Finite Group Databasebeta

← #10697067

Subgroups — #10697067

Order 768.

476 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 107 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#83207C211yesmaximal, characteristic
2C2×#5078C211yesmaximal, characteristic
3#79272C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic
5#70524C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic
7#67739C211yesmaximal, characteristic
8#10165—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5081—22no—
15#5080—21no—
16#5078—22no—
17#5078—21no—
18#5078—22no—
19#4964—22no—
20C2×#807—22no—
21C4×#208C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4850—22no—
24#4722—22no—
25#4682—21no—
26#4681—21no—
27C3×#333—22no—
28#4554—22no—
29#4484—22no—
30#3230—31no—
31#3165—31no—
32C2×#500—31no—
33#1997—31no—
34#1548D311yescharacteristic
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47#926—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D412yes—
50#924—21no—
51#924—21no—
52C2×#208D412yes—
53C22×C3⋊C8—21no—
54#870—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8D412yes—
66C3×C22⋊C8—21no—
67C3×C4×C8—21no—
68#818—24no—
69#816—24no—
70#808—24no—
71#807—28no—
72#807—24no—
73#803—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yes—
77#533—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#501—62no—
81#500—61no—
82#500—62no—
83#500—61no—
84#500—62no—
85#498—61no—
86#487—61no—
87#484—62no—
88#477—62no—
89#431C2×D311yescharacteristic
90#428—62no—
91C2×#107—62no—
92#423C2×D311yescharacteristic
93C4×M16—31no—
94C2×C4×C8—31no—
95#343—62no—
96#333—22no—
97Q8⋊C8—62no—
98C2×C3×Q8—22no—
99C2×C3×Q8—41no—
100C2×C3×Q8—82no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—22no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C2×Dic6—41no—
109C2×Dic6—41no—
110C2×Dic6—82no—
111C3×M16—42no—
112C2×C3×C8—22no—
113C2×C3×C8—41no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122#217—41no—
123C4⋊Dic3—22no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—22no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133C4×Dic3—41no—
134#208—22no—
135#208Q1614yes—
136C2×C3⋊C8—24no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—22no—
139C22×Q8—21no—
140C22×Q8—61no—
141C2×M16—21no—
142C2×M16—32no—
143C22×C8—61no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—31no—
148C4⋊Q8—31no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8D1212yes—
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—32no—
163C2×C4⋊C4—61no—
164C2×C42C22×D311yescharacteristic
165#107—68no—
166#107—64no—
167#107—64no—
168#106—64no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8#6712yes—
172#103—64no—
173C22⋊C8—21no—
174#97—61no—
175C4×C8—21no—
176C4×C8—32no—
177C4×C8—61no—
178C4×C8—61no—
179C23×C3#12111yescharacteristic
180C3×Q8—82no—
181C3×Q8—82no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—42no—
185C2×C3×C4—82no—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12711yescharacteristic
192C2×C3×C4C2×Q1612yes—
193C2×C3×C4#13712yes—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—42no—
197C2×Dic3—82no—
198Dic6—42no—
199Dic6—42no—
200Dic6—82no—
201Dic6—82no—
202C3×C8—42no—
203C3×C8—81no—
204C3⋊C8—24no—
205C3⋊C8—24no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—82no—
211C2×Q8—121no—
212C2×Q8—121no—
213C2×Q8—242no—
214C22×C4—41no—
215C22×C4—62no—
216C22×C4#23511yescharacteristic
217C22×C4D3×D411yescharacteristic
218C22×C4D3×D411yescharacteristic
219M16—34no—
220M16—42no—
221M16—62no—
222C2×C8—22no—
223C2×C8—41no—
224C2×C8—64no—
225C2×C8—62no—
226C2×C8—62no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—61no—
232C4⋊C4—62no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—62no—
236C4⋊C4—61no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C22⋊C4—121no—
242C42—121no—
243C42C2×D1211yescharacteristic
244C42D3×D411yescharacteristic
245C42D3×D411yescharacteristic
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#45112yes—
255C3×C4#45612yes—
256C3×C4#47212yes—
257C3×C4#49411yescharacteristic
258C3×C4#49711yescharacteristic
259Dic3—81no—
260Dic3—81no—
261Dic3—82no—
262C23#88411yescharacteristic
263Q8—82no—
264Q8—82no—
265Q8—122no—
266Q8—122no—
267Q8—242no—
268Q8—242no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—82no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—122no—
276C2×C4—242no—
277C2×C4#88211yescharacteristic
278C2×C4#88211yescharacteristic
279C2×C4#88411yescharacteristic
280C2×C4#91512yes—
281C2×C4D3×Q1612yes—
282C2×C4#93011yescharacteristic
283C2×C4#93711yescharacteristic
284C2×C4#94011yescharacteristic
285C8—42no—
286C8—64no—
287C8—64no—
288C8—81no—
289C2×C3—22no—
290C2×C3#200511yescharacteristic
291C2×C3#201611yescharacteristic
292C2×C3#201811yescharacteristic
293C22—21no—
294C22—21no—
295C22#479512yes—
296C22#496011yescharacteristic
297C4—21no—
298C4—21no—
299C4—82no—
300C4—242no—
301C4—241no—
302C4—241no—
303C4#474912yes—
304C4#480912yes—
305C4#516711yescharacteristic
306C4#518311yescharacteristic
307C4#520512yes—
308C3#1016511yescharacteristic, complemented, Sylow
309C2—22no—
310C2#7009011yescharacteristic
311C2#7028311yescharacteristic
312C2#7034611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1548D3
hypocenterC3#10165
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
308. C3#10165semidirect

Maximal subgroups

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

subgroupindex
#832072
C2×#50782
#792722
C3×#15482
#705242
#705012
#677392
#101653