Finite Group Databasebeta

← #46845577

Subgroups — #46845577

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×#22576179C212yesmaximal, complemented
3#31774407C211yesmaximal, characteristic, complemented
4#31774060C212yesmaximal, complemented
5#31774055C211yesmaximal, characteristic, complemented
6#31774048C211yesmaximal, characteristic, complemented
7#31774039C212yesmaximal, complemented
8#31774035C211yesmaximal, characteristic, complemented
9#31773890C212yesmaximal, complemented
10C22⋊C8×D271C211yesmaximal, characteristic, complemented
11#447×C271C211yesmaximal, characteristic, complemented
12C2×#16864653—28no—
13C2×#16864653—21no—
14C2×#16864653—21no—
15C2×#16864653C2212yes—
16C2×#16864653C2212yes—
17C2×#16864653C2211yescharacteristic, complemented
18C22×Dic542C2211yescharacteristic
19C22×C22⋊D271—21no—
20C23×Dic271—21no—
21C22×C4×D271C2211yescharacteristic
22#22576179—28no—
23C2×Dic1084C2212yescomplemented
24C2×#16864642C2212yescomplemented
25C2×#16864642C2212yescomplemented
26C2×C8×D271C2212yescomplemented
27#22576142—21no—
28#22576140C2211yescharacteristic, complemented
29#22576139C2211yescharacteristic
30#22576131C2211yescharacteristic, complemented
31#22576128—21no—
32D4×Dic271—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×D8×C271C2212yescomplemented
42#121×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#16864653—24no—
52#16864653D418yescomplemented
53C2×Dic542—24no—
54C2×Dic542—21no—
55C2×Dic542C2311yescharacteristic
56C2×Dic542C2312yes—
57C2×C22⋊D271—28no—
58C2×C22⋊D271—22no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—21no—
61C2×C22⋊D271—21no—
62C22×Dic271—28no—
63C22×Dic271—22no—
64C22×Dic271—22no—
65C22×Dic271—21no—
66C22×Dic271—21no—
67C22×Dic271D411yescharacteristic
68C2×C4×D271—24no—
69C2×C4×D271D411yescharacteristic
70C2×C4×D271D411yescharacteristic
71C2×C4×D271C2311yescharacteristic
72C2×C4×D271C2311yescharacteristic
73Dic1084—24no—
74#16864642—24no—
75#16864642—24no—
76C2×C271⋊C8C2312yes—
77C8×D271—24no—
78C4⋊Dic271—21no—
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81C22⋊Dic271—21no—
82C22⋊Dic271—21no—
83(C2×D271)⋊C4—21no—
84(C2×D271)⋊C4—21no—
85C4×Dic271—21no—
86C24×C271—21no—
87C2×D4×C271—24no—
88C2×D4×C271—21no—
89C2×D4×C271—21no—
90C2×D4×C271C2311yescharacteristic
91C2×D4×C271C2312yes—
92C22×C4×C271C2311yescharacteristic
93D8×C271—24no—
94C2×C8×C271C2312yes—
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—
105Dic542C2×D414yes—
106C22⋊D271—28no—
107C22⋊D271—28no—
108C22⋊D271—42no—
109C22⋊D271—42no—
110C2×Dic271—24no—
111C2×Dic271—28no—
112C2×Dic271—22no—
113C2×Dic271—22no—
114C2×Dic271—28no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—42no—
118C2×Dic271C2×D411yescharacteristic
119C2×Dic271C2×D411yescharacteristic
120C4×D271—22no—
121C4×D271—22no—
122C4×D271—22no—
123C4×D271C2×D414yes—
124C271⋊C8—22no—
125C23×C271—24no—
126C23×C271—22no—
127C23×C271—21no—
128C23×C271—42no—
129C23×C271—42no—
130C23×C271C2×D411yescharacteristic
131D4×C271—24no—
132D4×C271—22no—
133D4×C271—22no—
134D4×C271—42no—
135D4×C271C2×D414yes—
136C2×C4×C271—22no—
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—41no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271#13512yescomplemented
151C22×C271C22×D411yescharacteristic
152C2×D271—21no—
153C2×D271—21no—
154C2×D271—21no—
155C2×D271—21no—
156C2×D271—21no—
157C2×D271—44no—
158C2×D271#12012yes—
159C4×C271—21no—
160C4×C271—41no—
161C4×C271C22×D412yes—
162Dic271—21no—
163Dic271—24no—
164Dic271—41no—
165Dic271—41no—
166Dic271#12012yes—
167C2×C271—22no—
168C2×C271—44no—
169C2×C271—81no—
170C2×C271C2×#12011yescharacteristic
171C2×C271C2×#13511yescharacteristic
172C2×C271#57511yescharacteristic
173D271—22no—
174D271—41no—
175C271#300211yescharacteristic, complemented, Sylow
176#3002—2711nomaximal, Sylow
177C22×#41—2711no—
178C2×#135—2712no—
179#525—2711no—
180#452—2712no—
181#451—2711no—
182#450—2711no—
183#449—2712no—
184#448—2711no—
185#447D27111yescharacteristic, complemented
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×Q8—2711no—
195C22×D4—5421no—
196C22×D4C2×D27111yescharacteristic
197C23×C4—2711no—
198C23×C4—5421no—
199#135—5428no—
200C2×Q16—2712no—
201C2×SD16—2712no—
202C2×SD16—2712no—
203C2×D8C2×D27112yescomplemented
204C22×C8—2712no—
205#123—5421no—
206C22⋊Q8—2711no—
207C22⋊Q8—2711no—
208#121—2711no—
209#121C2×D27111yescharacteristic
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—2711no—
232C2×Q8—5424no—
233C2×Q8—5421no—
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—5422no—
238C2×D4—5422no—
239C2×D4—5428no—
240C2×D4—5421no—
241C2×D4—5421no—
242C2×D4C22×D27111yescharacteristic
243C2×D4C22×D27112yes—
244C22×C4—2711no—
245C22×C4—2711no—
246C22×C4—2711no—
247C22×C4—2711no—
248C22×C4—2711no—
249C22×C4—5424no—
250C22×C4—5422no—
251C22×C4—5422no—
252C22×C4—5421no—
253C22×C4—5428no—
254C22×C4—5421no—
255C22×C4C22×D27111yescharacteristic
256Q16—5424no—
257SD16—5424no—
258SD16—5424no—
259D8—24no—
260C2×C8—2712no—
261C2×C8—5424no—
262C2×C8C22×D27112yes—
263C4⋊C4—2711no—
264C4⋊C4—5421no—
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—21no—
276C23—42no—
277C23—42no—
278C23—2711no—
279C23—2711no—
280C23—5422no—
281C23—5422no—
282C23—10841no—
283C23D4×D27111yescharacteristic
284Q8—2714no—
285Q8—5424no—
286Q8—5422no—
287Q8—5422no—
288D4—24no—
289D4—22no—
290D4—22no—
291D4—42no—
292D4—5428no—
293D4—5428no—
294D4—10842no—
295D4—10842no—
296D4D4×D27114yes—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—2711no—
300C2×C4—2714no—
301C2×C4—2711no—
302C2×C4—5422no—
303C2×C4—5422no—
304C2×C4—5422no—
305C2×C4—5428no—
306C2×C4—5424no—
307C2×C4—5428no—
308C2×C4—5422no—
309C2×C4—5422no—
310C2×C4—5421no—
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—41no—
324C22—41no—
325C22—41no—
326C22—2712no—
327C22—5421no—
328C22—5421no—
329C22—5421no—
330C22—5421no—
331C22—5421no—
332C22—10844no—
333C22#2257617912yescomplemented
334C22C2×D4×D27111yescharacteristic
335C4—21no—
336C4—41no—
337C4—2712no—
338C4—5421no—
339C4—5424no—
340C4—10841no—
341C4—10841no—
342C4C2×D4×D27112yes—
343C2—22no—
344C2—44no—
345C2—81no—
346C2—5422no—
347C2—10841no—
348C2#120×D27111yescharacteristic
349C2C2×#2257617911yescharacteristic
350C2#3177469311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#447×C271C2
socleC22×C271C22×D4
hypercenter#447D271
hypocenterC271#3002
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×#16864653C2semidirect
2. C2×#22576179C2semidirect
3. #31774407C2semidirect
4. #31774060C2semidirect
5. #31774055C2semidirect
6. #31774048C2semidirect
7. #31774039C2semidirect
8. #31774035C2semidirect
9. #31773890C2semidirect
10. C22⋊C8×D271C2semidirect
11. #447×C271C2semidirect
17. C2×#16864653C22semidirect
23. C2×Dic1084C22semidirect
24. C2×#16864642C22semidirect
25. C2×#16864642C22semidirect
26. C2×C8×D271C22semidirect
28. #22576140C22semidirect
30. #22576131C22semidirect
34. #22576118C22semidirect
35. #22576085C22semidirect
36. #22576083C22semidirect
37. #22576081C22semidirect
39. (C2×D271)⋊C8C22semidirect
41. C2×D8×C271C22semidirect
43. #102×C271C22semidirect
52. #16864653D4semidirect
150. C22×C271#135semidirect
175. C271#3002semidirect
185. #447D271semidirect
203. C2×D8C2×D271semidirect
217. #102C2×D271semidirect
333. C22#22576179semidirect

Maximal subgroups

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

subgroupindex
C22×#168646532
C2×#225761792
#317744072
#317740602
#317740552
#317740482
#317740392
#317740352
#317738902
C22⋊C8×D2712
#447×C2712
#3002271