Finite Group Databasebeta

← #10705809

Subgroups — #10705809

Order 768.

516 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 87 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#82379C211yesmaximal, characteristic
2C3×#1465C211yesmaximal, characteristic
3C4×#834C211yesmaximal, characteristic
4#67451C214yesmaximal, complemented
5#11788—31nomaximal, Sylow
6C2×C3×C4×D4C2211yescharacteristic
7C2×#924—21no—
8C2×#924—21no—
9C2×#869C411yescharacteristic, complemented
10C3×#407—21no—
11#5221—21no—
12C2×#834—21no—
13C4×#217—21no—
14#4975—21no—
15#4949C412yescomplemented
16#4949C2212yes—
17C4×C4⋊Dic3C411yescharacteristic, complemented
18C3×#324—21no—
19C3×#323C2212yes—
20#4526C2212yes—
21#2542—31no—
22C4×#99—31no—
23#1465D311yescharacteristic
24#1352—34no—
25C22×C3×D4D411yescharacteristic
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C22×Dic6—21no—
29C2×C3×M16—41no—
30C3×C4×D4—41no—
31C3×C4×D4—41no—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C22⋊C4C2×C411yescharacteristic
34C2×C3×C22⋊C4C2311yescharacteristic
35C2×C3×C42C2×C411yescharacteristic
36C2×#217—22no—
37C2×#217—22no—
38C2×#217—22no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3C2×C412yes—
42#924—44no—
43#924—44no—
44C2×Dic3⋊C4—22no—
45C2×Dic3⋊C4—22no—
46C2×C4×Dic3—21no—
47C2×C4×Dic3—41no—
48#869—41no—
49#869—41no—
50C3×C22⋊C8—22no—
51C3×C22⋊C8—41no—
52#834—24no—
53C3⋊#95—41no—
54C3⋊#95C2×C412yes—
55C2×C4⋊Q8—31no—
56C2×C22⋊Q8—61no—
57C2×C22⋊Q8—61no—
58C2×C4×D4C2×D311yescharacteristic
59C2×#99—61no—
60#407—21no—
61#399—32no—
62#399—32no—
63#390—61no—
64#384—61no—
65C4×C4⋊C4—31no—
66C4×C22⋊C4—61no—
67#342—32no—
68#324—21no—
69#323C2×D312yes—
70C24×C3C22⋊C411yescharacteristic
71C24×C3C2×D411yescharacteristic
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—41no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—42no—
80C22×C3×C4—42no—
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—42no—
89C2×Dic6—42no—
90C2×Dic6—42no—
91C2×Dic6—41no—
92C2×Dic6—41no—
93C3×M16—82no—
94C2×C3×C8—42no—
95C3×C4⋊C4—41no—
96C3×C22⋊C4—41no—
97C3×C22⋊C4—41no—
98C3×C42—21no—
99C3×C42—21no—
100#217—24no—
101#217—44no—
102#217—44no—
103#217—42no—
104C4⋊Dic3—24no—
105C4⋊Dic3—42no—
106C4⋊Dic3—42no—
107Dic3⋊C4—44no—
108Dic3⋊C4—44no—
109C4×Dic3—42no—
110C4×Dic3—82no—
111C22×Q8—61no—
112C22×D4D1211yescharacteristic
113C23×C4D1211yescharacteristic
114C23×C4#6711yescharacteristic
115C2×M16—41no—
116C4⋊Q8—121no—
117C4⋊Q8—121no—
118C22⋊Q8—124no—
119C22⋊Q8—124no—
120C4×D4—41no—
121C4×D4—41no—
122C2×C4⋊C4—32no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—62no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4#6711yescharacteristic
128C2×C22⋊C4—62no—
129C2×C22⋊C4—62no—
130C2×C22⋊C4—62no—
131C2×C22⋊C4C4×D311yescharacteristic
132C2×C22⋊C4C22×D311yescharacteristic
133C2×C42—61no—
134C2×C42—121no—
135C2×C42C4×D311yescharacteristic
136#99—64no—
137C22⋊C8—22no—
138C22⋊C8—41no—
139#95—32no—
140#95—121no—
141C23×C3—21no—
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—41no—
147C23×C3—41no—
148C23×C3—41no—
149C23×C3—41no—
150C23×C3C2×C22⋊C411yescharacteristic
151C23×C3#13712yes—
152C3×D4—44no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—22no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—81no—
165C2×C3×C4—81no—
166C2×C3×C4#9911yescharacteristic
167C2×C3×C4#9911yescharacteristic
168C2×C3×C4#10412yes—
169C2×Dic3—24no—
170C2×Dic3—42no—
171C2×Dic3—42no—
172C2×Dic3—42no—
173C2×Dic3—42no—
174C2×Dic3—44no—
175C2×Dic3—42no—
176C2×Dic3—82no—
177C2×Dic3—82no—
178Dic6—82no—
179Dic6—82no—
180C3×C8—82no—
181C24(C2×D3)⋊C411yescharacteristic
182C24C2×D1211yescharacteristic
183C2×Q8—122no—
184C2×Q8—122no—
185C2×Q8—121no—
186C2×Q8—121no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—41no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—62no—
197C22×C4—62no—
198C22×C4—122no—
199C22×C4(C2×D3)⋊C411yescharacteristic
200C22×C4(C2×D3)⋊C411yescharacteristic
201C22×C4(C2×D3)⋊C411yescharacteristic
202C22×C4C2×C4×D311yescharacteristic
203C22×C4C2×#6711yescharacteristic
204M16—82no—
205C2×C8—42no—
206C4⋊C4—41no—
207C4⋊C4—64no—
208C4⋊C4—122no—
209C4⋊C4—122no—
210C4⋊C4—124no—
211C4⋊C4—124no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—64no—
215C22⋊C4—124no—
216C22⋊C4—124no—
217C22⋊C4—122no—
218C42—21no—
219C42—21no—
220C42—122no—
221C42—242no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—22no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—81no—
233C22×C3—81no—
234C22×C3C2×#9911yescharacteristic
235C22×C3#41911yescharacteristic
236C22×C3C2×#10411yescharacteristic
237C3×C4—21no—
238C3×C4—22no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4#35312yes—
243Dic3—44no—
244Dic3—84no—
245Dic3—82no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23#90912yes—
256C23C2×(C2×D3)⋊C411yescharacteristic
257Q8—242no—
258Q8—242no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—64no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—124no—
277C2×C4—122no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—242no—
281C2×C4—242no—
282C2×C4#80611yescharacteristic
283C2×C4#80611yescharacteristic
284C2×C4#82512yes—
285C8—82no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—42no—
289C2×C3—81no—
290C2×C3#150811yescharacteristic
291C2×C3#212311yescharacteristic
292C2×C3C2×#35311yescharacteristic
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×#80611yescharacteristic
306C22#512011yescharacteristic
307C22C2×#82511yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—124no—
314C4—244no—
315C4—242no—
316C4#447412yes—
317C3#1178811yescharacteristic, complemented, Sylow
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2#6998011yescharacteristic
323C2C2×#447411yescharacteristic
324C2#7014711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1465C2
socleC22×C3C2×#99
prime radicalC22×C3×D4D4
cosocleC2×C3×C22⋊C4C23
hypercenter#1465D3
hypocenterC3#11788
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
4. #67451C2semidirect
9. C2×#869C4semidirect
15. #4949C4semidirect
17. C4×C4⋊Dic3C4semidirect
317. C3#11788semidirect

Maximal subgroups

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

subgroupindex
#823792
C3×#14652
C4×#8342
#674512
#117883