Finite Group Databasebeta

← #10708253

Subgroups — #10708253

Order 768.

352 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2612C211yesmaximal, characteristic
2#69719C211yesmaximal, characteristic
3#69707C211yesmaximal, characteristic
4#69543C211yesmaximal, characteristic
5#67652C211yesmaximal, characteristic
6#67648C211yesmaximal, characteristic
7#67632C211yesmaximal, characteristic
8#12258—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C3×#400C2211yescharacteristic
11C3×#398—21no—
12C3×#395C2211yescharacteristic
13C3×#391—21no—
14C3×#389—21no—
15C3×#387—21no—
16C3×#387—21no—
17C3×#382C2211yescharacteristic
18C3×#376—21no—
19#5228—21no—
20#4970—21no—
21#4964—21no—
22#4530C2211yescharacteristic
23#4528C2211yescharacteristic
24#4528C2211yescharacteristic
25#4523—21no—
26#4452—21no—
27#4452—21no—
28#2612D311yescharacteristic
29#2061—31no—
30#1617—31no—
31#1605—31no—
32#1354—31no—
33#1353—31no—
34#1324—31no—
35C22×C3×Q8D411yescharacteristic
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×C3×C42—21no—
44C2×C3×C42—21no—
45C2×C3×C42D411yescharacteristic
46C2×#216—41no—
47#926—21no—
48#924—21no—
49#924—21no—
50C2×#208—21no—
51#890—41no—
52#888—41no—
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×#95D411yescharacteristic
62C3×#95D411yescharacteristic
63#835—41no—
64#830—21no—
65#830—21no—
66#810—41no—
67#808—41no—
68#807—41no—
69#807—41no—
70#544—31no—
71#484—61no—
72#451—61no—
73#428—61no—
74#400C2×D311yescharacteristic
75#398—21no—
76#395C2×D311yescharacteristic
77#391—21no—
78#389—21no—
79#387—21no—
80#387—21no—
81#382C2×D311yescharacteristic
82#376—21no—
83#343—31no—
84#339—61no—
85#328—61no—
86#328—61no—
87#324—31no—
88#324—31no—
89C2×C3×Q8—42no—
90C2×C3×Q8—41no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C2×Dic6—21no—
101C2×Dic6—21no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—42no—
106C3×C4⋊C4—42no—
107C3×C4⋊C4—42no—
108C3×C4⋊C4—81no—
109C3×C42—21no—
110C3×C42—42no—
111C3×C42—42no—
112C3×C42Q1612yescomplemented
113#217—41no—
114#216—82no—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119Dic3⋊C4—41no—
120Dic3⋊C4—41no—
121C4×Dic3—41no—
122#208—22no—
123#208—41no—
124C2×C3⋊C8—41no—
125C2×C3⋊C8—41no—
126C2×C3⋊C8—41no—
127C22×Q8#6711yescharacteristic
128C2×Q16—121no—
129C2×M16—61no—
130C4⋊Q8—121no—
131#125—121no—
132C22⋊Q8—61no—
133C22⋊Q8—61no—
134#117—61no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4#6711yescharacteristic
140C2×C4⋊C4#6711yescharacteristic
141C2×C4⋊C4C22×D311yescharacteristic
142C2×C42—21no—
143C2×C42—21no—
144C2×C42#6711yescharacteristic
145#107—121no—
146#107—121no—
147#106—121no—
148#103—121no—
149#103—121no—
150C22⋊C8—61no—
151C22⋊C8—61no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95#6711yescharacteristic
161#95#6711yescharacteristic
162C23×C3#12011yescharacteristic
163C3×Q8—42no—
164C3×Q8—81no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
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—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4C2×Q1611yescharacteristic
190C2×C3×C4#13711yescharacteristic
191C2×Dic3—41no—
192C2×Dic3—41no—
193Dic6—81no—
194Dic6—81no—
195C3⋊C8—42no—
196C3⋊C8—81no—
197C3⋊C8—81no—
198C2×Q8—42no—
199C2×Q8—41no—
200C2×Q8—61no—
201C2×Q8—61no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4C2×#6711yescharacteristic
209C22×C4C2×#6711yescharacteristic
210C22×C4C2×#6711yescharacteristic
211Q16—242no—
212M16—62no—
213M16—121no—
214C2×C8—121no—
215C2×C8—121no—
216C2×C8—121no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—81no—
228C4⋊C4—121no—
229C4⋊C4—121no—
230C22⋊C4—121no—
231C42—21no—
232C42—42no—
233C42—42no—
234C42—121no—
235C42#21612yescomplemented
236C22×C3—21no—
237C22×C3—21no—
238C22×C3#45111yescharacteristic
239C22×C3#45611yescharacteristic
240C22×C3B(4,𝔽2)11yescharacteristic
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—42no—
244C3×C4—42no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—42no—
248C3×C4—81no—
249C3×C4—81no—
250C3×C4—81no—
251Dic3—81no—
252Dic3—81no—
253C23#95311yescharacteristic
254Q8—42no—
255Q8—81no—
256Q8—241no—
257Q8—241no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4#94211yescharacteristic
285C2×C4C2×#21611yescharacteristic
286C8—122no—
287C8—241no—
288C8—241no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3#160511yescharacteristic
292C2×C3#219711yescharacteristic
293C2×C3#220411yescharacteristic
294C22—21no—
295C22—21no—
296C22#522811yescharacteristic
297C22#524411yescharacteristic
298C22#524611yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—41no—
304C4—41no—
305C4—42no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—241no—
310C4—241no—
311C3#1225811yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2#7190411yescharacteristic
315C2#7192211yescharacteristic
316C2#7193011yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2612C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#2612D3
hypocenterC3#12258
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2612

Complements

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

NH ≅ G/Ntype
112. C3×C42Q16semidirect
235. C42#216semidirect
311. C3#12258semidirect

Maximal subgroups

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

subgroupindex
C3×#26122
#697192
#697072
#695432
#676522
#676482
#676322
#122583