Finite Group Databasebeta

← #10697009

Subgroups — #10697009

Order 768.

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

normal subgroups only

This group has 476 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×#487C211yesmaximal, characteristic
2#83224C211yesmaximal, characteristic
3#83207C211yesmaximal, characteristic
4#71790C211yesmaximal, characteristic
5#71775C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic
8#10157—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#487—22no—
13C3×#487—22no—
14C2×C3×#103—22no—
15C2×C3×C4×C8C2211yescharacteristic
16#5093—21no—
17#5091—22no—
18#5081—22no—
19#5080—21no—
20#5078—21no—
21#5078—21no—
22#4970—22no—
23#4964—22no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4547—22no—
28#4530—22no—
29#4456—22no—
30#3230—31no—
31#3165—31no—
32C2×#487D311yescharacteristic
33#1997—31no—
34#1549—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×#103—28no—
64C3×C4×C8—21no—
65C3×C4×C8D412yes—
66#830—21no—
67#810—24no—
68#808—24no—
69#807—24no—
70#807—24no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D412yes—
74#803—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#533—31no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4⋊Q8C2×D311yescharacteristic
80#501—62no—
81#500—61no—
82#500—61no—
83#498—61no—
84#487—22no—
85#487—22no—
86#487—61no—
87#484—62no—
88#477—62no—
89#431—31no—
90#428—62no—
91#423—31no—
92C2×#103—22no—
93C4×M16—31no—
94C2×C4×C8C2×D311yescharacteristic
95#343—62no—
96#332—62no—
97Q8⋊C8—62no—
98C2×C3×Q8—22no—
99C2×C3×Q8—44no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—82no—
104C22×C3×C4—22no—
105C22×C3×C4—41no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×Dic6—21no—
110C2×Dic6—21no—
111C2×C3×C8—22no—
112C2×C3×C8—42no—
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×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125#217—41no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132C4×Dic3—41no—
133#208—22no—
134#208—42no—
135#208Q1614yes—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—41no—
139C22×Q8—21no—
140C22×Q8—21no—
141C2×M16—32no—
142C2×M16—61no—
143C22×C8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153C4⋊Q8#6712yes—
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—21no—
163C2×C4⋊C4#6712yes—
164C2×C42C22×D311yescharacteristic
165#107—64no—
166#107—64no—
167#106—64no—
168C4⋊C8—32no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171#103—28no—
172#103—64no—
173C22⋊C8—61no—
174#97—61no—
175C4×C8—21no—
176C4×C8—61no—
177C4×C8—61no—
178C4×C8#6712yes—
179C23×C3#12111yescharacteristic
180C3×Q8—44no—
181C3×Q8—82no—
182C3×Q8—82no—
183C3×Q8—82no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—82no—
189C2×C3×C4—82no—
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×Q1612yes—
197C2×C3×C4#13712yes—
198C2×Dic3—41no—
199C2×Dic3—41no—
200Dic6—42no—
201Dic6—42no—
202C3×C8—42no—
203C3⋊C8—24no—
204C3⋊C8—42no—
205C3⋊C8—81no—
206C2×Q8—22no—
207C2×Q8—44no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—61no—
212C2×Q8—61no—
213C2×Q8—82no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4D3×D411yescharacteristic
217C22×C4#23711yescharacteristic
218C22×C4C2×#6711yescharacteristic
219M16—34no—
220M16—62no—
221M16—122no—
222C2×C8—22no—
223C2×C8—42no—
224C2×C8—62no—
225C2×C8—62no—
226C2×C8—121no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C22⋊C4—121no—
242C42—121no—
243C42D3×D411yescharacteristic
244C42D3×D411yescharacteristic
245C42C2×#6711yescharacteristic
246C42C2×#6711yescharacteristic
247C22×C3—21no—
248C22×C3—21no—
249C22×C3#39211yescharacteristic
250C22×C3#46212yes—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—82no—
254C3×C4—82no—
255C3×C4#45112yes—
256C3×C4#45412yes—
257C3×C4#47112yes—
258C3×C4#49411yescharacteristic
259C3×C4#49711yescharacteristic
260Dic3—81no—
261Dic3—81no—
262C23#93911yescharacteristic
263Q8—44no—
264Q8—82no—
265Q8—82no—
266Q8—82no—
267Q8—122no—
268Q8—122no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4D3×Q1612yes—
278C2×C4#91812yes—
279C2×C4#93711yescharacteristic
280C2×C4#93711yescharacteristic
281C2×C4#93811yescharacteristic
282C2×C4#93911yescharacteristic
283C2×C4#94011yescharacteristic
284C2×C4#95311yescharacteristic
285C8—42no—
286C8—64no—
287C8—122no—
288C8—241no—
289C2×C3—22no—
290C2×C3#200311yescharacteristic
291C2×C3#201611yescharacteristic
292C2×C3#201911yescharacteristic
293C22—21no—
294C22—21no—
295C22#518212yes—
296C22#522311yescharacteristic
297C4—21no—
298C4—21no—
299C4—82no—
300C4—82no—
301C4—241no—
302C4—241no—
303C4#506212yes—
304C4#517712yes—
305C4#518311yescharacteristic
306C4#518612yes—
307C4#519011yescharacteristic
308C3#1015711yescharacteristic, complemented, Sylow
309C2—22no—
310C2#7173011yescharacteristic
311C2#7205911yescharacteristic
312C2#7207311yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#487C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenterC2×#487D3
hypocenterC3#10157
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
308. C3#10157semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4872
#832242
#832072
#717902
#717752
#697382
#676562
#101573