Finite Group Databasebeta

← #10697084

Subgroups — #10697084

Order 768.

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

normal subgroups only

This group has 492 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#83341C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic
3C2×#4680C211yesmaximal, characteristic
4C3×#1493C211yesmaximal, characteristic
5#70522C211yesmaximal, characteristic
6#70507C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic
8#10167—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C2×#824—22no—
15C2×#817—22no—
16#5080—21no—
17#5080—21no—
18#5079—21no—
19#5078—21no—
20#5064—22no—
21#5043—22no—
22#5020—22no—
23#4995—22no—
24#4925C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4680—22no—
27#4680—22no—
28C3×Q8⋊C8—22no—
29#4481—22no—
30#3229—31no—
31#3165—31no—
32C2×#498—31no—
33#1966—31no—
34#1548—31no—
35#1493D311yescharacteristic
36#1278—31no—
37C22×Dic6—21no—
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C42C2311yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D412yes—
54C2×#208D412yes—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869D412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4×C8—21no—
66C3×C4×C8D412yes—
67#835—24no—
68#830—21no—
69#824—44no—
70#817—28no—
71#808—24no—
72#808—24no—
73#807—24no—
74C4⋊(C3⋊C8)D412yes—
75#803—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#533C2×D311yescharacteristic
79C2×C4⋊Q8—31no—
80C2×C4⋊Q8—31no—
81#501—62no—
82#500—61no—
83#498—62no—
84#498—61no—
85#498—61no—
86#498—62no—
87#487—61no—
88#483—62no—
89#475—62no—
90#431—31no—
91#428—62no—
92C2×#106—62no—
93#423C2×D311yescharacteristic
94C2×#101—62no—
95C4×M16—31no—
96C2×C4×C8C2×D311yescharacteristic
97#333—62no—
98Q8⋊C8—22no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—22no—
105C22×Dic3—41no—
106C2×Dic6—22no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C2×Dic6—41no—
110C2×Dic6—42no—
111C2×Dic6—42no—
112C2×Dic6—82no—
113C3×M16—42no—
114C2×C3×C8—24no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C42—41no—
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C4⋊Dic3—22no—
130C4⋊Dic3—22no—
131C4⋊Dic3—22no—
132C4⋊Dic3—22no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137#208—22no—
138#208SD1614yes—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C22×Q8—61no—
142C22×Q8—61no—
143C2×M16—21no—
144C2×M16—32no—
145C22×C8—21no—
146C4⋊Q8—32no—
147C4⋊Q8—61no—
148C4⋊Q8—61no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8D1211yescharacteristic
155C4⋊Q8D1211yescharacteristic
156#125—21no—
157C22⋊Q8—21no—
158C22⋊Q8—21no—
159C4×Q8—21no—
160C4×Q8—21no—
161#117—21no—
162C2×C4⋊C4—32no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C42C22×D311yescharacteristic
167#107—64no—
168#106—64no—
169#106—64no—
170#106—68no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—32no—
174#103—64no—
175#101—124no—
176C22⋊C8—61no—
177#97—61no—
178C4×C8—21no—
179C4×C8—61no—
180C4×C8—61no—
181C4×C8#6712yes—
182C23×C3#12111yescharacteristic
183C3×Q8—42no—
184C3×Q8—42no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12711yescharacteristic
195C2×C3×C4C2×SD1612yes—
196C2×C3×C4#13712yes—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—44no—
200C2×Dic3—82no—
201Dic6—82no—
202Dic6—82no—
203Dic6—81no—
204Dic6—81no—
205Dic6—82no—
206C3×C8—24no—
207C3×C8—81no—
208C3⋊C8—24no—
209C3⋊C8—42no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—62no—
213C2×Q8—121no—
214C2×Q8—122no—
215C2×Q8—122no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8—242no—
219C22×C4—62no—
220C22×C4—121no—
221C22×C4#23511yescharacteristic
222C22×C4D3×D411yescharacteristic
223C22×C4D3×D411yescharacteristic
224M16—34no—
225M16—42no—
226M16—62no—
227C2×C8—24no—
228C2×C8—22no—
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—62no—
241C4⋊C4—62no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C22⋊C4—41no—
247C42—41no—
248C42C2×D1211yescharacteristic
249C42D3×D411yescharacteristic
250C42D3×D411yescharacteristic
251C42C2×#6711yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3#39211yescharacteristic
255C22×C3#46112yes—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—81no—
259C3×C4—81no—
260C3×C4#44812yes—
261C3×C4#45612yes—
262C3×C4#47112yes—
263C3×C4#49211yescharacteristic
264C3×C4#49711yescharacteristic
265Dic3—82no—
266Dic3—82no—
267C23#88411yescharacteristic
268Q8—42no—
269Q8—42no—
270Q8—242no—
271Q8—242no—
272Q8—241no—
273Q8—241no—
274Q8—242no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—124no—
282C2×C4—242no—
283C2×C4#88211yescharacteristic
284C2×C4#88211yescharacteristic
285C2×C4#88411yescharacteristic
286C2×C4D3×SD1612yes—
287C2×C4#91812yes—
288C2×C4#93011yescharacteristic
289C2×C4#93711yescharacteristic
290C2×C4#94011yescharacteristic
291C8—24no—
292C8—64no—
293C8—81no—
294C8—122no—
295C2×C3—22no—
296C2×C3#200511yescharacteristic
297C2×C3#201311yescharacteristic
298C2×C3#201911yescharacteristic
299C22—21no—
300C22—21no—
301C22#481812yes—
302C22#496011yescharacteristic
303C4—21no—
304C4—21no—
305C4—81no—
306C4—81no—
307C4—242no—
308C4—242no—
309C4#480612yes—
310C4#482612yes—
311C4#514112yes—
312C4#517511yescharacteristic
313C4#519011yescharacteristic
314C3#1016711yescharacteristic, complemented, Sylow
315C2—22no—
316C2#7000711yescharacteristic
317C2#7028211yescharacteristic
318C2#7038911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1493D3
hypocenterC3#10167
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
314. C3#10167semidirect

Maximal subgroups

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

subgroupindex
#833412
#833232
C2×#46802
C3×#14932
#705222
#705072
#676562
#101673