Finite Group Databasebeta

← #10697135

Subgroups — #10697135

Order 768.

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

normal subgroups only

This group has 492 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
1C2×C3×#500C211yesmaximal, characteristic
2#83223C211yesmaximal, characteristic
3#83198C211yesmaximal, characteristic
4#71789C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic
8#10175—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#500—22no—
13C3×#500—22no—
14C2×C3×#107—22no—
15C2×C3×C4×C8C2211yescharacteristic
16C2×#836—22no—
17#5225—22no—
18#5094—22no—
19#5093—21no—
20#5082—22no—
21#5080—21no—
22#5079—21no—
23#5079—21no—
24#4971—22no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27C4×#208C2211yescharacteristic
28#4547—22no—
29#4456—22no—
30#3229—31no—
31C2×#500D311yescharacteristic
32#3139—31no—
33#1966—31no—
34#1546—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×#107—28no—
64C3×C4×C8—21no—
65C3×C4×C8D412yes—
66#836—44no—
67#835—24no—
68#835—24no—
69#830—21no—
70#810—24no—
71#808—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#500—22no—
82#500—22no—
83#498—61no—
84#491—62no—
85#487—61no—
86#487—61no—
87#487—61no—
88#483—62no—
89#475—62no—
90#431—31no—
91C2×#107—22no—
92#423—31no—
93#419—62no—
94C2×#101—62no—
95C4×M16—31no—
96C2×C4×C8C2×D311yescharacteristic
97#332—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—42no—
106C2×C3×Q8—42no—
107C22×C3×C4—22no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×C3×C8—24no—
115C2×C3×C8—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128#217—41no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133Dic3⋊C4—41no—
134Dic3⋊C4—41no—
135C4×Dic3—41no—
136#208—22no—
137#208—22no—
138#208—42no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—41no—
142C22×Q8—21no—
143C22×Q8—21no—
144C2×M16—32no—
145C2×M16—61no—
146C22×C8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—31no—
155C4⋊Q8—31no—
156C4⋊Q8#6712yes—
157#125—61no—
158C22⋊Q8—61no—
159C22⋊Q8—61no—
160C4×Q8—61no—
161C4×Q8—61no—
162#117—61no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4#6712yes—
167C2×C42C22×D311yescharacteristic
168#107—28no—
169#106—64no—
170C4⋊C8—32no—
171C4⋊C8—61no—
172C4⋊C8—61no—
173#103—64no—
174#103—64no—
175#103—64no—
176#101—124no—
177C22⋊C8—61no—
178#97—61no—
179C4×C8—21no—
180C4×C8—61no—
181C4×C8—61no—
182C4×C8#6712yes—
183C23×C3#12111yescharacteristic
184C3×Q8—44no—
185C3×Q8—82no—
186C3×Q8—81no—
187C3×Q8—81no—
188C3×Q8—81no—
189C3×Q8—81no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—44no—
195C2×C3×C4—82no—
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12711yescharacteristic
202C2×C3×C4#13512yes—
203C2×C3×C4AGL(1,ℤ/8)12yes—
204C2×Dic3—41no—
205C2×Dic3—41no—
206Dic6—42no—
207Dic6—42no—
208C3×C8—24no—
209C3⋊C8—42no—
210C3⋊C8—42no—
211C3⋊C8—81no—
212C2×Q8—22no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—61no—
221C2×Q8—61no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4D3×D411yescharacteristic
225C22×C4#23711yescharacteristic
226C22×C4C2×#6711yescharacteristic
227M16—62no—
228M16—62no—
229M16—122no—
230C2×C8—24no—
231C2×C8—22no—
232C2×C8—62no—
233C2×C8—62no—
234C2×C8—121no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C22⋊C4—121no—
250C42—121no—
251C42D3×D411yescharacteristic
252C42D3×D411yescharacteristic
253C42C2×#6711yescharacteristic
254C42C2×#6711yescharacteristic
255C22×C3—21no—
256C22×C3—21no—
257C22×C3#39211yescharacteristic
258C22×C3#46312yes—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—82no—
262C3×C4—82no—
263C3×C4#44912yes—
264C3×C4#45612yes—
265C3×C4#47312yes—
266C3×C4#49511yescharacteristic
267C3×C4#49611yescharacteristic
268Dic3—81no—
269Dic3—81no—
270C23#93911yescharacteristic
271Q8—44no—
272Q8—81no—
273Q8—81no—
274Q8—81no—
275Q8—81no—
276Q8—82no—
277Q8—122no—
278Q8—122no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—82no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4#91112yes—
288C2×C4#91212yes—
289C2×C4#93711yescharacteristic
290C2×C4#93711yescharacteristic
291C2×C4#93811yescharacteristic
292C2×C4#93911yescharacteristic
293C2×C4#94011yescharacteristic
294C2×C4#95311yescharacteristic
295C8—24no—
296C8—122no—
297C8—122no—
298C8—241no—
299C2×C3—22no—
300C2×C3#200511yescharacteristic
301C2×C3#201211yescharacteristic
302C2×C3#202011yescharacteristic
303C22—21no—
304C22—21no—
305C22#515312yes—
306C22#522311yescharacteristic
307C4—21no—
308C4—21no—
309C4—82no—
310C4—82no—
311C4—241no—
312C4—241no—
313C4#509512yes—
314C4#515411yescharacteristic
315C4#515511yescharacteristic
316C4#515812yes—
317C4#519212yes—
318C3#1017511yescharacteristic, complemented, Sylow
319C2—22no—
320C2#7179511yescharacteristic
321C2#7204511yescharacteristic
322C2#7207411yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#500C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenterC2×#500D3
hypocenterC3#10175
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
318. C3#10175semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5002
#832232
#831982
#717892
#717732
#697382
#676562
#101753