Finite Group Databasebeta

← #10691868

Subgroups — #10691868

Order 768.

426 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 109 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#83223C211yesmaximal, characteristic
2#78480C211yesmaximal, characteristic
3C3×#1960C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic
5#69546C211yesmaximal, characteristic
6#67660C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#9292—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×#103—21no—
12C2×C3×#103—21no—
13C2×C3×#97C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5102—21no—
16#5101—21no—
17#5094—22no—
18#5093C412yescomplemented
19#5080C412yescomplemented
20#4971—22no—
21#4925C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4701—22no—
25#4701—22no—
26#4547—22no—
27#4452—22no—
28#3229—31no—
29#2961—31no—
30#1960D311yescharacteristic
31#1546—31no—
32#1518—31no—
33#1286—31no—
34#1275—31no—
35C22×C3×Q8—21no—
36C22×C3×C8D411yescharacteristic
37C22×C3×C8D411yescharacteristic
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8C2×C412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47#924—21no—
48#924D412yes—
49C2×#208—21no—
50#870—21no—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6D412yes—
55C3×#103—42no—
56C3×#103—42no—
57C3×#97—21no—
58C3×#97—21no—
59C3×#95—21no—
60C3×#95—21no—
61#830—21no—
62#830—21no—
63#815—22no—
64Dic3⋊C8—24no—
65C8×Dic3—22no—
66#810—24no—
67#808—24no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)D412yes—
72C4×C3⋊C8—21no—
73C4×C3⋊C8C2×C412yes—
74#533—31no—
75C2×C4⋊Q8C2×D311yescharacteristic
76#498—32no—
77#487—32no—
78#483—62no—
79#475—62no—
80#446—62no—
81#446—62no—
82#431—31no—
83#431—31no—
84#424—61no—
85#423—31no—
86C2×#103—21no—
87C2×#103—21no—
88#405—61no—
89C2×#97C2×D311yescharacteristic
90#382C2×D311yescharacteristic
91#332—62no—
92#328—62no—
93C2×C3×Q8—22no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×Dic6—21no—
105C2×Dic6C22⋊C412yes—
106C2×C3×C8—21no—
107C2×C3×C8—21no—
108C2×C3×C8—21no—
109C2×C3×C8—21no—
110C2×C3×C8—21no—
111C2×C3×C8SD1612yes—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C42C22×C411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122#217—22no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3C22⋊C412yes—
127Dic3⋊C4—22no—
128Dic3⋊C4—22no—
129C4×Dic3—22no—
130#208—41no—
131#208—41no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—41no—
135C22×Q8—21no—
136C2×M16—61no—
137C22×C8#6711yescharacteristic
138C22×C8#6711yescharacteristic
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144C4⋊Q8C4×D312yes—
145#125—61no—
146C22⋊Q8—32no—
147C22⋊Q8—61no—
148C4×Q8—32no—
149C4×Q8—61no—
150#117—61no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C42C22×D311yescharacteristic
155#106—64no—
156C4⋊C8—32no—
157C4⋊C8—64no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161#103—42no—
162#103—42no—
163#103—64no—
164C22⋊C8—61no—
165C22⋊C8—61no—
166#97—21no—
167#97—21no—
168#97—62no—
169C4×C8—32no—
170C4×C8—62no—
171C4×C8—61no—
172#95—21no—
173#95—21no—
174C23×C3#12111yescharacteristic
175C3×Q8—42no—
176C3×Q8—42no—
177C3×Q8—81no—
178C3×Q8—81no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—81no—
183C2×C3×C4—81no—
184C2×C3×C4—81no—
185C2×C3×C4—81no—
186C2×C3×C4—81no—
187C2×C3×C4C2×C22⋊C411yescharacteristic
188C2×C3×C4C4×D411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12311yescharacteristic
193C2×C3×C4C2×SD1611yescharacteristic
194C2×C3×C4#13511yescharacteristic
195C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
196C2×C3×C4#13711yescharacteristic
197C2×Dic3—22no—
198C2×Dic3—22no—
199Dic6—42no—
200Dic6—81no—
201C3×C8—22no—
202C3×C8—41no—
203C3⋊C8—42no—
204C3⋊C8—42no—
205C3⋊C8—81no—
206C2×Q8—22no—
207C2×Q8—32no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—61no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4D3×D411yescharacteristic
217C22×C4#23711yescharacteristic
218C22×C4C2×#6711yescharacteristic
219M16—121no—
220M16—121no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—62no—
227C2×C8—62no—
228C2×C8—121no—
229C2×C8#21412yes—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—32no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—61no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C22⋊C4—62no—
243C42—62no—
244C42C2×C4×D311yescharacteristic
245C42#23511yescharacteristic
246C42D3×D411yescharacteristic
247C42D3×D411yescharacteristic
248C22×C3—21no—
249C22×C3—21no—
250C22×C3#38611yescharacteristic
251C22×C3#46011yescharacteristic
252C22×C3#46411yescharacteristic
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—82no—
258C3×C4—161no—
259C3×C4C4×SD1611yescharacteristic
260C3×C4#44011yescharacteristic
261C3×C4#44811yescharacteristic
262C3×C4#44911yescharacteristic
263Dic3—42no—
264Dic3—42no—
265C23#93811yescharacteristic
266Q8—42no—
267Q8—42no—
268Q8—81no—
269Q8—81no—
270Q8—122no—
271Q8—241no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4D3×C22⋊C411yescharacteristic
283C2×C4#67⋊C411yescharacteristic
284C2×C4#88311yescharacteristic
285C2×C4#88411yescharacteristic
286C2×C4C4×#6711yescharacteristic
287C2×C4#93211yescharacteristic
288C2×C4C2×#21411yescharacteristic
289C2×C4#93711yescharacteristic
290C2×C4#95011yescharacteristic
291C2×C4#95111yescharacteristic
292C8—22no—
293C8—41no—
294C8—122no—
295C8—122no—
296C8—241no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3#188011yescharacteristic
300C2×C3#190211yescharacteristic
301C2×C3#190911yescharacteristic
302C22—21no—
303C22—21no—
304C22#495611yescharacteristic
305C22#505411yescharacteristic
306C22#507511yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—82no—
312C4—122no—
313C4—122no—
314C4—161no—
315C4C4×#21411yescharacteristic
316C4#501811yescharacteristic
317C4#503711yescharacteristic
318C4#504011yescharacteristic
319C3#929211yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2#6966011yescharacteristic
323C2#7034911yescharacteristic
324C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1960C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1960D3
hypocenterC3#9292
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
18. #5093C4semidirect
19. #5080C4semidirect
319. C3#9292semidirect

Maximal subgroups

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

subgroupindex
#832232
#784802
C3×#19602
#717732
#695462
#676602
#676312
#92923