Finite Group Databasebeta

← #10692525

Subgroups — #10692525

Order 768.

532 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#82261C211yesmaximal, characteristic
2#82202C211yesmaximal, characteristic
3C3×#1464C211yesmaximal, characteristic
4#70812C212yesmaximal, complemented
5#67748C212yesmaximal, complemented
6#9388—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C22×#208—21no—
9C2×#870C2211yescharacteristic
10C3×#432C2212yes—
11#5210—21no—
12#4997—24no—
13#4986C411yescharacteristic
14#4981C411yescharacteristic
15#4976—21no—
16#4953C411yescharacteristic
17#4949C411yescharacteristic
18#4945—21no—
19#4933—21no—
20#4932C2211yescharacteristic
21C4×#208—21no—
22C4×#208C2212yes—
23C3×#324—21no—
24C3×#323—21no—
25#4465—24no—
26#2873—31no—
27#2611—31no—
28#1528—32no—
29#1464D311yescharacteristic
30#1300—32no—
31C23×C3×C4D411yescharacteristic
32C3×#117—22no—
33C3×#117—22no—
34C2×C3×C4⋊C4C2×C411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C22⋊C4C2×C411yescharacteristic
37C2×C3×C22⋊C4D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42C2311yescharacteristic
40C2×#217—21no—
41C2×#217—21no—
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×#208—22no—
50C2×#208—24no—
51C2×#208—24no—
52C2×#208—22no—
53C22×C3⋊C8—22no—
54#870—22no—
55#870—22no—
56C3×C4⋊C8—22no—
57C3×C22⋊C8—22no—
58C3×#97—22no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95C2×C411yescharacteristic
65C3⋊#95C2×C411yescharacteristic
66#803—22no—
67#803D414yes—
68C4×C3⋊C8—22no—
69C4×C3⋊C8—22no—
70C22×M16—61no—
71C2×#125—31no—
72C2×#117C2×D311yescharacteristic
73#432C2×D312yes—
74#405—64no—
75C4×M16—32no—
76C4×M16—61no—
77#400—31no—
78#399—31no—
79#398—31no—
80#397—31no—
81#389—61no—
82#388—61no—
83#387—61no—
84#385—61no—
85#376—31no—
86#330—64no—
87#324—21no—
88#323—21no—
89C24×C3C22⋊C411yescharacteristic
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C2×C3×C8—42no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C22⋊C4—22no—
108C3×C22⋊C4—22no—
109C3×C42—21no—
110C3×C42—21no—
111C3×C42C2×D412yes—
112C3×C42#4112yes—
113#217—42no—
114#217—42no—
115C4⋊Dic3—22no—
116C4⋊Dic3—22no—
117Dic3⋊C4—42no—
118Dic3⋊C4—42no—
119Dic3⋊C4—41no—
120Dic3⋊C4—42no—
121Dic3⋊C4—41no—
122C4×Dic3—42no—
123#208—28no—
124#208—24no—
125#208—24no—
126C2×C3⋊C8—28no—
127C2×C3⋊C8—24no—
128C23×C4D1211yescharacteristic
129C2×M16—62no—
130C2×M16—64no—
131C2×M16—64no—
132C2×M16—62no—
133C22×C8—62no—
134#125—62no—
135#125—62no—
136#117—22no—
137#117—22no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C4⋊C4#6711yescharacteristic
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4C4×D311yescharacteristic
149C2×C22⋊C4#6711yescharacteristic
150C2×C42—61no—
151C2×C42D1211yescharacteristic
152C2×C42C22×D311yescharacteristic
153C4⋊C8—22no—
154C22⋊C8—22no—
155#97—22no—
156#97—34no—
157#97—62no—
158C4×C8—62no—
159C4×C8—62no—
160#95—31no—
161#95—31no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3#10412yes—
172C23×C3C2×C22⋊C411yescharacteristic
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—24no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4#10412yes—
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12112yes—
190C2×C3×C4#12311yescharacteristic
191C2×Dic3—22no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C3×C8—82no—
204C3⋊C8—28no—
205C24(C2×D3)⋊C411yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4(C2×D3)⋊C411yescharacteristic
215C22×C4(C2×D3)⋊C411yescharacteristic
216C22×C4(C2×D3)⋊C411yescharacteristic
217C22×C4C2×C4×D311yescharacteristic
218C22×C4C2×D1211yescharacteristic
219C22×C4C2×#6711yescharacteristic
220M16—68no—
221M16—64no—
222M16—64no—
223C2×C8—42no—
224C2×C8—68no—
225C2×C8—64no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—62no—
229C4⋊C4—62no—
230C4⋊C4—122no—
231C4⋊C4—122no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4—122no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—122no—
238C22⋊C4—122no—
239C42—21no—
240C42—21no—
241C42—122no—
242C42D3×D412yes—
243C42#23712yes—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—42no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—81no—
251C22×C3#38611yescharacteristic
252C22×C3C2×#10411yescharacteristic
253C22×C3C2×#10411yescharacteristic
254C3×C4—22no—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—41no—
260C3×C4—41no—
261Dic3—42no—
262Dic3—81no—
263Dic3—81no—
264Dic3—81no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23#82512yes—
270C23C2×(C2×D3)⋊C411yescharacteristic
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—62no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4#82512yes—
297C2×C4#67⋊C412yes—
298C2×C4#88211yescharacteristic
299C2×C4#88611yescharacteristic
300C2×C4#93912yes—
301C8—68no—
302C8—82no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—42no—
306C2×C3#188411yescharacteristic
307C2×C3#189611yescharacteristic
308C2×C3#191511yescharacteristic
309C22—21no—
310C22—21no—
311C22—22no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22#495811yescharacteristic
317C22C2×#82511yescharacteristic
318C22C2×#82511yescharacteristic
319C4—21no—
320C4—22no—
321C4—21no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—41no—
326C4—122no—
327C4—241no—
328C4—241no—
329C4—241no—
330C3#938811yescharacteristic, complemented, Sylow
331C2—21no—
332C2—21no—
333C2—42no—
334C2#6996511yescharacteristic
335C2#7004511yescharacteristic
336C2#7013811yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1464C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1464D3
hypocenterC3#9388
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
4. #70812C2semidirect
5. #67748C2semidirect
330. C3#9388semidirect

Maximal subgroups

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

subgroupindex
#822612
#822022
C3×#14642
#708122
#677482
#93883