Finite Group Databasebeta

← #10692831

Subgroups — #10692831

Order 768.

474 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2921C211yesmaximal, characteristic
2#82256C211yesmaximal, characteristic
3#71594C211yesmaximal, characteristic
4#69981C212yesmaximal
5#67682C212yesmaximal
6#9444—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#924C2211yescharacteristic
9C3×#433—24no—
10C3×#423—21no—
11C3×#407C2212yes—
12C3×C4×M16—21no—
13#5211C2212yes—
14C4×#217C2211yescharacteristic
15#4977C411yescharacteristic, complemented
16#4975C411yescharacteristic, complemented
17#4973C411yescharacteristic, complemented
18#4973C411yescharacteristic, complemented
19#4952—21no—
20#4952—21no—
21#4933—21no—
22#4933—21no—
23#4526—21no—
24#4526—21no—
25#4467—24no—
26#2921D311yescharacteristic
27#2499—31no—
28#1820—31no—
29#1511—32no—
30#1284—32no—
31C23×C3×C4C2311yescharacteristic
32C22×Dic6C2×C411yescharacteristic
33C2×C3×M16—22no—
34C3×#117—22no—
35C3×#117—22no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×#217—21no—
43C2×#217—21no—
44C2×#217—22no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46#924—22no—
47#924—22no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3C2×C411yescharacteristic
52C2×C4×Dic3C2×C411yescharacteristic
53C3×C4⋊C8—24no—
54C3×C22⋊C8—22no—
55C3×C22⋊C8D414yes—
56C3×#97—22no—
57C3×C4×C8—22no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—22no—
66C3⋊#95—22no—
67C2×C22⋊Q8—31no—
68C2×#117C2×D311yescharacteristic
69#433—24no—
70#423—21no—
71#407C2×D312yes—
72C4×M16—21no—
73#397—61no—
74#397—61no—
75#391—31no—
76#391—31no—
77#387—61no—
78#387—61no—
79#384—31no—
80#382—31no—
81#381—32no—
82C4×C22⋊C4—31no—
83#342—61no—
84#342—61no—
85#328—64no—
86C24×C3C2×D411yescharacteristic
87C22×C3×C4—41no—
88C22×C3×C4—41no—
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D412yes—
96C22×C3×C4#4112yes—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—42no—
102C2×Dic6—22no—
103C2×Dic6—41no—
104C2×Dic6—41no—
105C3×M16—42no—
106C3×M16—42no—
107C2×C3×C8—24no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C42—21no—
113C3×C42—21no—
114C3×C42—21no—
115C3×C42—21no—
116#217—42no—
117#217—42no—
118#217—42no—
119#217—42no—
120C4⋊Dic3—22no—
121Dic3⋊C4—42no—
122Dic3⋊C4—42no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125C4×Dic3—41no—
126C4×Dic3—41no—
127C22×Q8—31no—
128C23×C4C22×D311yescharacteristic
129C2×M16—22no—
130C22⋊Q8—62no—
131C22⋊Q8—62no—
132#117—22no—
133#117—22no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4C4×D311yescharacteristic
139C2×C4⋊C4C4×D311yescharacteristic
140C2×C22⋊C4—61no—
141C2×C22⋊C4—61no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4D1211yescharacteristic
144C2×C22⋊C4D1211yescharacteristic
145C2×C42—31no—
146C2×C42—31no—
147C2×C42#6711yescharacteristic
148C2×C42#6711yescharacteristic
149C4⋊C8—24no—
150C22⋊C8—22no—
151C22⋊C8#6714yes—
152#97—22no—
153C4×C8—22no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95—61no—
161#95—62no—
162#95—62no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3C2×C22⋊C411yescharacteristic
168C23×C3#12112yes—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4C4×D412yes—
184C2×C3×C4#12011yescharacteristic
185C2×C3×C4#12311yescharacteristic
186C2×Dic3—22no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—82no—
199C2×Dic3—82no—
200C2×Dic3—82no—
201Dic6—82no—
202C3×C8—44no—
203C24C2×D1211yescharacteristic
204C2×Q8—62no—
205C2×Q8—121no—
206C2×Q8—121no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213C22×C4—122no—
214C22×C4(C2×D3)⋊C411yescharacteristic
215C22×C4(C2×D3)⋊C411yescharacteristic
216C22×C4(C2×D3)⋊C411yescharacteristic
217C22×C4(C2×D3)⋊C411yescharacteristic
218C22×C4C2×C4×D311yescharacteristic
219C22×C4#23512yes—
220C22×C4C2×#6711yescharacteristic
221C22×C4C2×#6712yes—
222M16—42no—
223M16—42no—
224C2×C8—24no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—62no—
228C4⋊C4—122no—
229C4⋊C4—122no—
230C4⋊C4—121no—
231C4⋊C4—121no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—122no—
235C22⋊C4—122no—
236C22⋊C4—122no—
237C22⋊C4—122no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—121no—
243C42—121no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—81no—
250C22×C3—81no—
251C22×C3#38611yescharacteristic
252C22×C3#41011yescharacteristic
253C22×C3#41011yescharacteristic
254C3×C4—21no—
255C3×C4—22no—
256C3×C4—21no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261Dic3—42no—
262Dic3—81no—
263Dic3—81no—
264Dic3—81no—
265Dic3—162no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23C2×(C2×D3)⋊C411yescharacteristic
271C23#93012yes—
272Q8—242no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—62no—
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—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—242no—
300C2×C4—242no—
301C2×C4—242no—
302C2×C4C4×#6712yes—
303C2×C4#92911yescharacteristic
304C2×C4#95311yescharacteristic
305C8—44no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—42no—
309C2×C3#188411yescharacteristic
310C2×C3#189911yescharacteristic
311C2×C3#190911yescharacteristic
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—81no—
318C22—81no—
319C22#500611yescharacteristic
320C22#500611yescharacteristic
321C22#521311yescharacteristic
322C4—21no—
323C4—22no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—122no—
330C4—241no—
331C4—241no—
332C4—241no—
333C4—482no—
334C3#944411yescharacteristic, complemented, Sylow
335C2—21no—
336C2—21no—
337C2—42no—
338C2#7159511yescharacteristic
339C2#7202111yescharacteristic
340C2#7202411yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2921C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2921D3
hypocenterC3#9444
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2921

Complements

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

NH ≅ G/Ntype
15. #4977C4semidirect
16. #4975C4semidirect
17. #4973C4semidirect
18. #4973C4semidirect
334. C3#9444semidirect

Maximal subgroups

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

subgroupindex
C3×#29212
#822562
#715942
#699812
#676822
#94443