Finite Group Databasebeta

← #46839673

Subgroups — #46839673

Order 34688.

1564 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 181 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×D271C212yesmaximal, complemented
2C2×#22576130C212yesmaximal, complemented
3C2×#22576128C212yesmaximal, complemented
4#31773806C211yesmaximal, characteristic, complemented
5#31773805C212yesmaximal, complemented
6#31773775C212yesmaximal, complemented
7#31773773C212yesmaximal, complemented
8#31773752C211yesmaximal, characteristic, complemented
9#392×C271C211yesmaximal, characteristic, complemented
10C24×D271C2211yescharacteristic
11C24×D271C2212yescomplemented
12C22×D1084C2212yes—
13C22×D1084C2211yescharacteristic, complemented
14C22×C22⋊D271C2212yes—
15C22×C22⋊D271C2212yescomplemented
16C22×C22⋊D271C2212yescomplemented
17C2×D4×D271—28no—
18C2×D4×D271—28no—
19C2×D4×D271—28no—
20C22×C4×D271C2212yescomplemented
21#22576130—28no—
22#22576128—28no—
23C2×C22⋊Dic271C2212yescomplemented
24C2×C22⋊Dic271C2211yescharacteristic, complemented
25C2×(C2×D271)⋊C4C2212yescomplemented
26C2×(C2×D271)⋊C4C2212yescomplemented
27C2×(C2×D271)⋊C4C2212yescomplemented
28C2×(C2×D271)⋊C4C2212yescomplemented
29C271⋊#95C2211yescharacteristic, complemented
30C271⋊#95C2212yescomplemented
31C22×D4×C271C2212yescomplemented
32C22×D4×C271C2211yescharacteristic, complemented
33C2×C22⋊C4×C271C2212yescomplemented
34C2×C22⋊C4×C271C2211yescharacteristic, complemented
35#95×C271C2211yescharacteristic, complemented
36C23×D271—24no—
37C23×D271—24no—
38C23×D271—24no—
39C23×D271—24no—
40C23×D271—24no—
41C23×D271—24no—
42C23×D271—24no—
43C23×D271—41no—
44C23×D271—41no—
45C23×D271—42no—
46C23×D271—42no—
47C23×D271D414yescomplemented
48C23×D271C2312yes—
49C23×D271C2312yes—
50C2×D1084—24no—
51C2×D1084—24no—
52C2×D1084—24no—
53C2×D1084—41no—
54C2×D1084D418yescomplemented
55C2×C22⋊D271—24no—
56C2×C22⋊D271—24no—
57C2×C22⋊D271—24no—
58C2×C22⋊D271—24no—
59C2×C22⋊D271—24no—
60C2×C22⋊D271—24no—
61C2×C22⋊D271—24no—
62C2×C22⋊D271—42no—
63C2×C22⋊D271D418yescomplemented
64D4×D271—264no—
65C22×Dic271C2312yes—
66C22×Dic271C2312yes—
67C2×C4×D271—24no—
68C2×C4×D271—24no—
69C2×C4×D271—24no—
70C22⋊Dic271—24no—
71C22⋊Dic271—41no—
72(C2×D271)⋊C4—24no—
73(C2×D271)⋊C4—24no—
74(C2×D271)⋊C4—24no—
75(C2×D271)⋊C4—24no—
76C24×C271C2311yescharacteristic
77C24×C271C2312yes—
78C2×D4×C271—24no—
79C2×D4×C271—24no—
80C2×D4×C271—24no—
81C2×D4×C271—24no—
82C2×D4×C271—41no—
83C22×C4×C271C2311yescharacteristic
84C22×C4×C271C2311yescharacteristic
85C22×C4×C271C2312yes—
86C22⋊C4×C271—24no—
87C22⋊C4×C271—41no—
88C22×D271—22no—
89C22×D271—22no—
90C22×D271—22no—
91C22×D271—22no—
92C22×D271—216no—
93C22×D271—22no—
94C22×D271—22no—
95C22×D271—22no—
96C22×D271—22no—
97C22×D271—22no—
98C22×D271—22no—
99C22×D271—216no—
100C22×D271—44no—
101C22×D271—44no—
102C22×D271—48no—
103C22×D271—44no—
104C22×D271—48no—
105C22×D271—48no—
106C22×D271—48no—
107C22×D271—48no—
108C22×D271—48no—
109C22×D271—48no—
110C22×D271C2×D414yes—
111C22×D271C2×D414yes—
112C22×D271C2×D414yes—
113C22×D271C2×D414yescomplemented
114D1084—216no—
115D1084—44no—
116C22⋊D271—216no—
117C22⋊D271—216no—
118C22⋊D271—48no—
119C2×Dic271—22no—
120C2×Dic271—22no—
121C2×Dic271—22no—
122C2×Dic271—22no—
123C2×Dic271—22no—
124C2×Dic271C2×D414yescomplemented
125C4×D271—216no—
126C23×C271—22no—
127C23×C271—22no—
128C23×C271—22no—
129C23×C271—22no—
130C23×C271—22no—
131C23×C271—22no—
132C23×C271—21no—
133C23×C271—22no—
134C23×C271—21no—
135C23×C271—22no—
136C23×C271C2×D412yes—
137C23×C271C2×D414yescomplemented
138C23×C271C2411yescharacteristic
139D4×C271—216no—
140D4×C271—44no—
141C2×C4×C271—22no—
142C2×C4×C271—22no—
143C2×C4×C271—22no—
144C2×C4×C271—22no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271C2×D414yescomplemented
148C22×C271—24no—
149C22×C271—24no—
150C22×C271—22no—
151C22×C271—24no—
152C22×C271—24no—
153C22×C271—42no—
154C22×C271—42no—
155C22×C271—41no—
156C22×C271—42no—
157C22×C271—42no—
158C22×C271—42no—
159C22×C271—41no—
160C22×C271C22×D411yescharacteristic
161C22×C271C22×D412yes—
162C22×C271C22×D412yes—
163C22×C271H(5,𝔽2)11yescharacteristic
164C22×C271H(5,𝔽2)11yescharacteristic
165C2×D271—24no—
166C2×D271—24no—
167C2×D271—24no—
168C2×D271—24no—
169C2×D271—24no—
170C2×D271—24no—
171C2×D271—24no—
172C2×D271—24no—
173C2×D271—24no—
174C2×D271—416no—
175C2×D271—432no—
176C2×D271—42no—
177C2×D271—42no—
178C2×D271—42no—
179C2×D271—42no—
180C2×D271#12018yescomplemented
181C4×C271—24no—
182C4×C271—41no—
183C4×C271—41no—
184Dic271—24no—
185Dic271—42no—
186C2×C271—42no—
187C2×C271—44no—
188C2×C271C2×#12012yes—
189C2×C271#53411yescharacteristic
190C2×C271D4211yescharacteristic
191C2×C271D4212yes—
192C2×C271D4211yescharacteristic
193D271—28no—
194D271—44no—
195C271#240311yescharacteristic, complemented, Sylow
196#2403—2711nomaximal, Sylow
197C23×D4—2712no—
198C2×#120—2712no—
199C2×#120—2712no—
200#392—2712no—
201#392—2711no—
202#392D27111yescharacteristic, complemented
203#386—2712no—
204#386—2712no—
205#379—2711no—
206C25—2711no—
207C25—2712no—
208C22×D4—2712no—
209C22×D4—2712no—
210C22×D4—2712no—
211C22×D4—2712no—
212C22×D4—2711no—
213C22×D4—5428no—
214C22×D4—5428no—
215C22×D4—5428no—
216C22×D4C2×D27112yescomplemented
217C22×D4C2×D27111yescharacteristic, complemented
218C23×C4—2712no—
219#120—5428no—
220#120—5428no—
221C2×C22⋊C4—2712no—
222C2×C22⋊C4—2712no—
223C2×C22⋊C4—2712no—
224C2×C22⋊C4—2712no—
225C2×C22⋊C4—2712no—
226C2×C22⋊C4—2711no—
227C2×C22⋊C4C2×D27112yescomplemented
228C2×C22⋊C4C2×D27111yescharacteristic, complemented
229#95—2711no—
230#95—2712no—
231#95C2×D27111yescharacteristic, complemented
232C24—2712no—
233C24—2712no—
234C24—2714no—
235C24—5424no—
236C24—5424no—
237C24—5424no—
238C24—5424no—
239C24—5424no—
240C24—5424no—
241C24—5424no—
242C24—10841no—
243C24—10842no—
244C24—10841no—
245C24—10842no—
246C24C22×D27111yescharacteristic
247C24C22×D27112yes—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—41no—
253C2×D4—2718no—
254C2×D4—2718no—
255C2×D4—5424no—
256C2×D4—5424no—
257C2×D4—5424no—
258C2×D4—5424no—
259C2×D4—54264no—
260C2×D4—5424no—
261C2×D4—5424no—
262C2×D4—5424no—
263C2×D4—5424no—
264C2×D4—5424no—
265C2×D4—5424no—
266C2×D4—10841no—
267C2×D4—10842no—
268C22×C4—2712no—
269C22×C4—2712no—
270C22×C4—5424no—
271C22×C4—5424no—
272C22×C4—5424no—
273C22×C4C22×D27111yescharacteristic
274C22×C4C22×D27111yescharacteristic
275C22×C4C22×D27112yes—
276C22⋊C4—24no—
277C22⋊C4—41no—
278C22⋊C4—5424no—
279C22⋊C4—5424no—
280C22⋊C4—5424no—
281C22⋊C4—5424no—
282C22⋊C4—5424no—
283C22⋊C4—10841no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—21no—
291C23—22no—
292C23—21no—
293C23—22no—
294C23—2714no—
295C23—2714no—
296C23—2714no—
297C23—2714no—
298C23—54216no—
299C23—54216no—
300C23—5422no—
301C23—5422no—
302C23—5422no—
303C23—5422no—
304C23—5422no—
305C23—5422no—
306C23—5422no—
307C23—5422no—
308C23—5422no—
309C23—5422no—
310C23—10844no—
311C23—10848no—
312C23—10848no—
313C23—10848no—
314C23—10844no—
315C23—10848no—
316C23—10848no—
317C23—10844no—
318C23—10848no—
319C23—10848no—
320C23D4×D27112yes—
321C23D4×D27114yescomplemented
322C23C23×D27111yescharacteristic
323D4—216no—
324D4—44no—
325D4—54216no—
326D4—54216no—
327D4—54216no—
328D4—10844no—
329D4—10848no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—2714no—
337C2×C4—54216no—
338C2×C4—5422no—
339C2×C4—5422no—
340C2×C4—5422no—
341C2×C4—5422no—
342C2×C4—5422no—
343C2×C4D4×D27114yescomplemented
344C22—24no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—22no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—42no—
356C22—2718no—
357C22—5424no—
358C22—5424no—
359C22—5424no—
360C22—5424no—
361C22—5424no—
362C22—5424no—
363C22—5424no—
364C22—5424no—
365C22—5424no—
366C22—108416no—
367C22—108432no—
368C22—10842no—
369C22—10842no—
370C22—10842no—
371C22—10842no—
372C22C2×D4×D27112yes—
373C22C2×D4×D27112yes—
374C22C2×D4×D27111yescharacteristic
375C22#2257620511yescharacteristic
376C22#2257620611yescharacteristic
377C4—24no—
378C4—41no—
379C4—41no—
380C4—5424no—
381C4—10842no—
382C2—44no—
383C2—42no—
384C2—5428no—
385C2—10844no—
386C2#120×D27112yes—
387C2#3177445511yescharacteristic
388C2#3177452911yescharacteristic
389C2#3177453112yes—
390C2#3177453211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#392×C271C2
hypercenter#392D271
hypocenterC271#2403
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D271C2semidirect
2. C2×#22576130C2semidirect
3. C2×#22576128C2semidirect
4. #31773806C2semidirect
5. #31773805C2semidirect
6. #31773775C2semidirect
7. #31773773C2semidirect
8. #31773752C2semidirect
9. #392×C271C2semidirect
11. C24×D271C22semidirect
13. C22×D1084C22semidirect
15. C22×C22⋊D271C22semidirect
16. C22×C22⋊D271C22semidirect
20. C22×C4×D271C22semidirect
23. C2×C22⋊Dic271C22semidirect
24. C2×C22⋊Dic271C22semidirect
25. C2×(C2×D271)⋊C4C22semidirect
26. C2×(C2×D271)⋊C4C22semidirect
27. C2×(C2×D271)⋊C4C22semidirect
28. C2×(C2×D271)⋊C4C22semidirect
29. C271⋊#95C22semidirect
30. C271⋊#95C22semidirect
31. C22×D4×C271C22semidirect
32. C22×D4×C271C22semidirect
33. C2×C22⋊C4×C271C22semidirect
34. C2×C22⋊C4×C271C22semidirect
35. #95×C271C22semidirect
47. C23×D271D4semidirect
54. C2×D1084D4semidirect
63. C2×C22⋊D271D4semidirect
113. C22×D271C2×D4semidirect
124. C2×Dic271C2×D4semidirect
137. C23×C271C2×D4semidirect
147. C2×C4×C271C2×D4semidirect
180. C2×D271#120semidirect
195. C271#2403semidirect
202. #392D271semidirect
216. C22×D4C2×D271semidirect
217. C22×D4C2×D271semidirect
227. C2×C22⋊C4C2×D271semidirect
228. C2×C22⋊C4C2×D271semidirect
231. #95C2×D271semidirect
321. C23D4×D271semidirect
343. C2×C4D4×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×#225761302
C2×#225761282
#317738062
#317738052
#317737752
#317737732
#317737522
#392×C2712
#2403271