Finite Group Databasebeta

← #10696617

Subgroups — #10696617

Order 768.

474 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3165C211yesmaximal, characteristic
2#83326C211yesmaximal, characteristic
3#83246C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic
5#71776C211yesmaximal, characteristic
6#69863C211yesmaximal, characteristic
7#67665C211yesmaximal, characteristic
8#10101—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#501—22no—
13C3×#500—21no—
14C3×#498—21no—
15C3×#428—22no—
16C3×C4×M16C2211yescharacteristic
17#5230—22no—
18#5082—21no—
19#5081—21no—
20#5044—21no—
21#5043—21no—
22#5022—22no—
23#4983—22no—
24#4977—21no—
25#4975—21no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4547—22no—
30#4538—22no—
31#4465—22no—
32#3236—31no—
33#3165D311yescharacteristic
34#2660—31no—
35#1556—31no—
36#1528—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8—21no—
40C2×C3×M16—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D412yes—
55C2×C3×C42C2311yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—24no—
61C2×C4×Dic3—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65#870—21no—
66C3×#107—24no—
67C3×#106—24no—
68C3×#97D412yes—
69C3×C4×C8—21no—
70#835—22no—
71#835—22no—
72C3⋊#95—24no—
73C3⋊#95—21no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82#803—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125—31no—
86#501—22no—
87#501—61no—
88#500—21no—
89#498—21no—
90#491—61no—
91#484—61no—
92#483—61no—
93#479—62no—
94#432—31no—
95#432—31no—
96#428—22no—
97C4×M16C2×D311yescharacteristic
98#398—62no—
99#395—62no—
100#384—61no—
101#382—61no—
102#376—31no—
103#332—62no—
104#330—62no—
105#328—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C22×C3×C4—22no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—22no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C3×M16—24no—
119C2×C3×C8—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—84no—
145C4×Dic3—24no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C22×Q8—21no—
149C2×M16—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8#6711yescharacteristic
155C4⋊Q8#6711yescharacteristic
156#125—21no—
157#125—61no—
158#125—61no—
159#125—61no—
160#125—61no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165#117—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—64no—
172C2×C4⋊C4#6712yes—
173C2×C42—61no—
174C2×C42C22×D311yescharacteristic
175#107—24no—
176#107—62no—
177#106—24no—
178#106—62no—
179C4⋊C8—32no—
180C4⋊C8—61no—
181#103—62no—
182#103—62no—
183C22⋊C8—61no—
184C22⋊C8—61no—
185#97—61no—
186#97—61no—
187#97#6712yes—
188C4×C8—21no—
189#95—32no—
190#95—64no—
191#95—61no—
192C23×C3#12011yescharacteristic
193C3×Q8—42no—
194C3×Q8—81no—
195C3×Q8—82no—
196C3×Q8—82no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—41no—
200C2×C3×C4—44no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12711yescharacteristic
210C2×Dic3—24no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217C3×C8—24no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—82no—
225C22×C4—22no—
226C22×C4—62no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4D3×D411yescharacteristic
230C22×C4D3×D411yescharacteristic
231C22×C4C2×#6711yescharacteristic
232M16—24no—
233C2×C8—22no—
234C2×C8—62no—
235C2×C8—121no—
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—22no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—244no—
255C22⋊C4—41no—
256C42—41no—
257C42—64no—
258C42D3×D411yescharacteristic
259C42#23711yescharacteristic
260C42C2×#6711yescharacteristic
261C42C2×#6711yescharacteristic
262C22×C3—21no—
263C22×C3—21no—
264C22×C3#39211yescharacteristic
265C22×C3#45612yes—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—81no—
272C3×C4—82no—
273C3×C4—81no—
274C3×C4#45612yes—
275Dic3—44no—
276Dic3—81no—
277Dic3—81no—
278C23#93711yescharacteristic
279Q8—42no—
280Q8—81no—
281Q8—82no—
282Q8—82no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—44no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—64no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—122no—
297C2×C4#93711yescharacteristic
298C2×C4#93811yescharacteristic
299C2×C4#93911yescharacteristic
300C2×C4#93911yescharacteristic
301C2×C4#94011yescharacteristic
302C2×C4#95311yescharacteristic
303C8—24no—
304C8—122no—
305C8—241no—
306C2×C3—22no—
307C2×C3#200511yescharacteristic
308C2×C3#200511yescharacteristic
309C2×C3#200811yescharacteristic
310C22—21no—
311C22—21no—
312C22#509512yes—
313C22#522311yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—82no—
320C4—81no—
321C4—81no—
322C4—124no—
323C4—241no—
324C4—241no—
325C4#520512yes—
326C3#1010111yescharacteristic, complemented, Sylow
327C2—22no—
328C2#7179511yescharacteristic
329C2#7209911yescharacteristic
330C2#7210811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3165D3
hypocenterC3#10101
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
326. C3#10101semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#833262
#832462
#718362
#717762
#698632
#676652
#101013