Finite Group Databasebeta

← #10697079

Subgroups — #10697079

Order 768.

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

normal subgroups only

This group has 492 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
1C2×C3×#498C211yesmaximal, characteristic
2#83223C211yesmaximal, characteristic
3#83207C211yesmaximal, characteristic
4#71789C211yesmaximal, characteristic
5#71774C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic
8#10167—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#498—22no—
13C3×#498—22no—
14C2×C3×#106—22no—
15C2×C3×C4×C8C2211yescharacteristic
16C2×#836—22no—
17#5094—22no—
18#5093—21no—
19#5081—22no—
20#5080—21no—
21#5080—21no—
22#5078—21no—
23#4971—22no—
24#4964—22no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27C4×#208C2211yescharacteristic
28#4546—22no—
29#4456—22no—
30#3229—31no—
31#3165—31no—
32C2×#498D311yescharacteristic
33#1966—31no—
34#1548—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×C8—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—
47C3×C4⋊Q8D412yes—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D412yes—
52C2×C3×C42C2311yescharacteristic
53#926—21no—
54#924—21no—
55#924—21no—
56C2×#208—21no—
57C2×#208D412yes—
58#870—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×#106—28no—
64C3×C4×C8—21no—
65C3×C4×C8D412yes—
66#836—44no—
67#830—21no—
68#810—24no—
69#808—24no—
70#808—24no—
71#807—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75#803—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#533—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×C4⋊Q8C2×D311yescharacteristic
81#501—62no—
82#500—61no—
83#498—22no—
84#498—22no—
85#498—61no—
86#498—61no—
87#487—61no—
88#483—62no—
89#475—62no—
90#431—31no—
91#428—62no—
92C2×#106—22no—
93#423—31no—
94C2×#101—62no—
95C4×M16—31no—
96C2×C4×C8C2×D311yescharacteristic
97#333—62no—
98Q8⋊C8—62no—
99C2×C3×Q8—22no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—42no—
104C2×C3×Q8—42no—
105C2×C3×Q8—82no—
106C22×C3×C4—22no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×C3×C8—24no—
114C2×C3×C8—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127#217—41no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134C4×Dic3—41no—
135#208—22no—
136#208—42no—
137#208SD1614yes—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—41no—
141C22×Q8—21no—
142C22×Q8—21no—
143C2×M16—32no—
144C2×M16—61no—
145C22×C8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—31no—
154C4⋊Q8—31no—
155C4⋊Q8#6712yes—
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—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4#6712yes—
166C2×C42C22×D311yescharacteristic
167#107—64no—
168#106—28no—
169#106—64no—
170#106—64no—
171C4⋊C8—32no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174#103—64no—
175#101—124no—
176C22⋊C8—61no—
177#97—61no—
178C4×C8—21no—
179C4×C8—61no—
180C4×C8—61no—
181C4×C8#6712yes—
182C23×C3#12111yescharacteristic
183C3×Q8—82no—
184C3×Q8—82no—
185C3×Q8—82no—
186C3×Q8—81no—
187C3×Q8—81no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—44no—
193C2×C3×C4—82no—
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12711yescharacteristic
200C2×C3×C4C2×SD1612yes—
201C2×C3×C4#13712yes—
202C2×Dic3—41no—
203C2×Dic3—41no—
204Dic6—42no—
205Dic6—42no—
206C3×C8—24no—
207C3⋊C8—24no—
208C3⋊C8—42no—
209C3⋊C8—81no—
210C2×Q8—22no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—61no—
217C2×Q8—61no—
218C2×Q8—82no—
219C22×C4—22no—
220C22×C4—41no—
221C22×C4D3×D411yescharacteristic
222C22×C4#23711yescharacteristic
223C22×C4C2×#6711yescharacteristic
224M16—34no—
225M16—62no—
226M16—122no—
227C2×C8—24no—
228C2×C8—22no—
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—121no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C22⋊C4—121no—
247C42—121no—
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—82no—
259C3×C4—82no—
260C3×C4#44812yes—
261C3×C4#45612yes—
262C3×C4#47112yes—
263C3×C4#49211yescharacteristic
264C3×C4#49711yescharacteristic
265Dic3—81no—
266Dic3—81no—
267C23#93911yescharacteristic
268Q8—82no—
269Q8—81no—
270Q8—82no—
271Q8—81no—
272Q8—82no—
273Q8—122no—
274Q8—122no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4D3×SD1612yes—
284C2×C4#91512yes—
285C2×C4#93711yescharacteristic
286C2×C4#93711yescharacteristic
287C2×C4#93811yescharacteristic
288C2×C4#93911yescharacteristic
289C2×C4#94011yescharacteristic
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#516212yes—
302C22#522311yescharacteristic
303C4—21no—
304C4—21no—
305C4—82no—
306C4—82no—
307C4—241no—
308C4—241no—
309C4#509512yes—
310C4#516711yescharacteristic
311C4#517012yes—
312C4#517511yescharacteristic
313C4#517712yes—
314C3#1016711yescharacteristic, complemented, Sylow
315C2—22no—
316C2#7179511yescharacteristic
317C2#7205711yescharacteristic
318C2#7205911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#498C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenterC2×#498D3
hypocenterC3#10167
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#498

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
C2×C3×#4982
#832232
#832072
#717892
#717742
#697382
#676562
#101673