Finite Group Databasebeta

← #46845670

Subgroups — #46845670

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×#22576206C211yesmaximal, characteristic, complemented
2C2×#22576188C211yesmaximal, characteristic, complemented
3C2×#22576180C211yesmaximal, characteristic, complemented
4#31774478C211yesmaximal, characteristic, complemented
5#31774062C212yesmaximal, complemented
6#31774049C212yesmaximal, complemented
7#31774045C212yesmaximal, complemented
8#31774032C212yesmaximal, complemented
9C2×#22576080C211yesmaximal, characteristic, complemented
10#31773865C211yesmaximal, characteristic, complemented
11#417×C271C211yesmaximal, characteristic, complemented
12#22576206—24no—
13#22576206—24no—
14C2×#16864655C2212yes—
15C2×#16864654—21no—
16C2×#16864654C2211yescharacteristic
17C2×#16864654C2211yescharacteristic, complemented
18C22×D1084—21no—
19C22×D1084C2211yescharacteristic
20C2×D4×D271—22no—
21C2×D4×D271C2212yes—
22#22576188—24no—
23#22576180—24no—
24C2×#16864644C2211yescharacteristic, complemented
25C2×#16864643C2212yescomplemented
26C2×#16864642C2211yescharacteristic, complemented
27C2×#16864641C2211yescharacteristic, complemented
28C2×D2168C2212yescomplemented
29C2×#16864639C2211yescharacteristic, complemented
30C22×C271⋊C8C2211yescharacteristic, complemented
31#22576140—21no—
32#22576140C2212yescomplemented
33#22576135—21no—
34#22576135C2212yescomplemented
35#22576134—22no—
36#22576080C2212yescomplemented
37#22576080C2212yescomplemented
38(C2×D271)⋊C8C2212yescomplemented
39(C2×D271)⋊C8C2212yescomplemented
40C2×#41×C271C2211yescharacteristic
41C2×M16×C271C2211yescharacteristic, complemented
42C2×C4⋊C4×C271C2211yescharacteristic, complemented
43#103×C271C2212yescomplemented
44#102×C271C2212yescomplemented
45C23×D271—22no—
46C23×D271—21no—
47#16864655—24no—
48#16864655—24no—
49#16864654—28no—
50#16864654—22no—
51#16864654—22no—
52#16864654—22no—
53#16864654—22no—
54C2×Dic542D411yescharacteristic
55C2×Dic542C2311yescharacteristic
56C2×D1084—28no—
57C2×D1084—22no—
58C2×D1084—22no—
59C2×D1084—21no—
60C2×D1084—22no—
61C2×D1084D411yescharacteristic
62C2×D1084D414yescomplemented
63C2×D1084C2311yescharacteristic
64C2×D1084C2312yes—
65C2×C22⋊D271—22no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271D412yes—
68D4×D271—216no—
69D4×D271—24no—
70D4×D271—24no—
71C2×C4×D271—22no—
72C2×C4×D271—21no—
73C2×C4×D271C2312yes—
74#16864644—22no—
75#16864643—24no—
76#16864642—22no—
77#16864641—22no—
78D2168—24no—
79#16864639—22no—
80C2×C271⋊C8—22no—
81C2×C271⋊C8C2311yescharacteristic
82C2×C271⋊C8C2311yescharacteristic
83C4⋊Dic271—22no—
84(C2×D271)⋊C4—22no—
85(C2×D271)⋊C4—22no—
86#41×C271—22no—
87#41×C271D414yescomplemented
88C2×Q8×C271C2311yescharacteristic
89C2×D4×C271—21no—
90C2×D4×C271C2311yescharacteristic
91C22×C4×C271—21no—
92C22×C4×C271—21no—
93C22×C4×C271C2311yescharacteristic
94M16×C271—22no—
95C2×C8×C271C2312yes—
96C4⋊C4×C271—21no—
97C4⋊C4×C271C2312yes—
98C22×D271—22no—
99C22×D271—28no—
100C22×D271—24no—
101C22×D271—21no—
102C22×D271—82no—
103C22×D271—82no—
104C22×D271—81no—
105C22×D271—81no—
106C22×D271C2×D412yes—
107Dic542—22no—
108Dic542—21no—
109Dic542—21no—
110D1084—28no—
111D1084—22no—
112D1084—24no—
113D1084—22no—
114D1084—21no—
115D1084—21no—
116D1084—82no—
117D1084C2×D414yes—
118C22⋊D271—28no—
119C22⋊D271—24no—
120C22⋊D271—42no—
121C2×Dic271—21no—
122C2×Dic271C2×D412yes—
123C4×D271—28no—
124C4×D271—22no—
125C4×D271—22no—
126C4×D271—42no—
127C271⋊C8—21no—
128C271⋊C8—21no—
129C23×C271—21no—
130C23×C271C2×D411yescharacteristic
131Q8×C271—21no—
132Q8×C271C2×D412yes—
133D4×C271—24no—
134D4×C271—21no—
135D4×C271C2×D412yes—
136C2×C4×C271—21no—
137C2×C4×C271—24no—
138C2×C4×C271—22no—
139C2×C4×C271—81no—
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—22no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271#12012yes—
151C22×C271C22×D411yescharacteristic
152C2×D271—22no—
153C2×D271—24no—
154C2×D271—22no—
155C2×D271—22no—
156C2×D271—42no—
157C2×D271—42no—
158C2×D271—41no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—84no—
162C2×D271—88no—
163C4×C271—22no—
164C4×C271—42no—
165C4×C271#12012yes—
166C4×C271C22×D411yescharacteristic
167C4×C271C22×D411yescharacteristic
168Dic271—22no—
169Dic271—41no—
170C2×C271—22no—
171C2×C271—42no—
172C2×C271C2×#12011yescharacteristic
173C2×C271#57611yescharacteristic
174C2×C271#57711yescharacteristic
175D271—42no—
176D271—44no—
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#417D27111yescharacteristic, complemented
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
198C22×D4—2712no—
199C22×D4—2711no—
200C22×D4—5421no—
201C22×D4—5422no—
202AGL(1,ℤ/8)—5424no—
203#135—5424no—
204C2×Q16—2711no—
205C2×SD16—2711no—
206C2×SD16—2711no—
207C2×SD16—2712no—
208C2×D8—2711no—
209C2×D8—2712no—
210C2×M16C2×D27111yescharacteristic, complemented
211C22×C8—2711no—
212C22⋊Q8—2712no—
213C22⋊Q8—5421no—
214#121—2712no—
215#121—5421no—
216#121—5422no—
217C2×C4⋊C4C2×D27111yescharacteristic, complemented
218#103C2×D27112yescomplemented
219#102—2712no—
220#102—2712no—
221#102C2×D27112yescomplemented
222C22⋊C8—2712no—
223C22⋊C8—2712no—
224C24—5422no—
225C24—5421no—
226#41—22no—
227#41—5428no—
228#41—5424no—
229#41—5424no—
230#41—5422no—
231#41—5422no—
232#41—5422no—
233#41—5422no—
234#41D108414yescomplemented
235C2×Q8—2711no—
236C2×Q8—2711no—
237C2×Q8C22×D27111yescharacteristic
238C2×D4—21no—
239C2×D4—2712no—
240C2×D4—2714no—
241C2×D4—2711no—
242C2×D4—2712no—
243C2×D4—2711no—
244C2×D4—5422no—
245C2×D4—5428no—
246C2×D4—5422no—
247C2×D4—5422no—
248C2×D4—5424no—
249C2×D4—54216no—
250C2×D4—5424no—
251C2×D4—5421no—
252C2×D4—5421no—
253C2×D4—5422no—
254C2×D4C22×D27111yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—2712no—
258C22×C4—5422no—
259C22×C4—5421no—
260C22×C4C22×D27111yescharacteristic
261Q16—5422no—
262SD16—5424no—
263SD16—5422no—
264SD16—5422no—
265D8—5424no—
266D8—5422no—
267M16—22no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8—5422no—
271C2×C8C22×D27112yes—
272C4⋊C4—21no—
273C4⋊C4—5422no—
274C4⋊C4C22×D27112yes—
275C22⋊C4—5422no—
276C22⋊C4—5422no—
277C23—21no—
278C23—2712no—
279C23—5428no—
280C23—5422no—
281C23—5424no—
282C23—5421no—
283C23—21681no—
284C23—21681no—
285C23—21682no—
286C23—21682no—
287C23D4×D27111yescharacteristic
288Q8—21no—
289Q8—5422no—
290Q8—5421no—
291Q8—5421no—
292Q8C2×D108412yes—
293D4—24no—
294D4—21no—
295D4—2714no—
296D4—5428no—
297D4—5428no—
298D4—5424no—
299D4—5424no—
300D4—5422no—
301D4—5421no—
302D4—5422no—
303D4—5421no—
304D4—10842no—
305D4—21682no—
306D4C2×D108412yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—2712no—
312C2×C4—5428no—
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—22no—
322C8—5421no—
323C8—5421no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—5424no—
330C22—5422no—
331C22—5422no—
332C22—5422no—
333C22—10842no—
334C22—10842no—
335C22—10841no—
336C22—10841no—
337C22—10841no—
338C22—21684no—
339C22—21688no—
340C22#2257613012yes—
341C22C2×D4×D27111yescharacteristic
342C4—22no—
343C4—42no—
344C4—5422no—
345C4—10841no—
346C4#2257613012yes—
347C4C2×D4×D27111yescharacteristic
348C4C22×D108411yescharacteristic
349C2—22no—
350C2—42no—
351C2—10842no—
352C2—10844no—
353C2—21681no—
354C2C2×#2257613011yescharacteristic
355C2#3177470111yescharacteristic
356C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#417×C271C2
socleC22×C271C22×D4
hypercenter#417D271
hypocenterC271#3012
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#22576206C2semidirect
2. C2×#22576188C2semidirect
3. C2×#22576180C2semidirect
4. #31774478C2semidirect
5. #31774062C2semidirect
6. #31774049C2semidirect
7. #31774045C2semidirect
8. #31774032C2semidirect
9. C2×#22576080C2semidirect
10. #31773865C2semidirect
11. #417×C271C2semidirect
17. C2×#16864654C22semidirect
24. C2×#16864644C22semidirect
25. C2×#16864643C22semidirect
26. C2×#16864642C22semidirect
27. C2×#16864641C22semidirect
28. C2×D2168C22semidirect
29. C2×#16864639C22semidirect
30. C22×C271⋊C8C22semidirect
32. #22576140C22semidirect
34. #22576135C22semidirect
36. #22576080C22semidirect
37. #22576080C22semidirect
38. (C2×D271)⋊C8C22semidirect
39. (C2×D271)⋊C8C22semidirect
41. C2×M16×C271C22semidirect
42. C2×C4⋊C4×C271C22semidirect
43. #103×C271C22semidirect
44. #102×C271C22semidirect
62. C2×D1084D4semidirect
87. #41×C271D4semidirect
178. C271#3012semidirect
188. #417D271semidirect
210. C2×M16C2×D271semidirect
217. C2×C4⋊C4C2×D271semidirect
218. #103C2×D271semidirect
221. #102C2×D271semidirect
234. #41D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225762062
C2×#225761882
C2×#225761802
#317744782
#317740622
#317740492
#317740452
#317740322
C2×#225760802
#317738652
#417×C2712
#3012271