Finite Group Databasebeta

← #10692119

Subgroups — #10692119

Order 768.

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

normal subgroups only

This group has 444 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
1C3×#3113C211yesmaximal, characteristic
2#82454C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic
5#71623C211yesmaximal, characteristic
6#67750C211yesmaximal, characteristic
7#67748C211yesmaximal, characteristic
8#9328—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C3×#483—21no—
12C3×#475C412yescomplemented
13C3×#462C412yescomplemented
14C3×#451—21no—
15C3×#441—22no—
16C3×#432C2211yescharacteristic
17#5229—21no—
18#4984—21no—
19#4980—22no—
20#4937—21no—
21#4936—21no—
22C4×C4⋊Dic3C2211yescharacteristic
23#4932C2211yescharacteristic
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4538—21no—
27#4538—21no—
28#4535—22no—
29#4532—22no—
30#3113D311yescharacteristic
31#2320—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
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—22no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3D412yes—
59C2×#208—21no—
60C3×#106—22no—
61C3×C4⋊C8C2×C412yes—
62C3×#103—22no—
63C3×#103—22no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3⋊#95—22no—
67C3⋊#95—22no—
68C3⋊#95—22no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73#830—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76#803—21no—
77C4×C3⋊C8—21no—
78#533C2×D311yescharacteristic
79C2×C4×Q8C2×D311yescharacteristic
80#483—21no—
81#475Dic312yescomplemented
82#462Dic312yescomplemented
83#451—21no—
84#441—22no—
85#432—31no—
86#432C2×D311yescharacteristic
87C4×M16—31no—
88#391—61no—
89#389—61no—
90#387—62no—
91#383—61no—
92#382—61no—
93C4×C4⋊C4—31no—
94#376—31no—
95#339—62no—
96#328—61no—
97#328—61no—
98Q8⋊C8—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—42no—
103C2×C3×Q8C22⋊C412yes—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—22no—
111C22×Dic3—21no—
112C3×Q16—44no—
113C2×C3×C8—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—21no—
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×C4⋊C4C22⋊C412yes—
124C3×C22⋊C4—41no—
125C3×C42—22no—
126C3×C42—22no—
127C3×C42—41no—
128C3×C42C22×C411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131C3×C42#4111yescharacteristic
132C4⋊Dic3—42no—
133C4⋊Dic3—82no—
134C4×Dic3—42no—
135C4×Dic3—42no—
136#208—22no—
137#208—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C22×Q8#6711yescharacteristic
141C2×Q16—22no—
142C2×M16—61no—
143C4⋊Q8C2×Dic311yescharacteristic
144C4⋊Q8C2×Dic311yescharacteristic
145#125—21no—
146C22⋊Q8—21no—
147C22⋊Q8—21no—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8—21no—
152C4×Q8C2×Dic312yes—
153C4×Q8#6712yes—
154#117—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—62no—
158C2×C4⋊C4#6711yescharacteristic
159C2×C42—21no—
160C2×C42—32no—
161C2×C42—61no—
162C2×C42C22×D311yescharacteristic
163#106—22no—
164C4⋊C8—61no—
165C4⋊C8C2×Dic312yes—
166#103—22no—
167#103—22no—
168C22⋊C8—21no—
169C22⋊C8—61no—
170#97—21no—
171#97—61no—
172#97—61no—
173C4×C8—61no—
174#95—31no—
175#95—31no—
176#95—62no—
177#95—62no—
178#95—62no—
179#95—61no—
180#95—61no—
181C23×C3#12011yescharacteristic
182C3×Q8—22no—
183C3×Q8—22no—
184C3×Q8—42no—
185C3×Q8—42no—
186C3×Q8—81no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4C2×C22⋊C411yescharacteristic
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12311yescharacteristic
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—42no—
207C2×Dic3—41no—
208C2×Dic3—42no—
209C2×Dic3—41no—
210C2×Dic3—42no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C3×C8—42no—
214C3⋊C8—42no—
215C3⋊C8—81no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—42no—
220C2×Q8#21712yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—62no—
226C22×C4D3×D411yescharacteristic
227C22×C4D3×D411yescharacteristic
228C22×C4C2×#6711yescharacteristic
229Q16—44no—
230M16—62no—
231M16—121no—
232C2×C8—22no—
233C2×C8—121no—
234C2×C8—121no—
235C4⋊C4—22no—
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—122no—
245C4⋊C4—242no—
246C4⋊C4#21712yes—
247C22⋊C4—41no—
248C42—22no—
249C42—22no—
250C42—41no—
251C42—122no—
252C42—122no—
253C42#23711yescharacteristic
254C42#23711yescharacteristic
255C42C22×Dic311yescharacteristic
256C42C2×#6711yescharacteristic
257C22×C3—21no—
258C22×C3—21no—
259C22×C3#38611yescharacteristic
260C22×C3#45511yescharacteristic
261C22×C3#45611yescharacteristic
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—42no—
268C3×C4—42no—
269C3×C4—81no—
270C3×C4—81no—
271C3×C4#42112yes—
272Dic3—81no—
273Dic3—82no—
274Dic3—81no—
275C23#93711yescharacteristic
276Q8—22no—
277Q8—22no—
278Q8—42no—
279Q8—42no—
280Q8—81no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—122no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4D4×Dic311yescharacteristic
302C2×C4D4×Dic311yescharacteristic
303C2×C4#93511yescharacteristic
304C2×C4#93911yescharacteristic
305C2×C4#93911yescharacteristic
306C2×C4C2×#21711yescharacteristic
307C8—42no—
308C8—122no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#188211yescharacteristic
313C2×C3#189711yescharacteristic
314C2×C3#190511yescharacteristic
315C22—21no—
316C22—21no—
317C22#520211yescharacteristic
318C22#520511yescharacteristic
319C22#522011yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—42no—
327C4—81no—
328C4—81no—
329C4—241no—
330C4—242no—
331C4—241no—
332C4#523612yes—
333C3#932811yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#7171211yescharacteristic
337C2#7210011yescharacteristic
338C2#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#9328
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
12. C3×#475C4semidirect
13. C3×#462C4semidirect
81. #475Dic3semidirect
82. #462Dic3semidirect
333. C3#9328semidirect

Maximal subgroups

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

subgroupindex
C3×#31132
#824542
#718412
#718362
#716232
#677502
#677482
#93283