Finite Group Databasebeta

← #10706014

Subgroups — #10706014

Order 768.

382 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 79 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#82503C211yesmaximal, characteristic
2C3×#1467C211yesmaximal, characteristic, complemented
3#71637C211yesmaximal, characteristic, complemented
4#67457C212yesmaximal, complemented
5#67456C212yesmaximal
6#11833—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#875C411yescharacteristic
9C3×#432—21no—
10#5231—21no—
11C2×#836—21no—
12#4993—21no—
13#4991C411yescharacteristic
14#4991C2211yescharacteristic
15#4979C411yescharacteristic
16#4979C2211yescharacteristic
17#4940—21no—
18#4940—21no—
19#4937C411yescharacteristic
20#4924—21no—
21C3×#324—21no—
22C3×#324C2212yes—
23#4530C2212yes—
24#2632—31no—
25#1801—31no—
26#1467D311yescharacteristic, complemented
27#1354—32no—
28#1345—32no—
29C22×C3×Q8D411yescharacteristic
30C22×D12C2×C411yescharacteristic
31C22×Dic6C2×C411yescharacteristic
32C3×C4×Q8—41no—
33C3×C4×Q8—41no—
34C2×C3×C4⋊C4C2×C411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C4⋊C4C2311yescharacteristic
37C2×C3×C42C2×C411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×(C2×D3)⋊C4—21no—
41C2×(C2×D3)⋊C4—21no—
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×#208—22no—
49#875—41no—
50#875—41no—
51C3×C4⋊C8—41no—
52C3×C22⋊C8—22no—
53C3×#97—41no—
54#836—42no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95C2×C411yescharacteristic
60C3⋊#95C2×C411yescharacteristic
61C4⋊(C3⋊C8)—41no—
62C4⋊(C3⋊C8)—41no—
63C2×#124—31no—
64C2×C4×Q8C2×D311yescharacteristic
65#432—21no—
66#423—61no—
67C2×#101—61no—
68#399—61no—
69#395—31no—
70#395—31no—
71#394—31no—
72#394—31no—
73#393—61no—
74#388—61no—
75#388—61no—
76#383—31no—
77#343—32no—
78#324—21no—
79#324C2×D312yes—
80C23×D3—21no—
81C2×C3×Q8—21no—
82C2×C3×Q8—21no—
83C2×C3×Q8—41no—
84C2×C3×Q8—41no—
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22×C411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×Dic3—21no—
93C22×Dic3—21no—
94C22×Dic3—21no—
95C2×D12—41no—
96C2×D12—81no—
97C2×Dic6—41no—
98C2×Dic6—81no—
99C2×C3×C8—42no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C42—21no—
104C3×C42—21no—
105C3×C42—21no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42—21no—
109(C2×D3)⋊C4—42no—
110(C2×D3)⋊C4—42no—
111(C2×D3)⋊C4—41no—
112(C2×D3)⋊C4—41no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—81no—
116(C2×D3)⋊C4—81no—
117Dic3⋊C4—81no—
118Dic3⋊C4—81no—
119#208—42no—
120#208—42no—
121C2×C3⋊C8—42no—
122C22×Q8—31no—
123C22×Q8D1211yescharacteristic
124C22×D4—31no—
125C2×M16—62no—
126#124—121no—
127#124—121no—
128C4×Q8—41no—
129C4×Q8—41no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4C4×D311yescharacteristic
133C2×C4⋊C4#6711yescharacteristic
134C2×C4⋊C4C22×D311yescharacteristic
135C2×C22⋊C4—61no—
136C2×C22⋊C4—61no—
137C2×C22⋊C4—61no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4—61no—
141C2×C42C4×D311yescharacteristic
142C2×C42D1211yescharacteristic
143C2×C42#6711yescharacteristic
144C4⋊C8—41no—
145C4⋊C8—121no—
146C4⋊C8—121no—
147#101—122no—
148C22⋊C8—22no—
149#97—41no—
150#95—31no—
151#95—31no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156C23×C3C2×C22⋊C411yescharacteristic
157C22×D3—41no—
158C22×D3—41no—
159C22×D3—41no—
160C22×D3—41no—
161C22×D3—41no—
162C22×D3—41no—
163C22×D3—41no—
164C3×Q8—42no—
165C3×Q8—42no—
166C2×C3×C4—21no—
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—21no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4#9911yescharacteristic
182C2×C3×C4#9911yescharacteristic
183C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
184C2×C3×C4#13711yescharacteristic
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—
194D12—82no—
195Dic6—82no—
196C3×C8—82no—
197C3⋊C8—82no—
198C24—61no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—121no—
204C2×Q8—241no—
205C2×D4—121no—
206C2×D4—241no—
207C22×C4—61no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4(C2×D3)⋊C411yescharacteristic
211C22×C4(C2×D3)⋊C411yescharacteristic
212C22×C4(C2×D3)⋊C411yescharacteristic
213C22×C4(C2×D3)⋊C411yescharacteristic
214C22×C4C2×C4×D311yescharacteristic
215C22×C4C2×D1211yescharacteristic
216C22×C4C2×#6711yescharacteristic
217M16—122no—
218M16—122no—
219C2×C8—42no—
220C2×C8—122no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—241no—
225C4⋊C4—241no—
226C22⋊C4—122no—
227C22⋊C4—122no—
228C22⋊C4—121no—
229C22⋊C4—121no—
230C22⋊C4—121no—
231C22⋊C4—121no—
232C22⋊C4—241no—
233C22⋊C4—241no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3C2×#9911yescharacteristic
243C22×C3#41711yescharacteristic
244C22×C3#42111yescharacteristic
245C2×D3—81no—
246C2×D3—81no—
247C2×D3—81no—
248C2×D3—81no—
249C2×D3—81no—
250C2×D3—81no—
251C2×D3—81no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261Dic3—81no—
262Dic3—81no—
263Dic3—81no—
264C23—121no—
265C23—121no—
266C23—121no—
267C23—121no—
268C23—121no—
269C23—121no—
270C23—121no—
271C23C2×(C2×D3)⋊C411yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—242no—
275D4—242no—
276C2×C4—21no—
277C2×C4—21no—
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—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4#80611yescharacteristic
301C2×C4#80611yescharacteristic
302C2×C4#90811yescharacteristic
303C2×C4#90911yescharacteristic
304C8—82no—
305C8—242no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3#151311yescharacteristic
309C2×C3#212211yescharacteristic
310C2×C3#213311yescharacteristic
311D3—161no—
312C22—21no—
313C22—21no—
314C22—241no—
315C22—241no—
316C22—241no—
317C22—241no—
318C22—241no—
319C22—241no—
320C22—241no—
321C22C2×#80611yescharacteristic
322C22#513311yescharacteristic
323C22#513911yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—241no—
334C4—241no—
335C4—241no—
336C3#1183311yescharacteristic, complemented, Sylow
337C2—21no—
338C2—21no—
339C2—481no—
340C2#6997911yescharacteristic
341C2#7005711yescharacteristic
342C2#7012811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1467C2
socleC22×C3C2×#99
prime radicalC22×D12C2×C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1467D3
hypocenterC3#11833
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. C3×#1467C2semidirect
3. #71637C2semidirect
4. #67457C2semidirect
26. #1467D3semidirect
336. C3#11833semidirect

Maximal subgroups

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

subgroupindex
#825032
C3×#14672
#716372
#674572
#674562
#118333