Finite Group Databasebeta

← #10691903

Subgroups — #10691903

Order 768.

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

normal subgroups only

This group has 454 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#83370C211yesmaximal, characteristic
2#73391C211yesmaximal, characteristic
3C3×#1995C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic
5#70507C211yesmaximal, characteristic
6#67713C211yesmaximal, characteristic
7#67614C211yesmaximal, characteristic
8#9297—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×#376C2211yescharacteristic
13#5093C412yescomplemented
14#5092C412yescomplemented
15#5050—22no—
16#5031—22no—
17#4936C2211yescharacteristic
18C4×C4⋊Dic3—21no—
19#4928—21no—
20#4925C2211yescharacteristic
21#4925C2211yescharacteristic
22#4654—22no—
23#4646—22no—
24C3×#339—21no—
25C3×#339—21no—
26#4538—22no—
27#4481—22no—
28#3209—31no—
29#2331—31no—
30#1995D311yescharacteristic
31#1548—31no—
32#1518—31no—
33#1286—31no—
34#1274—31no—
35C22×Dic6D412yes—
36C2×C3×M16—21no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#125—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×#117—21no—
45C2×C3×C42D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C2×#216—24no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×Dic3⋊C4—22no—
53C2×C4×Dic3—22no—
54#869—21no—
55#869—21no—
56#869C2×C412yes—
57#869D412yes—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3⋊#95—22no—
63C3⋊#95—22no—
64C3⋊#95—21no—
65C3⋊#95D412yes—
66#830—21no—
67#830—21no—
68#810—24no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4×C3⋊C8—21no—
72C4×C3⋊C8C2×C412yes—
73#533C2×D311yescharacteristic
74C2×C4⋊Q8—31no—
75#494—32no—
76#487—32no—
77#462—62no—
78#451—62no—
79#431—31no—
80#431—31no—
81#423C2×D311yescharacteristic
82#391—62no—
83#389—61no—
84#387—62no—
85#382—31no—
86C4×C4⋊C4—61no—
87#376C2×D311yescharacteristic
88#339—21no—
89#339—21no—
90#333—62no—
91#328—62no—
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—22no—
100C22×Dic3—22no—
101C2×Dic6—22no—
102C2×Dic6—82no—
103C2×Dic6C22⋊C414yes—
104C3×M16—41no—
105C3×M16—41no—
106C2×C3×C8—41no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C42—21no—
115C3×C42—21no—
116C3×C42—21no—
117C3×C42—41no—
118C3×C42Q1612yescomplemented
119C3×C42C22×C411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122C3×C42#4111yescharacteristic
123#216—84no—
124C4⋊Dic3—22no—
125C4⋊Dic3—22no—
126C4⋊Dic3—22no—
127C4⋊Dic3—22no—
128C4⋊Dic3—41no—
129C4⋊Dic3—41no—
130Dic3⋊C4—82no—
131C4×Dic3—82no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—41no—
134C22×Q8—32no—
135C2×Q16—64no—
136C2×M16—21no—
137C4⋊Q8—32no—
138C4⋊Q8—32no—
139C4⋊Q8—61no—
140C4⋊Q8—61no—
141C4⋊Q8C4×D311yescharacteristic
142C4⋊Q8C4×D311yescharacteristic
143#125—21no—
144C22⋊Q8—21no—
145C22⋊Q8—21no—
146C4×Q8—21no—
147C4×Q8—21no—
148#117—21no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—62no—
153C2×C42—62no—
154C2×C42#6711yescharacteristic
155C2×C42#6711yescharacteristic
156C2×C42C22×D311yescharacteristic
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161#103—64no—
162C22⋊C8—61no—
163C22⋊C8—61no—
164C4×C8—32no—
165C4×C8—61no—
166#95—21no—
167#95—21no—
168#95—32no—
169#95—62no—
170#95—62no—
171#95—61no—
172C23×C3#12011yescharacteristic
173C3×Q8—42no—
174C3×Q8—81no—
175C2×C3×C4—22no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4C2×C22⋊C411yescharacteristic
186C2×C3×C4C4×D411yescharacteristic
187C2×C3×C4C4×D411yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12311yescharacteristic
191C2×C3×C4C2×Q1611yescharacteristic
192C2×C3×C4#13511yescharacteristic
193C2×C3×C4#13711yescharacteristic
194C2×C3×C4#13711yescharacteristic
195C2×Dic3—24no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201Dic6—44no—
202Dic6—82no—
203C3×C8—81no—
204C3⋊C8—42no—
205C3⋊C8—81no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—34no—
209C2×Q8—62no—
210C2×Q8—242no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—62no—
214C22×C4—62no—
215C22×C4D3×D411yescharacteristic
216C22×C4D3×D411yescharacteristic
217C22×C4C2×#6711yescharacteristic
218Q16—244no—
219M16—41no—
220M16—41no—
221C2×C8—41no—
222C2×C8—62no—
223C2×C8—121no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C4⋊C4—62no—
233C4⋊C4—62no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C4⋊C4—242no—
237C22⋊C4—41no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—41no—
242C42—242no—
243C42#21612yescomplemented
244C42C2×C4×D311yescharacteristic
245C42#23511yescharacteristic
246C42D3×D411yescharacteristic
247C42#23711yescharacteristic
248C22×C3—21no—
249C22×C3—21no—
250C22×C3#38611yescharacteristic
251C22×C3#45111yescharacteristic
252C22×C3#45211yescharacteristic
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—42no—
258C3×C4—81no—
259C3×C4—81no—
260C3×C4—81no—
261C3×C4C4×Q1611yescharacteristic
262C3×C4#44111yescharacteristic
263C3×C4#46211yescharacteristic
264C3×C4#46411yescharacteristic
265Dic3—44no—
266Dic3—82no—
267C23#93711yescharacteristic
268Q8—42no—
269Q8—81no—
270Q8—124no—
271Q8—242no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—64no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4D3×C22⋊C411yescharacteristic
289C2×C4#67⋊C411yescharacteristic
290C2×C4#88311yescharacteristic
291C2×C4#88411yescharacteristic
292C2×C4C4×#6711yescharacteristic
293C2×C4#93811yescharacteristic
294C2×C4#94211yescharacteristic
295C2×C4C2×#21611yescharacteristic
296C2×C4#95011yescharacteristic
297C2×C4#95111yescharacteristic
298C8—81no—
299C8—122no—
300C8—241no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3#187811yescharacteristic
304C2×C3#190511yescharacteristic
305C2×C3#191111yescharacteristic
306C22—21no—
307C22—21no—
308C22#495611yescharacteristic
309C22#504711yescharacteristic
310C22#505011yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—124no—
320C4—242no—
321C4C4×#21611yescharacteristic
322C4#503011yescharacteristic
323C4#505411yescharacteristic
324C4#509011yescharacteristic
325C3#929711yescharacteristic, complemented, Sylow
326C2—21no—
327C2—21no—
328C2#6969011yescharacteristic
329C2#7040811yescharacteristic
330C2#7041011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1995C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1995D3
hypocenterC3#9297
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
13. #5093C4semidirect
14. #5092C4semidirect
118. C3×C42Q16semidirect
243. C42#216semidirect
325. C3#9297semidirect

Maximal subgroups

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

subgroupindex
#833702
#733912
C3×#19952
#718382
#705072
#677132
#676142
#92973