Finite Group Databasebeta

← #10695057

Subgroups — #10695057

Order 768.

516 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 121 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#83343C211yesmaximal, characteristic
2C3×#1493C211yesmaximal, characteristic, complemented
3#71839C211yesmaximal, characteristic, complemented
4#70512C211yesmaximal, characteristic
5#70490C211yesmaximal, characteristic, complemented
6C4×C8×Dic3C211yesmaximal, characteristic, complemented
7#69735C211yesmaximal, characteristic
8#9843—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C2×#825—22no—
14C2×C8×Dic3—22no—
15#5088C411yescharacteristic, complemented
16#5085C411yescharacteristic, complemented
17#5080C411yescharacteristic, complemented
18#5079C411yescharacteristic, complemented
19#5048—22no—
20C4×#215—22no—
21C42×Dic3C2211yescharacteristic
22#4924C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×Q8⋊C8—22no—
25#4532—22no—
26#4480—22no—
27#4459—22no—
28#3167—31no—
29#1994—31no—
30#1962—31no—
31C42×C8—31no—
32#1566—31no—
33#1493D311yescharacteristic, complemented
34#1492—31no—
35C2×#235—21no—
36C2×C3×M16—21no—
37C22×C3×C8—21no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×#125—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×#117—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#215—22no—
48#930—21no—
49#930—21no—
50C2×C4×Dic3—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3D412yes—
53C2×#208—21no—
54C22×C3⋊C8D412yes—
55#875—21no—
56#874C2×C411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59#869C2×C411yescharacteristic
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C4×C8—21no—
63C3×C4×C8D412yes—
64#835—22no—
65#825—44no—
66C8×Dic3—28no—
67#809—22no—
68#808—22no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4×C3⋊C8C2×C411yescharacteristic
72C4×C3⋊C8C2×C411yescharacteristic
73C4×C3⋊C8D412yes—
74#533C2×D311yescharacteristic
75#532—31no—
76#498—31no—
77#492—31no—
78#487—31no—
79#486—31no—
80#465—62no—
81C4×SD16—62no—
82#423—31no—
83#423C2×D311yescharacteristic
84C2×#104—62no—
85C2×C4×C8—31no—
86C2×C4×C8—62no—
87C2×C4×C8C2×D311yescharacteristic
88C43—31no—
89#339—62no—
90#334—62no—
91Q8⋊C8—22no—
92D4⋊C8—62no—
93C2×C3×Q8—21no—
94C2×C3×Q8—21no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4111yescharacteristic
98C2×#67—41no—
99C22×Dic3—22no—
100#235—42no—
101#235—42no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12—21no—
105C2×C4×D3—41no—
106C2×Dic6—21no—
107C3×M16—42no—
108C2×C3×C8—24no—
109C2×C3×C8—22no—
110C2×C3×C8—41no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C42—41no—
119C3×C42C22×C411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123#215—44no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4×Dic3—24no—
129C4×Dic3—24no—
130C4×Dic3—22no—
131C4×Dic3SD1614yescomplemented
132#208—42no—
133C2×C3⋊C8—24no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8C22⋊C412yes—
136C2×C3⋊C8C22⋊C412yes—
137C2×#41—61no—
138C2×SD16—62no—
139C2×M16—21no—
140C2×M16—61no—
141C22×C8—21no—
142C22×C8—32no—
143C4⋊Q8—31no—
144C4⋊Q8C4×D311yescharacteristic
145C4⋊Q8C4×D311yescharacteristic
146#127—31no—
147#125—21no—
148#124—61no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151#121—61no—
152#121—61no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×D4—61no—
156C4×D4—61no—
157#117—21no—
158C2×C42—32no—
159C2×C42—61no—
160C2×C42—61no—
161C2×C42C22×D311yescharacteristic
162#106—62no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167#104—124no—
168#103—62no—
169#102—62no—
170C4×C8—21no—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—32no—
174C4×C8—68no—
175C4×C8#6712yes—
176C23×C3#12111yescharacteristic
177C22×D3—41no—
178C3×Q8—42no—
179C3×Q8—42no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4C2×C22⋊C411yescharacteristic
185C2×C3×C4C4×D411yescharacteristic
186C2×C3×C4C4×D411yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12411yescharacteristic
189C2×C3×C4#12711yescharacteristic
190C2×C3×C4C2×SD1612yes—
191C2×C3×C4#13512yes—
192#67—82no—
193C2×Dic3—24no—
194C2×Dic3—24no—
195C2×Dic3—24no—
196C2×Dic3—41no—
197D12—42no—
198D12—42no—
199D12—42no—
200C4×D3—81no—
201C4×D3—81no—
202Dic6—42no—
203C3×C8—24no—
204C3×C8—81no—
205C3⋊C8—22no—
206C3⋊C8—81no—
207C3⋊C8#10414yescomplemented
208#41—122no—
209#41—122no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—61no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—121no—
217C22×C4—62no—
218C22×C4—121no—
219C22×C4#23511yescharacteristic
220C22×C4D3×D411yescharacteristic
221C22×C4D3×D411yescharacteristic
222SD16—124no—
223M16—42no—
224M16—122no—
225C2×C8—24no—
226C2×C8—22no—
227C2×C8—32no—
228C2×C8—32no—
229C2×C8—41no—
230C2×C8—64no—
231C2×C8—121no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C22⋊C4—41no—
241C22⋊C4—121no—
242C22⋊C4—121no—
243C42—34no—
244C42—41no—
245C42—64no—
246C42—64no—
247C42—62no—
248C42C2×C4×D311yescharacteristic
249C42D3×D411yescharacteristic
250C42#23711yescharacteristic
251C42C2×#6711yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3#39011yescharacteristic
255C22×C3#46512yes—
256C2×D3—81no—
257C2×D3—81no—
258C2×D3—81no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—81no—
262C3×C4—81no—
263C3×C4C2×#10412yes—
264C3×C4C4×SD1612yes—
265C3×C4#44312yes—
266C3×C4#49211yescharacteristic
267C3×C4#49511yescharacteristic
268Dic3—28no—
269Dic3—81no—
270C23—121no—
271C23#88411yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—122no—
275D4—122no—
276D4—122no—
277D4—122no—
278D4—242no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—64no—
284C2×C4—64no—
285C2×C4—64no—
286C2×C4—121no—
287C2×C4—241no—
288C2×C4—241no—
289C2×C4D3×C22⋊C411yescharacteristic
290C2×C4#67⋊C411yescharacteristic
291C2×C4#88511yescharacteristic
292C2×C4D3×SD1612yes—
293C2×C4#91912yes—
294C2×C4C4×#6711yescharacteristic
295C2×C4#93911yescharacteristic
296C2×C4#94011yescharacteristic
297C8—24no—
298C8—34no—
299C8—62no—
300C8—81no—
301C8—241no—
302C2×C3—22no—
303C2×C3#196211yescharacteristic
304C2×C3#196811yescharacteristic
305C2×C3#197411yescharacteristic
306D3—161no—
307C22—21no—
308C22—21no—
309C22—241no—
310C22—241no—
311C22—241no—
312C22#480712yes—
313C22#495711yescharacteristic
314C4—21no—
315C4—21no—
316C4—68no—
317C4—81no—
318C4—81no—
319C4—241no—
320C4#478812yes—
321C4D3×#10412yes—
322C4#514412yes—
323C4#517511yescharacteristic
324C4#519111yescharacteristic
325C3#984311yescharacteristic, complemented, Sylow
326C2—22no—
327C2—481no—
328C2#6999911yescharacteristic
329C2#7025711yescharacteristic
330C2#7035611yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1493D3
hypocenterC3#9843
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
2. C3×#1493C2semidirect
3. #71839C2semidirect
5. #70490C2semidirect
6. C4×C8×Dic3C2semidirect
15. #5088C4semidirect
16. #5085C4semidirect
17. #5080C4semidirect
18. #5079C4semidirect
33. #1493D3semidirect
131. C4×Dic3SD16semidirect
207. C3⋊C8#104semidirect
325. C3#9843semidirect

Maximal subgroups

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

subgroupindex
#833432
C3×#14932
#718392
#705122
#704902
C4×C8×Dic32
#697352
#98433