Finite Group Databasebeta

← #10697933

Subgroups — #10697933

Order 768.

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

normal subgroups only

This group has 576 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×#3229C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic
3#83253C211yesmaximal, characteristic
4#71839C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic
6#70778C211yesmaximal, characteristic
7#67736C211yesmaximal, characteristic
8#10299—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#944—22no—
12C2×#869C2211yescharacteristic
13C3×#498—21no—
14C3×#487—21no—
15C3×#483—22no—
16C3×#475—22no—
17C3×#431C2211yescharacteristic
18#5080—21no—
19#5079—21no—
20#5043—22no—
21#5020—22no—
22#4975—22no—
23#4975—22no—
24C42×Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4547—22no—
28#4532—22no—
29#4486—22no—
30#3229—31no—
31#3229D311yescharacteristic
32#2868—31no—
33#1994—31no—
34#1546—31no—
35#1546—31no—
36#1276—31no—
37C22×C3×Q8—21no—
38C22×Dic6—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C42C2311yescharacteristic
54#944—44no—
55#944—44no—
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×#106—24no—
68C3×C4⋊C8D412yes—
69C3×#103—24no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72#835—24no—
73#830—21no—
74#808—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79#533C2×D311yescharacteristic
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8—62no—
82C2×C4⋊Q8C2×D311yescharacteristic
83#498—21no—
84#498—61no—
85#487—21no—
86#487—61no—
87#483—22no—
88#483—62no—
89#475—22no—
90#475—62no—
91#431—31no—
92#431C2×D311yescharacteristic
93#423—31no—
94#384—62no—
95#384—62no—
96C43—31no—
97#339—62no—
98#332—62no—
99#332—62no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C22×C3×C4—22no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—22no—
111C22×Dic3—22no—
112C2×Dic6—41no—
113C2×Dic6—41no—
114C2×Dic6—42no—
115C2×Dic6—42no—
116C2×C3×C8—22no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C42—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C4⋊Dic3—22no—
134C4⋊Dic3—22no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137Dic3⋊C4—416no—
138C4×Dic3—24no—
139C4×Dic3—24no—
140C4×Dic3—22no—
141C4×Dic3SD1614yes—
142#208—42no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C22×Q8—21no—
146C22×Q8—61no—
147C2×M16—61no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8—124no—
157C4⋊Q8—124no—
158C4⋊Q8#6711yescharacteristic
159C4⋊Q8#6711yescharacteristic
160#125—21no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165#117—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—32no—
168C2×C4⋊C4—68no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4#6712yes—
171C2×C42—32no—
172C2×C42—61no—
173C2×C42—61no—
174C2×C42C22×D311yescharacteristic
175#106—24no—
176#106—64no—
177C4⋊C8—32no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180C4⋊C8#6712yes—
181#103—24no—
182#103—64no—
183C22⋊C8—21no—
184C22⋊C8—61no—
185C4×C8—21no—
186C4×C8—61no—
187C23×C3#12011yescharacteristic
188C3×Q8—42no—
189C3×Q8—42no—
190C3×Q8—82no—
191C3×Q8—81no—
192C3×Q8—81no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—44no—
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12711yescharacteristic
205C2×C3×C4C2×SD1612yes—
206C2×C3×C4AGL(1,ℤ/8)12yes—
207C2×Dic3—24no—
208C2×Dic3—24no—
209C2×Dic3—24no—
210C2×Dic3—42no—
211C2×Dic3—44no—
212Dic6—82no—
213Dic6—81no—
214Dic6—81no—
215C3×C8—42no—
216C3⋊C8—42no—
217C3⋊C8—81no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—122no—
225C2×Q8—122no—
226C2×Q8—121no—
227C2×Q8—121no—
228C22×C4—22no—
229C22×C4—62no—
230C22×C4—62no—
231C22×C4D3×D411yescharacteristic
232C22×C4D3×D411yescharacteristic
233C22×C4C2×#6711yescharacteristic
234M16—122no—
235C2×C8—22no—
236C2×C8—62no—
237C2×C8—121no—
238C4⋊C4—21no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—22no—
243C4⋊C4—21no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—1216no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C22⋊C4—41no—
254C42—34no—
255C42—41no—
256C42—64no—
257C42—64no—
258C42—62no—
259C42D3×D411yescharacteristic
260C42#23711yescharacteristic
261C42C2×#6711yescharacteristic
262C42C2×#6711yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#39211yescharacteristic
266C22×C3#45012yes—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—81no—
270C3×C4—81no—
271C3×C4—82no—
272C3×C4#45612yes—
273C3×C4#45612yes—
274C3×C4#46012yes—
275C3×C4#49211yescharacteristic
276C3×C4#49611yescharacteristic
277Dic3—28no—
278Dic3—82no—
279C23#93711yescharacteristic
280Q8—42no—
281Q8—42no—
282Q8—82no—
283Q8—81no—
284Q8—81no—
285Q8—242no—
286Q8—241no—
287Q8—241no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—44no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—64no—
295C2×C4—64no—
296C2×C4—64no—
297C2×C4—124no—
298C2×C4—122no—
299C2×C4D3×SD1612yes—
300C2×C4#91412yes—
301C2×C4#93711yescharacteristic
302C2×C4#93811yescharacteristic
303C2×C4#93911yescharacteristic
304C2×C4#93911yescharacteristic
305C2×C4#94011yescharacteristic
306C2×C4#95311yescharacteristic
307C8—42no—
308C8—122no—
309C8—241no—
310C2×C3—22no—
311C2×C3#200511yescharacteristic
312C2×C3#200511yescharacteristic
313C2×C3#201411yescharacteristic
314C22—21no—
315C22—21no—
316C22#516912yes—
317C22#522311yescharacteristic
318C4—21no—
319C4—21no—
320C4—68no—
321C4—82no—
322C4—81no—
323C4—81no—
324C4—242no—
325C4#516312yes—
326C4#517511yescharacteristic
327C4#517611yescharacteristic
328C4#520512yes—
329C4#524612yes—
330C3#1029911yescharacteristic, complemented, Sylow
331C2—22no—
332C2#7186811yescharacteristic
333C2#7205611yescharacteristic
334C2#7210811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3229C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3229D3
hypocenterC3#10299
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
330. C3#10299semidirect

Maximal subgroups

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

subgroupindex
C3×#32292
#833232
#832532
#718392
#717732
#707782
#677362
#102993