Finite Group Databasebeta

← #46845619

Subgroups — #46845619

Order 34688.

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

normal subgroups only

This group has 754 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
2C2×#22576189C212yesmaximal, complemented
3C22⋊Q8×D271C211yesmaximal, characteristic, complemented
4#31774060C212yesmaximal, complemented
5#31774056C212yesmaximal, complemented
6#31774055C211yesmaximal, characteristic, complemented
7#31774052C211yesmaximal, characteristic, complemented
8#31774048C211yesmaximal, characteristic, complemented
9#31773895C212yesmaximal, complemented
10#31773859C211yesmaximal, characteristic, complemented
11#450×C271C211yesmaximal, characteristic, complemented
12C2×#16864653—28no—
13C2×#16864653—21no—
14C2×#16864653—21no—
15C2×#16864653C2212yes—
16C2×#16864653C2212yes—
17C22×Dic542C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic, complemented
19C22×C22⋊D271—21no—
20C23×Dic271—21no—
21C22×C4×D271C2211yescharacteristic
22#22576189—28no—
23C2×Dic1084C2212yescomplemented
24C2×#16864644C2212yescomplemented
25C2×#16864642C2212yescomplemented
26C2×#16864637C2212yescomplemented
27#22576140C2211yescharacteristic, complemented
28#22576139C2211yescharacteristic
29#22576138—21no—
30#22576137—21no—
31#22576136C2211yescharacteristic, complemented
32C4⋊C4×D271—21no—
33C4⋊C4×D271C2211yescharacteristic, complemented
34C22⋊C4×D271—21no—
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#16864653—28no—
47#16864653—28no—
48#16864653—24no—
49#16864653—24no—
50#16864653—24no—
51#16864653D418yes—
52C2×Dic542—24no—
53C2×Dic542—21no—
54C2×Dic542—21no—
55C2×Dic542C2311yescharacteristic
56C2×Dic542C2312yes—
57Q8×D271—24no—
58C2×C22⋊D271—28no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C22×Dic271—28no—
62C22×Dic271—22no—
63C22×Dic271—22no—
64C22×Dic271—21no—
65C22×Dic271D411yescharacteristic
66C2×C4×D271—24no—
67C2×C4×D271—21no—
68C2×C4×D271—21no—
69C2×C4×D271D411yescharacteristic
70C2×C4×D271D411yescharacteristic
71C2×C4×D271C2311yescharacteristic
72C2×C4×D271C2311yescharacteristic
73Dic1084—24no—
74#16864644—24no—
75#16864642—24no—
76#16864637—24no—
77C2×C271⋊C8C2312yes—
78C4⋊Dic271—21no—
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83C22⋊Dic271—21no—
84(C2×D271)⋊C4—21no—
85(C2×D271)⋊C4—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—41no—
100C22×D271C2×D411yescharacteristic
101C22×D271C2×D411yescharacteristic
102Dic542—24no—
103Dic542—22no—
104Dic542—22no—
105Dic542—42no—
106Dic542C2×D414yes—
107C22⋊D271—28no—
108C22⋊D271—28no—
109C2×Dic271—24no—
110C2×Dic271—28no—
111C2×Dic271—22no—
112C2×Dic271—22no—
113C2×Dic271—28no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271C2×D411yescharacteristic
117C2×Dic271C2×D411yescharacteristic
118C4×D271—22no—
119C4×D271—22no—
120C4×D271—22no—
121C4×D271—42no—
122C4×D271—42no—
123C4×D271C2×D414yes—
124C271⋊C8—22no—
125C23×C271—24no—
126C23×C271—22no—
127C23×C271—42no—
128C23×C271—42no—
129C23×C271C2×D411yescharacteristic
130Q8×C271—22no—
131D4×C271—24no—
132D4×C271—22no—
133D4×C271C2×D414yes—
134C2×C4×C271—22no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271C2×D411yescharacteristic
138C2×C4×C271C2411yescharacteristic
139C8×C271—22no—
140C22×C271—21no—
141C22×C271—21no—
142C22×C271—24no—
143C22×C271—48no—
144C22×C271—42no—
145C22×C271—42no—
146C22×C271#13712yescomplemented
147C22×C271C22×D411yescharacteristic
148C2×D271—21no—
149C2×D271—21no—
150C2×D271—21no—
151C2×D271—21no—
152C2×D271—21no—
153C2×D271—44no—
154C2×D271#12012yes—
155C4×C271—21no—
156C4×C271—41no—
157C4×C271—41no—
158C4×C271C22×D412yes—
159Dic271—21no—
160Dic271—24no—
161Dic271—41no—
162Dic271—41no—
163Dic271#12012yes—
164C2×C271—22no—
165C2×C271—44no—
166C2×C271C2×#12011yescharacteristic
167C2×C271C2×#13711yescharacteristic
168C2×C271#57511yescharacteristic
169D271—22no—
170D271—41no—
171C271#300711yescharacteristic, complemented, Sylow
172#3007—2711nomaximal, Sylow
173C22×#41—2711no—
174C2×#137—2712no—
175C2×C22⋊Q8—2711no—
176#452—2712no—
177#452—2712no—
178#451—2711no—
179#451—2711no—
180#450—2711no—
181#450D27111yescharacteristic, complemented
182#417—2712no—
183#407—2711no—
184C2×#41—2712no—
185C2×#41—2712no—
186C2×#41—5421no—
187C2×#41—5421no—
188C2×#41—5428no—
189C22×Q8—2711no—
190C22×Q8—2711no—
191C22×D4—5421no—
192C22×D4C2×D27111yescharacteristic
193C23×C4—2711no—
194C23×C4—5421no—
195#137—5428no—
196C2×Q16—2712no—
197C2×Q16—2712no—
198C2×SD16—2712no—
199C2×SD16C2×D27112yescomplemented
200C2×M16—2712no—
201C22⋊Q8—2711no—
202C22⋊Q8—2711no—
203C22⋊Q8—2711no—
204C22⋊Q8—5421no—
205C22⋊Q8—5421no—
206C22⋊Q8C2×D27111yescharacteristic
207C2×C4⋊C4—2711no—
208C2×C4⋊C4—5421no—
209C2×C22⋊C4—5421no—
210#103—2712no—
211#103—2712no—
212#102—2712no—
213#102C2×D27112yescomplemented
214C22⋊C8—2711no—
215C22⋊C8—2712no—
216C22⋊C8C2×D27111yescharacteristic
217C24—21no—
218C24—2711no—
219#41—2718no—
220#41—5424no—
221#41—5424no—
222#41—5428no—
223#41—5428no—
224#41—5424no—
225C2×Q8—2712no—
226C2×Q8—2711no—
227C2×Q8—5421no—
228C2×Q8—5424no—
229C2×Q8—5424no—
230C2×Q8—5421no—
231C2×Q8C22×D27111yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—5422no—
235C2×D4—5422no—
236C2×D4—5428no—
237C2×D4C22×D27112yes—
238C22×C4—2711no—
239C22×C4—2711no—
240C22×C4—2711no—
241C22×C4—2711no—
242C22×C4—2711no—
243C22×C4—5424no—
244C22×C4—5421no—
245C22×C4—5421no—
246C22×C4—5422no—
247C22×C4—5422no—
248C22×C4—5428no—
249C22×C4—5421no—
250C22×C4C22×D27111yescharacteristic
251Q16—5424no—
252Q16—5424no—
253SD16—24no—
254SD16—5424no—
255M16—5424no—
256C2×C8—2712no—
257C2×C8C22×D27112yes—
258C4⋊C4—21no—
259C4⋊C4—2711no—
260C4⋊C4—5421no—
261C4⋊C4—5421no—
262C4⋊C4—5421no—
263C4⋊C4—5421no—
264C4⋊C4C22×D27111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—5421no—
267C22⋊C4—5421no—
268C22⋊C4—5421no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—2711no—
274C23—2711no—
275C23—5422no—
276C23—5422no—
277C23—10841no—
278C23D4×D27111yescharacteristic
279Q8—22no—
280Q8—2714no—
281Q8—5424no—
282Q8—5422no—
283Q8—5422no—
284Q8—10842no—
285D4—24no—
286D4—22no—
287D4—5428no—
288D4—5428no—
289D4D4×D27114yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—2711no—
294C2×C4—2714no—
295C2×C4—2711no—
296C2×C4—5422no—
297C2×C4—5422no—
298C2×C4—5422no—
299C2×C4—5428no—
300C2×C4—5424no—
301C2×C4—5428no—
302C2×C4—5422no—
303C2×C4—5422no—
304C2×C4—5421no—
305C2×C4—5421no—
306C2×C4—10842no—
307C2×C4—10842no—
308C2×C4D4×D27111yescharacteristic
309C2×C4C23×D27111yescharacteristic
310C8—22no—
311C8—5422no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—2712no—
319C22—5421no—
320C22—5421no—
321C22—5421no—
322C22—5421no—
323C22—5421no—
324C22—10844no—
325C22#2257618912yescomplemented
326C22C2×D4×D27111yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—2712no—
331C4—5421no—
332C4—5424no—
333C4—10841no—
334C4—10841no—
335C4C2×D4×D27112yes—
336C2—22no—
337C2—44no—
338C2—5422no—
339C2—10841no—
340C2#120×D27111yescharacteristic
341C2C2×#2257618911yescharacteristic
342C2#3177469311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#450×C271C2
socleC22×C271C22×D4
prime radicalC22×#16864653C2
hypercenter#450D271
hypocenterC271#3007
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
2. C2×#22576189C2semidirect
3. C22⋊Q8×D271C2semidirect
4. #31774060C2semidirect
5. #31774056C2semidirect
6. #31774055C2semidirect
7. #31774052C2semidirect
8. #31774048C2semidirect
9. #31773895C2semidirect
10. #31773859C2semidirect
11. #450×C271C2semidirect
18. C2×Q8×D271C22semidirect
23. C2×Dic1084C22semidirect
24. C2×#16864644C22semidirect
25. C2×#16864642C22semidirect
26. C2×#16864637C22semidirect
27. #22576140C22semidirect
31. #22576136C22semidirect
33. C4⋊C4×D271C22semidirect
35. #22576085C22semidirect
36. #22576083C22semidirect
37. #22576081C22semidirect
39. (C2×D271)⋊C8C22semidirect
41. C2×SD16×C271C22semidirect
43. #102×C271C22semidirect
146. C22×C271#137semidirect
171. C271#3007semidirect
181. #450D271semidirect
199. C2×SD16C2×D271semidirect
213. #102C2×D271semidirect
325. C22#22576189semidirect

Maximal subgroups

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

subgroupindex
C22×#168646532
C2×#225761892
C22⋊Q8×D2712
#317740602
#317740562
#317740552
#317740522
#317740482
#317738952
#317738592
#450×C2712
#3007271