Finite Group Databasebeta

← #10692902

Subgroups — #10692902

Order 768.

536 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C3×#2548C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3#71760C212yesmaximal
4#71625C211yesmaximal, characteristic, complemented
5#67621C212yesmaximal, complemented
6#9458—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C2×C3×#124C412yes—
9C2×C3×C4×Q8C2211yescharacteristic
10C3×#394—21no—
11C3×#394—21no—
12C3×#390—21no—
13C3×#390—21no—
14C3×#390C412yes—
15C3×C4×C4⋊C4C2211yescharacteristic
16#5024—24no—
17#4926—21no—
18#4926C2212yes—
19#4926C2212yes—
20#4925—21no—
21#4544—24no—
22#4528—21no—
23#4528—21no—
24#2983—31no—
25#2548D311yescharacteristic
26#1555—32no—
27#1467—31no—
28#1285—32no—
29C22×C3×Q8C2×C411yescharacteristic
30C22×C3×D4—21no—
31C22×C3×D4—21no—
32C22×C3×D4C2×C412yes—
33C3×#127—22no—
34C3×#127—22no—
35C3×#124—44no—
36C3×C4×Q8—22no—
37C3×C4×Q8—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C42C2×C412yes—
47C2×C3×C42D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×C3×C42C2311yescharacteristic
50C3×#95—21no—
51#830—22no—
52#830—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)D414yes—
57#803—22no—
58#803—22no—
59#803—22no—
60C4×C3⋊C8—22no—
61C2×#127C2×D311yescharacteristic
62C2×#124Dic312yes—
63C2×C4×Q8C2×D311yescharacteristic
64#446—64no—
65#432—32no—
66#432—32no—
67#432—61no—
68#431—61no—
69#394—21no—
70#394—21no—
71#390—21no—
72#390—21no—
73#390Dic312yes—
74C4×C4⋊C4C2×D311yescharacteristic
75#331—64no—
76#324—61no—
77#324—61no—
78C24×C3—22no—
79C2×C3×Q8—22no—
80C2×C3×Q8—22no—
81C2×C3×Q8—41no—
82C2×C3×D4—24no—
83C2×C3×D4—42no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—42no—
87C2×C3×D4—41no—
88C2×C3×D4—42no—
89C2×C3×D4—41no—
90C2×C3×D4—82no—
91C22×C3×C4—22no—
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C3×C4⋊C4—22no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—42no—
103C3×C22⋊C4—44no—
104C3×C22⋊C4—44no—
105C3×C22⋊C4—82no—
106C3×C22⋊C4—82no—
107C3×C42—21no—
108C3×C42—21no—
109C3×C42—21no—
110C3×C42—21no—
111C3×C42—42no—
112C3×C42C2×D412yes—
113C3×C42#4112yes—
114C2×C3⋊C8—24no—
115C2×C3⋊C8—42no—
116C22×Q8C2×Dic311yescharacteristic
117C22×D4—21no—
118C22×D4—21no—
119C22×D4C2×Dic312yes—
120#127—22no—
121#127—22no—
122#124—44no—
123C4×Q8—22no—
124C4×Q8—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4C2×Dic311yescharacteristic
127C2×C4⋊C4#6711yescharacteristic
128C2×C4⋊C4#6711yescharacteristic
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C42C2×Dic312yes—
134C2×C42#6711yescharacteristic
135C2×C42#6711yescharacteristic
136C2×C42C22×D311yescharacteristic
137C4⋊C8—34no—
138C4⋊C8—62no—
139C4⋊C8—62no—
140C4⋊C8—62no—
141C22⋊C8—62no—
142C22⋊C8—62no—
143#97—62no—
144#97—62no—
145#97—62no—
146C4×C8—62no—
147#95—21no—
148C23×C3—42no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—42no—
152C23×C3—42no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3C2×C22⋊C411yescharacteristic
156C3×Q8—44no—
157C3×D4—84no—
158C3×D4—82no—
159C3×D4—84no—
160C3×D4—82no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—22no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4#10012yes—
178C2×C3×C4C4×D412yes—
179C2×C3×C4#12011yescharacteristic
180C2×C3×C4#12112yes—
181C2×C3×C4#12311yescharacteristic
182C3⋊C8—44no—
183C3⋊C8—82no—
184C24—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—41no—
188C2×D4—24no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—42no—
196C2×D4—82no—
197C22×C4—22no—
198C22×C4#21711yescharacteristic
199C22×C4#21711yescharacteristic
200C22×C4#21711yescharacteristic
201C22×C4#21711yescharacteristic
202C22×C4C22×Dic311yescharacteristic
203C22×C4C2×#6711yescharacteristic
204C22×C4C2×#6711yescharacteristic
205C2×C8—64no—
206C2×C8—122no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—42no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—82no—
214C22⋊C4—82no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—42no—
220C42D3×D412yes—
221C42#23712yes—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—82no—
225C22×C3—82no—
226C22×C3—82no—
227C22×C3—82no—
228C22×C3—82no—
229C22×C3—82no—
230C22×C3—82no—
231C22×C3#38611yescharacteristic
232C22×C3C2×#10011yescharacteristic
233C22×C3#41311yescharacteristic
234C3×C4—21no—
235C3×C4—22no—
236C3×C4—22no—
237C3×C4—22no—
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—82no—
242C3×C4#45412yes—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23C2×#21711yescharacteristic
251Q8—44no—
252D4—84no—
253D4—82no—
254D4—84no—
255D4—82no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#83312yes—
273C2×C4D4×Dic312yes—
274C2×C4#93511yescharacteristic
275C2×C4#93711yescharacteristic
276C2×C4#93912yes—
277C8—124no—
278C8—242no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—162no—
282C2×C3#188811yescharacteristic
283C2×C3#190011yescharacteristic
284C2×C3#191011yescharacteristic
285C22—21no—
286C22—21no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22C2×#83311yescharacteristic
295C22#522011yescharacteristic
296C22#523711yescharacteristic
297C4—22no—
298C4—22no—
299C4—21no—
300C4—22no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4#506212yes—
306C3#945811yescharacteristic, complemented, Sylow
307C2—21no—
308C2—21no—
309C2—162no—
310C2#7168311yescharacteristic
311C2#7172411yescharacteristic
312C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2548C2
socleC22×C3#386
prime radicalC2×C3×#127C22
cosocleC2×C3×C42C23
hypercenter#2548D3
hypocenterC3#9458
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
4. #71625C2semidirect
5. #67621C2semidirect
306. C3#9458semidirect

Maximal subgroups

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

subgroupindex
C3×#25482
#824362
#717602
#716252
#676212
#94583