Finite Group Databasebeta

← #46848078

Subgroups — #46848078

Order 34688.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576182C211yesmaximal, characteristic, complemented
2C2×SD16×D271C211yesmaximal, characteristic, complemented
3#31774557C211yesmaximal, characteristic, complemented
4#31774544C211yesmaximal, characteristic, complemented
5#31774278C211yesmaximal, characteristic, complemented
6#31774127C211yesmaximal, characteristic, complemented
7#31774118C211yesmaximal, characteristic, complemented
8#31774115C211yesmaximal, characteristic, complemented
9#31774057C211yesmaximal, characteristic, complemented
10#31774054C211yesmaximal, characteristic, complemented
11#31774040C211yesmaximal, characteristic, complemented
12#31774032C211yesmaximal, characteristic, complemented
13#31773984C211yesmaximal, characteristic, complemented
14C8×C22⋊D271C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×#16864654C2211yescharacteristic, complemented
18C2×Q8×D271C2211yescharacteristic
19C2×D4×D271C2211yescharacteristic
20#22576182—24no—
21C2×Dic1084C2211yescharacteristic, complemented
22C2×#16864644C2211yescharacteristic, complemented
23C2×#16864643C2211yescharacteristic
24C2×#16864643C2211yescharacteristic, complemented
25C2×#16864642C2211yescharacteristic
26C2×#16864641C2211yescharacteristic
27SD16×D271—24no—
28C2×D2168C2211yescharacteristic
29C2×#16864639C2211yescharacteristic
30C2×C8×D271C2211yescharacteristic
31#22576149C2211yescharacteristic, complemented
32#22576147C2211yescharacteristic
33#22576146—21no—
34#22576142—21no—
35#22576140C2211yescharacteristic
36#22576138—21no—
37#22576135C2211yescharacteristic
38#22576133—21no—
39#22576131—21no—
40#22576128—21no—
41#22576125C2211yescharacteristic, complemented
42#22576122—21no—
43#22576122C2211yescharacteristic
44C4×C22⋊D271C2211yescharacteristic
45C22⋊C4×D271—21no—
46Dic271⋊C8C2211yescharacteristic, complemented
47#22576083C2211yescharacteristic, complemented
48#22576083C2211yescharacteristic, complemented
49#22576080C2211yescharacteristic
50#22576080C2211yescharacteristic, complemented
51C22⋊(C271⋊C8)C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic
53C8×Dic271C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic
55C22×C8×C271C2211yescharacteristic, complemented
56C22⋊Q8×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic
58#106×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic
61C23×D271—21no—
62#16864654—22no—
63#16864654—22no—
64#16864654—41no—
65#16864654—41no—
66C2×Dic542—21no—
67C2×Dic542C2311yescharacteristic
68C2×Dic542C2311yescharacteristic
69Q8×D271—22no—
70Q8×D271—41no—
71C2×D1084—21no—
72C2×D1084C2311yescharacteristic
73C2×D1084C2311yescharacteristic
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77C2×C22⋊D271D411yescharacteristic
78D4×D271—22no—
79D4×D271—41no—
80C2×C4×D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271—21no—
83C2×C4×D271—21no—
84C2×C4×D271C2311yescharacteristic
85Dic1084—22no—
86#16864644—41no—
87#16864643—22no—
88#16864643D414yescomplemented
89#16864642—22no—
90#16864641—22no—
91D2168—22no—
92#16864639—41no—
93C2×C271⋊C8C2311yescharacteristic
94C2×C271⋊C8C2311yescharacteristic
95C8×D271—22no—
96C4⋊Dic271—21no—
97Dic271⋊C4—21no—
98Dic271⋊C4D411yescharacteristic
99C22⋊Dic271—21no—
100C22⋊Dic271—21no—
101C22⋊Dic271D411yescharacteristic
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
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×D4×C271—21no—
112C2×D4×C271C2311yescharacteristic
113C22×C4×C271C2311yescharacteristic
114SD16×C271—22no—
115C2×C8×C271—22no—
116C2×C8×C271C2311yescharacteristic
117C2×C8×C271C2311yescharacteristic
118C4⋊C4×C271—21no—
119C4⋊C4×C271C2311yescharacteristic
120C4⋊C4×C271C2311yescharacteristic
121C22⋊C4×C271—21no—
122C22⋊C4×C271—21no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—81no—
128C22×D271C2×D411yescharacteristic
129Dic542—21no—
130Dic542—21no—
131Dic542—21no—
132Dic542—42no—
133Dic542C2×D412yes—
134D1084—21no—
135D1084—21no—
136D1084—21no—
137D1084—42no—
138D1084C2×D412yes—
139C22⋊D271—22no—
140C22⋊D271—42no—
141C22⋊D271—42no—
142C22⋊D271—42no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271C2×D411yescharacteristic
146C2×Dic271C2×D411yescharacteristic
147C2×Dic271C2×D411yescharacteristic
148C4×D271—21no—
149C4×D271—21no—
150C4×D271—42no—
151C4×D271—42no—
152C4×D271—42no—
153C4×D271—81no—
154C271⋊C8—21no—
155C271⋊C8—21no—
156C23×C271—21no—
157C23×C271C2×D411yescharacteristic
158Q8×C271—21no—
159Q8×C271—21no—
160D4×C271—21no—
161D4×C271—21no—
162D4×C271—81no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2411yescharacteristic
170C8×C271—21no—
171C8×C271C2×D412yes—
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—21no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271C22×D411yescharacteristic
179C2×D271—21no—
180C2×D271—21no—
181C2×D271—22no—
182C2×D271—41no—
183C2×D271—41no—
184C2×D271—41no—
185C2×D271—41no—
186C2×D271—41no—
187C2×D271—84no—
188C2×D271#13512yescomplemented
189C4×C271—21no—
190C4×C271—41no—
191C4×C271—41no—
192C4×C271—41no—
193C4×C271C22×D411yescharacteristic
194C4×C271H(5,𝔽2)11yescharacteristic
195Dic271—21no—
196Dic271—22no—
197Dic271—21no—
198Dic271—21no—
199Dic271—41no—
200C2×C271—41no—
201C2×C271—81no—
202C2×C271D4211yescharacteristic
203C2×C271C2×#13511yescharacteristic
204C2×C271#57711yescharacteristic
205D271—22no—
206D271—42no—
207D271—81no—
208C271#328211yescharacteristic, complemented, Sylow
209#3282—2711nomaximal, Sylow
210C2×#135—2711no—
211C22×SD16—2711no—
212#548—2711no—
213#546—2711no—
214#495—2711no—
215#465—2711no—
216#465D27111yescharacteristic, complemented
217#464—2711no—
218#463—2711no—
219#452—2711no—
220#451—2711no—
221#449—2711no—
222#447—2711no—
223C4×SD16—2711no—
224C8×D4—2711no—
225C2×#41—2711no—
226C2×#41—2711no—
227C22×Q8—2711no—
228C22×D4—2711no—
229#135—5424no—
230C2×Q16—2711no—
231C2×Q16—2711no—
232C2×SD16—2711no—
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—2711no—
236C2×SD16—5424no—
237C2×SD16C2×D27111yescharacteristic
238C2×D8—2711no—
239C2×D8—2711no—
240C22×C8—2711no—
241C22×C8C2×D27111yescharacteristic, complemented
242#124—2711no—
243#124—2711no—
244#124—5421no—
245#123—5421no—
246C22⋊Q8—2711no—
247C22⋊Q8—5421no—
248C22⋊Q8C2×D27111yescharacteristic, complemented
249#121—2711no—
250#121—5421no—
251#121—5421no—
252#121C2×D27111yescharacteristic
253#120—5421no—
254C4×Q8—2711no—
255C4×D4—2711no—
256C4×D4—2711no—
257C4×D4—5421no—
258C2×C22⋊C4—5421no—
259#106C2×D27111yescharacteristic, complemented
260C4⋊C8—2711no—
261#103—2711no—
262#103—2711no—
263#103C2×D27111yescharacteristic, complemented
264#102—2711no—
265#102—2711no—
266#102C2×D27111yescharacteristic
267C22⋊C8—2711no—
268C22⋊C8—2711no—
269C4×C8—2711no—
270C24—5421no—
271#41—5422no—
272#41—5422no—
273#41—10841no—
274#41—10841no—
275C2×Q8—2711no—
276C2×Q8—2711no—
277C2×Q8—5421no—
278C2×Q8—5422no—
279C2×Q8—10841no—
280C2×Q8C22×D27111yescharacteristic
281C2×D4—21no—
282C2×D4—2711no—
283C2×D4—2711no—
284C2×D4—2711no—
285C2×D4—5421no—
286C2×D4—5421no—
287C2×D4—5422no—
288C2×D4—5421no—
289C2×D4—5421no—
290C2×D4—10841no—
291C2×D4C22×D27111yescharacteristic
292C22×C4—2711no—
293C22×C4—5421no—
294C22×C4—5421no—
295C22×C4—5421no—
296C22×C4—5421no—
297C22×C4C22×D27111yescharacteristic
298Q16—5422no—
299Q16—10841no—
300SD16—22no—
301SD16—2714no—
302SD16—5422no—
303SD16—5422no—
304SD16—5422no—
305D8—5422no—
306D8—10841no—
307C2×C8—22no—
308C2×C8—2711no—
309C2×C8—2711no—
310C2×C8—5422no—
311C2×C8C22×D27111yescharacteristic
312C2×C8C22×D27111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2711no—
315C4⋊C4—5421no—
316C4⋊C4—5421no—
317C4⋊C4C22×D27111yescharacteristic
318C4⋊C4C22×D27111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—2711no—
322C22⋊C4—2711no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C22⋊C4—5421no—
327C22⋊C4—5421no—
328C22⋊C4—5421no—
329C22⋊C4—5421no—
330C42—2711no—
331C42—5421no—
332C23—21no—
333C23—2711no—
334C23—5421no—
335C23—5421no—
336C23—10842no—
337C23—10842no—
338C23—21681no—
339C23D4×D27111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2712no—
343Q8—5421no—
344Q8—5421no—
345Q8—5421no—
346Q8—10842no—
347D4—21no—
348D4—21no—
349D4—81no—
350D4—2712no—
351D4—5422no—
352D4—5421no—
353D4—5421no—
354D4—5421no—
355D4—10842no—
356D4—10842no—
357D4—10842no—
358D4—10842no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—2711no—
365C2×C4—2711no—
366C2×C4—2711no—
367C2×C4—5421no—
368C2×C4—5421no—
369C2×C4—5421no—
370C2×C4—5421no—
371C2×C4—10842no—
372C2×C4—10842no—
373C2×C4—10842no—
374C2×C4—21681no—
375C2×C4D4×D27111yescharacteristic
376C2×C4C23×D27111yescharacteristic
377C8—21no—
378C8—5421no—
379C8—5421no—
380C8D4×D27112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—2712no—
388C22—5421no—
389C22—5421no—
390C22—5422no—
391C22—10841no—
392C22—10841no—
393C22—10841no—
394C22—10841no—
395C22—10841no—
396C22—21684no—
397C22C2×D4×D27111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—5421no—
403C4—5422no—
404C4—5421no—
405C4—5421no—
406C4—10841no—
407C4C2×D4×D27111yescharacteristic
408C4#2257620611yescharacteristic
409C2—41no—
410C2—81no—
411C2—5422no—
412C2—10842no—
413C2—21681no—
414C2#3177453111yescharacteristic
415C2#135×D27111yescharacteristic
416C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
hypercenter#465D271
hypocenterC271#3282
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×#22576182C2semidirect
2. C2×SD16×D271C2semidirect
3. #31774557C2semidirect
4. #31774544C2semidirect
5. #31774278C2semidirect
6. #31774127C2semidirect
7. #31774118C2semidirect
8. #31774115C2semidirect
9. #31774057C2semidirect
10. #31774054C2semidirect
11. #31774040C2semidirect
12. #31774032C2semidirect
13. #31773984C2semidirect
14. C8×C22⋊D271C2semidirect
15. #465×C271C2semidirect
17. C2×#16864654C22semidirect
21. C2×Dic1084C22semidirect
22. C2×#16864644C22semidirect
24. C2×#16864643C22semidirect
31. #22576149C22semidirect
41. #22576125C22semidirect
46. Dic271⋊C8C22semidirect
47. #22576083C22semidirect
48. #22576083C22semidirect
50. #22576080C22semidirect
51. C22⋊(C271⋊C8)C22semidirect
53. C8×Dic271C22semidirect
55. C22×C8×C271C22semidirect
56. C22⋊Q8×C271C22semidirect
58. #106×C271C22semidirect
59. #103×C271C22semidirect
88. #16864643D4semidirect
188. C2×D271#135semidirect
208. C271#3282semidirect
216. #465D271semidirect
241. C22×C8C2×D271semidirect
248. C22⋊Q8C2×D271semidirect
259. #106C2×D271semidirect
263. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761822
C2×SD16×D2712
#317745572
#317745442
#317742782
#317741272
#317741182
#317741152
#317740572
#317740542
#317740402
#317740322
#317739842
C8×C22⋊D2712
#465×C2712
#3282271