Finite Group Databasebeta

← #10697895

Subgroups — #10697895

Order 768.

588 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 109 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×#3209C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic
3#83253C211yesmaximal, characteristic
4#71839C211yesmaximal, characteristic
5#71774C211yesmaximal, characteristic
6#70780C211yesmaximal, characteristic
7#67736C211yesmaximal, characteristic
8#10293—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#944—22no—
12C2×#869C2211yescharacteristic
13C3×#494—21no—
14C3×#487—21no—
15C3×#462—22no—
16C3×#451—22no—
17C3×#431C2211yescharacteristic
18#5079—21no—
19#5078—21no—
20#5044—22no—
21#5021—22no—
22#4975—22no—
23#4975—22no—
24C42×Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4546—22no—
28#4532—22no—
29#4482—22no—
30#3230—31no—
31#3209D311yescharacteristic
32#2868—31no—
33#1994—31no—
34#1549—31no—
35#1548—31no—
36#1276—31no—
37C22×C3×Q8D412yes—
38C22×Dic6—21no—
39C2×C3×Q16—24no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55#944—48no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D412yes—
58C2×Dic3⋊C4—28no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3D412yes—
62C2×#208—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67C3×C4⋊C8D412yes—
68C3×#103—24no—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71#835—24no—
72#830—21no—
73#807—24no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78#533C2×D311yescharacteristic
79C2×C4⋊Q8—31no—
80C2×C4⋊Q8—62no—
81C2×C4⋊Q8C2×D311yescharacteristic
82#500—61no—
83#494—21no—
84#487—21no—
85#487—61no—
86#484—62no—
87#477—62no—
88#462—22no—
89#451—22no—
90#431—31no—
91#431C2×D311yescharacteristic
92#423—31no—
93#384—62no—
94#384—62no—
95C43—31no—
96#339—62no—
97#333—62no—
98#332—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8—22no—
101C2×C3×Q8—21no—
102C2×C3×Q8—22no—
103C2×C3×Q8—44no—
104C22×C3×C4—22no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—22no—
109C22×Dic3—22no—
110C2×Dic6—44no—
111C2×Dic6—41no—
112C2×Dic6—41no—
113C3×Q16—84no—
114C2×C3×C8—22no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C42—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131C4⋊Dic3—22no—
132C4⋊Dic3—22no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135Dic3⋊C4—416no—
136C4×Dic3—24no—
137C4×Dic3—24no—
138C4×Dic3—22no—
139C4×Dic3Q1614yescomplemented
140#208—42no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—41no—
143C22×Q8—61no—
144C22×Q8#6712yes—
145C2×Q16—24no—
146C2×M16—61no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—128no—
156C4⋊Q8#6711yescharacteristic
157C4⋊Q8#6711yescharacteristic
158#125—21no—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163#117—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—32no—
167C2×C4⋊C4—68no—
168C2×C4⋊C4—61no—
169C2×C42—32no—
170C2×C42—61no—
171C2×C42—61no—
172C2×C42C22×D311yescharacteristic
173#107—64no—
174C4⋊C8—32no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177C4⋊C8#6712yes—
178#103—24no—
179#103—64no—
180C22⋊C8—21no—
181C22⋊C8—61no—
182C4×C8—21no—
183C4×C8—61no—
184C23×C3#12011yescharacteristic
185C3×Q8—42no—
186C3×Q8—44no—
187C3×Q8—42no—
188C3×Q8—82no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—44no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12711yescharacteristic
201C2×C3×C4C2×Q1612yes—
202C2×C3×C4#13712yes—
203C2×Dic3—24no—
204C2×Dic3—24no—
205C2×Dic3—24no—
206C2×Dic3—44no—
207C2×Dic3—42no—
208Dic6—82no—
209Dic6—82no—
210C3×C8—42no—
211C3⋊C8—42no—
212C3⋊C8—81no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—21no—
217C2×Q8—44no—
218C2×Q8—124no—
219C2×Q8—121no—
220C2×Q8—121no—
221C22×C4—22no—
222C22×C4—62no—
223C22×C4—62no—
224C22×C4D3×D411yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4C2×#6711yescharacteristic
227Q16—84no—
228M16—122no—
229C2×C8—22no—
230C2×C8—62no—
231C2×C8—121no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—62no—
243C4⋊C4—62no—
244C4⋊C4—1216no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C22⋊C4—41no—
248C42—34no—
249C42—41no—
250C42—64no—
251C42—64no—
252C42—62no—
253C42D3×D411yescharacteristic
254C42#23711yescharacteristic
255C42C2×#6711yescharacteristic
256C42C2×#6711yescharacteristic
257C22×C3—21no—
258C22×C3—21no—
259C22×C3#39211yescharacteristic
260C22×C3#45112yes—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—82no—
264C3×C4—81no—
265C3×C4—81no—
266C3×C4#45312yes—
267C3×C4#45612yes—
268C3×C4#46212yes—
269C3×C4#49411yescharacteristic
270C3×C4#49711yescharacteristic
271Dic3—28no—
272Dic3—82no—
273C23#93711yescharacteristic
274Q8—42no—
275Q8—42no—
276Q8—44no—
277Q8—82no—
278Q8—242no—
279Q8—242no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—44no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—64no—
287C2×C4—64no—
288C2×C4—64no—
289C2×C4—124no—
290C2×C4—122no—
291C2×C4D3×Q1612yes—
292C2×C4#91812yes—
293C2×C4#93711yescharacteristic
294C2×C4#93811yescharacteristic
295C2×C4#93911yescharacteristic
296C2×C4#93911yescharacteristic
297C2×C4#94011yescharacteristic
298C2×C4#95311yescharacteristic
299C8—42no—
300C8—122no—
301C8—241no—
302C2×C3—22no—
303C2×C3#200211yescharacteristic
304C2×C3#200511yescharacteristic
305C2×C3#201611yescharacteristic
306C22—21no—
307C22—21no—
308C22#518612yes—
309C22#522311yescharacteristic
310C4—21no—
311C4—21no—
312C4—68no—
313C4—81no—
314C4—82no—
315C4—81no—
316C4—242no—
317C4#518212yes—
318C4#518311yescharacteristic
319C4#519011yescharacteristic
320C4#520512yes—
321C4#524112yes—
322C3#1029311yescharacteristic, complemented, Sylow
323C2—22no—
324C2#7185611yescharacteristic
325C2#7207311yescharacteristic
326C2#7210811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3209C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3209D3
hypocenterC3#10293
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
139. C4×Dic3Q16semidirect
322. C3#10293semidirect

Maximal subgroups

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

subgroupindex
C3×#32092
#833242
#832532
#718392
#717742
#707802
#677362
#102933