Finite Group Databasebeta

← #46847990

Subgroups — #46847990

Order 34688.

498 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×#22576192C211yesmaximal, characteristic, complemented
2C2×#22576187C211yesmaximal, characteristic, complemented
3#31774559C211yesmaximal, characteristic, complemented
4#31774540C211yesmaximal, characteristic, complemented
5#31774279C211yesmaximal, characteristic, complemented
6#31774154C211yesmaximal, characteristic, complemented
7#31774149C211yesmaximal, characteristic, complemented
8#31774113C211yesmaximal, characteristic, complemented
9#31774058C211yesmaximal, characteristic, complemented
10#31774049C211yesmaximal, characteristic, complemented
11#31774045C211yesmaximal, characteristic, complemented
12#31774034C211yesmaximal, characteristic, complemented
13C4×#16864641C211yesmaximal, characteristic, complemented
14#31773974C211yesmaximal, characteristic, complemented
15#464×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864655C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic, complemented
19C2×D4×D271C2211yescharacteristic
20#22576192—24no—
21#22576187—24no—
22C2×#16864644C2211yescharacteristic, complemented
23C2×#16864643C2211yescharacteristic
24C2×#16864642C2211yescharacteristic
25C2×#16864641C2211yescharacteristic
26C2×#16864641C2211yescharacteristic, complemented
27C2×D2168C2211yescharacteristic
28C2×#16864639C2211yescharacteristic
29C2×#16864637C2211yescharacteristic
30C2×#16864637C2211yescharacteristic, complemented
31#22576148—21no—
32#22576148C2211yescharacteristic, complemented
33#22576142—21no—
34#22576140—21no—
35#22576140C2211yescharacteristic
36#22576136C2211yescharacteristic
37#22576135—21no—
38#22576135C2211yescharacteristic, complemented
39#22576133—21no—
40#22576131C2211yescharacteristic
41#22576130—21no—
42C4×D1084—21no—
43C4×D1084C2211yescharacteristic
44C4×D1084C2211yescharacteristic
45C22⋊C4×D271—21no—
46#22576091C2211yescharacteristic, complemented
47#22576085C2211yescharacteristic, complemented
48#22576084C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576080C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53C4×C271⋊C8C2211yescharacteristic, complemented
54C2×Q16×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic
57C4×Q8×C271C2211yescharacteristic, complemented
58C4⋊C8×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic
61C23×D271—21no—
62#16864655—22no—
63#16864655—22no—
64#16864655—41no—
65#16864655—41no—
66C2×Dic542—21no—
67C2×Dic542C2311yescharacteristic
68Q8×D271—22no—
69Q8×D271—41no—
70C2×D1084—21no—
71C2×D1084—21no—
72C2×D1084—21no—
73C2×D1084D411yescharacteristic
74C2×D1084C2311yescharacteristic
75C2×D1084C2311yescharacteristic
76C2×C22⋊D271—21no—
77C2×C22⋊D271—21no—
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
85C2×C4×D271C2311yescharacteristic
86#16864644—22no—
87#16864643—22no—
88#16864642—41no—
89#16864641—41no—
90#16864641D414yescomplemented
91D2168—22no—
92#16864639—22no—
93#16864637—22no—
94#16864637—22no—
95C2×C271⋊C8C2311yescharacteristic
96C2×C271⋊C8C2311yescharacteristic
97C4⋊Dic271—21no—
98C4⋊Dic271D411yescharacteristic
99C4⋊Dic271C2311yescharacteristic
100Dic271⋊C4—21no—
101C22⋊Dic271—21no—
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)⋊C4—21no—
108(C2×D271)⋊C4D411yescharacteristic
109(C2×D271)⋊C4D411yescharacteristic
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×D271—21no—
124C22×D271—21no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—81no—
128C22×D271C2×D411yescharacteristic
129C22×D271C2×D411yescharacteristic
130Dic542—21no—
131Dic542—21no—
132Dic542—42no—
133D1084—21no—
134D1084—21no—
135D1084—21no—
136D1084—41no—
137D1084—42no—
138D1084—42no—
139D1084—42no—
140D1084C2×D412yes—
141C22⋊D271—42no—
142C22⋊D271—81no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271C2×D411yescharacteristic
146C2×Dic271C2×D411yescharacteristic
147C4×D271—21no—
148C4×D271—21no—
149C4×D271—21no—
150C4×D271—21no—
151C4×D271—42no—
152C4×D271—42no—
153C4×D271—42no—
154C4×D271—81no—
155C271⋊C8—21no—
156C271⋊C8C2×D412yes—
157C23×C271—21no—
158Q8×C271—21no—
159Q8×C271—21no—
160Q8×C271—21no—
161Q8×C271C2×D412yes—
162D4×C271—21no—
163D4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2411yescharacteristic
170C8×C271—21no—
171C8×C271—21no—
172C22×C271—41no—
173C22×C271—41no—
174C22×C271—41no—
175C22×C271C22×D411yescharacteristic
176C2×D271—21no—
177C2×D271—21no—
178C2×D271—22no—
179C2×D271—21no—
180C2×D271—21no—
181C2×D271—21no—
182C2×D271—21no—
183C2×D271—41no—
184C2×D271—41no—
185C2×D271—41no—
186C2×D271—41no—
187C2×D271—41no—
188C2×D271—84no—
189C4×C271—22no—
190C4×C271—21no—
191C4×C271—21no—
192C4×C271—41no—
193C4×C271—41no—
194C4×C271C22×D411yescharacteristic
195C4×C271H(5,𝔽2)11yescharacteristic
196Dic271—21no—
197Dic271—21no—
198Dic271—41no—
199Dic271—41no—
200C2×C271—81no—
201C2×C271D4211yescharacteristic
202C2×C271#57511yescharacteristic
203C2×C271#57711yescharacteristic
204D271—41no—
205D271—41no—
206D271—42no—
207D271—81no—
208C271#327511yescharacteristic, complemented, Sylow
209#3275—2711nomaximal, Sylow
210C2×#137—2711no—
211C2×AGL(1,ℤ/8)—2711no—
212#548—2711no—
213#546—2711no—
214#495—2711no—
215#470—2711no—
216#469—2711no—
217#464D27111yescharacteristic, complemented
218#463—2711no—
219#452—2711no—
220#450—2711no—
221#449—2711no—
222#448—2711no—
223C4×SD16—2711no—
224#436—2711no—
225C2×#41—2711no—
226C2×#41—2711no—
227C22×Q8—2711no—
228C22×D4—2711no—
229#137—5424no—
230AGL(1,ℤ/8)—5424no—
231C2×Q16—2711no—
232C2×Q16C2×D27111yescharacteristic, complemented
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—2711no—
236C2×SD16—2711no—
237C2×SD16C2×D27111yescharacteristic
238C2×D8—2711no—
239C2×D8—2711no—
240C2×M16—2711no—
241C2×M16—2711no—
242#124—2711no—
243#124—5421no—
244#124C2×D27111yescharacteristic
245#123—5421no—
246C22⋊Q8—2711no—
247C22⋊Q8—2711no—
248C22⋊Q8—5421no—
249#121—2711no—
250#121—2711no—
251#121—5421no—
252#121—5421no—
253#120—5421no—
254C4×Q8C2×D27111yescharacteristic, complemented
255C4×D4—2711no—
256C4×D4—2711no—
257C4×D4—5421no—
258C2×C22⋊C4—5421no—
259#106—2711no—
260C4⋊C8C2×D27111yescharacteristic, complemented
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—5421no—
277C2×Q8—5422no—
278C2×Q8—10841no—
279C2×Q8C22×D27111yescharacteristic
280C2×Q8C22×D27111yescharacteristic
281C2×D4—2711no—
282C2×D4—2711no—
283C2×D4—2711no—
284C2×D4—5421no—
285C2×D4—5421no—
286C2×D4—5421no—
287C2×D4—5421no—
288C2×D4—5422no—
289C2×D4—5421no—
290C2×D4—10841no—
291C2×D4C22×D27111yescharacteristic
292C22×C4—2711no—
293C22×C4—2711no—
294C22×C4—5421no—
295C22×C4—5421no—
296C22×C4—5421no—
297C22×C4—5421no—
298Q16—22no—
299Q16—5422no—
300SD16—22no—
301SD16—2714no—
302SD16—5422no—
303SD16—10841no—
304SD16—10841no—
305D8—5422no—
306D8—5422no—
307M16—5422no—
308M16—5422no—
309C2×C8—2711no—
310C2×C8—2711no—
311C2×C8C22×D27111yescharacteristic
312C2×C8C22×D27111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2711no—
315C4⋊C4—2711no—
316C4⋊C4—5421no—
317C4⋊C4—5421no—
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—21no—
331C42C22×D27111yescharacteristic
332C23—21no—
333C23—2711no—
334C23—2711no—
335C23—5421no—
336C23—5421no—
337C23—10842no—
338C23—10842no—
339C23—21681no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—5421no—
344Q8—5421no—
345Q8—10842no—
346Q8D4×D27112yes—
347D4—21no—
348D4—21no—
349D4—2712no—
350D4—5421no—
351D4—5421no—
352D4—5421no—
353D4—10841no—
354D4—10842no—
355D4—10842no—
356D4—10842no—
357D4—10842no—
358D4—21681no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—2711no—
363C2×C4—2711no—
364C2×C4—5421no—
365C2×C4—5421no—
366C2×C4—5421no—
367C2×C4—5421no—
368C2×C4—5421no—
369C2×C4—5421no—
370C2×C4—10842no—
371C2×C4—10842no—
372C2×C4—10842no—
373C2×C4—21681no—
374C2×C4D4×D27111yescharacteristic
375C2×C4D4×D27111yescharacteristic
376C2×C4C23×D27111yescharacteristic
377C8—21no—
378C8—21no—
379C8—2712no—
380C8—5421no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—5421no—
385C22—5421no—
386C22—5422no—
387C22—5421no—
388C22—5421no—
389C22—5421no—
390C22—5421no—
391C22—10841no—
392C22—10841no—
393C22—10841no—
394C22—10841no—
395C22—10841no—
396C22—21684no—
397C22C2×D4×D27111yescharacteristic
398C4—22no—
399C4—21no—
400C4—21no—
401C4—41no—
402C4—41no—
403C4—5421no—
404C4—5421no—
405C4—10841no—
406C4—10841no—
407C4C2×D4×D27111yescharacteristic
408C4#2257620611yescharacteristic
409C2—81no—
410C2—10841no—
411C2—10841no—
412C2—10842no—
413C2—21681no—
414C2#3177453111yescharacteristic
415C2#3177469411yescharacteristic
416C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#464×C271C2
socleC22×C271C22×D4
hypercenter#464D271
hypocenterC271#3275
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×#22576192C2semidirect
2. C2×#22576187C2semidirect
3. #31774559C2semidirect
4. #31774540C2semidirect
5. #31774279C2semidirect
6. #31774154C2semidirect
7. #31774149C2semidirect
8. #31774113C2semidirect
9. #31774058C2semidirect
10. #31774049C2semidirect
11. #31774045C2semidirect
12. #31774034C2semidirect
13. C4×#16864641C2semidirect
14. #31773974C2semidirect
15. #464×C271C2semidirect
18. C2×Q8×D271C22semidirect
22. C2×#16864644C22semidirect
26. C2×#16864641C22semidirect
30. C2×#16864637C22semidirect
32. #22576148C22semidirect
38. #22576135C22semidirect
46. #22576091C22semidirect
47. #22576085C22semidirect
48. #22576084C22semidirect
49. #22576082C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. C4×C271⋊C8C22semidirect
54. C2×Q16×C271C22semidirect
57. C4×Q8×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #103×C271C22semidirect
90. #16864641D4semidirect
208. C271#3275semidirect
217. #464D271semidirect
232. C2×Q16C2×D271semidirect
254. C4×Q8C2×D271semidirect
260. C4⋊C8C2×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×#225761922
C2×#225761872
#317745592
#317745402
#317742792
#317741542
#317741492
#317741132
#317740582
#317740492
#317740452
#317740342
C4×#168646412
#317739742
#464×C2712
#3275271