Finite Group Databasebeta

← #46840013

Subgroups — #46840013

Order 34688.

1088 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×C22⋊D271C211yesmaximal, characteristic
2C2×#22576146C212yesmaximal, complemented
3C2×#22576137C211yesmaximal, characteristic, complemented
4#31773813C212yesmaximal, complemented
5#31773812C212yesmaximal, complemented
6#31773797C212yesmaximal, complemented
7#31773753C211yesmaximal, characteristic, complemented
8#31773752C211yesmaximal, characteristic, complemented
9#31773752C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic271C211yesmaximal, characteristic, complemented
11#379×C271C211yesmaximal, characteristic, complemented
12C24×D271C2211yescharacteristic
13C22×Dic542C2211yescharacteristic, complemented
14C22×C22⋊D271—28no—
15C22×C22⋊D271—24no—
16C22×C22⋊D271C2212yes—
17C22×C22⋊D271C2212yes—
18C23×Dic271C2211yescharacteristic
19#22576146—28no—
20#22576137—24no—
21C2×C4⋊Dic271C2211yescharacteristic, complemented
22C2×Dic271⋊C4C2211yescharacteristic, complemented
23C2×Dic271⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic271C2211yescharacteristic
25C2×C22⋊Dic271C2211yescharacteristic
26C2×C22⋊Dic271C2212yescomplemented
27C2×C22⋊Dic271C2212yescomplemented
28C2×(C2×D271)⋊C4C2212yescomplemented
29C2×(C2×D271)⋊C4C2212yescomplemented
30C2×(C2×D271)⋊C4C2212yescomplemented
31C2×(C2×D271)⋊C4C2212yescomplemented
32C2×C4×Dic271C2212yescomplemented
33C271⋊#95C2212yescomplemented
34C25×C271C2211yescharacteristic
35C2×C22⋊C4×C271C2211yescharacteristic
36C2×C22⋊C4×C271C2211yescharacteristic
37C2×C22⋊C4×C271C2212yescomplemented
38C2×C22⋊C4×C271C2212yescomplemented
39C23×D271—24no—
40C23×D271—22no—
41C23×D271—22no—
42C23×D271—22no—
43C23×D271—42no—
44C23×D271C2312yes—
45C2×Dic542—22no—
46C2×Dic542—24no—
47C2×C22⋊D271—232no—
48C2×C22⋊D271—24no—
49C2×C22⋊D271—24no—
50C2×C22⋊D271—24no—
51C2×C22⋊D271—24no—
52C2×C22⋊D271—24no—
53C2×C22⋊D271D418yes—
54C22×Dic271—24no—
55C22×Dic271D414yes—
56C22×Dic271C2311yescharacteristic
57C22×Dic271C2312yes—
58C22×Dic271C2311yescharacteristic
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C4⋊Dic271—22no—
62Dic271⋊C4—22no—
63Dic271⋊C4—22no—
64C22⋊Dic271—22no—
65C22⋊Dic271—24no—
66C22⋊Dic271—22no—
67C22⋊Dic271—42no—
68(C2×D271)⋊C4—24no—
69(C2×D271)⋊C4—24no—
70(C2×D271)⋊C4—42no—
71(C2×D271)⋊C4—42no—
72C4×Dic271—24no—
73C24×C271—24no—
74C24×C271—22no—
75C24×C271—22no—
76C24×C271—22no—
77C24×C271—42no—
78C24×C271C2312yes—
79C24×C271C2311yescharacteristic
80C22×C4×C271C2311yescharacteristic
81C22×C4×C271C2312yes—
82C22×C4×C271C2311yescharacteristic
83C22⋊C4×C271—22no—
84C22⋊C4×C271—24no—
85C22⋊C4×C271—22no—
86C22⋊C4×C271—42no—
87C22×D271—28no—
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—44no—
97C22×D271—44no—
98C22×D271C2×D414yes—
99Dic542—44no—
100C22⋊D271—216no—
101C22⋊D271—216no—
102C2×Dic271—22no—
103C2×Dic271—28no—
104C2×Dic271—22no—
105C2×Dic271—22no—
106C2×Dic271—21no—
107C2×Dic271—21no—
108C2×Dic271—22no—
109C2×Dic271—22no—
110C2×Dic271—22no—
111C2×Dic271C2×D412yes—
112C2×Dic271C2×D414yes—
113C2×Dic271C2×D412yes—
114C23×C271—22no—
115C23×C271—22no—
116C23×C271—28no—
117C23×C271—22no—
118C23×C271—22no—
119C23×C271—22no—
120C23×C271—22no—
121C23×C271—22no—
122C23×C271—22no—
123C23×C271—48no—
124C23×C271—48no—
125C23×C271—44no—
126C23×C271—44no—
127C23×C271C2×D414yes—
128C23×C271C2×D412yes—
129C23×C271#4112yes—
130C23×C271#4112yes—
131C23×C271C2411yescharacteristic
132C2×C4×C271—22no—
133C2×C4×C271—22no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—22no—
137C2×C4×C271—22no—
138C2×C4×C271—22no—
139C22×C271—22no—
140C22×C271—22no—
141C22×C271—24no—
142C22×C271—24no—
143C22×C271—24no—
144C22×C271—24no—
145C22×C271—24no—
146C22×C271—416no—
147C22×C271—42no—
148C22×C271—42no—
149C22×C271—42no—
150C22×C271—42no—
151C22×C271#12414yescomplemented
152C22×C271C22×D411yescharacteristic
153C22×C271C22×D412yes—
154C22×C271C2×#4111yescharacteristic
155C22×C271C2×#4111yescharacteristic
156C22×C271H(5,𝔽2)12yes—
157C2×D271—24no—
158C2×D271—24no—
159C2×D271—24no—
160C2×D271—416no—
161C2×D271—42no—
162C2×D271—42no—
163C2×D271—42no—
164C2×D271—42no—
165C4×C271—41no—
166C4×C271—41no—
167C4×C271—42no—
168Dic271—22no—
169Dic271—41no—
170Dic271—41no—
171Dic271—42no—
172Dic271#12014yes—
173C2×C271—24no—
174C2×C271—44no—
175C2×C271C2×#12011yescharacteristic
176C2×C271C2×#12411yescharacteristic
177C2×C271#53511yescharacteristic
178C2×C271#54612yes—
179C2×C271#54612yes—
180D271—44no—
181C271#243511yescharacteristic, complemented, Sylow
182#2435—2711nomaximal, Sylow
183C23×D4—2711no—
184C2×#124—2712no—
185C2×C22⋊Q8—2711no—
186#394—2712no—
187#394—2712no—
188#390—2712no—
189#379—2711no—
190#379—2711no—
191#379—2711no—
192#379D27111yescharacteristic, complemented
193C4×C22⋊C4—2711no—
194C25—2711no—
195C25C2×D27111yescharacteristic
196C22×Q8—2711no—
197C22×D4—2712no—
198C22×D4—2712no—
199C22×D4—5424no—
200C22×D4—5428no—
201C23×C4—2711no—
202#124—5428no—
203C22⋊Q8—5424no—
204C2×C4⋊C4—2711no—
205C2×C4⋊C4—2711no—
206C2×C4⋊C4—2711no—
207C2×C22⋊C4—2711no—
208C2×C22⋊C4—2712no—
209C2×C22⋊C4—2711no—
210C2×C22⋊C4—2712no—
211C2×C22⋊C4—2712no—
212C2×C22⋊C4—2712no—
213C2×C22⋊C4—2712no—
214C2×C22⋊C4—2712no—
215C2×C22⋊C4C2×D27111yescharacteristic
216C2×C22⋊C4C2×D27111yescharacteristic
217C2×C22⋊C4C2×D27112yescomplemented
218C2×C22⋊C4C2×D27112yescomplemented
219C2×C42—2712no—
220#95—2712no—
221C24—24no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—2712no—
227C24—5424no—
228C24—5422no—
229C24—5422no—
230C24—5422no—
231C24—10842no—
232C24C22×D27112yes—
233C24C22×D27111yescharacteristic
234C2×Q8—5422no—
235C2×Q8—5424no—
236C2×D4—2718no—
237C2×D4—5424no—
238C2×D4—5424no—
239C2×D4—5424no—
240C2×D4—5424no—
241C2×D4—54232no—
242C2×D4—5424no—
243C22×C4—2711no—
244C22×C4—2714no—
245C22×C4—2711no—
246C22×C4—2712no—
247C22×C4—2711no—
248C22×C4—2711no—
249C22×C4—5424no—
250C22×C4C22×D27112yes—
251C22×C4C22×D27111yescharacteristic
252C22×C4C22×D27111yescharacteristic
253C4⋊C4—5422no—
254C4⋊C4—5422no—
255C4⋊C4—5422no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—5424no—
261C22⋊C4—5424no—
262C22⋊C4—5422no—
263C22⋊C4—5424no—
264C22⋊C4—5422no—
265C22⋊C4—10842no—
266C22⋊C4—10842no—
267C22⋊C4—10842no—
268C42—5424no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—48no—
279C23—48no—
280C23—44no—
281C23—44no—
282C23—2714no—
283C23—5428no—
284C23—5422no—
285C23—5422no—
286C23—5422no—
287C23—5422no—
288C23—5422no—
289C23—5422no—
290C23—10848no—
291C23—10848no—
292C23—10844no—
293C23—10844no—
294C23D4×D27114yes—
295C23D4×D27112yes—
296C23#1686465312yes—
297C23#1686465412yes—
298C23C23×D27111yescharacteristic
299Q8—10844no—
300D4—54216no—
301D4—54216no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—2714no—
310C2×C4—2712no—
311C2×C4—2712no—
312C2×C4—5428no—
313C2×C4—5422no—
314C2×C4—5422no—
315C2×C4—5422no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5422no—
319C2×C4—5422no—
320C2×C4—5422no—
321C22—24no—
322C22—24no—
323C22—22no—
324C22—22no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—416no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—5424no—
334C22—5424no—
335C22—5424no—
336C22—108416no—
337C22—10842no—
338C22—10842no—
339C22—10842no—
340C22—10842no—
341C22#2257614614yescomplemented
342C22C2×D4×D27112yes—
343C22C2×D4×D27111yescharacteristic
344C22C2×#1686465311yescharacteristic
345C22C2×#1686465411yescharacteristic
346C22#2257620512yes—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—2714no—
351C4—5422no—
352C4—10842no—
353C4—10841no—
354C4—10841no—
355C2—24no—
356C2—44no—
357C2—10844no—
358C2#120×D27111yescharacteristic
359C2C2×#2257614611yescharacteristic
360C2#3177446211yescharacteristic
361C2#3177453412yes—
362C2#3177453712yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#379×C271C2
prime radicalC23×C22⋊D271C2
hypercenter#379D271
hypocenterC271#2435
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#22576146C2semidirect
3. C2×#22576137C2semidirect
4. #31773813C2semidirect
5. #31773812C2semidirect
6. #31773797C2semidirect
7. #31773753C2semidirect
8. #31773752C2semidirect
9. #31773752C2semidirect
10. C22⋊C4×Dic271C2semidirect
11. #379×C271C2semidirect
13. C22×Dic542C22semidirect
21. C2×C4⋊Dic271C22semidirect
22. C2×Dic271⋊C4C22semidirect
23. C2×Dic271⋊C4C22semidirect
26. C2×C22⋊Dic271C22semidirect
27. 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
37. C2×C22⋊C4×C271C22semidirect
38. C2×C22⋊C4×C271C22semidirect
151. C22×C271#124semidirect
181. C271#2435semidirect
192. #379D271semidirect
217. C2×C22⋊C4C2×D271semidirect
218. C2×C22⋊C4C2×D271semidirect
341. C22#22576146semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D2712
C2×#225761462
C2×#225761372
#317738132
#317738122
#317737972
#317737532
#317737522
C22⋊C4×Dic2712
#379×C2712
#2435271