Finite Group Databasebeta

← #10692782

Subgroups — #10692782

Order 768.

400 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 97 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
2#79267C211yesmaximal, characteristic
3C3×#1546C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic
5#70511C211yesmaximal, characteristic
6#67743C211yesmaximal, characteristic
7#67710C211yesmaximal, characteristic
8#9434—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5081C414yes—
14#5080—21no—
15#5078—21no—
16#4964—22no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4831—21no—
21#4830—21no—
22#4720—22no—
23#4679—21no—
24C8×Dic6—21no—
25C3×#332—21no—
26C3×#332—21no—
27#4546—21no—
28#4546—21no—
29#4486—21no—
30#4482—21no—
31#3165—31no—
32#2986—31no—
33#1561—31no—
34#1548—31no—
35#1546D311yescharacteristic
36#1299—31no—
37#1299—31no—
38C22×C3×Q8—21no—
39C2×C3×M16—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8C2×C412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47#926D412yes—
48#924—21no—
49#924—21no—
50C2×#208—21no—
51C2×#208—21no—
52#870C2×C412yes—
53#869D411yescharacteristic
54#869D411yescharacteristic
55C4×Dic6—21no—
56C4×Dic6—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8D411yescharacteristic
60C3×C4⋊C8D411yescharacteristic
61C3×C22⋊C8—21no—
62C3×C4×C8—21no—
63#830—21no—
64#815—22no—
65Dic3⋊C8—22no—
66Dic3⋊C8—22no—
67C8×Dic3—22no—
68#808—24no—
69#807—24no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)D411yescharacteristic
73C4⋊(C3⋊C8)D411yescharacteristic
74#803C2×C412yes—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#533—31no—
78C2×C4⋊Q8C2×D311yescharacteristic
79#501—34no—
80#500—61no—
81#498—61no—
82#446—61no—
83#446—61no—
84C8×Q8—61no—
85C8×Q8—61no—
86#433—62no—
87#431—31no—
88#431C2×D311yescharacteristic
89#428—62no—
90#423—31no—
91#423C2×D311yescharacteristic
92C4×M16—31no—
93#333—61no—
94#333—61no—
95#332—21no—
96#332—21no—
97#332—61no—
98#332—61no—
99C2×C3×Q8—22no—
100C2×C3×Q8—41no—
101C2×C3×Q8—81no—
102C2×C3×Q8—81no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×Dic6—21no—
108C2×Dic6—21no—
109C3×M16—81no—
110C2×C3×C8—21no—
111C2×C3×C8—21no—
112C2×C3×C8—41no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C42C22×C411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121#217—22no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3C22⋊C412yes—
125C4⋊Dic3C22⋊C412yes—
126Dic3⋊C4—22no—
127Dic3⋊C4—22no—
128C4×Dic3—22no—
129#208—22no—
130#208—22no—
131#208—81no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—41no—
136C22×Q8—21no—
137C2×M16—21no—
138C2×M16—61no—
139C2×M16—61no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—31no—
144C4⋊Q8—31no—
145C4⋊Q8C4×D312yes—
146#125—32no—
147C22⋊Q8—61no—
148C22⋊Q8—61no—
149C4×Q8—61no—
150C4×Q8—61no—
151#117—32no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C42C22×D311yescharacteristic
155#107—64no—
156#106—64no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161C4⋊C8—62no—
162C4⋊C8—62no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8#6711yescharacteristic
166C4⋊C8#6711yescharacteristic
167C22⋊C8—21no—
168C22⋊C8—61no—
169#97—32no—
170#97—62no—
171C4×C8—21no—
172C4×C8—62no—
173C4×C8—61no—
174C4×C8—61no—
175C23×C3#12311yescharacteristic
176C3×Q8—82no—
177C3×Q8—81no—
178C3×Q8—81no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—42no—
186C2×C3×C4—82no—
187C2×C3×C4C2×C22⋊C411yescharacteristic
188C2×C3×C4C4×D411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×Dic3—22no—
194C2×Dic3—22no—
195Dic6—41no—
196Dic6—41no—
197Dic6—41no—
198Dic6—41no—
199C3×C8—41no—
200C3×C8—41no—
201C3×C8—81no—
202C3⋊C8—24no—
203C3⋊C8—41no—
204C3⋊C8—41no—
205C3⋊C8—81no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—81no—
211C2×Q8—81no—
212C22×C4—41no—
213C22×C4#23511yescharacteristic
214C22×C4D3×D411yescharacteristic
215C22×C4#23711yescharacteristic
216M16—62no—
217M16—62no—
218M16—81no—
219M16—241no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—41no—
223C2×C8—62no—
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—121no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—32no—
230C4⋊C4—32no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—62no—
234C4⋊C4—61no—
235C4⋊C4—62no—
236C4⋊C4—61no—
237C22⋊C4—62no—
238C42—62no—
239C42C2×C4×D311yescharacteristic
240C42D3×D411yescharacteristic
241C42D3×D411yescharacteristic
242C42C2×#6711yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3#38611yescharacteristic
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—82no—
251C3×C4#45611yescharacteristic
252C3×C4#45611yescharacteristic
253C3×C4#47111yescharacteristic
254C3×C4#47111yescharacteristic
255C3×C4#47211yescharacteristic
256C3×C4#47311yescharacteristic
257Dic3—42no—
258Dic3—42no—
259C23#88311yescharacteristic
260Q8—82no—
261Q8—81no—
262Q8—81no—
263Q8—121no—
264Q8—121no—
265Q8—121no—
266Q8—121no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—82no—
277C2×C4D3×C22⋊C411yescharacteristic
278C2×C4#67⋊C411yescharacteristic
279C2×C4#88411yescharacteristic
280C2×C4C4×#6711yescharacteristic
281C2×C4#93711yescharacteristic
282C2×C4#93811yescharacteristic
283C8—41no—
284C8—41no—
285C8—64no—
286C8—81no—
287C8—121no—
288C8—121no—
289C8—241no—
290C2×C3—41no—
291C2×C3#188611yescharacteristic
292C2×C3#190711yescharacteristic
293C2×C3#190811yescharacteristic
294C22—21no—
295C22—21no—
296C22—21no—
297C22#495611yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—82no—
302C4—122no—
303C4—122no—
304C4#475411yescharacteristic
305C4#475511yescharacteristic
306C4#516011yescharacteristic
307C4#517711yescharacteristic
308C4#520511yescharacteristic
309C4#520511yescharacteristic
310C3#943411yescharacteristic, complemented, Sylow
311C2—41no—
312C2#7012311yescharacteristic
313C2#7021811yescharacteristic
314C2#7034511yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#1546D3
hypocenterC3#9434
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
310. C3#9434semidirect

Maximal subgroups

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

subgroupindex
#832072
#792672
C3×#15462
#717742
#705112
#677432
#677102
#94343