Finite Group Databasebeta

← #46845583

Subgroups — #46845583

Order 34688.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#16864653C211yesmaximal, characteristic, complemented
2C2×#22576181C212yesmaximal, complemented
3#31774410C211yesmaximal, characteristic, complemented
4#31774056C212yesmaximal, complemented
5#31774052C211yesmaximal, characteristic, complemented
6#31774043C212yesmaximal, complemented
7#31774038C211yesmaximal, characteristic, complemented
8#31774031C211yesmaximal, characteristic, complemented
9#31773890C212yesmaximal, complemented
10C22⋊C8×D271C211yesmaximal, characteristic, complemented
11#450×C271C211yesmaximal, characteristic, complemented
12C2×#16864655C2211yescharacteristic, complemented
13C2×#16864653—28no—
14C2×#16864653—21no—
15C2×#16864653—21no—
16C2×#16864653C2212yes—
17C2×#16864653C2212yes—
18C22×Dic542C2211yescharacteristic
19C22×C22⋊D271—21no—
20C23×Dic271—21no—
21C22×C4×D271C2211yescharacteristic
22#22576181—28no—
23C2×#16864644C2212yescomplemented
24C2×#16864643C2212yescomplemented
25C2×#16864639C2212yescomplemented
26C2×C8×D271C2212yescomplemented
27#22576145—21no—
28#22576136C2211yescharacteristic, complemented
29#22576135C2211yescharacteristic, complemented
30#22576134C2211yescharacteristic
31#22576130—21no—
32#22576122—21no—
33(C22⋊D271)⋊C4—21no—
34#22576118C2211yescharacteristic, complemented
35#22576085C2212yescomplemented
36#22576083C2212yescomplemented
37#22576081C2212yescomplemented
38C22⋊(C271⋊C8)C2211yescharacteristic
39(C2×D271)⋊C8C2212yescomplemented
40C22×D4×C271C2211yescharacteristic
41C2×SD16×C271C2212yescomplemented
42C22⋊Q8×C271C2211yescharacteristic
43#102×C271C2212yescomplemented
44C22⋊C8×C271C2211yescharacteristic
45C23×D271D411yescharacteristic
46#16864655—24no—
47#16864653—28no—
48#16864653—28no—
49#16864653—24no—
50#16864653—24no—
51#16864653—24no—
52#16864653D418yescomplemented
53C2×Dic542—24no—
54C2×Dic542—21no—
55C2×Dic542C2312yes—
56C2×D1084—21no—
57C2×D1084—21no—
58C2×D1084C2311yescharacteristic
59C2×C22⋊D271—28no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271—22no—
62C2×C22⋊D271—21no—
63C22×Dic271—28no—
64C22×Dic271—22no—
65C22×Dic271—22no—
66C22×Dic271—21no—
67C22×Dic271D411yescharacteristic
68C2×C4×D271—24no—
69C2×C4×D271—21no—
70C2×C4×D271D411yescharacteristic
71C2×C4×D271D411yescharacteristic
72C2×C4×D271C2311yescharacteristic
73C2×C4×D271C2311yescharacteristic
74#16864644—24no—
75#16864643—24no—
76#16864639—24no—
77C2×C271⋊C8C2312yes—
78C8×D271—24no—
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81(C2×D271)⋊C4—21no—
82(C2×D271)⋊C4—21no—
83(C2×D271)⋊C4—21no—
84(C2×D271)⋊C4—21no—
85C4×Dic271—21no—
86C24×C271—21no—
87C2×Q8×C271C2311yescharacteristic
88C2×D4×C271—24no—
89C2×D4×C271—21no—
90C2×D4×C271C2312yes—
91C22×C4×C271C2311yescharacteristic
92SD16×C271—24no—
93C2×C8×C271C2312yes—
94C4⋊C4×C271—21no—
95C4⋊C4×C271C2311yescharacteristic
96C22⋊C4×C271—21no—
97C22×D271—22no—
98C22×D271—22no—
99C22×D271—21no—
100C22×D271—41no—
101C22×D271C2×D411yescharacteristic
102C22×D271C2×D411yescharacteristic
103Dic542—24no—
104Dic542—22no—
105Dic542C2×D414yes—
106D1084—22no—
107D1084—42no—
108D1084—42no—
109C22⋊D271—28no—
110C22⋊D271—28no—
111C22⋊D271—42no—
112C2×Dic271—24no—
113C2×Dic271—28no—
114C2×Dic271—22no—
115C2×Dic271—22no—
116C2×Dic271—28no—
117C2×Dic271—21no—
118C2×Dic271C2×D411yescharacteristic
119C2×Dic271C2×D411yescharacteristic
120C4×D271—22no—
121C4×D271—22no—
122C4×D271—22no—
123C4×D271—42no—
124C4×D271C2×D414yes—
125C271⋊C8—22no—
126C23×C271—24no—
127C23×C271—22no—
128C23×C271—42no—
129C23×C271—42no—
130C23×C271C2×D411yescharacteristic
131Q8×C271—22no—
132D4×C271—24no—
133D4×C271—22no—
134D4×C271C2×D414yes—
135C2×C4×C271—22no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271C2×D411yescharacteristic
139C2×C4×C271C2411yescharacteristic
140C8×C271—22no—
141C22×C271—21no—
142C22×C271—21no—
143C22×C271—24no—
144C22×C271—48no—
145C22×C271—42no—
146C22×C271—42no—
147C22×C271#13512yescomplemented
148C22×C271C22×D411yescharacteristic
149C2×D271—21no—
150C2×D271—21no—
151C2×D271—21no—
152C2×D271—21no—
153C2×D271—21no—
154C2×D271—44no—
155C2×D271—41no—
156C2×D271—41no—
157C2×D271—41no—
158C2×D271#12012yes—
159C4×C271—21no—
160C4×C271—41no—
161C4×C271—41no—
162C4×C271C22×D412yes—
163Dic271—21no—
164Dic271—24no—
165Dic271—41no—
166Dic271#12012yes—
167C2×C271—22no—
168C2×C271—44no—
169C2×C271C2×#12011yescharacteristic
170C2×C271C2×#13511yescharacteristic
171C2×C271#57511yescharacteristic
172D271—22no—
173D271—41no—
174D271—81no—
175C271#300211yescharacteristic, complemented, Sylow
176#3002—2711nomaximal, Sylow
177C22×#41—2711no—
178C2×#135—2712no—
179#525—2711no—
180#452—2712no—
181#451—2711no—
182#450D27111yescharacteristic, complemented
183#449—2712no—
184#448—2711no—
185#447—2711no—
186#416—2712no—
187C2×C22⋊C8—2711no—
188C2×#41—2711no—
189C2×#41—2712no—
190C2×#41—2712no—
191C2×#41—5421no—
192C2×#41—5421no—
193C2×#41—5428no—
194C22×Q8—2711no—
195C22×D4—5421no—
196C22×D4C2×D27111yescharacteristic
197C23×C4—2711no—
198C23×C4—5421no—
199#135—5428no—
200C2×Q16—2712no—
201C2×SD16—2712no—
202C2×SD16C2×D27112yescomplemented
203C2×D8—2712no—
204C22×C8—2712no—
205#123—5421no—
206C22⋊Q8—2711no—
207C22⋊Q8C2×D27111yescharacteristic
208#121—2711no—
209#121—2711no—
210#120—5421no—
211C4×D4—5421no—
212C4×D4—5421no—
213#117—2711no—
214#103—2712no—
215#103—2712no—
216#102—2712no—
217#102C2×D27112yescomplemented
218C22⋊C8—2711no—
219C22⋊C8—2712no—
220C22⋊C8C2×D27111yescharacteristic
221C24—21no—
222C24—2711no—
223#41—2718no—
224#41—5424no—
225#41—5424no—
226#41—5424no—
227#41—5428no—
228#41—5428no—
229#41—5424no—
230C2×Q8—2712no—
231C2×Q8—5424no—
232C2×Q8—5421no—
233C2×Q8C22×D27111yescharacteristic
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—2711no—
237C2×D4—5421no—
238C2×D4—5421no—
239C2×D4—5422no—
240C2×D4—5422no—
241C2×D4—5428no—
242C2×D4—5421no—
243C2×D4C22×D27112yes—
244C22×C4—2711no—
245C22×C4—2711no—
246C22×C4—2711no—
247C22×C4—2711no—
248C22×C4—2711no—
249C22×C4—5424no—
250C22×C4—5421no—
251C22×C4—5422no—
252C22×C4—5422no—
253C22×C4—5428no—
254C22×C4—5421no—
255C22×C4C22×D27111yescharacteristic
256Q16—5424no—
257SD16—24no—
258SD16—5424no—
259D8—5424no—
260C2×C8—2712no—
261C2×C8—5424no—
262C2×C8C22×D27112yes—
263C4⋊C4—21no—
264C4⋊C4—2711no—
265C4⋊C4—5421no—
266C4⋊C4C22×D27111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—5421no—
269C22⋊C4—5421no—
270C22⋊C4—5421no—
271C22⋊C4—5421no—
272C42—5421no—
273C23—24no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—2711no—
278C23—2711no—
279C23—5422no—
280C23—5422no—
281C23—5421no—
282C23—10841no—
283C23D4×D27111yescharacteristic
284Q8—22no—
285Q8—2714no—
286Q8—5424no—
287Q8—5422no—
288D4—24no—
289D4—22no—
290D4—5428no—
291D4—5428no—
292D4—5422no—
293D4—10842no—
294D4—10842no—
295D4—10842no—
296D4D4×D27114yes—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—2711no—
301C2×C4—2714no—
302C2×C4—2711no—
303C2×C4—5422no—
304C2×C4—5422no—
305C2×C4—5422no—
306C2×C4—5428no—
307C2×C4—5424no—
308C2×C4—5428no—
309C2×C4—5422no—
310C2×C4—5422no—
311C2×C4—5421no—
312C2×C4—10842no—
313C2×C4D4×D27111yescharacteristic
314C2×C4C23×D27111yescharacteristic
315C8—22no—
316C8—5422no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—48no—
321C22—42no—
322C22—42no—
323C22—2712no—
324C22—5421no—
325C22—5421no—
326C22—5421no—
327C22—5421no—
328C22—5421no—
329C22—10844no—
330C22—10841no—
331C22—10841no—
332C22—10841no—
333C22#2257618112yescomplemented
334C22C2×D4×D27111yescharacteristic
335C4—21no—
336C4—41no—
337C4—41no—
338C4—2712no—
339C4—5421no—
340C4—5424no—
341C4—10841no—
342C4C2×D4×D27112yes—
343C2—22no—
344C2—44no—
345C2—5422no—
346C2—10841no—
347C2—21681no—
348C2#120×D27111yescharacteristic
349C2C2×#2257618111yescharacteristic
350C2#3177469311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#450×C271C2
socleC22×C271C22×D4
hypercenter#450D271
hypocenterC271#3002
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#16864653C2semidirect
2. C2×#22576181C2semidirect
3. #31774410C2semidirect
4. #31774056C2semidirect
5. #31774052C2semidirect
6. #31774043C2semidirect
7. #31774038C2semidirect
8. #31774031C2semidirect
9. #31773890C2semidirect
10. C22⋊C8×D271C2semidirect
11. #450×C271C2semidirect
12. C2×#16864655C22semidirect
23. C2×#16864644C22semidirect
24. C2×#16864643C22semidirect
25. C2×#16864639C22semidirect
26. C2×C8×D271C22semidirect
28. #22576136C22semidirect
29. #22576135C22semidirect
34. #22576118C22semidirect
35. #22576085C22semidirect
36. #22576083C22semidirect
37. #22576081C22semidirect
39. (C2×D271)⋊C8C22semidirect
41. C2×SD16×C271C22semidirect
43. #102×C271C22semidirect
52. #16864653D4semidirect
147. C22×C271#135semidirect
175. C271#3002semidirect
182. #450D271semidirect
202. C2×SD16C2×D271semidirect
217. #102C2×D271semidirect
333. C22#22576181semidirect

Maximal subgroups

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

subgroupindex
C22×#168646532
C2×#225761812
#317744102
#317740562
#317740522
#317740432
#317740382
#317740312
#317738902
C22⋊C8×D2712
#450×C2712
#3002271