Finite Group Databasebeta

← #46845667

Subgroups — #46845667

Order 34688.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576205C211yesmaximal, characteristic, complemented
2C2×#22576180C211yesmaximal, characteristic, complemented
3#31774475C211yesmaximal, characteristic, complemented
4#31774056C212yesmaximal, complemented
5#31774046C212yesmaximal, complemented
6#31774039C212yesmaximal, complemented
7#31774029C212yesmaximal, complemented
8#31773899C211yesmaximal, characteristic, complemented
9C2×#22576081C211yesmaximal, characteristic, complemented
10#31773865C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C271C211yesmaximal, characteristic, complemented
12#22576205—24no—
13#22576205—24no—
14C2×#16864654C2211yescharacteristic
15C2×#16864654C2211yescharacteristic
16C2×#16864653—21no—
17C2×#16864653C2212yes—
18C22×C22⋊D271—22no—
19C2×D4×D271—21no—
20C2×D4×D271C2212yes—
21#22576180—24no—
22C2×#16864644C2211yescharacteristic, complemented
23C2×#16864642C2211yescharacteristic, complemented
24C2×#16864641C2211yescharacteristic, complemented
25C2×#16864639C2211yescharacteristic, complemented
26C22×C271⋊C8C2211yescharacteristic, complemented
27#22576139—21no—
28#22576136C2212yescomplemented
29#22576134—21no—
30#22576132—22no—
31#22576131C2212yescomplemented
32C2×C4⋊Dic271C2211yescharacteristic, complemented
33#22576085C2212yescomplemented
34#22576082C2212yescomplemented
35#22576081C2212yescomplemented
36#22576081C2212yescomplemented
37(C2×D271)⋊C8C2212yescomplemented
38(C2×D271)⋊C8C2212yescomplemented
39C2×#41×C271C2211yescharacteristic, complemented
40C22×D4×C271C2211yescharacteristic
41AGL(1,ℤ/8)×C271—24no—
42C2×SD16×C271C2212yescomplemented
43C2×D8×C271C2212yescomplemented
44C2×M16×C271C2211yescharacteristic, complemented
45C23×D271—22no—
46#16864654—22no—
47#16864654—22no—
48#16864654—22no—
49#16864654D414yescomplemented
50#16864653—28no—
51#16864653—24no—
52#16864653—24no—
53C2×Dic542D411yescharacteristic
54C2×Dic542C2311yescharacteristic
55C2×D1084D411yescharacteristic
56C2×D1084C2311yescharacteristic
57C2×C22⋊D271—216no—
58C2×C22⋊D271—22no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271—21no—
62C2×C22⋊D271D412yes—
63D4×D271—28no—
64D4×D271—24no—
65D4×D271—24no—
66C22×Dic271—22no—
67C22×Dic271—21no—
68C2×C4×D271—21no—
69C2×C4×D271C2312yes—
70#16864644—22no—
71#16864642—22no—
72#16864641—22no—
73#16864639—22no—
74C2×C271⋊C8—22no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C4⋊Dic271—21no—
78C4⋊Dic271C2312yes—
79Dic271⋊C4—22no—
80C22⋊Dic271—22no—
81(C2×D271)⋊C4—22no—
82C24×C271—21no—
83#41×C271—22no—
84#41×C271—22no—
85C2×Q8×C271C2311yescharacteristic
86C2×D4×C271—21no—
87C2×D4×C271—21no—
88C2×D4×C271—22no—
89C2×D4×C271D414yescomplemented
90C2×D4×C271C2311yescharacteristic
91C2×D4×C271C2312yes—
92C22×C4×C271—21no—
93C22×C4×C271C2311yescharacteristic
94SD16×C271—24no—
95D8×C271—24no—
96M16×C271—22no—
97C2×C8×C271C2312yes—
98C22×D271—21no—
99C22×D271—28no—
100C22×D271—82no—
101C22×D271—82no—
102C22×D271C2×D412yes—
103Dic542—22no—
104Dic542—21no—
105Dic542C2×D412yes—
106D1084—22no—
107D1084—21no—
108D1084C2×D412yes—
109C22⋊D271—28no—
110C22⋊D271—28no—
111C22⋊D271—24no—
112C22⋊D271—24no—
113C22⋊D271—82no—
114C2×Dic271—28no—
115C2×Dic271—21no—
116C2×Dic271—22no—
117C2×Dic271—81no—
118C2×Dic271C2×D412yes—
119C4×D271—24no—
120C4×D271—22no—
121C4×D271—22no—
122C271⋊C8—21no—
123C271⋊C8—21no—
124C23×C271—22no—
125C23×C271—24no—
126C23×C271—21no—
127C23×C271—81no—
128C23×C271—81no—
129C23×C271C2×D411yescharacteristic
130Q8×C271—21no—
131Q8×C271—21no—
132D4×C271—24no—
133D4×C271—22no—
134D4×C271—21no—
135D4×C271—21no—
136D4×C271—42no—
137D4×C271C2×D414yes—
138C2×C4×C271—21no—
139C2×C4×C271—42no—
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2×D412yes—
142C2×C4×C271C2×D412yes—
143C2×C4×C271C2411yescharacteristic
144C8×C271—22no—
145C22×C271—21no—
146C22×C271—21no—
147C22×C271—24no—
148C22×C271—42no—
149C22×C271—42no—
150C22×C271—41no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271—84no—
154C22×C271#12012yes—
155C22×C271C22×D411yescharacteristic
156C2×D271—22no—
157C2×D271—22no—
158C2×D271—22no—
159C2×D271—22no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—88no—
163C4×C271—41no—
164C4×C271#12012yes—
165C4×C271C22×D411yescharacteristic
166C4×C271C22×D411yescharacteristic
167Dic271—22no—
168Dic271—22no—
169Dic271—42no—
170C2×C271—22no—
171C2×C271—44no—
172C2×C271—81no—
173C2×C271C2×#12011yescharacteristic
174C2×C271#57611yescharacteristic
175C2×C271#57711yescharacteristic
176D271—42no—
177D271—42no—
178C271#301211yescharacteristic, complemented, Sylow
179#3012—2711nomaximal, Sylow
180C2×H(5,𝔽2)—2711no—
181C2×AGL(1,ℤ/8)D27111yescharacteristic, complemented
182C2×#135—2711no—
183#537—2711no—
184#452—2712no—
185#450—2712no—
186#449—2712no—
187#447—2712no—
188#417—2711no—
189C2×#102—2711no—
190#408—2711no—
191H(5,𝔽2)—5424no—
192H(5,𝔽2)—5424no—
193C2×#41—2711no—
194C2×#41—2712no—
195C2×#41—2711no—
196C2×#41—5421no—
197C2×#41C2×D27111yescharacteristic, complemented
198C22×D4—2712no—
199C22×D4—5421no—
200C22×D4—5422no—
201C22×D4C2×D27111yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—5424no—
204C2×Q16—2711no—
205C2×SD16—2711no—
206C2×SD16—2711no—
207C2×SD16C2×D27112yescomplemented
208C2×D8—2711no—
209C2×D8C2×D27112yescomplemented
210C2×M16C2×D27111yescharacteristic, complemented
211C22×C8—2711no—
212C22⋊Q8—2712no—
213C22⋊Q8—5421no—
214#121—2712no—
215#121—5421no—
216#121—5422no—
217C2×C4⋊C4—2711no—
218#103—2712no—
219#102—2712no—
220#102—2712no—
221#102—2712no—
222C22⋊C8—2712no—
223C22⋊C8—2712no—
224C24—21no—
225C24—5422no—
226#41—22no—
227#41—22no—
228#41—2714no—
229#41—5424no—
230#41—5428no—
231#41—5424no—
232#41—5422no—
233#41—5422no—
234#41—5422no—
235C2×Q8—2711no—
236C2×Q8—2711no—
237C2×Q8C22×D27111yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—2712no—
242C2×D4—2711no—
243C2×D4—2711no—
244C2×D4—5422no—
245C2×D4—5424no—
246C2×D4—5428no—
247C2×D4—5422no—
248C2×D4—54216no—
249C2×D4—5421no—
250C2×D4—5424no—
251C2×D4—5422no—
252C2×D4C22⋊D27114yescomplemented
253C2×D4C22×D27111yescharacteristic
254C2×D4C22×D27112yes—
255C22×C4—21no—
256C22×C4—2712no—
257C22×C4—5422no—
258C22×C4—5421no—
259C22×C4—5421no—
260C22×C4C22×D27111yescharacteristic
261Q16—5422no—
262SD16—24no—
263SD16—5422no—
264SD16—5422no—
265D8—24no—
266D8—5422no—
267M16—22no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8—5422no—
271C2×C8C22×D27112yes—
272C4⋊C4—2712no—
273C4⋊C4—5422no—
274C4⋊C4—5421no—
275C22⋊C4—5422no—
276C22⋊C4—5422no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—2712no—
283C23—5428no—
284C23—5421no—
285C23—21682no—
286C23—21682no—
287C23D4×D27111yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2712no—
291Q8—5422no—
292Q8—5421no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—2712no—
299D4—5428no—
300D4—5428no—
301D4—5424no—
302D4—5424no—
303D4—5422no—
304D4—5421no—
305D4—21682no—
306D4C2×C22⋊D27114yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2712no—
310C2×C4—5428no—
311C2×C4—5421no—
312C2×C4—5424no—
313C2×C4—5422no—
314C2×C4—5422no—
315C2×C4—5422no—
316C2×C4—21681no—
317C2×C4D4×D27111yescharacteristic
318C2×C4D4×D27112yes—
319C2×C4C2×C22⋊D27112yes—
320C2×C4C23×D27111yescharacteristic
321C8—22no—
322C8—5421no—
323C8—5421no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—5422no—
334C22—5422no—
335C22—5422no—
336C22—5422no—
337C22—10841no—
338C22—10841no—
339C22—21688no—
340C22#2257612812yes—
341C22C2×D4×D27111yescharacteristic
342C4—41no—
343C4—5422no—
344C4—5422no—
345C4—10842no—
346C4#2257612812yes—
347C4C2×D4×D27111yescharacteristic
348C4C22×C22⋊D27111yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—10842no—
353C2—10842no—
354C2C2×#2257612811yescharacteristic
355C2#3177470111yescharacteristic
356C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×AGL(1,ℤ/8)×C271C2
socleC22×C271C22×D4
hypercenterC2×AGL(1,ℤ/8)D271
hypocenterC271#3012
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22576205C2semidirect
2. C2×#22576180C2semidirect
3. #31774475C2semidirect
4. #31774056C2semidirect
5. #31774046C2semidirect
6. #31774039C2semidirect
7. #31774029C2semidirect
8. #31773899C2semidirect
9. C2×#22576081C2semidirect
10. #31773865C2semidirect
11. C2×AGL(1,ℤ/8)×C271C2semidirect
22. C2×#16864644C22semidirect
23. C2×#16864642C22semidirect
24. C2×#16864641C22semidirect
25. C2×#16864639C22semidirect
26. C22×C271⋊C8C22semidirect
28. #22576136C22semidirect
31. #22576131C22semidirect
32. C2×C4⋊Dic271C22semidirect
33. #22576085C22semidirect
34. #22576082C22semidirect
35. #22576081C22semidirect
36. #22576081C22semidirect
37. (C2×D271)⋊C8C22semidirect
38. (C2×D271)⋊C8C22semidirect
39. C2×#41×C271C22semidirect
42. C2×SD16×C271C22semidirect
43. C2×D8×C271C22semidirect
44. C2×M16×C271C22semidirect
49. #16864654D4semidirect
89. C2×D4×C271D4semidirect
178. C271#3012semidirect
181. C2×AGL(1,ℤ/8)D271semidirect
197. C2×#41C2×D271semidirect
207. C2×SD16C2×D271semidirect
209. C2×D8C2×D271semidirect
210. C2×M16C2×D271semidirect
252. C2×D4C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225762052
C2×#225761802
#317744752
#317740562
#317740462
#317740392
#317740292
#317738992
C2×#225760812
#317738652
C2×AGL(1,ℤ/8)×C2712
#3012271