Finite Group Databasebeta

← #10692091

Subgroups — #10692091

Order 768.

536 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C3×#3113C211yesmaximal, characteristic
2#82450C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic
5#71623C211yesmaximal, characteristic
6#67750C211yesmaximal, characteristic
7#67748C211yesmaximal, characteristic
8#9324—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×Q8×Dic3—21no—
12C3×#483—21no—
13C3×#475C412yescomplemented
14C3×#462C412yescomplemented
15C3×#451—21no—
16C3×#441—22no—
17C3×#432C2211yescharacteristic
18Dic3×C4⋊C4—21no—
19Dic3×C4⋊C4—22no—
20C4×C4⋊Dic3—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22#4932—21no—
23#4932C2211yescharacteristic
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4538—21no—
27#4538—21no—
28#4535—22no—
29#4532—22no—
30#3113D311yescharacteristic
31#2276—31no—
32#1986—31no—
33#1506—31no—
34#1491—31no—
35#1300—31no—
36#1285—31no—
37C22×C3×Q8D411yescharacteristic
38C2×C3×Q16—22no—
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#125—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8C2×C412yes—
49C3×C4×Q8D412yes—
50C3×#117—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55Q8×Dic3—28no—
56C2×C4⋊Dic3—22no—
57C2×C4⋊Dic3—21no—
58C2×C4×Dic3—22no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—22no—
61C2×C4×Dic3D412yes—
62C2×#208—21no—
63C3×#106—22no—
64C3×C4⋊C8C2×C412yes—
65C3×#103—22no—
66C3×#103—22no—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95D411yescharacteristic
73C3⋊#95D411yescharacteristic
74#830—21no—
75C4⋊(C3⋊C8)—21no—
76#803—21no—
77#803—21no—
78C4×C3⋊C8—21no—
79#533C2×D311yescharacteristic
80C2×C4×Q8—61no—
81C2×C4×Q8C2×D311yescharacteristic
82#483—21no—
83#475Dic312yescomplemented
84#462Dic312yescomplemented
85#451—21no—
86#441—22no—
87#432—31no—
88#432C2×D311yescharacteristic
89C4×M16—31no—
90C4×C4⋊C4—31no—
91C4×C4⋊C4—61no—
92C4×C4⋊C4—62no—
93C4×C4⋊C4—61no—
94#376—31no—
95#376—61no—
96#339—62no—
97#328—61no—
98#328—61no—
99Q8⋊C8—62no—
100C2×C3×Q8—24no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8C22⋊C412yes—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—22no—
112C22×Dic3—21no—
113C3×Q16—44no—
114C2×C3×C8—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4C22⋊C412yes—
125C3×C22⋊C4—41no—
126C3×C42—22no—
127C3×C42—22no—
128C3×C42—41no—
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133C4⋊Dic3—28no—
134C4⋊Dic3—24no—
135C4×Dic3—28no—
136C4×Dic3—24no—
137C4×Dic3—42no—
138C4×Dic3—82no—
139#208—22no—
140#208—41no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C22×Q8#6711yescharacteristic
144C2×Q16—22no—
145C2×M16—61no—
146C4⋊Q8C2×Dic311yescharacteristic
147C4⋊Q8C2×Dic311yescharacteristic
148#125—21no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—68no—
156C4×Q8C2×Dic312yes—
157C4×Q8#6712yes—
158#117—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—62no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4#6711yescharacteristic
163C2×C42—21no—
164C2×C42—32no—
165C2×C42—62no—
166C2×C42—61no—
167C2×C42—62no—
168C2×C42C22×D311yescharacteristic
169#106—22no—
170C4⋊C8—61no—
171C4⋊C8C2×Dic312yes—
172#103—22no—
173#103—22no—
174C22⋊C8—21no—
175C22⋊C8—61no—
176#97—21no—
177#97—61no—
178#97—61no—
179C4×C8—61no—
180#95—31no—
181#95—31no—
182#95—62no—
183#95—61no—
184#95—61no—
185C23×C3#12011yescharacteristic
186C3×Q8—24no—
187C3×Q8—22no—
188C3×Q8—42no—
189C3×Q8—81no—
190C3×Q8#10414yes—
191C2×C3×C4—24no—
192C2×C3×C4—22no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4C2×C22⋊C411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12311yescharacteristic
208C2×Dic3—22no—
209C2×Dic3—24no—
210C2×Dic3—22no—
211C2×Dic3—41no—
212C2×Dic3—42no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C3×C8—42no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220C2×Q8—24no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8#21712yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—62no—
230C22×C4D3×D411yescharacteristic
231C22×C4D3×D411yescharacteristic
232C22×C4C2×#6711yescharacteristic
233Q16—44no—
234M16—62no—
235M16—121no—
236C2×C8—22no—
237C2×C8—121no—
238C2×C8—121no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—22no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—68no—
249C4⋊C4—64no—
250C4⋊C4#21712yes—
251C22⋊C4—41no—
252C42—22no—
253C42—22no—
254C42—41no—
255C42—68no—
256C42—64no—
257C42—122no—
258C42—242no—
259C42#23711yescharacteristic
260C42#23711yescharacteristic
261C42C22×Dic311yescharacteristic
262C42C2×#6711yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#38611yescharacteristic
266C22×C3#45511yescharacteristic
267C22×C3#45611yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—24no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—42no—
275C3×C4—81no—
276C3×C4—81no—
277C3×C4C2×#10412yes—
278Dic3—44no—
279Dic3—42no—
280Dic3—42no—
281C23#93711yescharacteristic
282Q8—24no—
283Q8—22no—
284Q8—42no—
285Q8—81no—
286Q8#83714yes—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—64no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4D4×Dic311yescharacteristic
308C2×C4D4×Dic311yescharacteristic
309C2×C4#93511yescharacteristic
310C2×C4#93911yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4C2×#21711yescharacteristic
313C8—42no—
314C8—122no—
315C8—241no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#188211yescharacteristic
319C2×C3#189611yescharacteristic
320C2×C3#190511yescharacteristic
321C22—21no—
322C22—21no—
323C22#520211yescharacteristic
324C22#520511yescharacteristic
325C22#522011yescharacteristic
326C4—24no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—42no—
333C4—81no—
334C4—81no—
335C4—124no—
336C4—122no—
337C4—122no—
338C4C2×#83712yes—
339C3#932411yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2#7171111yescharacteristic
343C2#7210011yescharacteristic
344C2#7210111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3113C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#3113D3
hypocenterC3#9324
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
13. C3×#475C4semidirect
14. C3×#462C4semidirect
83. #475Dic3semidirect
84. #462Dic3semidirect
339. C3#9324semidirect

Maximal subgroups

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

subgroupindex
C3×#31132
#824502
#718412
#718362
#716232
#677502
#677482
#93243