Finite Group Databasebeta

← #10696620

Subgroups — #10696620

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×#2660C211yesmaximal, characteristic
2#83207C211yesmaximal, characteristic
3#83021C211yesmaximal, characteristic
4#71776C211yesmaximal, characteristic
5#71735C211yesmaximal, characteristic
6#69433C211yesmaximal, characteristic
7#67615C211yesmaximal, characteristic
8#10101—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#398—22no—
13C3×#395—22no—
14C3×#384—21no—
15C3×#382—21no—
16C3×#376C2211yescharacteristic
17#5081—22no—
18#5080—21no—
19#5078—21no—
20#5069—21no—
21#5064—21no—
22#5063—22no—
23#4971—21no—
24#4970—21no—
25#4964—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4547—22no—
30#4543—22no—
31#4452—22no—
32#3236—31no—
33#3165—31no—
34#2660D311yescharacteristic
35#1556—31no—
36#1528—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—24no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42—21no—
56C2×C3×C42C2311yescharacteristic
57#926—21no—
58#924—21no—
59#924—21no—
60C2×#208—21no—
61#870—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×#95—24no—
67C3×#95—21no—
68C3×#95D412yes—
69#830—21no—
70#830—21no—
71#810—22no—
72#810—22no—
73#808—24no—
74#808—22no—
75#807—24no—
76#807—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79#803—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125C2×D311yescharacteristic
86#501—62no—
87#501—61no—
88#500—61no—
89#498—61no—
90#491—61no—
91#484—61no—
92#483—61no—
93#479—62no—
94#432—31no—
95#432—31no—
96#428—62no—
97C4×M16—31no—
98#398—22no—
99#395—22no—
100#384—21no—
101#382—21no—
102#376C2×D311yescharacteristic
103#332—62no—
104#330—62no—
105#328—62no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—82no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—21no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C2×Dic6—21no—
117C2×Dic6—21no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—84no—
131C3×C42—24no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#217—41no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143C4×Dic3—41no—
144#208—24no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C22×Q8—21no—
149C2×M16—61no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—31no—
155C4⋊Q8—31no—
156#125—21no—
157#125—21no—
158#125—21no—
159#125—21no—
160#125—61no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163C4×Q8—61no—
164C4×Q8—61no—
165#117—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4#6711yescharacteristic
171C2×C4⋊C4#6711yescharacteristic
172C2×C4⋊C4#6712yes—
173C2×C42—21no—
174C2×C42C22×D311yescharacteristic
175#107—64no—
176#107—62no—
177#106—64no—
178#106—62no—
179C4⋊C8—32no—
180C4⋊C8—61no—
181#103—62no—
182#103—62no—
183C22⋊C8—61no—
184C22⋊C8—61no—
185#97—32no—
186#97—61no—
187#97—61no—
188C4×C8—61no—
189#95—24no—
190#95—21no—
191#95#6712yes—
192C23×C3#12011yescharacteristic
193C3×Q8—82no—
194C3×Q8—82no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—44no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12711yescharacteristic
213C2×Dic3—41no—
214C2×Dic3—41no—
215Dic6—42no—
216Dic6—81no—
217C3⋊C8—24no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—82no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×#6711yescharacteristic
231C22×C4C2×#6711yescharacteristic
232M16—64no—
233C2×C8—62no—
234C2×C8—62no—
235C2×C8—121no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—22no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—84no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C22⋊C4—121no—
256C42—24no—
257C42—121no—
258C42D3×D411yescharacteristic
259C42D3×D411yescharacteristic
260C42D3×D411yescharacteristic
261C42#23711yescharacteristic
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—44no—
272C3×C4—81no—
273C3×C4—81no—
274C3×C4—82no—
275C3×C4#45612yes—
276Dic3—81no—
277Dic3—81no—
278C23#95311yescharacteristic
279Q8—82no—
280Q8—82no—
281Q8—122no—
282Q8—241no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4#93711yescharacteristic
298C2×C4#93711yescharacteristic
299C2×C4#93811yescharacteristic
300C2×C4#93911yescharacteristic
301C2×C4#93911yescharacteristic
302C2×C4#94011yescharacteristic
303C8—64no—
304C8—122no—
305C8—241no—
306C2×C3—22no—
307C2×C3#200511yescharacteristic
308C2×C3#200511yescharacteristic
309C2×C3#200811yescharacteristic
310C22—21no—
311C22—21no—
312C22#522311yescharacteristic
313C22#524612yes—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—44no—
320C4—82no—
321C4—81no—
322C4—81no—
323C4—241no—
324C4—241no—
325C4#509512yes—
326C3#1010111yescharacteristic, complemented, Sylow
327C2—22no—
328C2#7179511yescharacteristic
329C2#7185811yescharacteristic
330C2#7186811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2660C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2660D3
hypocenterC3#10101
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2660

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×#26602
#832072
#830212
#717762
#717352
#694332
#676152
#101013