Finite Group Databasebeta

← #46840010

Subgroups — #46840010

Order 34688.

1088 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 161 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×D4×D271C211yesmaximal, characteristic
2C2×#22576146C212yesmaximal, complemented
3C2×#22576138C211yesmaximal, characteristic, complemented
4#31773815C211yesmaximal, characteristic, complemented
5#31773814C211yesmaximal, characteristic, complemented
6#31773811C211yesmaximal, characteristic, complemented
7#31773797C212yesmaximal, complemented
8#31773752C212yesmaximal, complemented
9#31773752C212yesmaximal, complemented
10#31773744C211yesmaximal, characteristic, complemented
11#394×C271C211yesmaximal, characteristic, complemented
12C24×D271C2212yes—
13C22×Dic542C2211yescharacteristic, complemented
14C22×D1084C2211yescharacteristic
15C22×C22⋊D271C2212yes—
16C2×D4×D271—28no—
17C2×D4×D271—24no—
18C22×C4×D271C2211yescharacteristic
19#22576146—28no—
20#22576138—24no—
21C2×Dic271⋊C4C2211yescharacteristic, complemented
22C2×Dic271⋊C4C2211yescharacteristic, complemented
23C2×C22⋊Dic271C2212yescomplemented
24C2×C22⋊Dic271C2212yescomplemented
25C2×(C2×D271)⋊C4C2212yes—
26C2×(C2×D271)⋊C4C2211yescharacteristic
27C2×(C2×D271)⋊C4C2211yescharacteristic
28C2×(C2×D271)⋊C4C2212yescomplemented
29C2×(C2×D271)⋊C4C2212yescomplemented
30C2×(C2×D271)⋊C4C2212yescomplemented
31C2×(C2×D271)⋊C4C2212yescomplemented
32C2×C4×Dic271C2212yescomplemented
33C271⋊#95C2211yescharacteristic, complemented
34C22×D4×C271C2211yescharacteristic
35C2×C4⋊C4×C271C2211yescharacteristic, complemented
36C2×C22⋊C4×C271C2212yescomplemented
37C2×C22⋊C4×C271C2212yescomplemented
38#95×C271C2211yescharacteristic, complemented
39C23×D271—24no—
40C23×D271—24no—
41C23×D271—24no—
42C23×D271—24no—
43C23×D271—24no—
44C23×D271—42no—
45C23×D271—42no—
46C23×D271C2312yes—
47C23×D271C2311yescharacteristic
48C2×Dic542—22no—
49C2×Dic542—24no—
50C2×D1084—24no—
51C2×D1084—22no—
52C2×C22⋊D271—24no—
53C2×C22⋊D271—24no—
54C2×C22⋊D271D418yes—
55D4×D271—232no—
56C22×Dic271C2311yescharacteristic
57C22×Dic271C2312yes—
58C22×Dic271C2311yescharacteristic
59C22×Dic271C2311yescharacteristic
60C2×C4×D271—24no—
61C2×C4×D271D414yes—
62Dic271⋊C4—22no—
63Dic271⋊C4—22no—
64C22⋊Dic271—24no—
65C22⋊Dic271—42no—
66(C2×D271)⋊C4—22no—
67(C2×D271)⋊C4—22no—
68(C2×D271)⋊C4—24no—
69(C2×D271)⋊C4—24no—
70(C2×D271)⋊C4—24no—
71(C2×D271)⋊C4—42no—
72(C2×D271)⋊C4—42no—
73C4×Dic271—24no—
74C24×C271C2312yes—
75C2×D4×C271—24no—
76C2×D4×C271—22no—
77C22×C4×C271C2311yescharacteristic
78C22×C4×C271C2312yes—
79C22×C4×C271C2311yescharacteristic
80C22×C4×C271C2311yescharacteristic
81C4⋊C4×C271—22no—
82C22⋊C4×C271—24no—
83C22⋊C4×C271—42no—
84C22×D271—22no—
85C22×D271—21no—
86C22×D271—21no—
87C22×D271—216no—
88C22×D271—22no—
89C22×D271—22no—
90C22×D271—22no—
91C22×D271—22no—
92C22×D271—22no—
93C22×D271—22no—
94C22×D271—48no—
95C22×D271—48no—
96C22×D271—48no—
97C22×D271—48no—
98C22×D271—48no—
99C22×D271—48no—
100C22×D271C2×D414yes—
101C22×D271C2×D412yes—
102C22×D271#4114yes—
103Dic542—44no—
104D1084—28no—
105C22⋊D271—216no—
106C2×Dic271—22no—
107C2×Dic271—22no—
108C2×Dic271—21no—
109C2×Dic271—21no—
110C2×Dic271—22no—
111C2×Dic271—21no—
112C2×Dic271—22no—
113C2×Dic271—21no—
114C2×Dic271C2×D414yes—
115C2×Dic271C2×D412yes—
116C4×D271—28no—
117C23×C271—22no—
118C23×C271—22no—
119C23×C271—22no—
120C23×C271—22no—
121C23×C271—22no—
122C23×C271—22no—
123C23×C271C2×D414yes—
124C23×C271C2411yescharacteristic
125D4×C271—28no—
126C2×C4×C271—22no—
127C2×C4×C271—22no—
128C2×C4×C271—22no—
129C2×C4×C271—21no—
130C2×C4×C271—21no—
131C2×C4×C271—22no—
132C2×C4×C271—21no—
133C2×C4×C271—22no—
134C2×C4×C271—21no—
135C2×C4×C271C2×D412yes—
136C22×C271—24no—
137C22×C271—24no—
138C22×C271—24no—
139C22×C271—42no—
140C22×C271—42no—
141C22×C271—42no—
142C22×C271—42no—
143C22×C271C22×D411yescharacteristic
144C22×C271C22×D412yes—
145C22×C271C2×#4112yes—
146C22×C271H(5,𝔽2)11yescharacteristic
147C22×C271H(5,𝔽2)11yescharacteristic
148C2×D271—22no—
149C2×D271—22no—
150C2×D271—24no—
151C2×D271—24no—
152C2×D271—24no—
153C2×D271—24no—
154C2×D271—24no—
155C2×D271—432no—
156C2×D271—42no—
157C2×D271—42no—
158C2×D271—42no—
159C2×D271—42no—
160C2×D271#12414yescomplemented
161C4×C271—22no—
162C4×C271—41no—
163C4×C271—41no—
164C4×C271—42no—
165Dic271—41no—
166Dic271—41no—
167Dic271—42no—
168Dic271#12014yes—
169C2×C271—44no—
170C2×C271C2×#12011yescharacteristic
171C2×C271C2×#12411yescharacteristic
172C2×C271#53511yescharacteristic
173C2×C271#54612yes—
174C2×C271#54612yes—
175D271—24no—
176D271—44no—
177C271#243511yescharacteristic, complemented, Sylow
178#2435—2711nomaximal, Sylow
179C23×D4—2711no—
180C2×#124—2712no—
181C2×C22⋊Q8—2711no—
182#394—2711no—
183#394—2711no—
184#394—2711no—
185#394D27111yescharacteristic, complemented
186#390—2712no—
187#379—2712no—
188#379—2712no—
189C4×C22⋊C4—2711no—
190C25—2712no—
191C22×Q8—2711no—
192C22×D4—2712no—
193C22×D4—2711no—
194C22×D4—5424no—
195C22×D4—5428no—
196C22×D4C2×D27111yescharacteristic
197C23×C4—2711no—
198#124—5428no—
199C22⋊Q8—5424no—
200C2×C4⋊C4—2711no—
201C2×C4⋊C4—2711no—
202C2×C4⋊C4C2×D27111yescharacteristic, complemented
203C2×C22⋊C4—2712no—
204C2×C22⋊C4—2711no—
205C2×C22⋊C4—2711no—
206C2×C22⋊C4—2712no—
207C2×C22⋊C4—2712no—
208C2×C22⋊C4—2712no—
209C2×C22⋊C4—2712no—
210C2×C22⋊C4—2712no—
211C2×C22⋊C4—2712no—
212C2×C22⋊C4C2×D27112yescomplemented
213C2×C22⋊C4C2×D27112yescomplemented
214C2×C42—2712no—
215#95—2711no—
216#95C2×D27111yescharacteristic, complemented
217C24—2711no—
218C24—2712no—
219C24—5424no—
220C24—5424no—
221C24—5424no—
222C24—5424no—
223C24—5424no—
224C24—10842no—
225C24—10842no—
226C24C22×D27112yes—
227C2×Q8—5422no—
228C2×Q8—5424no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—2718no—
232C2×D4—5424no—
233C2×D4—5424no—
234C2×D4—5424no—
235C2×D4—54232no—
236C2×D4—5422no—
237C22×C4—2711no—
238C22×C4—2714no—
239C22×C4—2712no—
240C22×C4—2711no—
241C22×C4—2711no—
242C22×C4—5424no—
243C22×C4C22×D27112yes—
244C22×C4C22×D27111yescharacteristic
245C22×C4C22×D27111yescharacteristic
246C22×C4C22×D27111yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—5422no—
249C4⋊C4—5422no—
250C22⋊C4—24no—
251C22⋊C4—42no—
252C22⋊C4—5424no—
253C22⋊C4—5422no—
254C22⋊C4—5422no—
255C22⋊C4—5424no—
256C22⋊C4—5424no—
257C22⋊C4—5424no—
258C22⋊C4—10842no—
259C22⋊C4—10842no—
260C22⋊C4—10842no—
261C42—5424no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—2712no—
269C23—2714no—
270C23—2714no—
271C23—54216no—
272C23—5422no—
273C23—5421no—
274C23—5421no—
275C23—5422no—
276C23—5422no—
277C23—5422no—
278C23—5422no—
279C23—5422no—
280C23—5422no—
281C23—10848no—
282C23—10848no—
283C23—10848no—
284C23—10848no—
285C23—10848no—
286C23—10848no—
287C23D4×D27114yes—
288C23C23×D27111yescharacteristic
289Q8—10844no—
290D4—28no—
291D4—5428no—
292D4—54216no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—2714no—
303C2×C4—2712no—
304C2×C4—5428no—
305C2×C4—5422no—
306C2×C4—5422no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4—5422no—
310C2×C4—5421no—
311C2×C4—5422no—
312C2×C4—5421no—
313C2×C4D4×D27112yes—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—2714no—
322C22—5424no—
323C22—5424no—
324C22—5422no—
325C22—5422no—
326C22—5424no—
327C22—5424no—
328C22—5424no—
329C22—108432no—
330C22—10842no—
331C22—10842no—
332C22—10842no—
333C22—10842no—
334C22C2×D4×D27112yes—
335C22C2×D4×D27111yescharacteristic
336C22#41×D27112yes—
337C22#2257620511yescharacteristic
338C22#2257620611yescharacteristic
339C4—22no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—2714no—
344C4—10842no—
345C4—10841no—
346C4—10841no—
347C2—44no—
348C2—5424no—
349C2—10844no—
350C2#120×D27111yescharacteristic
351C2#124×D27111yescharacteristic
352C2#3177446311yescharacteristic
353C2#3177453612yes—
354C2#3177453912yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#394×C271C2
prime radicalC22×D4×D271C2
hypercenter#394D271
hypocenterC271#2435
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#22576146C2semidirect
3. C2×#22576138C2semidirect
4. #31773815C2semidirect
5. #31773814C2semidirect
6. #31773811C2semidirect
7. #31773797C2semidirect
8. #31773752C2semidirect
9. #31773752C2semidirect
10. #31773744C2semidirect
11. #394×C271C2semidirect
13. C22×Dic542C22semidirect
21. C2×Dic271⋊C4C22semidirect
22. C2×Dic271⋊C4C22semidirect
23. C2×C22⋊Dic271C22semidirect
24. C2×C22⋊Dic271C22semidirect
28. C2×(C2×D271)⋊C4C22semidirect
29. C2×(C2×D271)⋊C4C22semidirect
30. C2×(C2×D271)⋊C4C22semidirect
31. C2×(C2×D271)⋊C4C22semidirect
32. C2×C4×Dic271C22semidirect
33. C271⋊#95C22semidirect
35. C2×C4⋊C4×C271C22semidirect
36. C2×C22⋊C4×C271C22semidirect
37. C2×C22⋊C4×C271C22semidirect
38. #95×C271C22semidirect
160. C2×D271#124semidirect
177. C271#2435semidirect
185. #394D271semidirect
202. C2×C4⋊C4C2×D271semidirect
212. C2×C22⋊C4C2×D271semidirect
213. C2×C22⋊C4C2×D271semidirect
216. #95C2×D271semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2712
C2×#225761462
C2×#225761382
#317738152
#317738142
#317738112
#317737972
#317737522
#317737442
#394×C2712
#2435271