Finite Group Databasebeta

← #10696631

Subgroups — #10696631

Order 768.

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

normal subgroups only

This group has 458 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#83245C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic
5#71776C211yesmaximal, characteristic
6#69863C211yesmaximal, characteristic
7#67665C211yesmaximal, characteristic
8#10103—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#5229—21no—
18#5229—21no—
19#5082—21no—
20#5081—21no—
21#5044—21no—
22#5043—21no—
23#5022—22no—
24#4985—22no—
25#4985—22no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4547—22no—
30#4538—22no—
31#4465—22no—
32#3236—31no—
33#3165D311yescharacteristic
34#2642—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×C4×Dic3—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65C3×#107—24no—
66C3×#106—24no—
67C3×#97D412yes—
68C3×C4×C8—21no—
69#835—22no—
70#835—22no—
71C3⋊#95—24no—
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#400—62no—
99#400—62no—
100#391—61no—
101#391—61no—
102#376—31no—
103#332—62no—
104#330—62no—
105#328—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—44no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
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—
144C4×Dic3—42no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147C22×Q8—21no—
148C2×M16—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8#6711yescharacteristic
154C4⋊Q8#6711yescharacteristic
155#125—21no—
156#125—61no—
157#125—61no—
158#125—61no—
159#125—61no—
160C22⋊Q8—21no—
161C22⋊Q8—21no—
162C4×Q8—21no—
163C4×Q8—21no—
164#117—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4#6712yes—
171C2×C42—61no—
172C2×C42C22×D311yescharacteristic
173#107—24no—
174#107—62no—
175#106—24no—
176#106—62no—
177C4⋊C8—32no—
178C4⋊C8—61no—
179#103—62no—
180#103—62no—
181C22⋊C8—61no—
182C22⋊C8—61no—
183#97—61no—
184#97—61no—
185#97#6712yes—
186C4×C8—21no—
187#95—32no—
188#95—64no—
189#95—64no—
190#95—61no—
191C23×C3#12011yescharacteristic
192C3×Q8—42no—
193C3×Q8—81no—
194C3×Q8—82no—
195C3×Q8—82no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—44no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12711yescharacteristic
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C3×C8—24no—
217C3⋊C8—42no—
218C3⋊C8—81no—
219C2×Q8—21no—
220C2×Q8—21no—
221C2×Q8—44no—
222C2×Q8—41no—
223C2×Q8—41no—
224C22×C4—22no—
225C22×C4—62no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4D3×D411yescharacteristic
229C22×C4D3×D411yescharacteristic
230C22×C4C2×#6711yescharacteristic
231M16—24no—
232C2×C8—22no—
233C2×C8—62no—
234C2×C8—121no—
235C4⋊C4—21no—
236C4⋊C4—22no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C22⋊C4—41no—
254C42—41no—
255C42—122no—
256C42D3×D411yescharacteristic
257C42#23711yescharacteristic
258C42C2×#6711yescharacteristic
259C42C2×#6711yescharacteristic
260C22×C3—21no—
261C22×C3—21no—
262C22×C3#39211yescharacteristic
263C22×C3#45612yes—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—82no—
270C3×C4—81no—
271C3×C4—81no—
272C3×C4#45412yes—
273Dic3—82no—
274Dic3—81no—
275Dic3—81no—
276C23#93711yescharacteristic
277Q8—42no—
278Q8—81no—
279Q8—82no—
280Q8—82no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4#93711yescharacteristic
296C2×C4#93811yescharacteristic
297C2×C4#93911yescharacteristic
298C2×C4#93911yescharacteristic
299C2×C4#94011yescharacteristic
300C2×C4#95311yescharacteristic
301C8—24no—
302C8—122no—
303C8—241no—
304C2×C3—22no—
305C2×C3#200311yescharacteristic
306C2×C3#200511yescharacteristic
307C2×C3#200911yescharacteristic
308C22—21no—
309C22—21no—
310C22#509512yes—
311C22#522311yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—82no—
318C4—81no—
319C4—81no—
320C4—242no—
321C4—241no—
322C4—241no—
323C4#520112yes—
324C3#1010311yescharacteristic, complemented, Sylow
325C2—22no—
326C2#7179511yescharacteristic
327C2#7209611yescharacteristic
328C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3165D3
hypocenterC3#10103
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
324. C3#10103semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#833262
#832452
#718362
#717762
#698632
#676652
#101033