Finite Group Databasebeta

← #10705711

Subgroups — #10705711

Order 768.

420 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 83 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#82261C211yesmaximal, characteristic
2C3×#1742C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic
4#67496C212yesmaximal, complemented
5#67495C212yesmaximal, complemented
6#11767—31nomaximal, Sylow
7C22×C3×M16—21no—
8C2×C3×#117C2211yescharacteristic
9C2×#870C411yescharacteristic
10#5210—21no—
11#4986C2211yescharacteristic
12#4981C2211yescharacteristic
13#4976—21no—
14#4953C411yescharacteristic
15#4949C411yescharacteristic
16#4945—21no—
17#4933—21no—
18#4932C411yescharacteristic
19#4926—21no—
20C3×#343—21no—
21C3×#343C2212yes—
22#4528C2212yes—
23#4525—21no—
24#2611—31no—
25#1742D311yescharacteristic
26#1464—31no—
27#1353—32no—
28#1346—32no—
29C23×C3×C4D411yescharacteristic
30C2×C3×M16—24no—
31C2×C3×M16—22no—
32C2×C3×M16—41no—
33C2×C3×M16—42no—
34C22×C3×C8—41no—
35C3×#117—41no—
36C3×#117—41no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×C3×C22⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C42C2×C411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×#217—21no—
44C2×#217—21no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52#870—41no—
53#870—41no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95C2×C411yescharacteristic
60C3⋊#95C2×C411yescharacteristic
61#830—22no—
62C4⋊(C3⋊C8)—41no—
63#803—41no—
64C22×M16—21no—
65C2×#125—31no—
66C2×#117C2×D311yescharacteristic
67#432—61no—
68#400—31no—
69#399—31no—
70#398—31no—
71#397—31no—
72#389—61no—
73#388—61no—
74#387—61no—
75#385—61no—
76#376—31no—
77#343—21no—
78#343C2×D312yes—
79#324—32no—
80#323—61no—
81C24×C3C22⋊C411yescharacteristic
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×Dic3—21no—
94C22×Dic3—21no—
95C22×Dic3—21no—
96C22×Dic3—21no—
97C3×M16—24no—
98C3×M16—44no—
99C3×M16—42no—
100C2×C3×C8—44no—
101C2×C3×C8—42no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42—21no—
109C3×C42—21no—
110#217—42no—
111#217—42no—
112C4⋊Dic3—41no—
113C4⋊Dic3—41no—
114Dic3⋊C4—42no—
115Dic3⋊C4—42no—
116Dic3⋊C4—41no—
117Dic3⋊C4—41no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120C4×Dic3—42no—
121C2×C3⋊C8—42no—
122C23×C4#6711yescharacteristic
123C2×M16—24no—
124C2×M16—22no—
125C2×M16—41no—
126C2×M16—42no—
127C22×C8—41no—
128#125—121no—
129#125—121no—
130#117—41no—
131#117—41no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4#6711yescharacteristic
139C2×C4⋊C4C22×D311yescharacteristic
140C2×C22⋊C4—61no—
141C2×C22⋊C4—61no—
142C2×C22⋊C4C4×D311yescharacteristic
143C2×C22⋊C4D1211yescharacteristic
144C2×C42—61no—
145C2×C42C4×D311yescharacteristic
146C2×C42D1211yescharacteristic
147C4⋊C8—121no—
148C22⋊C8—62no—
149#97—121no—
150#95—31no—
151#95—31no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3#10412yes—
162C23×C3C2×C22⋊C411yescharacteristic
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—21no—
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—21no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4#9911yescharacteristic
180C2×C3×C4#9911yescharacteristic
181C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
182C2×C3×C4#13711yescharacteristic
183C2×Dic3—22no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C3×C8—44no—
196C3⋊C8—82no—
197C24(C2×D3)⋊C411yescharacteristic
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—61no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4(C2×D3)⋊C411yescharacteristic
208C22×C4(C2×D3)⋊C411yescharacteristic
209C22×C4(C2×D3)⋊C411yescharacteristic
210C22×C4C2×C4×D311yescharacteristic
211C22×C4C2×D1211yescharacteristic
212C22×C4C2×#6711yescharacteristic
213M16—24no—
214M16—44no—
215M16—42no—
216C2×C8—44no—
217C2×C8—42no—
218C2×C8—122no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—121no—
222C4⋊C4—121no—
223C4⋊C4—122no—
224C4⋊C4—122no—
225C4⋊C4—121no—
226C4⋊C4—121no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—122no—
232C22⋊C4—122no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—122no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—42no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—81no—
245C22×C3C2×#9911yescharacteristic
246C22×C3#41711yescharacteristic
247C22×C3C2×#10411yescharacteristic
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—41no—
256Dic3—42no—
257Dic3—81no—
258Dic3—81no—
259Dic3—81no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23#80512yes—
265C23C2×(C2×D3)⋊C411yescharacteristic
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—62no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4#80611yescharacteristic
295C2×C4#80611yescharacteristic
296C2×C4#93211yescharacteristic
297C2×C4#94211yescharacteristic
298C8—44no—
299C8—242no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—42no—
303C2×C3#150411yescharacteristic
304C2×C3#212311yescharacteristic
305C2×C3#213211yescharacteristic
306C22—21no—
307C22—21no—
308C22—22no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—81no—
313C22C2×#80511yescharacteristic
314C22C2×#80611yescharacteristic
315C22#496611yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—122no—
325C4—241no—
326C4—241no—
327C4—241no—
328C3#1176711yescharacteristic, complemented, Sylow
329C2—21no—
330C2—21no—
331C2—42no—
332C2#6939811yescharacteristic
333C2#7014711yescharacteristic
334C2#7014811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1742C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1742D3
hypocenterC3#11767
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
4. #67496C2semidirect
5. #67495C2semidirect
328. C3#11767semidirect

Maximal subgroups

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

subgroupindex
#822612
C3×#17422
#715722
#674962
#674952
#117673