebeta

← #46845940

Subgroups — #46845940

Order 34688.

520 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576181C211yesmaximal, characteristic, complemented
2C22×C8×D271C211yesmaximal, characteristic, complemented
3#31774410C211yesmaximal, characteristic, complemented
4#31774407C211yesmaximal, characteristic, complemented
5#31774137C211yesmaximal, characteristic, complemented
6#31774136C211yesmaximal, characteristic, complemented
7#31774132C211yesmaximal, characteristic, complemented
8#31774131C211yesmaximal, characteristic, complemented
9#31774127C211yesmaximal, characteristic, complemented
10#31774126C211yesmaximal, characteristic, complemented
11#31774123C211yesmaximal, characteristic, complemented
12#31773932C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14#31773890C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C22×C4×D271C2211yescharacteristic
19#22576181—24no—
20C2×#16864644C2211yescharacteristic, complemented
21C2×#16864643C2211yescharacteristic, complemented
22C2×#16864639C2211yescharacteristic, complemented
23C22×C271⋊C8C2211yescharacteristic
24C2×C8×D271—24no—
25C2×C8×D271C2211yescharacteristic
26C2×C8×D271C2211yescharacteristic
27C2×C8×D271C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576145—21no—
30#22576142—21no—
31#22576140C2211yescharacteristic
32#22576139C2211yescharacteristic
33#22576136C2211yescharacteristic
34#22576135C2211yescharacteristic
35#22576134C2211yescharacteristic
36#22576131C2211yescharacteristic
37#22576130—21no—
38#22576128—21no—
39D4×Dic271—21no—
40#22576122—21no—
41(C22⋊D271)⋊C4—21no—
42(C22⋊D271)⋊C4—21no—
43#22576118C2211yescharacteristic
44#22576118C2211yescharacteristic
45#22576093C2211yescharacteristic, complemented
46#22576093C2211yescharacteristic, complemented
47#22576092C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576084C2211yescharacteristic, complemented
50#22576083C2211yescharacteristic, complemented
51#22576082C2211yescharacteristic, complemented
52#22576081C2211yescharacteristic, complemented
53#22576080C2211yescharacteristic, 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#16864655—22no—
63#16864655—22no—
64#16864653—22no—
65#16864653—22no—
66C2×Dic542C2311yescharacteristic
67C2×D1084—21no—
68C2×D1084—21no—
69C2×D1084C2311yescharacteristic
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C22×Dic271—21no—
74C22×Dic271D411yescharacteristic
75C2×C4×D271—22no—
76C2×C4×D271—22no—
77C2×C4×D271—21no—
78C2×C4×D271D411yescharacteristic
79C2×C4×D271D411yescharacteristic
80C2×C4×D271C2311yescharacteristic
81C2×C4×D271C2311yescharacteristic
82#16864644—22no—
83#16864643—22no—
84#16864639—22no—
85C2×C271⋊C8—22no—
86C2×C271⋊C8C2311yescharacteristic
87C2×C271⋊C8C2311yescharacteristic
88C8×D271—22no—
89C8×D271—22no—
90C8×D271—22no—
91C8×D271D414yescomplemented
92C4⋊Dic271—21no—
93C4⋊Dic271C2311yescharacteristic
94C4⋊Dic271C2311yescharacteristic
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97C22⋊Dic271—21no—
98C22⋊Dic271—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105C4×Dic271—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—22no—
123C22×D271—21no—
124C22×D271C2×D411yescharacteristic
125C22×D271C2×D411yescharacteristic
126Dic542—21no—
127Dic542—21no—
128D1084—21no—
129D1084—21no—
130D1084—42no—
131D1084—42no—
132C22⋊D271—42no—
133C22⋊D271—42no—
134C22⋊D271—42no—
135C2×Dic271—22no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—42no—
140C2×Dic271C2×D411yescharacteristic
141C2×Dic271C2×D411yescharacteristic
142C4×D271—22no—
143C4×D271—21no—
144C4×D271—21no—
145C4×D271—22no—
146C4×D271—42no—
147C4×D271C2×D412yes—
148C4×D271#4112yes—
149C271⋊C8—21no—
150C271⋊C8C2×D412yes—
151C23×C271—21no—
152C23×C271C2×D411yescharacteristic
153Q8×C271—21no—
154Q8×C271—21no—
155D4×C271—21no—
156D4×C271—21no—
157D4×C271—42no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2411yescharacteristic
165C8×C271—21no—
166C8×C271C2×D412yes—
167C22×C271—21no—
168C22×C271—21no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271—41no—
172C22×C271#13512yescomplemented
173C22×C271C22×D411yescharacteristic
174C2×D271—28no—
175C2×D271—21no—
176C2×D271—21no—
177C2×D271—21no—
178C2×D271—21no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271#12112yes—
183C2×D271#13512yescomplemented
184C4×C271—21no—
185C4×C271—41no—
186C4×C271—41no—
187C4×C271—41no—
188C4×C271C22×D411yescharacteristic
189C4×C271C2×#4111yescharacteristic
190Dic271—21no—
191Dic271—41no—
192Dic271—41no—
193Dic271—41no—
194Dic271#12112yes—
195C2×C271—22no—
196C2×C271—81no—
197C2×C271C2×#12111yescharacteristic
198C2×C271C2×#13511yescharacteristic
199C2×C271C2×#13511yescharacteristic
200D271—22no—
201D271—22no—
202D271—81no—
203C271#305011yescharacteristic, complemented, Sylow
204#3050—2711nomaximal, Sylow
205C2×#135—2711no—
206C23×C8—2711no—
207#525—2711no—
208#525—2711no—
209#467—2711no—
210#467—2711no—
211#466—2711no—
212#466—2711no—
213#465—2711no—
214#465—2711no—
215#465—2711no—
216#465D27111yescharacteristic, complemented
217#427—2711no—
218#416—2711no—
219#416—2711no—
220C2×#41—2711no—
221C2×#41—2711no—
222C23×C4—2711no—
223#135—5424no—
224C2×Q16—2711no—
225C2×SD16—2711no—
226C2×SD16C2×D27111yescharacteristic, complemented
227C2×D8—2711no—
228C22×C8—2711no—
229C22×C8—2711no—
230C22×C8—2711no—
231C22×C8—2711no—
232C22×C8—2711no—
233C22×C8—5424no—
234C22×C8C2×D27111yescharacteristic
235#123—5421no—
236#123—5421no—
237C22⋊Q8—2711no—
238C22⋊Q8—2711no—
239C22⋊Q8—2711no—
240C22⋊Q8C2×D27111yescharacteristic
241#121—2711no—
242#121—2711no—
243#121—2711no—
244#121C2×D27111yescharacteristic
245#120—5421no—
246#120—5421no—
247C4×D4—5421no—
248C4×D4—5421no—
249C4×D4—5421no—
250C4×D4—5421no—
251#117—2711no—
252#117—2711no—
253#107—2711no—
254#107—2711no—
255#106—2711no—
256#106C2×D27111yescharacteristic, complemented
257#103—2711no—
258#103—2711no—
259#103—2711no—
260#103C2×D27111yescharacteristic, complemented
261#102—2711no—
262#102—2711no—
263#102—2711no—
264#102C2×D27111yescharacteristic, complemented
265C24—2711no—
266#41—5422no—
267#41—5422no—
268#41—5422no—
269#41—5422no—
270C2×Q8—2711no—
271C2×Q8C22×D27111yescharacteristic
272C2×D4—21no—
273C2×D4—2711no—
274C2×D4—5421no—
275C2×D4—5421no—
276C2×D4—5421no—
277C2×D4—5421no—
278C2×D4—5421no—
279C2×D4C22×D27111yescharacteristic
280C22×C4—2711no—
281C22×C4—2711no—
282C22×C4—2711no—
283C22×C4—2711no—
284C22×C4—2711no—
285C22×C4—5422no—
286C22×C4—5422no—
287C22×C4—5421no—
288C22×C4—5421no—
289C22×C4C22×D27111yescharacteristic
290Q16—5422no—
291SD16—22no—
292SD16—5422no—
293D8—5422no—
294C2×C8—22no—
295C2×C8—2711no—
296C2×C8—2714no—
297C2×C8—2711no—
298C2×C8—5422no—
299C2×C8—5422no—
300C2×C8—5422no—
301C2×C8—5422no—
302C2×C8C22×D27111yescharacteristic
303C2×C8C22×D27111yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—2711no—
306C4⋊C4—2711no—
307C4⋊C4—5421no—
308C4⋊C4—5421no—
309C4⋊C4—5421no—
310C4⋊C4C22×D27111yescharacteristic
311C4⋊C4C22×D27111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C22⋊C4—5421no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C42—5421no—
323C42—5421no—
324C23—21no—
325C23—2711no—
326C23—2711no—
327C23—5422no—
328C23—5422no—
329C23—5422no—
330C23—5421no—
331C23D4×D27111yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—5421no—
335Q8—5421no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—5421no—
340D4—5421no—
341D4—10842no—
342D4—10842no—
343D4—10842no—
344D4—10842no—
345D4—10842no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—2711no—
352C2×C4—2712no—
353C2×C4—2712no—
354C2×C4—2711no—
355C2×C4—5422no—
356C2×C4—5422no—
357C2×C4—5422no—
358C2×C4—5421no—
359C2×C4—5421no—
360C2×C4—5421no—
361C2×C4—5421no—
362C2×C4—5421no—
363C2×C4—10842no—
364C2×C4—10842no—
365C2×C4D4×D27111yescharacteristic
366C2×C4C23×D27111yescharacteristic
367C8—21no—
368C8—2712no—
369C8—5421no—
370C8D4×D27112yes—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—2712no—
377C22—2712no—
378C22—5428no—
379C22—5421no—
380C22—5421no—
381C22—5421no—
382C22—5421no—
383C22—10841no—
384C22—10841no—
385C22—10841no—
386C22#2257618112yescomplemented
387C22C2×D4×D27111yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—2712no—
393C4—5421no—
394C4—10841no—
395C4—10841no—
396C4—10841no—
397C4C2×D4×D27111yescharacteristic
398C4#41×D27111yescharacteristic
399C2—22no—
400C2—81no—
401C2—5422no—
402C2—5422no—
403C2—21681no—
404C2#121×D27111yescharacteristic
405C2#135×D27111yescharacteristic
406C2C2×#2257618111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
hypercenter#465D271
hypocenterC271#3050
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×#22576181C2semidirect
2. C22×C8×D271C2semidirect
3. #31774410C2semidirect
4. #31774407C2semidirect
5. #31774137C2semidirect
6. #31774136C2semidirect
7. #31774132C2semidirect
8. #31774131C2semidirect
9. #31774127C2semidirect
10. #31774126C2semidirect
11. #31774123C2semidirect
12. #31773932C2semidirect
13. #31773892C2semidirect
14. #31773890C2semidirect
15. #465×C271C2semidirect
20. C2×#16864644C22semidirect
21. C2×#16864643C22semidirect
22. C2×#16864639C22semidirect
45. #22576093C22semidirect
46. #22576093C22semidirect
47. #22576092C22semidirect
48. #22576085C22semidirect
49. #22576084C22semidirect
50. #22576083C22semidirect
51. #22576082C22semidirect
52. #22576081C22semidirect
53. #22576080C22semidirect
54. C2×SD16×C271C22semidirect
58. #106×C271C22semidirect
59. #103×C271C22semidirect
60. #102×C271C22semidirect
91. C8×D271D4semidirect
172. C22×C271#135semidirect
183. C2×D271#135semidirect
203. C271#3050semidirect
216. #465D271semidirect
226. C2×SD16C2×D271semidirect
256. #106C2×D271semidirect
260. #103C2×D271semidirect
264. #102C2×D271semidirect
386. C22#22576181semidirect

Maximal subgroups

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

subgroupindex
C2×#225761812
C22×C8×D2712
#317744102
#317744072
#317741372
#317741362
#317741322
#317741312
#317741272
#317741262
#317741232
#317739322
#317738922
#317738902
#465×C2712
#3050271