Finite Group Databasebeta

← #10697080

Subgroups — #10697080

Order 768.

492 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 113 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#83323C211yesmaximal, characteristic
3C2×#5080C211yesmaximal, characteristic
4#71826C211yesmaximal, characteristic
5#71774C211yesmaximal, characteristic
6#69877C211yesmaximal, characteristic
7#67667C211yesmaximal, characteristic
8#10167—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#501—22no—
13C3×#500—21no—
14C3×#498—21no—
15C3×#428—22no—
16C3×C4×M16C2211yescharacteristic
17C2×#824—22no—
18#5080—22no—
19#5080—21no—
20#5080—22no—
21#5079—21no—
22#5043—22no—
23#5020—22no—
24C2×#808—22no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27C2×C4×C3⋊C8C2211yescharacteristic
28#4546—22no—
29#4535—22no—
30#3229—31no—
31#3165D311yescharacteristic
32C2×#498—31no—
33#1966—31no—
34#1548—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×Dic6—21no—
39C2×C3×M16D412yes—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D412yes—
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
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D412yes—
57C2×#208—21no—
58C22×C3⋊C8—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63C3×#107—24no—
64C3×#106—24no—
65C3×#97—21no—
66C3×C4×C8—21no—
67#835—24no—
68#830—21no—
69#824—44no—
70#808—28no—
71#808—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8D412yes—
78#533C2×D311yescharacteristic
79C2×C4⋊Q8—31no—
80C2×C4⋊Q8C2×D311yescharacteristic
81#501—22no—
82#500—21no—
83#498—21no—
84#498—62no—
85#498—62no—
86#498—61no—
87#487—61no—
88#483—62no—
89#475—62no—
90#431—31no—
91#428—22no—
92C2×#106—62no—
93#423—31no—
94C2×#101—62no—
95C4×M16C2×D311yescharacteristic
96C2×C4×C8—31no—
97#333—62no—
98Q8⋊C8—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—22no—
102C2×C3×Q8—41no—
103C2×C3×Q8—82no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×Dic3—22no—
109C2×Dic6—41no—
110C2×Dic6—41no—
111C2×Dic6—42no—
112C2×Dic6—42no—
113C3×M16—22no—
114C3×M16SD1614yes—
115C2×C3×C8—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C42—41no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C4⋊Dic3—22no—
133C4⋊Dic3—22no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136#208—42no—
137C2×C3⋊C8—24no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—41no—
141C22×Q8—21no—
142C22×Q8—61no—
143C2×M16—61no—
144C2×M16#6712yes—
145C22×C8—61no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8#6712yes—
154C4⋊Q8#6711yescharacteristic
155C4⋊Q8#6711yescharacteristic
156#125—21no—
157C22⋊Q8—21no—
158C22⋊Q8—21no—
159C4×Q8—21no—
160C4×Q8—21no—
161#117—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—32no—
165C2×C4⋊C4—61no—
166C2×C42C22×D311yescharacteristic
167#107—24no—
168#106—24no—
169#106—68no—
170#106—64no—
171C4⋊C8—32no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174#103—64no—
175#101—124no—
176C22⋊C8—61no—
177#97—21no—
178C4×C8—21no—
179C4×C8—32no—
180C4×C8—61no—
181C4×C8—61no—
182C23×C3#12111yescharacteristic
183C3×Q8—42no—
184C3×Q8—42no—
185C3×Q8—82no—
186C3×Q8—82no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—82no—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12711yescharacteristic
199C2×C3×C4C2×SD1612yes—
200C2×C3×C4#13712yes—
201C2×Dic3—42no—
202C2×Dic3—44no—
203Dic6—82no—
204Dic6—81no—
205Dic6—81no—
206C3×C8—24no—
207C3⋊C8—24no—
208C3⋊C8—42no—
209C3⋊C8—81no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—82no—
215C2×Q8—121no—
216C2×Q8—122no—
217C2×Q8—122no—
218C2×Q8—121no—
219C22×C4—41no—
220C22×C4—62no—
221C22×C4D3×D411yescharacteristic
222C22×C4#23711yescharacteristic
223C22×C4C2×#6711yescharacteristic
224M16—22no—
225M16—122no—
226M16#21514yes—
227C2×C8—22no—
228C2×C8—64no—
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—121no—
232C4⋊C4—22no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—22no—
237C4⋊C4—21no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—62no—
243C4⋊C4—62no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C22⋊C4—41no—
247C42—41no—
248C42D3×D411yescharacteristic
249C42D3×D411yescharacteristic
250C42C2×#6711yescharacteristic
251C42C2×#6711yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3#39211yescharacteristic
255C22×C3#46112yes—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—81no—
259C3×C4—82no—
260C3×C4—81no—
261C3×C4#44812yes—
262C3×C4#45612yes—
263C3×C4#47112yes—
264C3×C4#49211yescharacteristic
265C3×C4#49711yescharacteristic
266Dic3—82no—
267C23#93811yescharacteristic
268Q8—42no—
269Q8—42no—
270Q8—82no—
271Q8—82no—
272Q8—242no—
273Q8—241no—
274Q8—241no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—122no—
282C2×C4—124no—
283C2×C4#93711yescharacteristic
284C2×C4#93711yescharacteristic
285C2×C4#93911yescharacteristic
286C2×C4#93911yescharacteristic
287C2×C4#94011yescharacteristic
288C2×C4C2×#21512yes—
289C2×C4#94212yes—
290C2×C4#95311yescharacteristic
291C8—24no—
292C8—64no—
293C8—122no—
294C8—241no—
295C2×C3—22no—
296C2×C3#200511yescharacteristic
297C2×C3#201311yescharacteristic
298C2×C3#201911yescharacteristic
299C22—21no—
300C22—21no—
301C22#508312yes—
302C22#522311yescharacteristic
303C4—21no—
304C4—21no—
305C4—82no—
306C4—81no—
307C4—81no—
308C4—242no—
309C4#508511yescharacteristic
310C4#508611yescharacteristic
311C4#520012yes—
312C4#520512yes—
313C4#522712yes—
314C3#1016711yescharacteristic, complemented, Sylow
315C2—22no—
316C2#7179111yescharacteristic
317C2#7209711yescharacteristic
318C2#7210811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3165D3
hypocenterC3#10167
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
314. C3#10167semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#833232
C2×#50802
#718262
#717742
#698772
#676672
#101673