Finite Group Databasebeta

← #46848002

Subgroups — #46848002

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×#22576193C211yesmaximal, characteristic, complemented
2C2×#22576184C211yesmaximal, characteristic, complemented
3#31774553C211yesmaximal, characteristic, complemented
4#31774544C211yesmaximal, characteristic, complemented
5#31774280C211yesmaximal, characteristic, complemented
6#31774156C211yesmaximal, characteristic, complemented
7#31774118C211yesmaximal, characteristic, complemented
8#31774111C211yesmaximal, characteristic, complemented
9#31774060C211yesmaximal, characteristic, complemented
10#31774049C211yesmaximal, characteristic, complemented
11#31774040C211yesmaximal, characteristic, complemented
12#31774035C211yesmaximal, characteristic, complemented
13#31773984C211yesmaximal, characteristic, complemented
14#31773973C211yesmaximal, characteristic, complemented
15#469×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C22×Dic542C2211yescharacteristic, complemented
19C2×D4×D271C2211yescharacteristic
20#22576193—24no—
21#22576184—24no—
22C2×Dic1084C2211yescharacteristic, complemented
23C2×Dic1084C2211yescharacteristic, complemented
24C2×#16864643C2211yescharacteristic
25C2×#16864643C2211yescharacteristic
26C2×#16864642C2211yescharacteristic
27C2×#16864642C2211yescharacteristic
28C2×#16864642C2211yescharacteristic, complemented
29C2×#16864639C2211yescharacteristic
30C2×#16864637C2211yescharacteristic
31#22576147C2211yescharacteristic
32#22576147C2211yescharacteristic, complemented
33#22576146—21no—
34#22576142—21no—
35#22576140C2211yescharacteristic
36#22576140C2211yescharacteristic
37#22576137—21no—
38#22576132—21no—
39#22576131—21no—
40#22576128—21no—
41#22576125C2211yescharacteristic, complemented
42D4×Dic271—21no—
43#22576122C2211yescharacteristic
44C4×C22⋊D271C2211yescharacteristic
45C22⋊C4×D271—21no—
46Dic271⋊C8C2211yescharacteristic, complemented
47#22576083C2211yescharacteristic, complemented
48#22576083C2211yescharacteristic, complemented
49#22576083C2211yescharacteristic, complemented
50#22576080C2211yescharacteristic
51C22⋊(C271⋊C8)C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic
53C8×Dic271C2211yescharacteristic, complemented
54C2×D8×C271C2211yescharacteristic
55C2×M16×C271C2211yescharacteristic, complemented
56#121×C271C2211yescharacteristic
57#121×C271C2211yescharacteristic, complemented
58#106×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864654—22no—
63#16864654—41no—
64#16864653—22no—
65#16864653—41no—
66C2×Dic542—22no—
67C2×Dic542—21no—
68C2×Dic542—41no—
69C2×Dic542C2311yescharacteristic
70C2×Dic542C2311yescharacteristic
71C2×Dic542C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271D411yescharacteristic
77D4×D271—22no—
78D4×D271—41no—
79C22×Dic271—21no—
80C22×Dic271—21no—
81C2×C4×D271—21no—
82C2×C4×D271—21no—
83C2×C4×D271C2311yescharacteristic
84Dic1084—22no—
85Dic1084—22no—
86#16864643—22no—
87#16864643D414yescomplemented
88#16864642—22no—
89#16864642—41no—
90#16864642—41no—
91#16864639—22no—
92#16864637—22no—
93C2×C271⋊C8C2311yescharacteristic
94C2×C271⋊C8C2311yescharacteristic
95C4⋊Dic271—21no—
96C4⋊Dic271—21no—
97Dic271⋊C4—21no—
98Dic271⋊C4D411yescharacteristic
99C22⋊Dic271—21no—
100C22⋊Dic271—21no—
101C22⋊Dic271—21no—
102C22⋊Dic271—21no—
103C22⋊Dic271D411yescharacteristic
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×D4×C271—21no—
111C2×D4×C271—21no—
112C2×D4×C271C2311yescharacteristic
113C2×D4×C271C2311yescharacteristic
114C22×C4×C271C2311yescharacteristic
115D8×C271—22no—
116M16×C271—22no—
117C2×C8×C271C2311yescharacteristic
118C2×C8×C271C2311yescharacteristic
119C4⋊C4×C271C2311yescharacteristic
120C4⋊C4×C271C2311yescharacteristic
121C22⋊C4×C271—21no—
122C22⋊C4×C271—21no—
123C22×D271—21no—
124C22×D271—42no—
125C22×D271—42no—
126C22×D271—81no—
127C22×D271C2×D411yescharacteristic
128Dic542—21no—
129Dic542—21no—
130Dic542—21no—
131Dic542—21no—
132Dic542—21no—
133Dic542—42no—
134Dic542C2×D412yes—
135D1084—21no—
136D1084C2×D412yes—
137C22⋊D271—41no—
138C22⋊D271—42no—
139C22⋊D271—42no—
140C22⋊D271—42no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271—42no—
145C2×Dic271—42no—
146C2×Dic271C2×D411yescharacteristic
147C2×Dic271C2×D411yescharacteristic
148C2×Dic271C2×D411yescharacteristic
149C4×D271—21no—
150C4×D271—21no—
151C4×D271—42no—
152C4×D271—81no—
153C271⋊C8—21no—
154C271⋊C8—21no—
155C23×C271—21no—
156C23×C271—21no—
157C23×C271C2×D411yescharacteristic
158D4×C271—21no—
159D4×C271—21no—
160D4×C271—21no—
161D4×C271—21no—
162D4×C271—42no—
163D4×C271—81no—
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×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271C22×D411yescharacteristic
182C2×D271—21no—
183C2×D271—22no—
184C2×D271—21no—
185C2×D271—21no—
186C2×D271—41no—
187C2×D271—41no—
188C2×D271—84no—
189C4×C271—21no—
190C4×C271—41no—
191C4×C271—41no—
192C4×C271C22×D411yescharacteristic
193C4×C271H(5,𝔽2)11yescharacteristic
194Dic271—21no—
195Dic271—22no—
196Dic271—21no—
197Dic271—21no—
198Dic271—41no—
199Dic271—41no—
200C2×C271—41no—
201C2×C271—81no—
202C2×C271—81no—
203C2×C271D4211yescharacteristic
204C2×C271#57511yescharacteristic
205C2×C271#57711yescharacteristic
206D271—41no—
207D271—42no—
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#469D27111yescharacteristic, complemented
217#464—2711no—
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×Q16—2711no—
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—2711no—
236C2×SD16—2711no—
237C2×SD16—2711no—
238C2×D8—2711no—
239C2×D8C2×D27111yescharacteristic
240C2×M16—2711no—
241C2×M16C2×D27111yescharacteristic, complemented
242#124—2711no—
243#124—2711no—
244#124—5421no—
245#123—5421no—
246C22⋊Q8—2711no—
247C22⋊Q8—2711no—
248C22⋊Q8—5421no—
249#121—5421no—
250#121—5421no—
251#121C2×D27111yescharacteristic
252#121C2×D27111yescharacteristic, complemented
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#103—2711no—
264#102—2711no—
265#102C2×D27111yescharacteristic
266#102C2×D27111yescharacteristic, complemented
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—2711no—
278C2×Q8—5422no—
279C2×Q8—5421no—
280C2×Q8—10841no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—2711no—
284C2×D4—2711no—
285C2×D4—5421no—
286C2×D4—5422no—
287C2×D4—5421no—
288C2×D4—5421no—
289C2×D4—10841no—
290C2×D4C22×D27111yescharacteristic
291C2×D4C22×D27111yescharacteristic
292C22×C4—2711no—
293C22×C4—5421no—
294C22×C4—5421no—
295C22×C4—5421no—
296C22×C4—5421no—
297C22×C4C22×D27111yescharacteristic
298Q16—5422no—
299Q16—5422no—
300SD16—2714no—
301SD16—5422no—
302SD16—5422no—
303SD16—10841no—
304SD16—10841no—
305D8—22no—
306D8—5422no—
307M16—22no—
308M16—5422no—
309C2×C8—2711no—
310C2×C8—2711no—
311C2×C8C22×D27111yescharacteristic
312C2×C8C22×D27111yescharacteristic
313C4⋊C4—2711no—
314C4⋊C4—5421no—
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—21no—
334C23—2711no—
335C23—5421no—
336C23—10842no—
337C23—10842no—
338C23—21681no—
339C23D4×D27111yescharacteristic
340Q8—2712no—
341Q8—5421no—
342Q8—5421no—
343Q8—5421no—
344Q8—5421no—
345Q8—5421no—
346Q8—10842no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—81no—
353D4—2712no—
354D4—5421no—
355D4—10841no—
356D4—10842no—
357D4—10842no—
358D4—10842no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—2711no—
364C2×C4—2711no—
365C2×C4—2711no—
366C2×C4—5421no—
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—41no—
388C22—41no—
389C22—41no—
390C22—5421no—
391C22—5422no—
392C22—5421no—
393C22—5421no—
394C22—10841no—
395C22—10841no—
396C22—21684no—
397C22C2×D4×D27111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—5421no—
402C4—5422no—
403C4—5421no—
404C4—5421no—
405C4—10841no—
406C4—10841no—
407C4C2×D4×D27111yescharacteristic
408C4#2257620611yescharacteristic
409C2—41no—
410C2—81no—
411C2—81no—
412C2—10841no—
413C2—10842no—
414C2#3177453111yescharacteristic
415C2#3177469311yescharacteristic
416C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#469×C271C2
socleC22×C271C22×D4
hypercenter#469D271
hypocenterC271#3275
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#22576193C2semidirect
2. C2×#22576184C2semidirect
3. #31774553C2semidirect
4. #31774544C2semidirect
5. #31774280C2semidirect
6. #31774156C2semidirect
7. #31774118C2semidirect
8. #31774111C2semidirect
9. #31774060C2semidirect
10. #31774049C2semidirect
11. #31774040C2semidirect
12. #31774035C2semidirect
13. #31773984C2semidirect
14. #31773973C2semidirect
15. #469×C271C2semidirect
18. C22×Dic542C22semidirect
22. C2×Dic1084C22semidirect
23. C2×Dic1084C22semidirect
28. C2×#16864642C22semidirect
32. #22576147C22semidirect
41. #22576125C22semidirect
46. Dic271⋊C8C22semidirect
47. #22576083C22semidirect
48. #22576083C22semidirect
49. #22576083C22semidirect
51. C22⋊(C271⋊C8)C22semidirect
53. C8×Dic271C22semidirect
55. C2×M16×C271C22semidirect
57. #121×C271C22semidirect
58. #106×C271C22semidirect
60. #102×C271C22semidirect
87. #16864643D4semidirect
208. C271#3275semidirect
216. #469D271semidirect
241. C2×M16C2×D271semidirect
252. #121C2×D271semidirect
259. #106C2×D271semidirect
266. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761932
C2×#225761842
#317745532
#317745442
#317742802
#317741562
#317741182
#317741112
#317740602
#317740492
#317740402
#317740352
#317739842
#317739732
#469×C2712
#3275271