Finite Group Databasebeta

← #10708113

Subgroups — #10708113

Order 768.

324 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 55 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#74352C211yesmaximal, characteristic
2#69573C211yesmaximal, characteristic
3#68141C211yesmaximal, characteristic
4#68140C211yesmaximal, characteristic
5C3×#1325C211yesmaximal, characteristic
6#67459C211yesmaximal, characteristic
7#67458C211yesmaximal, characteristic
8#12236—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C3×#384C2211yescharacteristic
11#4986C2211yescharacteristic
12#4985—21no—
13#4985—21no—
14#4981C2211yescharacteristic
15#4980—21no—
16#4980—21no—
17#4978—21no—
18#4964—21no—
19#4933—21no—
20#4722—21no—
21#4721—21no—
22C3×#324C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4530C2211yescharacteristic
25#4523—21no—
26#4467—21no—
27#4467—21no—
28#2794—31no—
29#2060—31no—
30#1617—31no—
31#1611—31no—
32#1353—31no—
33#1346—31no—
34#1325D311yescharacteristic
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×C3×C42D411yescharacteristic
40#926—21no—
41C2×C4⋊Dic3D411yescharacteristic
42C2×C4⋊Dic3D411yescharacteristic
43#924—21no—
44#924—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×C4×Dic3—21no—
48C2×#208—21no—
49#890—41no—
50#888—41no—
51C3×C22⋊C8—21no—
52C3×C22⋊C8—21no—
53C3⋊#95—21no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95D411yescharacteristic
63C3⋊#95D411yescharacteristic
64#818—41no—
65#817—41no—
66#816—41no—
67#816—41no—
68#808—41no—
69#807—41no—
70#544—31no—
71#484—61no—
72#483—61no—
73#428—61no—
74#400—31no—
75#400—61no—
76#400—61no—
77#398—31no—
78#387—61no—
79#387—61no—
80#387—61no—
81#387—61no—
82#384C2×D311yescharacteristic
83#343—31no—
84#339—61no—
85#328—61no—
86#328—61no—
87#324C2×D311yescharacteristic
88#324C2×D311yescharacteristic
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×Dic3—21no—
95C22×Dic3—21no—
96C22×Dic3—21no—
97C22×Dic3—21no—
98C2×Dic6—21no—
99C2×Dic6—21no—
100C2×C3×C8—41no—
101C2×C3×C8—41no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—42no—
105C3×C4⋊C4—42no—
106C3×C42—21no—
107C3×C42—21no—
108#217—41no—
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115Dic3⋊C4—41no—
116Dic3⋊C4—41no—
117Dic3⋊C4—42no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120C4×Dic3—41no—
121C4×Dic3—81no—
122#208—22no—
123#208—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—31no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4D1211yescharacteristic
138C2×C4⋊C4C22×D311yescharacteristic
139C2×C42—61no—
140C2×C42D1211yescharacteristic
141#107—121no—
142#107—121no—
143#106—121no—
144#106—121no—
145#103—121no—
146#103—121no—
147C22⋊C8—21no—
148C22⋊C8—21no—
149#95—31no—
150#95—31no—
151#95—61no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
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#13511yescharacteristic
174C2×C3×C4#13711yescharacteristic
175C2×Dic3—41no—
176C2×Dic3—41no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179C2×Dic3—42no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188Dic6—81no—
189Dic6—81no—
190C3×C8—81no—
191C3×C8—81no—
192C3⋊C8—42no—
193C2×Q8—61no—
194C2×Q8—61no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4—61no—
200C22×C4—61no—
201C22×C4C2×D1211yescharacteristic
202C22×C4D3×D411yescharacteristic
203C22×C4D3×D411yescharacteristic
204M16—62no—
205M16—121no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—121no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—61no—
214C4⋊C4—61no—
215C4⋊C4—61no—
216C4⋊C4—61no—
217C4⋊C4—121no—
218C4⋊C4—121no—
219C4⋊C4—122no—
220C4⋊C4—121no—
221C4⋊C4—121no—
222C4⋊C4—121no—
223C4⋊C4—121no—
224C22⋊C4—121no—
225C42—21no—
226C42—21no—
227C42—121no—
228C42—241no—
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—
240Dic3—81no—
241Dic3—81no—
242Dic3—81no—
243Dic3—81no—
244Dic3—81no—
245Dic3—81no—
246C23#88211yescharacteristic
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—121no—
262C2×C4—121no—
263C2×C4—121no—
264C2×C4—121no—
265C2×C4—122no—
266C2×C4—121no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—121no—
270C2×C4—121no—
271C2×C4—121no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4#90411yescharacteristic
275C2×C4#90911yescharacteristic
276C8—81no—
277C8—81no—
278C8—122no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3#160311yescharacteristic
282C2×C3#219811yescharacteristic
283C2×C3#220311yescharacteristic
284C22—21no—
285C22—21no—
286C22#473311yescharacteristic
287C22#474111yescharacteristic
288C22#475211yescharacteristic
289C4—21no—
290C4—21no—
291C4—42no—
292C4—41no—
293C4—41no—
294C4—81no—
295C4—241no—
296C4—241no—
297C4—241no—
298C4—241no—
299C4—241no—
300C4—241no—
301C3#1223611yescharacteristic, complemented, Sylow
302C2—21no—
303C2—21no—
304C2#6813211yescharacteristic
305C2#6914811yescharacteristic
306C2#6916711yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1325C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1325D3
hypocenterC3#12236
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

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
#743522
#695732
#681412
#681402
C3×#13252
#674592
#674582
#122363