Finite Group Databasebeta

← #10697083

Subgroups — #10697083

Order 768.

492 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#83223C211yesmaximal, characteristic
2C2×#5080C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic
4C3×#1966C211yesmaximal, characteristic
5#70507C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic
7#67667C211yesmaximal, characteristic
8#10167—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×#101—22no—
14C3×C4×M16C2211yescharacteristic
15#5121—22no—
16#5094—22no—
17#5093—21no—
18#5080—22no—
19#5080—21no—
20#5080—22no—
21#4971—22no—
22C2×#808—22no—
23#4925C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25#4702—22no—
26#4682—21no—
27#4680—21no—
28#4484—22no—
29#4481—22no—
30#3229—31no—
31#3165—31no—
32C2×#498—31no—
33#1966D311yescharacteristic
34#1548—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C2×C3×M16D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D412yes—
47C2×C3×C42C2311yescharacteristic
48#926—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51#924—21no—
52#924—21no—
53C22×C3⋊C8—21no—
54#870—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869D411yescharacteristic
59#869D411yescharacteristic
60#869D412yes—
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×#101—44no—
66C3×#97—21no—
67C3×C4×C8—21no—
68#830—21no—
69#818—24no—
70#817—24no—
71#810—24no—
72#808—28no—
73#808—24no—
74C4⋊(C3⋊C8)D412yes—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8D412yes—
78#533—31no—
79C2×C4⋊Q8—31no—
80C2×C4⋊Q8C2×D311yescharacteristic
81#501—62no—
82#500—61no—
83#498—61no—
84#498—62no—
85#498—62no—
86#498—61no—
87#487—61no—
88#483—62no—
89#475—62no—
90#431—31no—
91#428—62no—
92C2×#106—62no—
93#423C2×D311yescharacteristic
94C2×#101—22no—
95C4×M16C2×D311yescharacteristic
96C2×C4×C8—31no—
97#333—62no—
98Q8⋊C8—62no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—42no—
102C2×C3×Q8—42no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—41no—
108C2×Dic6—21no—
109C2×Dic6—21no—
110C2×Dic6—22no—
111C2×Dic6—41no—
112C2×Dic6—82no—
113C3×M16—22no—
114C3×M16—42no—
115C3×M16SD1614yes—
116C2×C3×C8—22no—
117C2×C3×C8—41no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126#217—41no—
127C4⋊Dic3—21no—
128C4⋊Dic3—22no—
129C4⋊Dic3—21no—
130C4⋊Dic3—22no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137C4×Dic3—41no—
138C2×C3⋊C8—24no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C22×Q8—21no—
142C22×Q8—61no—
143C2×M16—21no—
144C2×M16#6712yes—
145C22×C8—61no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—32no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156#125—61no—
157C22⋊Q8—61no—
158C22⋊Q8—61no—
159C4×Q8—61no—
160C4×Q8—61no—
161#117—61no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4D1212yes—
166C2×C42C22×D311yescharacteristic
167#107—64no—
168#106—68no—
169#106—64no—
170#106—64no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—32no—
174#103—64no—
175#101—44no—
176C22⋊C8—61no—
177#97—21no—
178C4×C8—21no—
179C4×C8—32no—
180C4×C8—61no—
181C4×C8—61no—
182C23×C3#12111yescharacteristic
183C3×Q8—82no—
184C3×Q8—81no—
185C3×Q8—81no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—42no—
189C2×C3×C4—44no—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12711yescharacteristic
196C2×C3×C4C2×SD1612yes—
197C2×C3×C4#13712yes—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—42no—
201C2×Dic3—82no—
202Dic6—42no—
203Dic6—42no—
204Dic6—82no—
205Dic6—82no—
206C3×C8—24no—
207C3×C8—81no—
208C3⋊C8—24no—
209C3⋊C8—42no—
210C2×Q8—42no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—62no—
217C2×Q8—121no—
218C2×Q8—242no—
219C22×C4—22no—
220C22×C4—121no—
221C22×C4C2×D1211yescharacteristic
222C22×C4#23511yescharacteristic
223C22×C4C2×#6711yescharacteristic
224M16—22no—
225M16—42no—
226M16#21414yes—
227C2×C8—22no—
228C2×C8—41no—
229C2×C8—64no—
230C2×C8—62no—
231C2×C8—62no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—62no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—62no—
241C4⋊C4—61no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C22⋊C4—121no—
247C42—121no—
248C42D3×D411yescharacteristic
249C42D3×D411yescharacteristic
250C42D3×D411yescharacteristic
251C42D3×D411yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3#39211yescharacteristic
255C22×C3#46112yes—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—82no—
259C3×C4#44812yes—
260C3×C4#45612yes—
261C3×C4#47112yes—
262C3×C4#49211yescharacteristic
263C3×C4#49711yescharacteristic
264Dic3—81no—
265Dic3—81no—
266Dic3—82no—
267C23#93011yescharacteristic
268Q8—82no—
269Q8—81no—
270Q8—81no—
271Q8—122no—
272Q8—122no—
273Q8—242no—
274Q8—242no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—122no—
282C2×C4—242no—
283C2×C4#88211yescharacteristic
284C2×C4#88211yescharacteristic
285C2×C4#88411yescharacteristic
286C2×C4#88411yescharacteristic
287C2×C4C2×#21412yes—
288C2×C4#93711yescharacteristic
289C2×C4#94011yescharacteristic
290C2×C4#95112yes—
291C8—24no—
292C8—64no—
293C8—81no—
294C8—122no—
295C2×C3—22no—
296C2×C3#200511yescharacteristic
297C2×C3#201311yescharacteristic
298C2×C3#201911yescharacteristic
299C22—21no—
300C22—21no—
301C22#496011yescharacteristic
302C22#501912yes—
303C4—21no—
304C4—21no—
305C4—82no—
306C4—242no—
307C4—241no—
308C4—241no—
309C4#475212yes—
310C4#475412yes—
311C4#504012yes—
312C4#506011yescharacteristic
313C4#507611yescharacteristic
314C3#1016711yescharacteristic, complemented, Sylow
315C2—22no—
316C2#6966111yescharacteristic
317C2#7011711yescharacteristic
318C2#7011911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1966C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1966D3
hypocenterC3#10167
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1966

Complements

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

NH ≅ G/Ntype
314. C3#10167semidirect

Maximal subgroups

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

subgroupindex
#832232
C2×#50802
#785062
C3×#19662
#705072
#705012
#676672
#101673