Finite Group Databasebeta

← #10699214

Subgroups — #10699214

Order 768.

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

normal subgroups only

This group has 374 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
1#83324C211yesmaximal, characteristic
2C3×#1986C211yesmaximal, characteristic
3#71830C211yesmaximal, characteristic
4#69879C211yesmaximal, characteristic
5#69850C211yesmaximal, characteristic
6#69489C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#10506—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×C4×M16C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5079—21no—
14#5078—21no—
15#5044—22no—
16#5021—21no—
17#5021—21no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×#339—21no—
22C3×#339—21no—
23#4554—21no—
24#4554—21no—
25#4547—21no—
26#4546—21no—
27#4529—22no—
28#4458—21no—
29#4457—21no—
30#3230—31no—
31#1997—31no—
32#1987—31no—
33#1986D311yescharacteristic
34#1565—31no—
35#1550—31no—
36#1303—31no—
37C22×Dic6—21no—
38C2×C3×M16D411yescharacteristic
39C2×C3×M16D411yescharacteristic
40C3×C4⋊Q8D411yescharacteristic
41C3×C4⋊Q8D411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D412yes—
53C2×#208—21no—
54C22×C3⋊C8—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59C3×#97—21no—
60C3×C4×C8—21no—
61C3×#95—21no—
62#835—22no—
63#835—22no—
64#830—21no—
65#807—22no—
66#807—22no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)D411yescharacteristic
72C4⋊(C3⋊C8)D411yescharacteristic
73C4×C3⋊C8—21no—
74#533C2×D311yescharacteristic
75C2×C4⋊Q8—31no—
76#500—61no—
77#487—61no—
78#484—62no—
79#477—61no—
80#477—61no—
81#431—31no—
82#423—31no—
83C2×C4⋊C8—31no—
84C4×M16C2×D311yescharacteristic
85C4×C4⋊C4C2×D311yescharacteristic
86#343—61no—
87#343—61no—
88#340—62no—
89#339—21no—
90#339—21no—
91#335—61no—
92#334—61no—
93#333—61no—
94#332—61no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C22×C3×C4—41no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×Dic3—22no—
102C2×Dic6—41no—
103C2×Dic6—41no—
104C2×Dic6—82no—
105C3×M16—21no—
106C3×M16—21no—
107C3×M16—21no—
108C3×M16Q1612yes—
109C2×C3×C8—21no—
110C2×C3×C8—21no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C42—41no—
121C3×C42—41no—
122C3×C42—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131#208—41no—
132#208—41no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—42no—
136C2×C3⋊C8—41no—
137C2×C3⋊C8—41no—
138C22×Q8—61no—
139C2×M16—61no—
140C2×M16#6711yescharacteristic
141C2×M16#6711yescharacteristic
142C22×C8—61no—
143C4⋊Q8—61no—
144C4⋊Q8—61no—
145C4⋊Q8—61no—
146C4⋊Q8—61no—
147C4⋊Q8D1211yescharacteristic
148C4⋊Q8D1211yescharacteristic
149#125—21no—
150C22⋊Q8—21no—
151C22⋊Q8—21no—
152C4×Q8—21no—
153C4×Q8—21no—
154#117—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—32no—
157C2×C4⋊C4—61no—
158C2×C42—21no—
159C2×C42C22×D311yescharacteristic
160#107—62no—
161#107—62no—
162C4⋊C8—31no—
163C4⋊C8—31no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168#103—62no—
169#103—62no—
170C22⋊C8—61no—
171#97—21no—
172C4×C8—21no—
173C4×C8—61no—
174#95—21no—
175C23×C3#12111yescharacteristic
176C3×Q8—41no—
177C3×Q8—41no—
178C3×Q8—81no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—41no—
182C2×C3×C4—42no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—81no—
186C2×C3×C4—81no—
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12311yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×C3×C4#12411yescharacteristic
193C2×C3×C4C2×Q1611yescharacteristic
194C2×C3×C4#13511yescharacteristic
195C2×C3×C4#13711yescharacteristic
196C2×C3×C4#13711yescharacteristic
197C2×Dic3—42no—
198C2×Dic3—82no—
199Dic6—82no—
200Dic6—82no—
201C3×C8—22no—
202C3×C8—41no—
203C3⋊C8—41no—
204C3⋊C8—41no—
205C3⋊C8—81no—
206C3⋊C8—81no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—121no—
210C2×Q8—121no—
211C2×Q8—242no—
212C22×C4—41no—
213C22×C4—62no—
214C22×C4D3×D411yescharacteristic
215C22×C4#23711yescharacteristic
216C22×C4C2×#6711yescharacteristic
217M16—21no—
218M16—21no—
219M16—21no—
220M16—121no—
221M16—121no—
222M16#21612yes—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—61no—
226C2×C8—61no—
227C2×C8—122no—
228C2×C8—121no—
229C2×C8—121no—
230C4⋊C4—22no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C22⋊C4—41no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42C2×D1211yescharacteristic
247C42#23511yescharacteristic
248C42D3×D411yescharacteristic
249C42#23711yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3#39411yescharacteristic
253C22×C3#46211yescharacteristic
254C22×C3#46411yescharacteristic
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—82no—
259C3×C4—81no—
260C3×C4—81no—
261C3×C4#45411yescharacteristic
262C3×C4#45611yescharacteristic
263C3×C4#46211yescharacteristic
264C3×C4#46411yescharacteristic
265C3×C4#46711yescharacteristic
266C3×C4#47011yescharacteristic
267Dic3—82no—
268C23#93811yescharacteristic
269Q8—41no—
270Q8—41no—
271Q8—81no—
272Q8—242no—
273Q8—242no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—122no—
283C2×C4—242no—
284C2×C4#88211yescharacteristic
285C2×C4#88311yescharacteristic
286C2×C4#88511yescharacteristic
287C2×C4#88611yescharacteristic
288C2×C4#93011yescharacteristic
289C2×C4#93911yescharacteristic
290C2×C4#94211yescharacteristic
291C2×C4C2×#21611yescharacteristic
292C2×C4#95011yescharacteristic
293C2×C4#95111yescharacteristic
294C8—22no—
295C8—41no—
296C8—121no—
297C8—121no—
298C8—241no—
299C8—241no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3#203811yescharacteristic
303C2×C3#204811yescharacteristic
304C2×C3#205611yescharacteristic
305C22—21no—
306C22—21no—
307C22#496111yescharacteristic
308C22#505411yescharacteristic
309C22#509011yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—82no—
314C4—81no—
315C4—81no—
316C4—242no—
317C4#482611yescharacteristic
318C4#482711yescharacteristic
319C4#502811yescharacteristic
320C4#503111yescharacteristic
321C4#505111yescharacteristic
322C4#505211yescharacteristic
323C3#1050611yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#6968111yescharacteristic
327C2#7036011yescharacteristic
328C2#7040211yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1986D3
hypocenterC3#10506
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
323. C3#10506semidirect

Maximal subgroups

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

subgroupindex
#833242
C3×#19862
#718302
#698792
#698502
#694892
#676202
#105063