Finite Group Databasebeta

← #10693018

Subgroups — #10693018

Order 768.

518 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 93 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#82254C211yesmaximal, characteristic
2#73445C211yesmaximal, characteristic
3C3×#1826C211yesmaximal, characteristic
4#71590C211yesmaximal, characteristic
5#69971C211yesmaximal, characteristic
6#67682C211yesmaximal, characteristic
7#67616C211yesmaximal, characteristic
8#9482—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5211C2211yescharacteristic
14C4×#217C2211yescharacteristic
15#5207C2211yescharacteristic
16#4985C412yescomplemented
17#4983C412yescomplemented
18#4946—21no—
19#4943—21no—
20#4943—21no—
21#4942—21no—
22#4933—22no—
23#4933—22no—
24#4652—22no—
25C3×#339—22no—
26#4532—22no—
27#4469—22no—
28#2484—31no—
29#2313—31no—
30#1826—31no—
31#1826D311yescharacteristic
32#1463—31no—
33#1284—31no—
34#1284—31no—
35C23×C3×C4C2311yescharacteristic
36C23×Dic3D411yescharacteristic
37C23×Dic3D411yescharacteristic
38C2×C3×M16—21no—
39C3×#117—21no—
40C3×#117—21no—
41C3×#117—21no—
42C3×#117—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C42—21no—
48C2×C3×C42D412yes—
49C2×#217—21no—
50C2×#217—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—22no—
56C2×Dic3⋊C4—22no—
57C2×Dic3⋊C4C2×C412yes—
58C2×Dic3⋊C4D412yes—
59C2×C4×Dic3D412yes—
60C2×#208—21no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×#95—21no—
64C3⋊#95—22no—
65C3⋊#95—24no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95C2×C412yes—
72#830—21no—
73#830—21no—
74C2×#117C2×D311yescharacteristic
75C22×C4⋊C4—31no—
76#407—31no—
77#407C2×D311yescharacteristic
78#400—32no—
79#398—32no—
80#387—62no—
81#387—62no—
82#387—62no—
83#385—61no—
84#381—31no—
85#381—61no—
86#381—61no—
87#380—61no—
88C4×C22⋊C4—31no—
89C4×C22⋊C4C2×D311yescharacteristic
90#339—22no—
91#339—62no—
92#324—62no—
93C24×C3C2×D411yescharacteristic
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—42no—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—22no—
109C22×Dic3—22no—
110C22×Dic3—42no—
111C22×Dic3—42no—
112C22×Dic3C22⋊C412yes—
113C22×Dic3C22⋊C412yes—
114C3×M16—42no—
115C2×C3×C8—41no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C22⋊C4—42no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—41no—
123C3×C42—42no—
124C3×C42—41no—
125C3×C42—41no—
126#217—42no—
127#217—42no—
128Dic3⋊C4—22no—
129Dic3⋊C4—22no—
130Dic3⋊C4—22no—
131Dic3⋊C4—22no—
132Dic3⋊C4—44no—
133Dic3⋊C4—82no—
134C4×Dic3—42no—
135#208—42no—
136C2×C3⋊C8—41no—
137C23×C4—31no—
138C23×C4—31no—
139C23×C4C22×D311yescharacteristic
140C2×M16—21no—
141C2×M16—61no—
142#117—21no—
143#117—21no—
144#117—21no—
145#117—21no—
146C2×C4⋊C4—32no—
147C2×C4⋊C4—32no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—62no—
153C2×C4⋊C4—62no—
154C2×C4⋊C4C4×D311yescharacteristic
155C2×C4⋊C4C4×D311yescharacteristic
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C42—21no—
161C2×C42—32no—
162C2×C42#6712yes—
163C22⋊C8—21no—
164C22⋊C8—21no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167#95—21no—
168#95—32no—
169#95—62no—
170#95—62no—
171#95—62no—
172#95—62no—
173#95—64no—
174#95—61no—
175#95—61no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—41no—
179C23×C3C2×C22⋊C411yescharacteristic
180C23×C3#12011yescharacteristic
181C23×C3#12311yescharacteristic
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×Dic3—22no—
200C2×Dic3—22no—
201C2×Dic3—22no—
202C2×Dic3—22no—
203C2×Dic3—42no—
204C2×Dic3—42no—
205C2×Dic3—42no—
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3—42no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—81no—
214C3×C8—81no—
215C3⋊C8—81no—
216C24D3×D411yescharacteristic
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—32no—
221C22×C4—32no—
222C22×C4—42no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—122no—
230C22×C4—122no—
231C22×C4C2×C4×D311yescharacteristic
232C22×C4#23511yescharacteristic
233C22×C4D3×D411yescharacteristic
234C22×C4D3×D411yescharacteristic
235C22×C4#23711yescharacteristic
236C22×C4C2×#6711yescharacteristic
237M16—42no—
238M16—122no—
239C2×C8—41no—
240C2×C8—121no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—62no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—124no—
250C4⋊C4—242no—
251C22⋊C4—42no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—122no—
255C22⋊C4—122no—
256C42—42no—
257C42—41no—
258C42—41no—
259C42—122no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—44no—
268C22×C3#38611yescharacteristic
269C22×C3#42112yes—
270C22×C3#45812yes—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—82no—
276C3×C4—81no—
277C3×C4—81no—
278Dic3—42no—
279Dic3—42no—
280Dic3—82no—
281Dic3—82no—
282C23—22no—
283C23—22no—
284C23—41no—
285C23D3×C22⋊C411yescharacteristic
286C23#88311yescharacteristic
287C23#93711yescharacteristic
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—62no—
302C2×C4—62no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—122no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—241no—
316C2×C4#67⋊C411yescharacteristic
317C2×C4#88411yescharacteristic
318C2×C4C4×#6711yescharacteristic
319C2×C4#93811yescharacteristic
320C8—81no—
321C8—241no—
322C2×C3—22no—
323C2×C3—22no—
324C2×C3—41no—
325C2×C3#189011yescharacteristic
326C2×C3#189711yescharacteristic
327C2×C3#191211yescharacteristic
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—44no—
336C22#482112yes—
337C22#495611yescharacteristic
338C22#505312yes—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—82no—
344C4—81no—
345C4—81no—
346C4—122no—
347C4—122no—
348C4—242no—
349C4—242no—
350C3#948211yescharacteristic, complemented, Sylow
351C2—22no—
352C2—22no—
353C2—41no—
354C2#6953811yescharacteristic
355C2#7038311yescharacteristic
356C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1826C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1826D3
hypocenterC3#9482
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
16. #4985C4semidirect
17. #4983C4semidirect
350. C3#9482semidirect

Maximal subgroups

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

subgroupindex
#822542
#734452
C3×#18262
#715902
#699712
#676822
#676162
#94823