Finite Group Databasebeta

← #10697917

Subgroups — #10697917

Order 768.

504 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2862C211yesmaximal, characteristic
2#83224C211yesmaximal, characteristic
3#83223C211yesmaximal, characteristic
4#71775C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic
6#69396C211yesmaximal, characteristic
7#67611C211yesmaximal, characteristic
8#10296—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#391—22no—
13C3×#391—22no—
14C3×#382—22no—
15C3×C43C2211yescharacteristic
16#5094—22no—
17#5093—21no—
18#5093—21no—
19#5091—22no—
20#5080—21no—
21#5078—21no—
22#4971—22no—
23#4970—22no—
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4547—22no—
28#4547—22no—
29#4523—22no—
30#3230—31no—
31#3229—31no—
32#2862D311yescharacteristic
33#1994—31no—
34#1549—31no—
35#1546—31no—
36#1276—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D412yes—
50C2×C3×C4⋊C4D412yes—
51C2×C3×C42—21no—
52C2×C3×C42—21no—
53C2×C3×C42D412yes—
54C2×C3×C42C2311yescharacteristic
55#926—21no—
56#924—21no—
57#924—21no—
58C2×#208—21no—
59#870—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×#95—28no—
65#830—21no—
66#830—21no—
67#810—24no—
68#810—24no—
69#808—24no—
70#807—24no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D412yes—
74C4⋊(C3⋊C8)D412yes—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#533—31no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4⋊Q8C2×D311yescharacteristic
80#500—61no—
81#498—61no—
82#487—61no—
83#487—61no—
84#484—62no—
85#483—62no—
86#477—62no—
87#475—62no—
88#431—31no—
89#431—31no—
90#423—31no—
91#391—22no—
92#391—22no—
93#382—22no—
94C43C2×D311yescharacteristic
95#339—62no—
96#332—62no—
97#332—62no—
98C2×C3×Q8—44no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—42no—
104C2×C3×Q8—42no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C2×Dic6—21no—
112C2×Dic6—21no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C42—24no—
122C3×C42—22no—
123C3×C42—22no—
124C3×C42—42no—
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129#217—41no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134Dic3⋊C4—41no—
135Dic3⋊C4—41no—
136C4×Dic3—41no—
137#208—42no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—41no—
141C22×Q8—21no—
142C22×Q8—21no—
143C2×M16—61no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—31no—
153C4⋊Q8—31no—
154#125—61no—
155C22⋊Q8—61no—
156C22⋊Q8—61no—
157C4×Q8—61no—
158C4×Q8—61no—
159#117—61no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4#6712yes—
163C2×C4⋊C4#6712yes—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42#6712yes—
167C2×C42C22×D311yescharacteristic
168#107—64no—
169#106—64no—
170C4⋊C8—32no—
171C4⋊C8—32no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174#103—64no—
175#103—64no—
176C22⋊C8—61no—
177C22⋊C8—61no—
178C4×C8—61no—
179C4×C8—61no—
180#95—28no—
181C23×C3#12011yescharacteristic
182C3×Q8—82no—
183C3×Q8—82no—
184C3×Q8—82no—
185C3×Q8—81no—
186C3×Q8—81no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—24no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—44no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—44no—
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12711yescharacteristic
202C2×C3×C4#13512yes—
203C2×C3×C4#13712yes—
204C2×Dic3—41no—
205C2×Dic3—41no—
206Dic6—42no—
207Dic6—42no—
208C3⋊C8—42no—
209C3⋊C8—42no—
210C3⋊C8—81no—
211C2×Q8—44no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—61no—
219C2×Q8—61no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4C2×#6711yescharacteristic
224C22×C4C2×#6711yescharacteristic
225C22×C4C2×#6711yescharacteristic
226M16—122no—
227C2×C8—62no—
228C2×C8—62no—
229C2×C8—121no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C22⋊C4—121no—
245C42—24no—
246C42—22no—
247C42—22no—
248C42—42no—
249C42—121no—
250C42D3×D411yescharacteristic
251C42D3×D411yescharacteristic
252C42D3×D411yescharacteristic
253C42#23711yescharacteristic
254C22×C3—21no—
255C22×C3—21no—
256C22×C3#39211yescharacteristic
257C22×C3#45212yes—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—44no—
261C3×C4—82no—
262C3×C4—82no—
263C3×C4#45412yes—
264C3×C4#45612yes—
265C3×C4#46412yes—
266C3×C4#49511yescharacteristic
267C3×C4#49711yescharacteristic
268Dic3—81no—
269Dic3—81no—
270C23#95311yescharacteristic
271Q8—82no—
272Q8—82no—
273Q8—81no—
274Q8—81no—
275Q8—82no—
276Q8—122no—
277Q8—122no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4#93711yescharacteristic
290C2×C4#93711yescharacteristic
291C2×C4#93811yescharacteristic
292C2×C4#93911yescharacteristic
293C2×C4#93911yescharacteristic
294C2×C4#94011yescharacteristic
295C2×C4#95012yes—
296C2×C4#95112yes—
297C8—122no—
298C8—122no—
299C8—241no—
300C2×C3—22no—
301C2×C3#200311yescharacteristic
302C2×C3#200511yescharacteristic
303C2×C3#201511yescharacteristic
304C22—21no—
305C22—21no—
306C22#522311yescharacteristic
307C22#523912yes—
308C4—21no—
309C4—21no—
310C4—44no—
311C4—82no—
312C4—82no—
313C4—241no—
314C4—241no—
315C4#505412yes—
316C4#505911yescharacteristic
317C4#506011yescharacteristic
318C4#506212yes—
319C4#509512yes—
320C3#1029611yescharacteristic, complemented, Sylow
321C2—22no—
322C2#7172911yescharacteristic
323C2#7173011yescharacteristic
324C2#7179511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2862C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2862D3
hypocenterC3#10296
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
320. C3#10296semidirect

Maximal subgroups

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

subgroupindex
C3×#28622
#832242
#832232
#717752
#717732
#693962
#676112
#102963