Finite Group Databasebeta

← #46839979

Subgroups — #46839979

Order 34688.

952 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 185 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
1C2×#22576156C211yesmaximal, characteristic, complemented
2C2×#22576149C211yesmaximal, characteristic, complemented
3C2×#22576134C211yesmaximal, characteristic, complemented
4#31773805C214yesmaximal, complemented
5#31773804C211yesmaximal, characteristic, complemented
6#31773800C211yesmaximal, characteristic, complemented
7#31773776C214yesmaximal, complemented
8C4×C22⋊Dic271C211yesmaximal, characteristic, complemented
9C22×#41×C271C211yesmaximal, characteristic, complemented
10C22×D1084C2211yescharacteristic
11C22×C22⋊D271C2214yes—
12C22×C22⋊D271C2212yes—
13#22576156—24no—
14#22576149—24no—
15#22576134—24no—
16C2×C4⋊Dic271C2211yescharacteristic
17C2×C22⋊Dic271C2212yes—
18C2×C22⋊Dic271C2214yes—
19C2×(C2×D271)⋊C4C2212yes—
20C2×(C2×D271)⋊C4C2214yes—
21C2×C4×Dic271C2211yescharacteristic
22C2×C4×Dic271C2211yescharacteristic
23C271⋊#95C2212yes—
24C271⋊#95C2214yes—
25C2×#41×C271—24no—
26C2×#41×C271—24no—
27C2×#41×C271—24no—
28C22×Q8×C271C2211yescharacteristic
29C22×D4×C271C2212yes—
30C22×D4×C271C2211yescharacteristic
31C23×C4×C271C2212yes—
32C23×C4×C271C2211yescharacteristic
33C23×D271C2312yes—
34C2×D1084—22no—
35C2×D1084—22no—
36C2×D1084—22no—
37C2×C22⋊D271—28no—
38C2×C22⋊D271—24no—
39C2×C22⋊D271—28no—
40C2×C22⋊D271—44no—
41C2×C22⋊D271—44no—
42C22×Dic271C2312yes—
43C22×Dic271C2312yes—
44C22×Dic271C2312yes—
45C4⋊Dic271—22no—
46C22⋊Dic271—24no—
47C22⋊Dic271—44no—
48(C2×D271)⋊C4—24no—
49(C2×D271)⋊C4—28no—
50C4×Dic271—22no—
51C4×Dic271—22no—
52C24×C271C2311yescharacteristic
53C24×C271C2312yes—
54#41×C271—232no—
55C2×Q8×C271—22no—
56C2×Q8×C271—22no—
57C2×Q8×C271D414yescomplemented
58C2×D4×C271—24no—
59C2×D4×C271—24no—
60C2×D4×C271—24no—
61C2×D4×C271—22no—
62C2×D4×C271—22no—
63C2×D4×C271D414yescomplemented
64C22×C4×C271—24no—
65C22×C4×C271—24no—
66C22×C4×C271—24no—
67C22×C4×C271—22no—
68C22×C4×C271—22no—
69C22×C4×C271D414yescomplemented
70C22×C4×C271C2311yescharacteristic
71C22×C4×C271C2312yes—
72C22×C4×C271C2311yescharacteristic
73C22×D271—22no—
74C22×D271—22no—
75C22×D271—22no—
76C22×D271—24no—
77C22×D271—24no—
78D1084—44no—
79C22⋊D271—416no—
80C22⋊D271—48no—
81C2×Dic271—22no—
82C2×Dic271—22no—
83C2×Dic271—22no—
84C2×Dic271—22no—
85C2×Dic271—22no—
86C2×Dic271—22no—
87C2×Dic271—24no—
88C2×Dic271#4114yescomplemented
89C23×C271—22no—
90C23×C271—21no—
91C23×C271—22no—
92C23×C271—21no—
93C23×C271—22no—
94C23×C271—24no—
95C23×C271—24no—
96C23×C271—22no—
97C23×C271C2×D414yes—
98C23×C271C2×D412yes—
99C23×C271C2411yescharacteristic
100Q8×C271—28no—
101D4×C271—216no—
102D4×C271—28no—
103C2×C4×C271—216no—
104C2×C4×C271—21no—
105C2×C4×C271—21no—
106C2×C4×C271—22no—
107C2×C4×C271—22no—
108C2×C4×C271—28no—
109C2×C4×C271C2×D412yes—
110C2×C4×C271C2×D414yes—
111C2×C4×C271C2×D412yes—
112C2×C4×C271C2×D412yes—
113C2×C4×C271C2×D412yes—
114C22×C271—24no—
115C22×C271—22no—
116C22×C271—22no—
117C22×C271—22no—
118C22×C271—28no—
119C22×C271—42no—
120C22×C271—42no—
121C22×C271—42no—
122C22×C271—42no—
123C22×C271—44no—
124C22×C271C22×D411yescharacteristic
125C22×C271C22×D411yescharacteristic
126C22×C271C22×D411yescharacteristic
127C22×C271C2×#4112yes—
128C22×C271H(5,𝔽2)12yes—
129C2×D271—42no—
130C2×D271—42no—
131C2×D271—42no—
132C2×D271—44no—
133C2×D271—44no—
134C4×C271—22no—
135C4×C271—24no—
136C4×C271#12014yes—
137Dic271—24no—
138Dic271—42no—
139Dic271—42no—
140C2×C271—42no—
141C2×C271—44no—
142C2×C271C2×#12011yescharacteristic
143C2×C271#53211yescharacteristic
144C2×C271#53511yescharacteristic
145C2×C271#54612yes—
146C2×C271#55012yes—
147D271—82no—
148C271#243111yescharacteristic, complemented, Sylow
149#2431—2711nomaximal, Sylow
150C22×#41D27111yescharacteristic, complemented
151C2×#127—2711no—
152C2×#124—2711no—
153C2×#121—2711no—
154#392—2714no—
155#391—2711no—
156#390—2711no—
157#386—2714no—
158C4×C22⋊C4—2711no—
159C2×#41—24no—
160C2×#41—24no—
161C2×#41—24no—
162C22×Q8C2×D27111yescharacteristic
163C22×D4—2714no—
164C22×D4—2712no—
165C22×D4—2711no—
166C22×D4C2×D27112yes—
167C22×D4C2×D27111yescharacteristic
168C23×C4C2×D27112yes—
169C23×C4C2×D27111yescharacteristic
170#127—5424no—
171#124—5424no—
172#121—5424no—
173C2×C4⋊C4—2711no—
174C2×C22⋊C4—2712no—
175C2×C22⋊C4—2712no—
176C2×C22⋊C4—2714no—
177C2×C22⋊C4—2714no—
178C2×C42—2711no—
179C2×C42—2711no—
180#95—2712no—
181#95—2714no—
182C24—2712no—
183C24C22×D27111yescharacteristic
184C24C22×D27112yes—
185#41—232no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8C22⋊D27114yescomplemented
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—5428no—
195C2×D4—5424no—
196C2×D4—5428no—
197C2×D4—5422no—
198C2×D4—5422no—
199C2×D4—5422no—
200C2×D4—10844no—
201C2×D4—10844no—
202C2×D4C22⋊D27114yescomplemented
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—2712no—
209C22×C4—2712no—
210C22×C4—2712no—
211C22×C4C22⋊D27114yescomplemented
212C22×C4C22×D27111yescharacteristic
213C22×C4C22×D27112yes—
214C22×C4C22×D27111yescharacteristic
215C4⋊C4—5422no—
216C22⋊C4—5424no—
217C22⋊C4—5424no—
218C22⋊C4—5428no—
219C22⋊C4—10844no—
220C42—5422no—
221C42—5422no—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—22no—
230C23—5422no—
231C23—5422no—
232C23—5422no—
233C23—5424no—
234C23—5424no—
235C23C2×C22⋊D27114yes—
236C23C2×C22⋊D27112yes—
237C23C23×D27111yescharacteristic
238Q8—28no—
239D4—216no—
240D4—28no—
241D4—108416no—
242D4—10848no—
243D4—10844no—
244C2×C4—216no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—28no—
250C2×C4—2714no—
251C2×C4—5422no—
252C2×C4—5422no—
253C2×C4—5422no—
254C2×C4—5422no—
255C2×C4—5422no—
256C2×C4—5422no—
257C2×C4—5424no—
258C2×C4C2×C22⋊D27112yes—
259C2×C4C2×C22⋊D27114yes—
260C2×C4C2×C22⋊D27112yes—
261C2×C4C2×C22⋊D27112yes—
262C2×C4C2×C22⋊D27112yes—
263C22—28no—
264C22—24no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—44no—
271C22—42no—
272C22—42no—
273C22—10844no—
274C22—10844no—
275C22—10842no—
276C22—10842no—
277C22—10842no—
278C22C22×C22⋊D27111yescharacteristic
279C22C22×C22⋊D27111yescharacteristic
280C22C22×C22⋊D27111yescharacteristic
281C22#41×D27112yes—
282C22#2257620612yes—
283C4—22no—
284C4—24no—
285C4—5424no—
286C4—10842no—
287C4—10842no—
288C4#2257612914yes—
289C2—44no—
290C2—42no—
291C2—21682no—
292C2C2×#2257612911yescharacteristic
293C2#3177444311yescharacteristic
294C2#3177446111yescharacteristic
295C2#3177453512yes—
296C2#3177456612yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC22×#41×C271C2
hypercenterC22×#41D271
hypocenterC271#2431
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#22576156C2semidirect
2. C2×#22576149C2semidirect
3. C2×#22576134C2semidirect
4. #31773805C2semidirect
5. #31773804C2semidirect
6. #31773800C2semidirect
7. #31773776C2semidirect
8. C4×C22⋊Dic271C2semidirect
9. C22×#41×C271C2semidirect
57. C2×Q8×C271D4semidirect
63. C2×D4×C271D4semidirect
69. C22×C4×C271D4semidirect
88. C2×Dic271#41semidirect
148. C271#2431semidirect
150. C22×#41D271semidirect
188. C2×Q8C22⋊D271semidirect
202. C2×D4C22⋊D271semidirect
211. C22×C4C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761562
C2×#225761492
C2×#225761342
#317738052
#317738042
#317738002
#317737762
C4×C22⋊Dic2712
C22×#41×C2712
#2431271