Finite Group Databasebeta

← #10697897

Subgroups — #10697897

Order 768.

588 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 111 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#83370C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic
3#73311C211yesmaximal, characteristic
4C3×#1994C211yesmaximal, characteristic
5#70508C211yesmaximal, characteristic
6#70507C211yesmaximal, characteristic
7#67611C211yesmaximal, characteristic
8#10293—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869—22no—
11C2×#869C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C43C2211yescharacteristic
15#5093—21no—
16#5092—21no—
17#5079—21no—
18#5078—21no—
19#5050—22no—
20#5044—22no—
21#5031—22no—
22#5021—22no—
23#4935—22no—
24#4935—22no—
25#4925C2211yescharacteristic
26#4925C2211yescharacteristic
27C3×#339—22no—
28#4486—22no—
29#4481—22no—
30#3230—31no—
31#3209—31no—
32#2868—31no—
33#1994D311yescharacteristic
34#1549—31no—
35#1548—31no—
36#1276—31no—
37C22×Dic6—21no—
38C22×Dic6D412yes—
39C2×C3×M16—21no—
40C3×C4⋊Q8D411yescharacteristic
41C3×C4⋊Q8D411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C42—21no—
49C2×C3×C42—21no—
50C2×C3×C42D412yes—
51C2×C3×C42C2311yescharacteristic
52C2×#216—24no—
53C2×C4⋊Dic3—28no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D412yes—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869—48no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69#835—24no—
70#830—21no—
71#830—21no—
72#810—24no—
73#807—24no—
74C4⋊(C3⋊C8)D412yes—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#533C2×D311yescharacteristic
79C2×C4⋊Q8—31no—
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8—62no—
82#500—61no—
83#494—61no—
84#487—61no—
85#487—61no—
86#484—62no—
87#477—62no—
88#462—62no—
89#451—62no—
90#431—31no—
91#431—31no—
92#423C2×D311yescharacteristic
93#384—62no—
94#384—62no—
95C43C2×D311yescharacteristic
96#339—22no—
97#333—62no—
98#332—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
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—22no—
109C2×Dic6—44no—
110C2×Dic6—44no—
111C2×Dic6—41no—
112C2×Dic6—41no—
113C3×M16—42no—
114C2×C3×C8—41no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C42—24no—
123C3×C42—24no—
124C3×C42—22no—
125C3×C42—41no—
126C3×C42Q1614yescomplemented
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131#216—84no—
132C4⋊Dic3—22no—
133C4⋊Dic3—22no—
134C4⋊Dic3—416no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C22×Q8—32no—
144C22×Q8—61no—
145C2×Q16—64no—
146C2×M16—21no—
147C4⋊Q8—61no—
148C4⋊Q8—61no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—128no—
156C4⋊Q8D1211yescharacteristic
157C4⋊Q8D1211yescharacteristic
158#125—21no—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163#117—21no—
164C2×C4⋊C4—32no—
165C2×C4⋊C4—68no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42#6712yes—
172C2×C42C22×D311yescharacteristic
173#107—64no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—32no—
177C4⋊C8—32no—
178#103—64no—
179#103—64no—
180C22⋊C8—61no—
181C22⋊C8—61no—
182C4×C8—61no—
183C4×C8—61no—
184C23×C3#12011yescharacteristic
185C3×Q8—42no—
186C3×Q8—42no—
187C2×C3×C4—24no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12711yescharacteristic
200C2×C3×C4C2×Q1612yes—
201C2×C3×C4#13712yes—
202C2×Dic3—44no—
203C2×Dic3—44no—
204C2×Dic3—42no—
205C2×Dic3—42no—
206Dic6—44no—
207Dic6—82no—
208Dic6—82no—
209Dic6—82no—
210C3×C8—81no—
211C3⋊C8—42no—
212C3⋊C8—42no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—62no—
216C2×Q8—62no—
217C2×Q8—124no—
218C2×Q8—124no—
219C2×Q8—121no—
220C2×Q8—121no—
221C22×C4—22no—
222C22×C4—62no—
223C22×C4—62no—
224C22×C4D3×D411yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4C2×#6711yescharacteristic
227Q16—244no—
228M16—42no—
229C2×C8—41no—
230C2×C8—62no—
231C2×C8—62no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—1216no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C22⋊C4—41no—
248C42—24no—
249C42—24no—
250C42—22no—
251C42—41no—
252C42#21614yescomplemented
253C42C2×D1211yescharacteristic
254C42#23511yescharacteristic
255C42D3×D411yescharacteristic
256C42D3×D411yescharacteristic
257C22×C3—21no—
258C22×C3—21no—
259C22×C3#39211yescharacteristic
260C22×C3#45112yes—
261C3×C4—28no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—81no—
265C3×C4—81no—
266C3×C4#45312yes—
267C3×C4#45612yes—
268C3×C4#46212yes—
269C3×C4#49411yescharacteristic
270C3×C4#49711yescharacteristic
271Dic3—82no—
272Dic3—82no—
273C23#93711yescharacteristic
274Q8—42no—
275Q8—42no—
276Q8—124no—
277Q8—242no—
278Q8—242no—
279Q8—242no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—124no—
288C2×C4—124no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4#88211yescharacteristic
292C2×C4#88211yescharacteristic
293C2×C4#88411yescharacteristic
294C2×C4#88411yescharacteristic
295C2×C4#93011yescharacteristic
296C2×C4#94011yescharacteristic
297C2×C4C2×#21612yes—
298C2×C4#95112yes—
299C8—81no—
300C8—122no—
301C8—122no—
302C2×C3—22no—
303C2×C3#200211yescharacteristic
304C2×C3#200511yescharacteristic
305C2×C3#201611yescharacteristic
306C22—21no—
307C22—21no—
308C22#496011yescharacteristic
309C22#505012yes—
310C4—28no—
311C4—21no—
312C4—21no—
313C4—81no—
314C4—81no—
315C4—242no—
316C4—242no—
317C4#482212yes—
318C4#482612yes—
319C4#503112yes—
320C4#506011yescharacteristic
321C4#509211yescharacteristic
322C3#1029311yescharacteristic, complemented, Sylow
323C2—22no—
324C2#6969111yescharacteristic
325C2#7038511yescharacteristic
326C2#7038911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1994C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1994D3
hypocenterC3#10293
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
126. C3×C42Q16semidirect
252. C42#216semidirect
322. C3#10293semidirect

Maximal subgroups

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

subgroupindex
#833702
#833242
#733112
C3×#19942
#705082
#705072
#676112
#102933