Finite Group Databasebeta

← #10705993

Subgroups — #10705993

Order 768.

416 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C3×#2398C211yesmaximal, characteristic
2#71637C211yesmaximal, characteristic, complemented
3#71625C211yesmaximal, characteristic, complemented
4#67652C212yesmaximal, complemented
5#67647C212yesmaximal
6#11829—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C3×C4×D4C411yescharacteristic
9C3×#400C411yescharacteristic
10C3×#400C2211yescharacteristic
11C3×#397—21no—
12C3×#394C411yescharacteristic
13C3×#394C2211yescharacteristic
14C3×#393—21no—
15C3×C4×C22⋊C4—21no—
16C3×C4×C22⋊C4—21no—
17C3×#376C411yescharacteristic
18C2×#836—21no—
19#4926—21no—
20#4924—21no—
21#4530C2212yes—
22#4528—21no—
23#4528C2212yes—
24#2398D311yescharacteristic
25#1801—31no—
26#1467—31no—
27#1353—32no—
28#1345—32no—
29C22×C3×Q8D411yescharacteristic
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4—21no—
32C3×C4×Q8—41no—
33C3×C4×Q8—41no—
34C3×C4×D4—41no—
35C3×C4×D4—41no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4C2311yescharacteristic
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C42—21no—
46C2×C3×C42C2×C411yescharacteristic
47C2×C3×C42D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×#208—22no—
50C3×#95—21no—
51C3×#95—21no—
52C3×#95—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95C2×C411yescharacteristic
56C3×#95C2×C411yescharacteristic
57#836—42no—
58#830—22no—
59C4⋊(C3⋊C8)—41no—
60C4⋊(C3⋊C8)—41no—
61C4⋊(C3⋊C8)—41no—
62#803—41no—
63C2×C4×Q8C2×D311yescharacteristic
64C2×C4×D4Dic311yescharacteristic
65#432—61no—
66#423—61no—
67C2×#101—61no—
68#400Dic311yescharacteristic
69#400C2×D311yescharacteristic
70#397—21no—
71#394Dic311yescharacteristic
72#394C2×D311yescharacteristic
73#393—21no—
74C4×C22⋊C4—21no—
75C4×C22⋊C4—21no—
76#376Dic311yescharacteristic
77#343—32no—
78#324—32no—
79#324—61no—
80C24×C3—21no—
81C2×C3×Q8—21no—
82C2×C3×Q8—21no—
83C2×C3×Q8—41no—
84C2×C3×Q8—41no—
85C2×C3×D4—41no—
86C2×C3×D4—42no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—41no—
93C22×C3×C4—41no—
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C22⋊C4—42no—
106C3×C22⋊C4—42no—
107C3×C22⋊C4—42no—
108C3×C22⋊C4—42no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—41no—
111C3×C42—21no—
112C3×C42—21no—
113C3×C42—21no—
114C3×C42—21no—
115C3×C42—21no—
116C3×C42—21no—
117C3×C42—42no—
118#208—42no—
119#208—42no—
120C2×C3⋊C8—42no—
121C2×C3⋊C8—42no—
122C22×Q8#6711yescharacteristic
123C22×D4C2×Dic311yescharacteristic
124C23×C4—21no—
125C2×M16—62no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×D4—41no—
129C4×D4—41no—
130C2×C4⋊C4C2×Dic311yescharacteristic
131C2×C4⋊C4C2×Dic311yescharacteristic
132C2×C4⋊C4#6711yescharacteristic
133C2×C4⋊C4C22×D311yescharacteristic
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C42—21no—
140C2×C42C2×Dic311yescharacteristic
141C2×C42#6711yescharacteristic
142C2×C42#6711yescharacteristic
143C4⋊C8—121no—
144C4⋊C8—121no—
145C4⋊C8—121no—
146#101—122no—
147C22⋊C8—62no—
148#97—121no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95C2×Dic311yescharacteristic
155#95C2×Dic311yescharacteristic
156C23×C3—22no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3C2×C22⋊C411yescharacteristic
164C3×Q8—42no—
165C3×Q8—42no—
166C3×D4—42no—
167C3×D4—42no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—22no—
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—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—82no—
193C2×C3×C4—82no—
194C2×C3×C4#9911yescharacteristic
195C2×C3×C4#9911yescharacteristic
196C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
197C2×C3×C4#13711yescharacteristic
198C3⋊C8—82no—
199C3⋊C8—82no—
200C24—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×D4—41no—
206C2×D4—42no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4#21711yescharacteristic
215C22×C4#21711yescharacteristic
216C22×C4#21711yescharacteristic
217C22×C4#21711yescharacteristic
218C22×C4C22×Dic311yescharacteristic
219C22×C4C2×#6711yescharacteristic
220C22×C4C2×#6711yescharacteristic
221M16—122no—
222M16—122no—
223C2×C8—122no—
224C2×C8—122no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—42no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—42no—
245C22×C3—42no—
246C22×C3—42no—
247C22×C3—81no—
248C22×C3—81no—
249C22×C3—81no—
250C22×C3—81no—
251C22×C3C2×#9911yescharacteristic
252C22×C3#41711yescharacteristic
253C22×C3#42111yescharacteristic
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4—81no—
265C3×C4#35512yes—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23C2×#21711yescharacteristic
274Q8—42no—
275Q8—42no—
276D4—42no—
277D4—42no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4#83411yescharacteristic
305C2×C4#83411yescharacteristic
306C2×C4#94911yescharacteristic
307C2×C4#95111yescharacteristic
308C8—242no—
309C8—242no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—82no—
313C2×C3#151311yescharacteristic
314C2×C3#212011yescharacteristic
315C2×C3C2×#35511yescharacteristic
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C2×#83411yescharacteristic
326C22#523211yescharacteristic
327C22#523611yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4#454212yes—
340C3#1182911yescharacteristic, complemented, Sylow
341C2—21no—
342C2—21no—
343C2—82no—
344C2#7170911yescharacteristic
345C2#7171611yescharacteristic
346C2C2×#454211yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2398C2
socleC22×C3C2×#99
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2398D3
hypocenterC3#11829
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2398

Complements

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

NH ≅ G/Ntype
2. #71637C2semidirect
3. #71625C2semidirect
4. #67652C2semidirect
340. C3#11829semidirect

Maximal subgroups

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

subgroupindex
C3×#23982
#716372
#716252
#676522
#676472
#118293