Finite Group Databasebeta

← #46845673

Subgroups — #46845673

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×#22576186C211yesmaximal, characteristic, complemented
3C2×#22576182C211yesmaximal, characteristic, complemented
4#31774478C211yesmaximal, characteristic, complemented
5#31774060C212yesmaximal, complemented
6#31774049C212yesmaximal, complemented
7#31774043C212yesmaximal, complemented
8#31774032C212yesmaximal, complemented
9#31773896C211yesmaximal, characteristic, complemented
10#31773863C211yesmaximal, characteristic, complemented
11C2×#102×C271C211yesmaximal, characteristic, complemented
12#22576205—24no—
13#22576205—24no—
14C2×#16864654C2211yescharacteristic
15C2×#16864654C2211yescharacteristic
16C2×#16864654C2211yescharacteristic, complemented
17C2×#16864653—21no—
18C2×#16864653C2212yes—
19C22×C22⋊D271—22no—
20C2×D4×D271—21no—
21C2×D4×D271C2212yes—
22#22576186—24no—
23#22576182—24no—
24C2×Dic1084C2211yescharacteristic, complemented
25C2×#16864643C2211yescharacteristic, complemented
26C2×#16864643C2211yescharacteristic, complemented
27C2×#16864642C2212yescomplemented
28C2×D2168C2211yescharacteristic, complemented
29C2×#16864639C2212yescomplemented
30C2×#16864638C2211yescharacteristic, complemented
31#22576140—21no—
32#22576140C2212yescomplemented
33#22576135—21no—
34#22576135C2212yescomplemented
35#22576134—22no—
36#22576083C2212yescomplemented
37#22576080C2212yescomplemented
38(C2×D271)⋊C8C2212yescomplemented
39(C2×D271)⋊C8C2212yescomplemented
40C22×D4×C271C2211yescharacteristic
41C22×C8×C271C2211yescharacteristic, complemented
42C2×C4⋊C4×C271C2211yescharacteristic, complemented
43#102×C271C2212yescomplemented
44#102×C271C2212yescomplemented
45C23×D271—22no—
46#16864654—22no—
47#16864654—22no—
48#16864654—22no—
49#16864654—22no—
50#16864654—22no—
51#16864654D414yescomplemented
52#16864653—28no—
53#16864653—24no—
54#16864653—24no—
55C2×Dic542D411yescharacteristic
56C2×Dic542C2311yescharacteristic
57C2×Dic542C2311yescharacteristic
58C2×D1084—22no—
59C2×D1084D411yescharacteristic
60C2×D1084C2311yescharacteristic
61C2×D1084C2311yescharacteristic
62C2×C22⋊D271—216no—
63C2×C22⋊D271—22no—
64C2×C22⋊D271—22no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271D412yes—
68D4×D271—28no—
69D4×D271—24no—
70D4×D271—24no—
71C22×Dic271—22no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271C2312yes—
75Dic1084—22no—
76#16864643—22no—
77#16864643—22no—
78#16864642—24no—
79D2168—22no—
80#16864639—24no—
81#16864638—22no—
82C2×C271⋊C8C2312yes—
83C4⋊Dic271—22no—
84(C2×D271)⋊C4—22no—
85(C2×D271)⋊C4—22no—
86C24×C271—21no—
87C2×D4×C271—21no—
88C2×D4×C271—22no—
89C2×D4×C271D414yescomplemented
90C2×D4×C271C2312yes—
91C22×C4×C271—21no—
92C22×C4×C271C2311yescharacteristic
93C2×C8×C271—22no—
94C2×C8×C271C2311yescharacteristic
95C2×C8×C271C2311yescharacteristic
96C4⋊C4×C271—21no—
97C4⋊C4×C271C2312yes—
98C22×D271—21no—
99C22×D271—28no—
100C22×D271—21no—
101C22×D271—82no—
102C22×D271—82no—
103C22×D271C2×D412yes—
104Dic542—22no—
105Dic542—21no—
106Dic542—21no—
107Dic542—21no—
108Dic542C2×D412yes—
109D1084—22no—
110D1084—21no—
111D1084—21no—
112D1084—21no—
113D1084—82no—
114D1084C2×D412yes—
115C22⋊D271—28no—
116C22⋊D271—28no—
117C22⋊D271—24no—
118C22⋊D271—24no—
119C22⋊D271—42no—
120C2×Dic271—28no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271C2×D412yes—
124C4×D271—24no—
125C4×D271—22no—
126C4×D271—22no—
127C4×D271—42no—
128C271⋊C8—22no—
129C23×C271—22no—
130C23×C271—24no—
131C23×C271—81no—
132C23×C271—81no—
133C23×C271C2×D411yescharacteristic
134D4×C271—24no—
135D4×C271—22no—
136D4×C271C2×D414yes—
137C2×C4×C271—22no—
138C2×C4×C271—81no—
139C2×C4×C271C2×D411yescharacteristic
140C2×C4×C271C2×D412yes—
141C2×C4×C271C2×D412yes—
142C2×C4×C271C2411yescharacteristic
143C8×C271—21no—
144C8×C271—21no—
145C22×C271—21no—
146C22×C271—21no—
147C22×C271—24no—
148C22×C271—42no—
149C22×C271—42no—
150C22×C271—84no—
151C22×C271#12012yes—
152C22×C271C22×D411yescharacteristic
153C2×D271—22no—
154C2×D271—22no—
155C2×D271—22no—
156C2×D271—22no—
157C2×D271—41no—
158C2×D271—41no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—88no—
163C4×C271—42no—
164C4×C271#12012yes—
165C4×C271C22×D411yescharacteristic
166C4×C271C22×D411yescharacteristic
167Dic271—22no—
168Dic271—22no—
169Dic271—41no—
170C2×C271—22no—
171C2×C271—44no—
172C2×C271C2×#12011yescharacteristic
173C2×C271#57611yescharacteristic
174C2×C271#57711yescharacteristic
175D271—42no—
176D271—42no—
177D271—81no—
178C271#301211yescharacteristic, complemented, Sylow
179#3012—2711nomaximal, Sylow
180C2×H(5,𝔽2)—2711no—
181C2×AGL(1,ℤ/8)—2711no—
182C2×#135—2711no—
183#537—2711no—
184#452—2712no—
185#450—2712no—
186#449—2712no—
187#447—2712no—
188#417—2711no—
189C2×#102D27111yescharacteristic, complemented
190#408—2711no—
191H(5,𝔽2)—5424no—
192H(5,𝔽2)—5424no—
193C2×#41—2711no—
194C2×#41—2712no—
195C2×#41—2711no—
196C2×#41—2711no—
197C2×#41—5421no—
198C22×D4—2712no—
199C22×D4—5421no—
200C22×D4—5422no—
201C22×D4C2×D27111yescharacteristic
202AGL(1,ℤ/8)—5424no—
203#135—5424no—
204C2×Q16—2711no—
205C2×SD16—2712no—
206C2×SD16—2711no—
207C2×SD16—2711no—
208C2×D8—2712no—
209C2×D8—2711no—
210C2×M16—2711no—
211C22×C8C2×D27111yescharacteristic, complemented
212C22⋊Q8—2712no—
213C22⋊Q8—5421no—
214#121—2712no—
215#121—5421no—
216#121—5422no—
217C2×C4⋊C4C2×D27111yescharacteristic, complemented
218#103—2712no—
219#102—2712no—
220#102C2×D27112yescomplemented
221#102C2×D27112yescomplemented
222C22⋊C8—2712no—
223C22⋊C8—2712no—
224C24—21no—
225C24—5422no—
226#41—2714no—
227#41—5424no—
228#41—5428no—
229#41—5424no—
230#41—5422no—
231#41—5422no—
232#41—5422no—
233#41—5422no—
234#41—5422no—
235C2×Q8—2711no—
236C2×Q8—2711no—
237C2×Q8—2711no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—2712no—
241C2×D4—2711no—
242C2×D4—2711no—
243C2×D4—2711no—
244C2×D4—5422no—
245C2×D4—5422no—
246C2×D4—5424no—
247C2×D4—5428no—
248C2×D4—5422no—
249C2×D4—54216no—
250C2×D4—5421no—
251C2×D4—5424no—
252C2×D4—5421no—
253C2×D4D108414yescomplemented
254C2×D4C22×D27112yes—
255C22×C4—21no—
256C22×C4—2712no—
257C22×C4—5422no—
258C22×C4—5421no—
259C22×C4—5421no—
260C22×C4C22×D27111yescharacteristic
261Q16—5422no—
262SD16—5422no—
263SD16—5422no—
264SD16—5424no—
265D8—5422no—
266D8—5424no—
267M16—5422no—
268C2×C8—22no—
269C2×C8—2712no—
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—5422no—
274C4⋊C4C22×D27112yes—
275C22⋊C4—5422no—
276C22⋊C4—5422no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—2712no—
282C23—5428no—
283C23—5421no—
284C23—5421no—
285C23—21682no—
286C23—21682no—
287C23D4×D27111yescharacteristic
288Q8—2712no—
289Q8—5422no—
290Q8—5421no—
291Q8—5421no—
292Q8—5421no—
293D4—24no—
294D4—22no—
295D4—2712no—
296D4—5428no—
297D4—5428no—
298D4—5424no—
299D4—5424no—
300D4—5422no—
301D4—5421no—
302D4—5421no—
303D4—5421no—
304D4—10842no—
305D4—21682no—
306D4C2×D108414yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—2712no—
310C2×C4—5428no—
311C2×C4—5421no—
312C2×C4—5424no—
313C2×C4—5422no—
314C2×C4—5422no—
315C2×C4—5421no—
316C2×C4—10842no—
317C2×C4D4×D27111yescharacteristic
318C2×C4D4×D27112yes—
319C2×C4C2×D108412yes—
320C2×C4C23×D27111yescharacteristic
321C8—21no—
322C8—21no—
323C8—5422no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—5422no—
331C22—5422no—
332C22—5422no—
333C22—5422no—
334C22—10841no—
335C22—10841no—
336C22—10841no—
337C22—10841no—
338C22—10841no—
339C22—21688no—
340C22#2257613012yes—
341C22C2×D4×D27111yescharacteristic
342C4—42no—
343C4—5422no—
344C4—5422no—
345C4—10841no—
346C4#2257613012yes—
347C4C2×D4×D27111yescharacteristic
348C4C22×D108411yescharacteristic
349C2—22no—
350C2—44no—
351C2—10842no—
352C2—10842no—
353C2—21681no—
354C2C2×#2257613011yescharacteristic
355C2#3177469811yescharacteristic
356C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#102×C271C2
socleC22×C271C22×D4
hypercenterC2×#102D271
hypocenterC271#3012
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#22576205C2semidirect
2. C2×#22576186C2semidirect
3. C2×#22576182C2semidirect
4. #31774478C2semidirect
5. #31774060C2semidirect
6. #31774049C2semidirect
7. #31774043C2semidirect
8. #31774032C2semidirect
9. #31773896C2semidirect
10. #31773863C2semidirect
11. C2×#102×C271C2semidirect
16. C2×#16864654C22semidirect
24. C2×Dic1084C22semidirect
25. C2×#16864643C22semidirect
26. C2×#16864643C22semidirect
27. C2×#16864642C22semidirect
28. C2×D2168C22semidirect
29. C2×#16864639C22semidirect
30. C2×#16864638C22semidirect
32. #22576140C22semidirect
34. #22576135C22semidirect
36. #22576083C22semidirect
37. #22576080C22semidirect
38. (C2×D271)⋊C8C22semidirect
39. (C2×D271)⋊C8C22semidirect
41. C22×C8×C271C22semidirect
42. C2×C4⋊C4×C271C22semidirect
43. #102×C271C22semidirect
44. #102×C271C22semidirect
51. #16864654D4semidirect
89. C2×D4×C271D4semidirect
178. C271#3012semidirect
189. C2×#102D271semidirect
211. C22×C8C2×D271semidirect
217. C2×C4⋊C4C2×D271semidirect
220. #102C2×D271semidirect
221. #102C2×D271semidirect
253. C2×D4D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225762052
C2×#225761862
C2×#225761822
#317744782
#317740602
#317740492
#317740432
#317740322
#317738962
#317738632
C2×#102×C2712
#3012271