Finite Group Databasebeta

← #46845611

Subgroups — #46845611

Order 34688.

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

normal subgroups only

This group has 806 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×#22576184C212yesmaximal, complemented
3#121×D271C211yesmaximal, characteristic, complemented
4#31774043C212yesmaximal, complemented
5#31774039C212yesmaximal, complemented
6#31774038C211yesmaximal, characteristic, complemented
7#31774035C211yesmaximal, characteristic, complemented
8#31774031C211yesmaximal, characteristic, complemented
9#31773895C212yesmaximal, complemented
10#31773859C211yesmaximal, characteristic, complemented
11#447×C271C211yesmaximal, characteristic, complemented
12C2×#16864653—28no—
13C2×#16864653—21no—
14C2×#16864653—21no—
15C2×#16864653C2212yes—
16C2×#16864653C2212yescomplemented
17C22×Dic542C2211yescharacteristic, complemented
18C22×C22⋊D271—21no—
19C2×D4×D271—21no—
20C2×D4×D271C2211yescharacteristic, complemented
21C23×Dic271—21no—
22C22×C4×D271C2211yescharacteristic
23#22576184—28no—
24C2×#16864643C2212yescomplemented
25C2×#16864642C2212yescomplemented
26C2×#16864639C2212yescomplemented
27C2×#16864637C2212yescomplemented
28#22576135C2211yescharacteristic, complemented
29#22576134C2211yescharacteristic
30#22576133—21no—
31#22576132—21no—
32#22576131C2211yescharacteristic, complemented
33C4⋊C4×D271C2211yescharacteristic, complemented
34C22⋊C4×D271—21no—
35#22576085C2212yescomplemented
36#22576083C2212yescomplemented
37#22576081C2212yescomplemented
38C22⋊(C271⋊C8)C2211yescharacteristic, complemented
39(C2×D271)⋊C8C2212yescomplemented
40C22×D4×C271C2211yescharacteristic, complemented
41C2×D8×C271C2212yescomplemented
42#121×C271C2211yescharacteristic
43#102×C271C2212yescomplemented
44C22⋊C8×C271C2211yescharacteristic, complemented
45C23×D271—21no—
46C23×D271D411yescharacteristic
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×D1084C2311yescharacteristic
58C2×C22⋊D271—28no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271—21no—
62C2×C22⋊D271—21no—
63C2×C22⋊D271—21no—
64D4×D271—24no—
65D4×D271—44no—
66C22×Dic271—28no—
67C22×Dic271—22no—
68C22×Dic271—22no—
69C22×Dic271—21no—
70C22×Dic271D411yescharacteristic
71C2×C4×D271—24no—
72C2×C4×D271—21no—
73C2×C4×D271D411yescharacteristic
74C2×C4×D271D411yescharacteristic
75C2×C4×D271C2311yescharacteristic
76C2×C4×D271C2311yescharacteristic
77#16864643—24no—
78#16864642—24no—
79#16864639—24no—
80#16864637—24no—
81C2×C271⋊C8C2312yes—
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4—21no—
84C22⋊Dic271—21no—
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87C24×C271—21no—
88C2×D4×C271—24no—
89C2×D4×C271—21no—
90C2×D4×C271—21no—
91C2×D4×C271C2311yescharacteristic
92C2×D4×C271C2312yes—
93C22×C4×C271C2311yescharacteristic
94D8×C271—24no—
95C2×C8×C271C2312yes—
96C4⋊C4×C271C2311yescharacteristic
97C22⋊C4×C271—21no—
98C22×D271—22no—
99C22×D271—22no—
100C22×D271—21no—
101C22×D271—41no—
102C22×D271—42no—
103C22×D271—42no—
104C22×D271—42no—
105C22×D271C2×D411yescharacteristic
106C22×D271C2×D411yescharacteristic
107Dic542—24no—
108Dic542—22no—
109Dic542C2×D414yescomplemented
110D1084—22no—
111D1084—42no—
112C22⋊D271—28no—
113C22⋊D271—28no—
114C22⋊D271—42no—
115C22⋊D271—42no—
116C22⋊D271—42no—
117C2×Dic271—24no—
118C2×Dic271—28no—
119C2×Dic271—22no—
120C2×Dic271—22no—
121C2×Dic271—28no—
122C2×Dic271—21no—
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C4×D271—22no—
126C4×D271—22no—
127C4×D271—22no—
128C4×D271—42no—
129C4×D271C2×D414yes—
130C271⋊C8—22no—
131C23×C271—24no—
132C23×C271—22no—
133C23×C271—21no—
134C23×C271—42no—
135C23×C271—42no—
136C23×C271C2×D411yescharacteristic
137D4×C271—24no—
138D4×C271—22no—
139D4×C271—22no—
140D4×C271—42no—
141D4×C271C2×D414yescomplemented
142C2×C4×C271—22no—
143C2×C4×C271—21no—
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2411yescharacteristic
146C8×C271—22no—
147C22×C271—21no—
148C22×C271—21no—
149C22×C271—24no—
150C22×C271—48no—
151C22×C271—42no—
152C22×C271—42no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271—41no—
156C22×C271AGL(1,ℤ/8)12yescomplemented
157C22×C271C22×D411yescharacteristic
158C2×D271—21no—
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—21no—
162C2×D271—21no—
163C2×D271—44no—
164C2×D271—41no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—84no—
168C2×D271#12012yes—
169C4×C271—21no—
170C4×C271—41no—
171C4×C271C22×D412yes—
172Dic271—21no—
173Dic271—24no—
174Dic271—41no—
175Dic271#12012yes—
176C2×C271—22no—
177C2×C271—44no—
178C2×C271—81no—
179C2×C271C2×#12011yescharacteristic
180C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C271#57511yescharacteristic
182D271—22no—
183D271—41no—
184D271—81no—
185C271#300611yescharacteristic, complemented, Sylow
186#3006—2711nomaximal, Sylow
187C22×#41—2711no—
188C2×AGL(1,ℤ/8)—2712no—
189C2×#121—2711no—
190#449—2712no—
191#449—2712no—
192#448—2711no—
193#448—2711no—
194#447—2711no—
195#447D27111yescharacteristic, complemented
196#417—2712no—
197#407—2711no—
198C2×#41—2712no—
199C2×#41—2712no—
200C2×#41—5421no—
201C2×#41—5421no—
202C2×#41—5428no—
203C22×Q8—2711no—
204C22×D4—2711no—
205C22×D4—5421no—
206C22×D4—5421no—
207C22×D4C2×D27111yescharacteristic, complemented
208C23×C4—2711no—
209C23×C4—5421no—
210AGL(1,ℤ/8)—5428no—
211C2×SD16—2712no—
212C2×SD16—2712no—
213C2×D8—2712no—
214C2×D8C2×D27112yescomplemented
215C2×M16—2712no—
216#121—2711no—
217#121—2711no—
218#121—2711no—
219#121—5421no—
220#121—5421no—
221#121C2×D27111yescharacteristic
222C2×C4⋊C4—2711no—
223C2×C22⋊C4—5421no—
224#103—2712no—
225#103—2712no—
226#102—2712no—
227#102C2×D27112yescomplemented
228C22⋊C8—2711no—
229C22⋊C8—2712no—
230C22⋊C8C2×D27111yescharacteristic, complemented
231C24—21no—
232C24—2711no—
233C24—5421no—
234#41—2718no—
235#41—5424no—
236#41—5424no—
237#41—5428no—
238#41—5428no—
239#41—5424no—
240C2×Q8—2712no—
241C2×Q8—5424no—
242C2×Q8—5421no—
243C2×D4—24no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—2711no—
247C2×D4—5421no—
248C2×D4—5424no—
249C2×D4—5421no—
250C2×D4—5422no—
251C2×D4—5422no—
252C2×D4—5428no—
253C2×D4—5421no—
254C2×D4—5421no—
255C2×D4—10844no—
256C2×D4C22×D27111yescharacteristic
257C2×D4C22×D27112yes—
258C22×C4—2711no—
259C22×C4—2711no—
260C22×C4—2711no—
261C22×C4—2711no—
262C22×C4—2711no—
263C22×C4—5424no—
264C22×C4—5421no—
265C22×C4—5422no—
266C22×C4—5422no—
267C22×C4—5428no—
268C22×C4—5421no—
269C22×C4C22×D27111yescharacteristic
270SD16—5424no—
271SD16—5424no—
272D8—24no—
273D8—5424no—
274M16—5424no—
275C2×C8—2712no—
276C2×C8C22×D27112yes—
277C4⋊C4—2711no—
278C4⋊C4—5421no—
279C4⋊C4C22×D27111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C23—24no—
285C23—22no—
286C23—21no—
287C23—42no—
288C23—42no—
289C23—2711no—
290C23—2711no—
291C23—5422no—
292C23—5422no—
293C23—5421no—
294C23—10841no—
295C23—10842no—
296C23—10842no—
297C23—10842no—
298C23D4×D27111yescharacteristic
299Q8—2714no—
300Q8—5424no—
301Q8—5422no—
302D4—24no—
303D4—22no—
304D4—22no—
305D4—42no—
306D4—5428no—
307D4—5428no—
308D4—5422no—
309D4—10842no—
310D4—10842no—
311D4—10842no—
312D4—10842no—
313D4D4×D27114yescomplemented
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—2711no—
317C2×C4—2714no—
318C2×C4—2711no—
319C2×C4—5422no—
320C2×C4—5422no—
321C2×C4—5422no—
322C2×C4—5428no—
323C2×C4—5424no—
324C2×C4—5428no—
325C2×C4—5422no—
326C2×C4—5422no—
327C2×C4—5421no—
328C2×C4—10842no—
329C2×C4D4×D27111yescharacteristic
330C2×C4C23×D27111yescharacteristic
331C8—22no—
332C8—5422no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—48no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—2712no—
343C22—5421no—
344C22—5421no—
345C22—5421no—
346C22—5421no—
347C22—5421no—
348C22—10844no—
349C22—10841no—
350C22—10841no—
351C22—10841no—
352C22—21684no—
353C22#2257618412yescomplemented
354C22C2×D4×D27111yescharacteristic
355C4—21no—
356C4—41no—
357C4—2712no—
358C4—5421no—
359C4—5424no—
360C4—10841no—
361C4C2×D4×D27112yes—
362C2—22no—
363C2—44no—
364C2—81no—
365C2—5422no—
366C2—10841no—
367C2—21681no—
368C2#120×D27111yescharacteristic
369C2C2×#2257618411yescharacteristic
370C2#3177469311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#447×C271C2
socleC22×C271C22×D4
hypercenter#447D271
hypocenterC271#3006
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×#22576184C2semidirect
3. #121×D271C2semidirect
4. #31774043C2semidirect
5. #31774039C2semidirect
6. #31774038C2semidirect
7. #31774035C2semidirect
8. #31774031C2semidirect
9. #31773895C2semidirect
10. #31773859C2semidirect
11. #447×C271C2semidirect
16. C2×#16864653C22semidirect
17. C22×Dic542C22semidirect
20. C2×D4×D271C22semidirect
24. C2×#16864643C22semidirect
25. C2×#16864642C22semidirect
26. C2×#16864639C22semidirect
27. C2×#16864637C22semidirect
28. #22576135C22semidirect
32. #22576131C22semidirect
33. C4⋊C4×D271C22semidirect
35. #22576085C22semidirect
36. #22576083C22semidirect
37. #22576081C22semidirect
38. C22⋊(C271⋊C8)C22semidirect
39. (C2×D271)⋊C8C22semidirect
40. C22×D4×C271C22semidirect
41. C2×D8×C271C22semidirect
43. #102×C271C22semidirect
44. C22⋊C8×C271C22semidirect
52. #16864653D4semidirect
109. Dic542C2×D4semidirect
141. D4×C271C2×D4semidirect
156. C22×C271AGL(1,ℤ/8)semidirect
185. C271#3006semidirect
195. #447D271semidirect
207. C22×D4C2×D271semidirect
214. C2×D8C2×D271semidirect
227. #102C2×D271semidirect
230. C22⋊C8C2×D271semidirect
313. D4D4×D271semidirect
353. C22#22576184semidirect

Maximal subgroups

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

subgroupindex
C22×#168646532
C2×#225761842
#121×D2712
#317740432
#317740392
#317740382
#317740352
#317740312
#317738952
#317738592
#447×C2712
#3006271