Finite Group Databasebeta

← #46845587

Subgroups — #46845587

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×#16864655C211yesmaximal, characteristic, complemented
2C2×#22576183C212yesmaximal, complemented
3#31774410C211yesmaximal, characteristic, complemented
4#31774058C212yesmaximal, complemented
5#31774047C211yesmaximal, characteristic, complemented
6#31774045C212yesmaximal, complemented
7#31774036C211yesmaximal, characteristic, complemented
8#31774033C211yesmaximal, characteristic, complemented
9#31773892C212yesmaximal, complemented
10C22⋊C8×D271C211yesmaximal, characteristic, complemented
11#451×C271C211yesmaximal, characteristic, complemented
12C2×#16864655—28no—
13C2×#16864655—21no—
14C2×#16864655—21no—
15C2×#16864655C2212yes—
16C2×#16864655C2212yes—
17C2×#16864655C2211yescharacteristic, complemented
18C22×D1084—21no—
19C22×D1084C2211yescharacteristic
20C22×C4×D271—21no—
21C22×C4×D271C2211yescharacteristic
22#22576183—28no—
23C2×#16864641C2212yescomplemented
24C2×#16864641C2212yescomplemented
25C2×D2168C2212yescomplemented
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#22576084C2212yescomplemented
36#22576082C2212yescomplemented
37#22576080C2212yescomplemented
38C22⋊(C271⋊C8)C2211yescharacteristic
39(C2×D271)⋊C8C2212yescomplemented
40C22×Q8×C271C2211yescharacteristic
41C2×Q16×C271C2212yescomplemented
42C22⋊Q8×C271C2211yescharacteristic
43#103×C271C2212yescomplemented
44C22⋊C8×C271C2211yescharacteristic
45C23×D271—21no—
46C23×D271D411yescharacteristic
47#16864655—28no—
48#16864655—28no—
49#16864655—24no—
50#16864655—24no—
51#16864655—24no—
52#16864655—24no—
53#16864655D418yescomplemented
54C2×D1084—28no—
55C2×D1084—22no—
56C2×D1084—22no—
57C2×D1084—24no—
58C2×D1084—21no—
59C2×D1084—21no—
60C2×D1084—21no—
61C2×D1084C2311yescharacteristic
62C2×D1084C2312yes—
63C2×C22⋊D271—21no—
64C22×Dic271D411yescharacteristic
65C2×C4×D271—28no—
66C2×C4×D271—22no—
67C2×C4×D271—22no—
68C2×C4×D271—24no—
69C2×C4×D271—21no—
70C2×C4×D271D411yescharacteristic
71C2×C4×D271D411yescharacteristic
72C2×C4×D271C2311yescharacteristic
73C2×C4×D271C2311yescharacteristic
74#16864641—24no—
75#16864641—24no—
76D2168—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—
86C2×Q8×C271—24no—
87C2×Q8×C271—21no—
88C2×Q8×C271C2311yescharacteristic
89C2×Q8×C271C2312yes—
90C22×C4×C271—21no—
91C22×C4×C271C2311yescharacteristic
92Q16×C271—24no—
93C2×C8×C271C2312yes—
94C4⋊C4×C271—21no—
95C4⋊C4×C271C2311yescharacteristic
96C22⋊C4×C271—21no—
97C22×D271—24no—
98C22×D271—22no—
99C22×D271—22no—
100C22×D271—22no—
101C22×D271—21no—
102C22×D271—41no—
103C22×D271—42no—
104C22×D271—42no—
105C22×D271C2×D411yescharacteristic
106C22×D271C2×D411yescharacteristic
107D1084—28no—
108D1084—24no—
109D1084—28no—
110D1084—22no—
111D1084—22no—
112D1084—42no—
113D1084—42no—
114D1084C2×D414yes—
115C22⋊D271—42no—
116C2×Dic271—22no—
117C2×Dic271—21no—
118C2×Dic271C2×D411yescharacteristic
119C2×Dic271C2×D411yescharacteristic
120C4×D271—28no—
121C4×D271—22no—
122C4×D271—22no—
123C4×D271—28no—
124C4×D271—22no—
125C4×D271—42no—
126C4×D271C2×D414yes—
127C271⋊C8—22no—
128C23×C271C2×D411yescharacteristic
129Q8×C271—24no—
130Q8×C271—22no—
131Q8×C271—22no—
132Q8×C271C2×D414yes—
133C2×C4×C271—24no—
134C2×C4×C271—22no—
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#13512yescomplemented
144C22×C271C22×D411yescharacteristic
145C2×D271—21no—
146C2×D271—21no—
147C2×D271—21no—
148C2×D271—24no—
149C2×D271—21no—
150C2×D271—21no—
151C2×D271—44no—
152C2×D271—48no—
153C2×D271—42no—
154C2×D271—42no—
155C2×D271—41no—
156C2×D271—41no—
157C2×D271—41no—
158C2×D271#12012yes—
159C4×C271—24no—
160C4×C271—21no—
161C4×C271—41no—
162C4×C271—41no—
163C4×C271C22×D412yes—
164Dic271—21no—
165Dic271—41no—
166Dic271#12012yes—
167C2×C271—22no—
168C2×C271C2×#12011yescharacteristic
169C2×C271C2×#13511yescharacteristic
170C2×C271#57511yescharacteristic
171D271—22no—
172D271—41no—
173D271—44no—
174D271—81no—
175C271#300211yescharacteristic, complemented, Sylow
176#3002—2711nomaximal, Sylow
177C22×#41—2711no—
178C2×#135—2712no—
179#525—2711no—
180#452—2712no—
181#451D27111yescharacteristic, complemented
182#450—2711no—
183#449—2712no—
184#448—2711no—
185#447—2711no—
186#416—2712no—
187C2×C22⋊C8—2711no—
188C2×#41—2712no—
189C2×#41—2711no—
190C2×#41—2712no—
191C2×#41—5421no—
192C2×#41—5421no—
193C2×#41—5428no—
194C22×Q8C2×D27111yescharacteristic
195C22×D4—2711no—
196C22×D4—5421no—
197C23×C4—2711no—
198C23×C4—5421no—
199#135—5428no—
200C2×Q16C2×D27112yescomplemented
201C2×SD16—2712no—
202C2×SD16—2712no—
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#103C2×D27112yescomplemented
216#102—2712no—
217#102—2712no—
218C22⋊C8—2711no—
219C22⋊C8—2712no—
220C22⋊C8C2×D27111yescharacteristic
221C24—2711no—
222C24—5421no—
223#41—2718no—
224#41—5424no—
225#41—5424no—
226#41—5424no—
227#41—5428no—
228#41—5428no—
229#41—5424no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8C22×D27111yescharacteristic
233C2×Q8C22×D27112yes—
234C2×D4—2712no—
235C2×D4—2711no—
236C2×D4—5424no—
237C2×D4—5422no—
238C2×D4—5422no—
239C2×D4—5428no—
240C2×D4—5421no—
241C2×D4—5421no—
242C2×D4—5421no—
243C2×D4—5421no—
244C22×C4—21no—
245C22×C4—2711no—
246C22×C4—2711no—
247C22×C4—2711no—
248C22×C4—2711no—
249C22×C4—2711no—
250C22×C4—5424no—
251C22×C4—5422no—
252C22×C4—5422no—
253C22×C4—5421no—
254C22×C4—5428no—
255C22×C4C22×D27111yescharacteristic
256Q16—24no—
257SD16—5424no—
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—2711no—
274C23—2711no—
275C23—5422no—
276C23—5424no—
277C23—5422no—
278C23—5422no—
279C23—5421no—
280C23—10841no—
281C23—10842no—
282C23—10842no—
283C23D4×D27111yescharacteristic
284Q8—24no—
285Q8—22no—
286Q8—22no—
287Q8D4×D27114yes—
288D4—2714no—
289D4—5428no—
290D4—5428no—
291D4—5424no—
292D4—5422no—
293D4—5422no—
294D4—10842no—
295D4—10842no—
296D4—10842no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—2711no—
303C2×C4—2714no—
304C2×C4—2711no—
305C2×C4—5422no—
306C2×C4—5428no—
307C2×C4—5422no—
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—2712no—
320C22—5421no—
321C22—5421no—
322C22—5424no—
323C22—5421no—
324C22—5421no—
325C22—5421no—
326C22—10844no—
327C22—10848no—
328C22—10842no—
329C22—10842no—
330C22—10841no—
331C22—10841no—
332C22—10841no—
333C22#2257618312yescomplemented
334C22C2×D4×D27111yescharacteristic
335C4—24no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—2712no—
340C4—5421no—
341C4—10841no—
342C4C2×D4×D27112yes—
343C2—22no—
344C2—5422no—
345C2—10841no—
346C2—10844no—
347C2—21681no—
348C2#120×D27111yescharacteristic
349C2C2×#2257618311yescharacteristic
350C2#3177469611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#451×C271C2
socleC22×C271C22×D4
hypercenter#451D271
hypocenterC271#3002
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#16864655C2semidirect
2. C2×#22576183C2semidirect
3. #31774410C2semidirect
4. #31774058C2semidirect
5. #31774047C2semidirect
6. #31774045C2semidirect
7. #31774036C2semidirect
8. #31774033C2semidirect
9. #31773892C2semidirect
10. C22⋊C8×D271C2semidirect
11. #451×C271C2semidirect
17. C2×#16864655C22semidirect
23. C2×#16864641C22semidirect
24. C2×#16864641C22semidirect
25. C2×D2168C22semidirect
26. C2×C8×D271C22semidirect
28. #22576136C22semidirect
29. #22576135C22semidirect
34. #22576118C22semidirect
35. #22576084C22semidirect
36. #22576082C22semidirect
37. #22576080C22semidirect
39. (C2×D271)⋊C8C22semidirect
41. C2×Q16×C271C22semidirect
43. #103×C271C22semidirect
53. #16864655D4semidirect
143. C22×C271#135semidirect
175. C271#3002semidirect
181. #451D271semidirect
200. C2×Q16C2×D271semidirect
215. #103C2×D271semidirect
333. C22#22576183semidirect

Maximal subgroups

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

subgroupindex
C22×#168646552
C2×#225761832
#317744102
#317740582
#317740472
#317740452
#317740362
#317740332
#317738922
C22⋊C8×D2712
#451×C2712
#3002271