Finite Group Databasebeta

← #10705696

Subgroups — #10705696

Order 768.

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

normal subgroups only

This group has 446 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#82176C211yesmaximal, characteristic
2C2×C3×#343C211yesmaximal, characteristic
3#71552C211yesmaximal, characteristic
4#67496C212yesmaximal, complemented
5#67495C212yesmaximal, complemented
6#11764—31nomaximal, Sylow
7C22×C3×M16—21no—
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#926C411yescharacteristic
10#5211—21no—
11#5211C411yescharacteristic
12#5210—21no—
13#5207—21no—
14#4986C411yescharacteristic
15#4986C2211yescharacteristic
16#4981C411yescharacteristic
17#4981C2211yescharacteristic
18#4978—21no—
19#4978—21no—
20C3×#343—21no—
21C3×#343C2212yes—
22#4528—21no—
23#4528C2212yes—
24#2484—31no—
25C2×#343D311yescharacteristic
26#1463—31no—
27#1353—32no—
28#1346—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C2×C3×M16—24no—
33C2×C3×M16—22no—
34C2×C3×M16—41no—
35C2×C3×M16—42no—
36C22×C3×C8—41no—
37C2×C3×C4⋊C4—41no—
38C2×C3×C4⋊C4—41no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×#217—21no—
44C2×#217—21no—
45#926—41no—
46#926—41no—
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C2×#208—41no—
53C3⋊#95—21no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95C2×C411yescharacteristic
61C3⋊#95C2×C411yescharacteristic
62#830—22no—
63#830—41no—
64C22×M16—21no—
65C2×#117—31no—
66C22×C4⋊C4C2×D311yescharacteristic
67#407—61no—
68#400—31no—
69#400—31no—
70#398—31no—
71#398—31no—
72#387—61no—
73#387—61no—
74#385—61no—
75#381—31no—
76#381—61no—
77#343—21no—
78#343C2×D312yes—
79#324—32no—
80#324—61no—
81C24×C3C22⋊C411yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C3×M16—24no—
104C3×M16—44no—
105C3×M16—42no—
106C2×C3×C8—44no—
107C2×C3×C8—42no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112#217—42no—
113#217—41no—
114#217—41no—
115C4⋊Dic3—41no—
116C4⋊Dic3—41no—
117Dic3⋊C4—42no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120C4×Dic3—42no—
121#208—82no—
122C2×C3⋊C8—42no—
123C23×C4C4×D311yescharacteristic
124C23×C4D1211yescharacteristic
125C23×C4#6711yescharacteristic
126C2×M16—24no—
127C2×M16—22no—
128C2×M16—41no—
129C2×M16—42no—
130C2×M16—121no—
131C22×C8—41no—
132#117—121no—
133#117—121no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4C4×D311yescharacteristic
141C2×C4⋊C4D1211yescharacteristic
142C2×C4⋊C4#6711yescharacteristic
143C2×C4⋊C4C22×D311yescharacteristic
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C42—61no—
147C22⋊C8—62no—
148C22⋊C8—121no—
149#95—31no—
150#95—31no—
151#95—61no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158C23×C3—22no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—41no—
162C23×C3#9911yescharacteristic
163C23×C3#9911yescharacteristic
164C23×C3C2×C22⋊C411yescharacteristic
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
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—21no—
176C2×C3×C4—21no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—81no—
185C2×C3×C4#10412yes—
186C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
187C2×C3×C4#13711yescharacteristic
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—42no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C3×C8—44no—
200C3⋊C8—82no—
201C24(C2×D3)⋊C411yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4(C2×D3)⋊C411yescharacteristic
218C22×C4(C2×D3)⋊C411yescharacteristic
219C22×C4(C2×D3)⋊C411yescharacteristic
220C22×C4C2×C4×D311yescharacteristic
221C22×C4C2×D1211yescharacteristic
222C22×C4C2×#6711yescharacteristic
223M16—24no—
224M16—44no—
225M16—42no—
226M16—242no—
227C2×C8—44no—
228C2×C8—42no—
229C2×C8—122no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C4⋊C4—122no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C22⋊C4—122no—
240C22⋊C4—121no—
241C22⋊C4—121no—
242C42—122no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3#35212yes—
253C22×C3C2×#9911yescharacteristic
254C22×C3#41711yescharacteristic
255C22×C3C2×#10411yescharacteristic
256C3×C4—21no—
257C3×C4—22no—
258C3×C4—21no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—41no—
263Dic3—81no—
264Dic3—81no—
265Dic3—81no—
266Dic3—81no—
267C23—21no—
268C23—21no—
269C23—22no—
270C23—41no—
271C23#80611yescharacteristic
272C23#80611yescharacteristic
273C23C2×(C2×D3)⋊C411yescharacteristic
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—122no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4#80512yes—
306C2×C4#93211yescharacteristic
307C2×C4#94211yescharacteristic
308C8—44no—
309C8—242no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—41no—
314C2×C3#150411yescharacteristic
315C2×C3C2×#35211yescharacteristic
316C2×C3#213311yescharacteristic
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22#447612yes—
327C22C2×#80511yescharacteristic
328C22C2×#80611yescharacteristic
329C22#496611yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—241no—
338C4—241no—
339C4—241no—
340C4—241no—
341C3#1176411yescharacteristic, complemented, Sylow
342C2—21no—
343C2—21no—
344C2—22no—
345C2—41no—
346C2#6939811yescharacteristic
347C2C2×#447611yescharacteristic
348C2#7012811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC2×C3×#343C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenterC2×#343D3
hypocenterC3#11764
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
4. #67496C2semidirect
5. #67495C2semidirect
341. C3#11764semidirect

Maximal subgroups

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

subgroupindex
#821762
C2×C3×#3432
#715522
#674962
#674952
#117643