Finite Group Databasebeta

← #10697924

Subgroups — #10697924

Order 768.

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

normal subgroups only

This group has 528 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#83252C211yesmaximal, characteristic
2#83222C211yesmaximal, characteristic
3#79313C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic
5#70514C211yesmaximal, characteristic
6#70507C211yesmaximal, characteristic
7#67736C211yesmaximal, characteristic
8#10297—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5229—22no—
15#5228—22no—
16#5092—21no—
17#5090—22no—
18#5079—21no—
19#4977—22no—
20#4973—22no—
21C42×Dic3C2211yescharacteristic
22#4925C2211yescharacteristic
23#4849—22no—
24#4739—22no—
25#4699—21no—
26#4681—21no—
27C3×#333—22no—
28#4555—22no—
29#4481—22no—
30#3209—31no—
31#3209—31no—
32#2862—31no—
33#1994—31no—
34#1548—31no—
35#1548D311yescharacteristic
36#1276—31no—
37C22×C3×Q8D412yes—
38C22×Dic6D412yes—
39C2×C3×M16—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#216—24no—
48#926—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51#924—21no—
52#924—21no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3D412yes—
56C2×Dic12—24no—
57#870—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71#835—24no—
72C3⋊#95—28no—
73#830—21no—
74#816—24no—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77#533—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#494—61no—
81#494—61no—
82#487—61no—
83#487—61no—
84#462—62no—
85#462—62no—
86#451—62no—
87#451—62no—
88#431—31no—
89#431C2×D311yescharacteristic
90#423C2×D311yescharacteristic
91#391—62no—
92#391—62no—
93#382—62no—
94C43—31no—
95#339—62no—
96#333—22no—
97#333—62no—
98C2×C3×Q8—22no—
99C2×C3×Q8—22no—
100C2×C3×Q8—44no—
101C22×C3×C4—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—22no—
106C22×Dic3—22no—
107C2×Dic6—22no—
108C2×Dic6—21no—
109C2×Dic6—21no—
110C2×Dic6—22no—
111C2×Dic6—44no—
112C3×M16—42no—
113C2×C3×C8—22no—
114C2×C3×C8—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123#217—41no—
124#216—84no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—41no—
135C4×Dic3—22no—
136C4×Dic3—42no—
137C4×Dic3—42no—
138C4×Dic3—41no—
139C4×Dic3Q1614yescomplemented
140C2×C3⋊C8—22no—
141Dic12—84no—
142C22×Q8—32no—
143C22×Q8D1212yes—
144C2×Q16—64no—
145C2×Q16—64no—
146C2×M16—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
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—61no—
166C2×C4⋊C4—61no—
167C2×C42—32no—
168C2×C42—61no—
169C2×C42—61no—
170C2×C42C22×D311yescharacteristic
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—32no—
174C4⋊C8#6712yes—
175#103—64no—
176#103—64no—
177C22⋊C8—21no—
178C22⋊C8—61no—
179C4×C8—21no—
180C4×C8—61no—
181#95—68no—
182C23×C3#12011yescharacteristic
183C3×Q8—44no—
184C3×Q8—82no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—44no—
188C2×C3×C4—42no—
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12711yescharacteristic
195C2×C3×C4C2×Q1612yes—
196C2×C3×C4#13712yes—
197C2×Dic3—24no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—44no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—42no—
204Dic6—42no—
205Dic6—44no—
206Dic6—42no—
207Dic6—82no—
208C3×C8—42no—
209C3×C8—81no—
210C3⋊C8—42no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—44no—
214C2×Q8—61no—
215C2×Q8—62no—
216C2×Q8—62no—
217C2×Q8—61no—
218C2×Q8—124no—
219C22×C4—22no—
220C22×C4—62no—
221C22×C4—62no—
222C22×C4C2×D1211yescharacteristic
223C22×C4D3×D411yescharacteristic
224C22×C4D3×D411yescharacteristic
225Q16—244no—
226Q16—244no—
227M16—42no—
228C2×C8—22no—
229C2×C8—41no—
230C2×C8—62no—
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—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C22⋊C4—121no—
246C42—34no—
247C42—62no—
248C42—122no—
249C42—122no—
250C42—121no—
251C42#23511yescharacteristic
252C42D3×D411yescharacteristic
253C42D3×D411yescharacteristic
254C42C2×#6711yescharacteristic
255C22×C3—21no—
256C22×C3—21no—
257C22×C3#39211yescharacteristic
258C22×C3#45112yes—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—82no—
262C3×C4#45512yes—
263C3×C4#45512yes—
264C3×C4#46212yes—
265C3×C4#49411yescharacteristic
266C3×C4#49711yescharacteristic
267Dic3—44no—
268Dic3—82no—
269Dic3—81no—
270Dic3—81no—
271C23#88211yescharacteristic
272Q8—44no—
273Q8—82no—
274Q8—122no—
275Q8—124no—
276Q8—122no—
277Q8—242no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—64no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—124no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—122no—
289C2×C4#88211yescharacteristic
290C2×C4#88411yescharacteristic
291C2×C4#88411yescharacteristic
292C2×C4#91512yes—
293C2×C4D3×Q1612yes—
294C2×C4#93011yescharacteristic
295C2×C4#93711yescharacteristic
296C2×C4#94011yescharacteristic
297C8—42no—
298C8—81no—
299C8—122no—
300C2×C3—22no—
301C2×C3#200411yescharacteristic
302C2×C3#200411yescharacteristic
303C2×C3#201611yescharacteristic
304C22—21no—
305C22—21no—
306C22#480912yes—
307C22#496011yescharacteristic
308C4—21no—
309C4—21no—
310C4—82no—
311C4—124no—
312C4—241no—
313C4—242no—
314C4—241no—
315C4#474612yes—
316C4#479512yes—
317C4#516711yescharacteristic
318C4#518311yescharacteristic
319C4#520212yes—
320C3#1029711yescharacteristic, complemented, Sylow
321C2—22no—
322C2#7008411yescharacteristic
323C2#7028311yescharacteristic
324C2#7034411yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1548D3
hypocenterC3#10297
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
139. C4×Dic3Q16semidirect
320. C3#10297semidirect

Maximal subgroups

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

subgroupindex
#832522
#832222
#793132
C3×#15482
#705142
#705072
#677362
#102973