Finite Group Databasebeta

← #10694896

Subgroups — #10694896

Order 768.

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

normal subgroups only

This group has 496 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
1C2×C3×#467C211yesmaximal, characteristic
2#71977C211yesmaximal, characteristic
3C8×#217C211yesmaximal, characteristic
4#71691C212yesmaximal
5#71688C212yesmaximal
6#9814—31nomaximal, Sylow
7C22×C3×Q16C411yescharacteristic, complemented
8C23×C3×C8C2211yescharacteristic
9C2×C3×C22⋊Q8C2212yes—
10C3×#467—21no—
11C3×#467—21no—
12C3×#467—21no—
13C3×#467—21no—
14C2×C3×#107C411yescharacteristic, complemented
15C2×C3×#103C412yescomplemented
16C4×#217C2211yescharacteristic
17#5207C2212yes—
18C2×#830C2211yescharacteristic
19C2×#819—21no—
20C2×C8×Dic3—21no—
21#4552—21no—
22#4550—21no—
23#4538—24no—
24#4528—24no—
25C2×#467D311yescharacteristic
26#1810—31no—
27C8×C22⋊C4—31no—
28#1508—32no—
29#1472—32no—
30C22×C3×Q8C2×C412yes—
31C23×C3×C4C2311yescharacteristic
32C2×C3×Q16—21no—
33C2×C3×Q16—21no—
34C2×C3×Q16—42no—
35C22×C3×C8—42no—
36C22×C3×C8C2×C411yescharacteristic
37C22×C3×C8C2×C411yescharacteristic
38C22×C3×C8D411yescharacteristic
39C22×C3×C8D411yescharacteristic
40C22×C3×C8D411yescharacteristic
41C22×C3×C8D411yescharacteristic
42C3×C22⋊Q8—22no—
43C3×C22⋊Q8—22no—
44C3×C22⋊Q8—22no—
45C3×C22⋊Q8—22no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4C2×C412yes—
48C2×C3×C22⋊C4—22no—
49C2×#217—21no—
50C2×C4×Dic3—21no—
51C2×#208—22no—
52C22×C3⋊C8—21no—
53C3×#107—21no—
54C3×#107—21no—
55C3×#103—22no—
56C3×#103—22no—
57C3⋊#95D412yes—
58#830—21no—
59#830—22no—
60#830—22no—
61#830D412yes—
62#819—42no—
63C8×Dic3—42no—
64C22×Q16Dic311yescharacteristic, complemented
65C23×C8C2×D311yescharacteristic
66C2×C22⋊Q8C2×D312yes—
67#467—21no—
68#467—21no—
69#467—21no—
70#467—21no—
71C2×#108—61no—
72C2×#107Dic311yescharacteristic, complemented
73C2×#103Dic312yescomplemented
74#407—32no—
75C2×C22⋊C8—31no—
76C2×C4×C8—61no—
77C4×C22⋊C4—31no—
78#341—61no—
79#336—61no—
80#328—64no—
81#324—64no—
82C24×C3#4111yescharacteristic
83C2×C3×Q8—22no—
84C2×C3×Q8—22no—
85C2×C3×Q8—42no—
86C2×C3×Q8—42no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C22×Dic3—22no—
98C3×Q16—42no—
99C3×Q16—42no—
100C2×C3×C8—22no—
101C2×C3×C8—22no—
102C2×C3×C8—22no—
103C2×C3×C8—41no—
104C2×C3×C8—41no—
105C2×C3×C8—41no—
106C2×C3×C8C22⋊C411yescharacteristic
107C2×C3×C8C22⋊C411yescharacteristic
108C2×C3×C8C22⋊C411yescharacteristic
109C2×C3×C8C22⋊C411yescharacteristic
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—42no—
113C3×C4⋊C4—42no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—42no—
116#217—24no—
117C4×Dic3—41no—
118C4×Dic3—41no—
119#208—42no—
120#208—42no—
121C2×C3⋊C8—22no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8—42no—
125C22×Q8C2×Dic312yes—
126C23×C4C22×D311yescharacteristic
127C2×Q16—21no—
128C2×Q16—21no—
129C2×Q16—42no—
130C2×M16—62no—
131C22×C8—42no—
132C22×C8—61no—
133C22×C8C2×Dic311yescharacteristic
134C22×C8C2×Dic311yescharacteristic
135C22×C8#6711yescharacteristic
136C22×C8#6711yescharacteristic
137C22×C8#6711yescharacteristic
138C22×C8#6711yescharacteristic
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C2×Dic312yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—61no—
147C2×C42—61no—
148#108—122no—
149#107—21no—
150#107—21no—
151#103—22no—
152#103—22no—
153C22⋊C8—32no—
154C22⋊C8—61no—
155C22⋊C8—62no—
156C22⋊C8—62no—
157C4×C8—122no—
158#95—32no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—41no—
162C23×C3C4×D411yescharacteristic
163C23×C3#12111yescharacteristic
164C23×C3#12411yescharacteristic
165C3×Q8—42no—
166C3×Q8—42no—
167C3×Q8—82no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—44no—
180C2×C3×C4—82no—
181C2×C3×C4C2×C22⋊C411yescharacteristic
182C2×C3×C4C4×D411yescharacteristic
183C2×C3×C4#12111yescharacteristic
184C2×C3×C4#12711yescharacteristic
185C2×Dic3—24no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C3×C8—21no—
189C3×C8—21no—
190C3×C8—21no—
191C3×C8#9912yes—
192C3⋊C8—42no—
193C3⋊C8—82no—
194C24#23711yescharacteristic
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×Q8—42no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—62no—
204C22×C4D3×D411yescharacteristic
205C22×C4D3×D411yescharacteristic
206C22×C4#23711yescharacteristic
207C22×C4C22×Dic311yescharacteristic
208C22×C4C2×#6711yescharacteristic
209C22×C4C2×#6711yescharacteristic
210Q16—42no—
211Q16—42no—
212M16—122no—
213M16—122no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—62no—
221C2×C8—121no—
222C2×C8—121no—
223C2×C8—122no—
224C2×C8#21711yescharacteristic
225C2×C8#21711yescharacteristic
226C2×C8#21711yescharacteristic
227C2×C8#21711yescharacteristic
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—64no—
235C42—121no—
236C42—121no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—44no—
245C22×C3#39011yescharacteristic
246C22×C3#44312yes—
247C22×C3#47312yes—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—82no—
252C3×C4—82no—
253C3×C4C2×#9911yescharacteristic
254C3×C4#41011yescharacteristic
255Dic3—44no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23D4×Dic311yescharacteristic
260C23#93611yescharacteristic
261C23#93811yescharacteristic
262Q8—42no—
263Q8—42no—
264Q8—82no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—64no—
278C2×C4—82no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4D4×Dic311yescharacteristic
282C2×C4#93811yescharacteristic
283C2×C4#94011yescharacteristic
284C2×C4C2×#21711yescharacteristic
285C8—21no—
286C8—21no—
287C8—21no—
288C8—122no—
289C8—242no—
290C8#83412yes—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—41no—
294C2×C3#196411yescharacteristic
295C2×C3#197411yescharacteristic
296C2×C3#197911yescharacteristic
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—44no—
305C22#519212yes—
306C22#519612yes—
307C22#522111yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—82no—
312C4—82no—
313C4—124no—
314C4C2×#83411yescharacteristic
315C4#524011yescharacteristic
316C3#981411yescharacteristic, complemented, Sylow
317C2—22no—
318C2—22no—
319C2—41no—
320C2#7168611yescharacteristic
321C2#7206911yescharacteristic
322C2#7208411yescharacteristic
subgroupG/H
centerC22#5221
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#467C2
socleC22×C3#390
prime radicalC24×C3#41
cosocleC23×C3×C4C23
hypercenterC2×#467D3
hypocenterC3#9814
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
7. C22×C3×Q16C4semidirect
14. C2×C3×#107C4semidirect
15. C2×C3×#103C4semidirect
64. C22×Q16Dic3semidirect
72. C2×#107Dic3semidirect
73. C2×#103Dic3semidirect
316. C3#9814semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4672
#719772
C8×#2172
#716912
#716882
#98143