Finite Group Databasebeta

← #10708111

Subgroups — #10708111

Order 768.

324 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 59 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×#2794C211yesmaximal, characteristic
2#69719C211yesmaximal, characteristic
3#69713C211yesmaximal, characteristic
4#69573C211yesmaximal, characteristic
5#67652C211yesmaximal, characteristic
6#67650C211yesmaximal, characteristic
7#67635C211yesmaximal, characteristic
8#12236—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C3×#400—21no—
11C3×#400—21no—
12C3×#400C2211yescharacteristic
13C3×#398C2211yescharacteristic
14C3×#387—21no—
15C3×#387—21no—
16C3×#387—21no—
17C3×#387—21no—
18C3×#384C2211yescharacteristic
19#4971—21no—
20#4970—21no—
21#4964—21no—
22#4530C2211yescharacteristic
23#4528C2211yescharacteristic
24#4528C2211yescharacteristic
25#4523—21no—
26#4452—21no—
27#4452—21no—
28#2794D311yescharacteristic
29#2060—31no—
30#1617—31no—
31#1611—31no—
32#1353—31no—
33#1346—31no—
34#1325—31no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×C3×C42—21no—
44C2×C3×C42D411yescharacteristic
45#926—21no—
46#924—21no—
47#924—21no—
48C2×#208—21no—
49#890—41no—
50#888—41no—
51C3×#95—21no—
52C3×#95—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95D411yescharacteristic
61C3×#95D411yescharacteristic
62#830—21no—
63#830—21no—
64#810—41no—
65#810—41no—
66#808—41no—
67#808—41no—
68#807—41no—
69#807—41no—
70#544—31no—
71#484—61no—
72#483—61no—
73#428—61no—
74#400—21no—
75#400—21no—
76#400C2×D311yescharacteristic
77#398C2×D311yescharacteristic
78#387—21no—
79#387—21no—
80#387—21no—
81#387—21no—
82#384C2×D311yescharacteristic
83#343—31no—
84#339—61no—
85#328—61no—
86#328—61no—
87#324—31no—
88#324—31no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C2×Dic6—21no—
99C2×Dic6—21no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—42no—
105C3×C4⋊C4—42no—
106C3×C4⋊C4—42no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C42—21no—
110C3×C42—21no—
111C3×C42—81no—
112#217—41no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117Dic3⋊C4—41no—
118Dic3⋊C4—41no—
119C4×Dic3—41no—
120#208—22no—
121#208—41no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8—41no—
125C2×M16—61no—
126C4⋊Q8—121no—
127#125—121no—
128C22⋊Q8—61no—
129C22⋊Q8—61no—
130#117—61no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4#6711yescharacteristic
136C2×C4⋊C4#6711yescharacteristic
137C2×C4⋊C4#6711yescharacteristic
138C2×C4⋊C4C22×D311yescharacteristic
139C2×C42—21no—
140C2×C42#6711yescharacteristic
141#107—121no—
142#107—121no—
143#106—121no—
144#106—121no—
145#103—121no—
146#103—121no—
147C22⋊C8—61no—
148C22⋊C8—61no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95#6711yescharacteristic
159#95#6711yescharacteristic
160C23×C3#12011yescharacteristic
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—22no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—42no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4#13511yescharacteristic
185C2×C3×C4#13711yescharacteristic
186C2×Dic3—41no—
187C2×Dic3—41no—
188Dic6—81no—
189Dic6—81no—
190C3⋊C8—42no—
191C3⋊C8—81no—
192C3⋊C8—81no—
193C2×Q8—61no—
194C2×Q8—61no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4C2×#6711yescharacteristic
202C22×C4C2×#6711yescharacteristic
203C22×C4C2×#6711yescharacteristic
204M16—62no—
205M16—121no—
206C2×C8—121no—
207C2×C8—121no—
208C2×C8—121no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—121no—
223C4⋊C4—121no—
224C22⋊C4—121no—
225C42—21no—
226C42—21no—
227C42—81no—
228C42—121no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3#45211yescharacteristic
232C22×C3#45611yescharacteristic
233C22×C3B(4,𝔽2)11yescharacteristic
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—42no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—81no—
240C3×C4—81no—
241C3×C4—81no—
242C3×C4—81no—
243C3×C4—81no—
244Dic3—81no—
245Dic3—81no—
246C23#95311yescharacteristic
247Q8—241no—
248Q8—241no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4#95011yescharacteristic
275C2×C4#95111yescharacteristic
276C8—122no—
277C8—241no—
278C8—241no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3#160311yescharacteristic
282C2×C3#219811yescharacteristic
283C2×C3#220311yescharacteristic
284C22—21no—
285C22—21no—
286C22#523911yescharacteristic
287C22#524411yescharacteristic
288C22#524611yescharacteristic
289C4—21no—
290C4—21no—
291C4—42no—
292C4—41no—
293C4—41no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—241no—
300C4—241no—
301C3#1223611yescharacteristic, complemented, Sylow
302C2—21no—
303C2—21no—
304C2#7188911yescharacteristic
305C2#7194611yescharacteristic
306C2#7195911yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2794C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#2794D3
hypocenterC3#12236
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2794

Complements

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

NH ≅ G/Ntype
301. C3#12236semidirect

Maximal subgroups

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

subgroupindex
C3×#27942
#697192
#697132
#695732
#676522
#676502
#676352
#122363