Finite Group Databasebeta

← #10705398

Subgroups — #10705398

Order 768.

552 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2619C211yesmaximal, characteristic
2#71587C211yesmaximal, characteristic, complemented
3#71571C211yesmaximal, characteristic, complemented
4#67647C214yesmaximal, complemented
5#11699—31nomaximal, Sylow
6C2×C3×#127C411yescharacteristic
7C2×C3×#121—21no—
8C2×C3×#117C2211yescharacteristic
9C22×#208—21no—
10C3×#394—21no—
11C3×#394C2212yes—
12C3×#392—21no—
13C3×#392C412yes—
14C3×#390—21no—
15C3×#382C411yescharacteristic
16#4925—21no—
17#4530—21no—
18#4530C2212yes—
19#4528C2212yes—
20#4525—21no—
21#2619D311yescharacteristic
22#1742—31no—
23#1458—31no—
24#1345—34no—
25C22×C3×D4—22no—
26C22×C3×D4—21no—
27C22×C3×D4—21no—
28C22×C3×D4C2×C412yes—
29C23×C3×C4D411yescharacteristic
30C3×#127—41no—
31C3×#127—41no—
32C3×#121—42no—
33C3×#121—42no—
34C3×#117—41no—
35C3×#117—41no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4D411yescharacteristic
44C2×C3×C42C2×C411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×#208—24no—
47C2×#208—22no—
48C2×#208—41no—
49C2×#208—42no—
50C22×C3⋊C8—41no—
51C3×#95—21no—
52C3×#95C2×C412yes—
53#830—22no—
54C4⋊(C3⋊C8)—41no—
55C4×C3⋊C8—41no—
56C22×M16—61no—
57C2×#127Dic311yescharacteristic
58C2×#121—21no—
59C2×#117C2×D311yescharacteristic
60#431—61no—
61#394—21no—
62#394C2×D312yes—
63#392—21no—
64#392Dic312yes—
65#390—21no—
66#382Dic311yescharacteristic
67#343—32no—
68#343—61no—
69#324—32no—
70#323—61no—
71C24×C3—22no—
72C24×C3C22⋊C411yescharacteristic
73C2×C3×D4—42no—
74C2×C3×D4—42no—
75C2×C3×D4—44no—
76C2×C3×D4—41no—
77C2×C3×D4—41no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C2×C3×D4—82no—
81C2×C3×D4—82no—
82C2×C3×D4—82no—
83C2×C3×D4—82no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4D812yes—
92C22×C3×C4SD1612yes—
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C3×C4⋊C4—41no—
97C3×C4⋊C4—41no—
98C3×C4⋊C4—42no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—41no—
101C3×C22⋊C4—42no—
102C3×C22⋊C4—42no—
103C3×C22⋊C4—82no—
104C3×C22⋊C4—82no—
105C3×C42—21no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42—21no—
109#208—24no—
110#208—44no—
111#208—42no—
112C2×C3⋊C8—42no—
113C2×C3⋊C8—44no—
114C2×C3⋊C8—42no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4C2×Dic312yes—
119C23×C4#6711yescharacteristic
120C2×M16—64no—
121C2×M16—62no—
122C2×M16—121no—
123C2×M16—122no—
124C22×C8—121no—
125#127—41no—
126#127—41no—
127#121—42no—
128#121—42no—
129#117—41no—
130#117—41no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4#6711yescharacteristic
133C2×C4⋊C4C22×D311yescharacteristic
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4C2×Dic311yescharacteristic
138C2×C22⋊C4#6711yescharacteristic
139C2×C42C2×Dic311yescharacteristic
140C2×C42#6711yescharacteristic
141C4⋊C8—121no—
142C22⋊C8—62no—
143C4×C8—121no—
144#95—21no—
145#95C2×Dic312yes—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—41no—
149C23×C3—41no—
150C23×C3—42no—
151C23×C3—42no—
152C23×C3—42no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3—42no—
156C23×C3—42no—
157C23×C3#10212yes—
158C23×C3C2×C22⋊C411yescharacteristic
159C3×D4—84no—
160C3×D4—84no—
161C3×D4—88no—
162C3×D4—82no—
163C3×D4—82no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4#9911yescharacteristic
182C2×C3×C4#9911yescharacteristic
183C2×C3×C4#10212yes—
184C2×C3×C4#10412yes—
185C2×C3×C4C2×D811yescharacteristic
186C2×C3×C4C2×SD1611yescharacteristic
187C3⋊C8—44no—
188C3⋊C8—82no—
189C24—22no—
190C24#21711yescharacteristic
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4—42no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4—82no—
199C2×D4—82no—
200C2×D4—82no—
201C2×D4—82no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4#21312yes—
207C22×C4#21412yes—
208C22×C4#21711yescharacteristic
209C22×C4#21711yescharacteristic
210C22×C4#21711yescharacteristic
211C22×C4C22×Dic311yescharacteristic
212C22×C4C2×#6711yescharacteristic
213C22×C4C2×#6711yescharacteristic
214M16—64no—
215M16—124no—
216M16—122no—
217C2×C8—122no—
218C2×C8—124no—
219C2×C8—122no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—22no—
236C22×C3—42no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—81no—
240C22×C3—82no—
241C22×C3—82no—
242C22×C3—82no—
243C22×C3—82no—
244C22×C3—82no—
245C22×C3—82no—
246C22×C3—82no—
247C22×C3C2×#9911yescharacteristic
248C22×C3C2×#10211yescharacteristic
249C22×C3C2×#10411yescharacteristic
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—82no—
257C3×C4#32712yes—
258C3×C4#32712yes—
259C23—21no—
260C23—21no—
261C23—41no—
262C23—41no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23#83212yes—
271C23C2×#21711yescharacteristic
272D4—84no—
273D4—84no—
274D4—88no—
275D4—82no—
276D4—82no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
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—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4#83212yes—
295C2×C4#83411yescharacteristic
296C2×C4#83411yescharacteristic
297C2×C4#83712yes—
298C2×C4C2×#21311yescharacteristic
299C2×C4C2×#21411yescharacteristic
300C8—124no—
301C8—242no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—42no—
305C2×C3—162no—
306C2×C3C2×#32711yescharacteristic
307C2×C3#212211yescharacteristic
308C2×C3#213411yescharacteristic
309C22—21no—
310C22—21no—
311C22—22no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22C2×#83211yescharacteristic
324C22C2×#83411yescharacteristic
325C22C2×#83711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—82no—
333C4#453712yes—
334C4#453712yes—
335C3#1169911yescharacteristic, complemented, Sylow
336C2—21no—
337C2—21no—
338C2—42no—
339C2—162no—
340C2C2×#453711yescharacteristic
341C2#7185911yescharacteristic
342C2#7186211yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
Fitting, prime radicalC3×#2619C2
socleC22×C3C2×#99
cosocleC2×C3×C4⋊C4C23
hypercenter#2619D3
hypocenterC3#11699
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
2. #71587C2semidirect
3. #71571C2semidirect
4. #67647C2semidirect
335. C3#11699semidirect

Maximal subgroups

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

subgroupindex
C3×#26192
#715872
#715712
#676472
#116993