Finite Group Databasebeta

← #10698308

Subgroups — #10698308

Order 768.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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#74222C211yesmaximal, characteristic
2C3×#1518C211yesmaximal, characteristic
3#71839C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic
5#67738C211yesmaximal, characteristic
6#67736C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#10369—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#431C2211yescharacteristic
11C3×#382C2211yescharacteristic
12#4984—21no—
13#4972—21no—
14#4936C2211yescharacteristic
15C42×Dic3C2211yescharacteristic
16#4925C2211yescharacteristic
17#4924C2211yescharacteristic
18#4655—22no—
19#4654—22no—
20C3×#328—22no—
21#4538—22no—
22#4532—21no—
23#4532—21no—
24#2864—31no—
25#1994—31no—
26#1518—31no—
27#1518D311yescharacteristic
28#1286—31no—
29#1286—31no—
30#1276—31no—
31C3×C4⋊Q8Q811yescharacteristic
32C3×C4⋊Q8Q811yescharacteristic
33C3×#125—21no—
34C3×C22⋊Q8—21no—
35C3×C22⋊Q8—21no—
36C3×C4×Q8—21no—
37C3×C4×Q8—21no—
38C3×#117—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42C2311yescharacteristic
41C2×C4⋊Dic3—22no—
42C2×C4⋊Dic3—21no—
43C2×Dic3⋊C4—22no—
44C2×C4×Dic3—21no—
45C2×C4×Dic3—21no—
46C2×C4×Dic3D411yescharacteristic
47C2×C4×Dic3D411yescharacteristic
48C2×#208—21no—
49C3×C4⋊C8—21no—
50C3×C22⋊C8—21no—
51C3×C4×C8—21no—
52C3×#95—21no—
53C3×#95D412yes—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95—21no—
57C3⋊#95D412yes—
58#830—21no—
59C4⋊(C3⋊C8)—21no—
60C4⋊(C3⋊C8)—21no—
61C4⋊(C3⋊C8)—21no—
62C4×C3⋊C8—21no—
63#533C2×D311yescharacteristic
64#431—31no—
65#431C2×D311yescharacteristic
66#423—31no—
67#389—61no—
68#389—61no—
69#387—62no—
70#387—62no—
71#382—31no—
72#382C2×D311yescharacteristic
73C43—31no—
74#339—61no—
75#339—61no—
76#328—22no—
77#328—62no—
78C2×C3×Q8—21no—
79C2×C3×Q8—21no—
80C22×C3×C4—22no—
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4C2×D411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×Dic3—21no—
85C22×Dic3—21no—
86C22×Dic3—22no—
87C2×C3×C8—41no—
88C3×C4⋊C4—21no—
89C3×C4⋊C4—21no—
90C3×C4⋊C4—21no—
91C3×C4⋊C4—21no—
92C3×C4⋊C4—41no—
93C3×C4⋊C4—41no—
94C3×C4⋊C4—41no—
95C3×C4⋊C4—41no—
96C3×C22⋊C4—41no—
97C3×C42—41no—
98C3×C42C2×Q811yescharacteristic
99C3×C42#4111yescharacteristic
100C3×C42#4111yescharacteristic
101C3×C42#4111yescharacteristic
102C4⋊Dic3—44no—
103C4⋊Dic3—41no—
104C4⋊Dic3—41no—
105Dic3⋊C4—44no—
106C4×Dic3—21no—
107C4×Dic3—21no—
108C4×Dic3—21no—
109C4×Dic3—21no—
110C4×Dic3—42no—
111C4×Dic3—42no—
112#208—41no—
113#208—41no—
114C2×C3⋊C8—41no—
115C2×C3⋊C8—41no—
116C2×M16—61no—
117C4⋊Q8Dic611yescharacteristic
118C4⋊Q8Dic611yescharacteristic
119#125—21no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C4×Q8—21no—
123C4×Q8—21no—
124#117—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—61no—
129C2×C42—31no—
130C2×C42—31no—
131C2×C42—61no—
132C2×C42—61no—
133C2×C42C22×D311yescharacteristic
134C4⋊C8—21no—
135C4⋊C8—61no—
136C4⋊C8—61no—
137C4⋊C8—61no—
138C22⋊C8—21no—
139C22⋊C8—61no—
140C4×C8—21no—
141C4×C8—61no—
142#95—21no—
143#95—32no—
144#95—62no—
145#95—62no—
146#95—61no—
147#95#6712yes—
148C23×C3#12011yescharacteristic
149C3×Q8—81no—
150C3×Q8—81no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4C22⋊Q811yescharacteristic
159C2×C3×C4C22⋊Q811yescharacteristic
160C2×C3×C4C22⋊Q811yescharacteristic
161C2×C3×C4#12411yescharacteristic
162C2×C3×C4#12411yescharacteristic
163C2×C3×C4#12411yescharacteristic
164C2×C3×C4#13511yescharacteristic
165C2×C3×C4#13511yescharacteristic
166C2×C3×C4#13711yescharacteristic
167C2×C3×C4#13711yescharacteristic
168C2×Dic3—22no—
169C2×Dic3—22no—
170C2×Dic3—41no—
171C2×Dic3—41no—
172C2×Dic3—41no—
173C2×Dic3—41no—
174C2×Dic3—42no—
175C2×Dic3—42no—
176C2×Dic3—42no—
177C3×C8—81no—
178C3⋊C8—81no—
179C3⋊C8—81no—
180C2×Q8—21no—
181C2×Q8—21no—
182C22×C4—22no—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4—62no—
186C22×C4D3×D411yescharacteristic
187C22×C4D3×D411yescharacteristic
188C22×C4C2×#6711yescharacteristic
189M16—121no—
190M16—121no—
191C2×C8—41no—
192C2×C8—121no—
193C2×C8—121no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—124no—
203C4⋊C4—124no—
204C4⋊C4—121no—
205C4⋊C4—121no—
206C22⋊C4—41no—
207C42—41no—
208C42—61no—
209C42—61no—
210C42—61no—
211C42—61no—
212C42—122no—
213C42—122no—
214C42C2×Dic611yescharacteristic
215C42#23511yescharacteristic
216C42#23711yescharacteristic
217C42#23711yescharacteristic
218C22×C3—21no—
219C22×C3—21no—
220C22×C3#39311yescharacteristic
221C22×C3#45211yescharacteristic
222C22×C3#45211yescharacteristic
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—82no—
228C3×C4—81no—
229C3×C4—81no—
230C3×C4#47911yescharacteristic
231C3×C4#47911yescharacteristic
232C3×C4#48811yescharacteristic
233C3×C4#49111yescharacteristic
234Dic3—42no—
235Dic3—42no—
236Dic3—82no—
237C23#93711yescharacteristic
238Q8—81no—
239Q8—81no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—121no—
250C2×C4—121no—
251C2×C4—121no—
252C2×C4—121no—
253C2×C4—122no—
254C2×C4—122no—
255C2×C4—122no—
256C2×C4#87811yescharacteristic
257C2×C4#87811yescharacteristic
258C2×C4#88511yescharacteristic
259C2×C4#88511yescharacteristic
260C2×C4#91511yescharacteristic
261C2×C4#91611yescharacteristic
262C2×C4#91811yescharacteristic
263C2×C4#91911yescharacteristic
264C2×C4#92411yescharacteristic
265C2×C4#93611yescharacteristic
266C8—81no—
267C8—241no—
268C8—241no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3#202611yescharacteristic
272C2×C3#203111yescharacteristic
273C2×C3#203111yescharacteristic
274C22—21no—
275C22—21no—
276C22#494711yescharacteristic
277C22#517311yescharacteristic
278C22#518711yescharacteristic
279C4—21no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—82no—
284C4—81no—
285C4—81no—
286C4—122no—
287C4—122no—
288C4—242no—
289C4#479611yescharacteristic
290C4#479711yescharacteristic
291C4#479911yescharacteristic
292C4#480011yescharacteristic
293C3#1036911yescharacteristic, complemented, Sylow
294C2—21no—
295C2—21no—
296C2#7003811yescharacteristic
297C2#7023211yescharacteristic
298C2#7039411yescharacteristic
subgroupG/H
centerC22#4947
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1518D3
hypocenterC3#10369
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
293. C3#10369semidirect

Maximal subgroups

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

subgroupindex
#742222
C3×#15182
#718392
#718382
#677382
#677362
#676312
#103693