Finite Group Databasebeta

← #10697930

Subgroups — #10697930

Order 768.

516 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 103 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#79271C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic
5#70514C211yesmaximal, characteristic
6#70506C211yesmaximal, characteristic
7#67736C211yesmaximal, characteristic
8#10298—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#4851—22no—
24#4721—22no—
25#4681—21no—
26#4680—21no—
27C3×#333—22no—
28#4555—22no—
29#4482—22no—
30#3229—31no—
31#3209—31no—
32#2862—31no—
33#1994—31no—
34#1548D311yescharacteristic
35#1546—31no—
36#1276—31no—
37C22×C3×Q8D412yes—
38C22×Dic6—21no—
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⋊Dic3D412yes—
51#924—21no—
52#924—21no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3D412yes—
56#870—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70#835—24no—
71C3⋊#95—28no—
72#830—21no—
73#817—24no—
74#816—24no—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77#533—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#498—61no—
81#494—61no—
82#487—61no—
83#487—61no—
84#483—62no—
85#475—62no—
86#462—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#332—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—21no—
108C2×Dic6—21no—
109C2×Dic6—41no—
110C2×Dic6—41no—
111C2×Dic6—42no—
112C2×Dic6—42no—
113C3×M16—42no—
114C2×C3×C8—22no—
115C2×C3×C8—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42#4111yescharacteristic
124#217—41no—
125#216—84no—
126C4⋊Dic3—22no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—22no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134Dic3⋊C4—41no—
135Dic3⋊C4—41no—
136C4×Dic3—24no—
137C4×Dic3—22no—
138C4×Dic3—22no—
139C4×Dic3—42no—
140C4×Dic3—41no—
141C2×C3⋊C8—22no—
142C22×Q8—61no—
143C22×Q8D1212yes—
144C2×Q16—64no—
145C2×M16—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—31no—
151C4⋊Q8—31no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
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—32no—
165C2×C4⋊C4—61no—
166C2×C42—32no—
167C2×C42—61no—
168C2×C42—61no—
169C2×C42C22×D311yescharacteristic
170#106—64no—
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×C4#13512yes—
196C2×C3×C4#13712yes—
197C2×Dic3—24no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—42no—
203C2×Dic3—44no—
204Dic6—42no—
205Dic6—42no—
206Dic6—82no—
207Dic6—81no—
208Dic6—81no—
209C3×C8—42no—
210C3×C8—81no—
211C3⋊C8—42no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—44no—
215C2×Q8—61no—
216C2×Q8—61no—
217C2×Q8—121no—
218C2×Q8—122no—
219C2×Q8—122no—
220C2×Q8—121no—
221C22×C4—22no—
222C22×C4—62no—
223C22×C4—62no—
224C22×C4C2×D1211yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4D3×D411yescharacteristic
227Q16—244no—
228M16—42no—
229C2×C8—22no—
230C2×C8—41no—
231C2×C8—62no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—61no—
237C4⋊C4—62no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—62no—
241C4⋊C4—61no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C22⋊C4—121no—
247C42—64no—
248C42—62no—
249C42—62no—
250C42—122no—
251C42—121no—
252C42#23511yescharacteristic
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C42C2×#6711yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#39211yescharacteristic
259C22×C3#45212yes—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—82no—
263C3×C4#45512yes—
264C3×C4#45612yes—
265C3×C4#46412yes—
266C3×C4#49511yescharacteristic
267C3×C4#49711yescharacteristic
268Dic3—44no—
269Dic3—81no—
270Dic3—81no—
271Dic3—82no—
272C23#88211yescharacteristic
273Q8—44no—
274Q8—82no—
275Q8—122no—
276Q8—122no—
277Q8—241no—
278Q8—241no—
279Q8—242no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—64no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—122no—
290C2×C4—124no—
291C2×C4#88211yescharacteristic
292C2×C4#88411yescharacteristic
293C2×C4#88411yescharacteristic
294C2×C4#91612yes—
295C2×C4#91812yes—
296C2×C4#93011yescharacteristic
297C2×C4#93711yescharacteristic
298C2×C4#94011yescharacteristic
299C8—42no—
300C8—81no—
301C8—122no—
302C2×C3—22no—
303C2×C3#200411yescharacteristic
304C2×C3#200511yescharacteristic
305C2×C3#201511yescharacteristic
306C22—21no—
307C22—21no—
308C22#480812yes—
309C22#496011yescharacteristic
310C4—21no—
311C4—21no—
312C4—82no—
313C4—124no—
314C4—242no—
315C4—241no—
316C4—241no—
317C4#474612yes—
318C4#479812yes—
319C4#516811yescharacteristic
320C4#519011yescharacteristic
321C4#520512yes—
322C3#1029811yescharacteristic, complemented, Sylow
323C2—22no—
324C2#7008411yescharacteristic
325C2#7027811yescharacteristic
326C2#7034611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1548D3
hypocenterC3#10298
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
322. C3#10298semidirect

Maximal subgroups

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

subgroupindex
#832522
#832222
#792712
C3×#15482
#705142
#705062
#677362
#102983