Finite Group Databasebeta

← #10695161

Subgroups — #10695161

Order 768.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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#83334C211yesmaximal, characteristic
2C3×#1493C211yesmaximal, characteristic, complemented
3#71826C211yesmaximal, characteristic, complemented
4#70516C211yesmaximal, characteristic, complemented
5#70498C211yesmaximal, characteristic
6#69735C211yesmaximal, characteristic
7C2×#4454C211yesmaximal, characteristic, complemented
8#9863—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13#5093C411yescharacteristic
14#5093C411yescharacteristic
15#5088C411yescharacteristic
16#5088C411yescharacteristic
17#5068—22no—
18#5002—22no—
19#4924C2211yescharacteristic
20C2×C4×C3⋊C8C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22C3×Q8⋊C8—22no—
23#4558—22no—
24#4535—22no—
25#4483—22no—
26#4459—22no—
27#4454—22no—
28#3131—31no—
29#1837—31no—
30#1493—31no—
31#1493D311yescharacteristic, complemented
32#1492—31no—
33#1492—31no—
34C2×#322—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
47#930—21no—
48#930—21no—
49C2×#208—21no—
50C22×C3⋊C8—21no—
51C22×C3⋊C8D412yes—
52#875—21no—
53#874C2×C411yescharacteristic
54C4×D12—21no—
55C4×D12—21no—
56#869C2×C411yescharacteristic
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4×C8—21no—
60C3×C4×C8D412yes—
61#810—22no—
62#810—22no—
63#809—22no—
64#809—22no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)—21no—
67C4×C3⋊C8—21no—
68C4×C3⋊C8C2×C411yescharacteristic
69C4×C3⋊C8C2×C411yescharacteristic
70C4×C3⋊C8D412yes—
71C4×C3⋊C8D412yes—
72#533C2×D311yescharacteristic
73#532—31no—
74#488—62no—
75#487—31no—
76#487—31no—
77#486—31no—
78#486—31no—
79#423—31no—
80#423C2×D311yescharacteristic
81#416—62no—
82C2×C4×C8—31no—
83C2×C4×C8—31no—
84C2×C4×C8C2×D311yescharacteristic
85#341—62no—
86Q8⋊C8—22no—
87Q8⋊C8—62no—
88D4⋊C8—62no—
89D4⋊C8—62no—
90#322—62no—
91C2×C3×Q8—21no—
92C2×C3×Q8—21no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4111yescharacteristic
95C22×C3×C4#4111yescharacteristic
96C2×#67—41no—
97#235—42no—
98#235—42no—
99C2×D12—21no—
100C2×D12—21no—
101C2×D12—21no—
102C2×C4×D3—41no—
103C2×Dic6—21no—
104C3×M16—42no—
105C2×C3×C8—22no—
106C2×C3×C8—42no—
107C2×C3×C8—41no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C42—41no—
116C3×C42C22×C411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124#208—42no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—42no—
129C2×C3⋊C8—41no—
130C2×C3⋊C8C22⋊C412yes—
131C2×C3⋊C8C22⋊C412yes—
132C2×#41—61no—
133C2×M16—21no—
134C2×M16—61no—
135C22×C8—21no—
136C22×C8—32no—
137C22×C8—61no—
138C4⋊Q8—31no—
139C4⋊Q8C4×D311yescharacteristic
140C4⋊Q8C4×D311yescharacteristic
141#127—31no—
142#125—21no—
143#124—61no—
144C22⋊Q8—21no—
145C22⋊Q8—21no—
146#121—61no—
147#121—61no—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×D4—61no—
151C4×D4—61no—
152#117—21no—
153C2×C42C22×D311yescharacteristic
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158#103—62no—
159#103—62no—
160#102—62no—
161#102—62no—
162C4×C8—21no—
163C4×C8—31no—
164C4×C8—31no—
165C4×C8—32no—
166C4×C8—32no—
167C4×C8—61no—
168C4×C8#6712yes—
169C23×C3#12411yescharacteristic
170C22×D3—41no—
171C3×Q8—42no—
172C3×Q8—42no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4C2×C22⋊C411yescharacteristic
178C2×C3×C4C4×D411yescharacteristic
179C2×C3×C4C4×D411yescharacteristic
180C2×C3×C4#12111yescharacteristic
181C2×C3×C4#12111yescharacteristic
182C2×C3×C4#12711yescharacteristic
183C2×C3×C4#13512yes—
184C2×C3×C4#13512yes—
185#67—82no—
186C2×Dic3—41no—
187D12—42no—
188D12—42no—
189D12—42no—
190C4×D3—81no—
191C4×D3—81no—
192Dic6—42no—
193C3×C8—42no—
194C3×C8—81no—
195C3⋊C8—22no—
196C3⋊C8—22no—
197C3⋊C8—42no—
198C3⋊C8—81no—
199#41—122no—
200#41—122no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—61no—
204C2×D4—61no—
205C2×D4—61no—
206C2×D4—61no—
207C2×D4—121no—
208C22×C4—121no—
209C22×C4#23511yescharacteristic
210C22×C4D3×D411yescharacteristic
211C22×C4#23711yescharacteristic
212M16—42no—
213M16—122no—
214C2×C8—22no—
215C2×C8—32no—
216C2×C8—32no—
217C2×C8—42no—
218C2×C8—41no—
219C2×C8—62no—
220C2×C8—62no—
221C2×C8—62no—
222C2×C8—122no—
223C2×C8—121no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C22⋊C4—41no—
233C22⋊C4—121no—
234C22⋊C4—121no—
235C42—41no—
236C42C2×C4×D311yescharacteristic
237C42D3×D411yescharacteristic
238C42D3×D411yescharacteristic
239C42C2×#6711yescharacteristic
240C22×C3—21no—
241C22×C3—21no—
242C22×C3#39011yescharacteristic
243C22×C3#48812yes—
244C2×D3—81no—
245C2×D3—81no—
246C2×D3—81no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—81no—
250C3×C4—81no—
251C3×C4#41612yes—
252C3×C4#44412yes—
253C3×C4#44412yes—
254C3×C4#49511yescharacteristic
255C3×C4#49511yescharacteristic
256Dic3—81no—
257C23—121no—
258C23#88511yescharacteristic
259Q8—42no—
260Q8—42no—
261Q8—122no—
262D4—122no—
263D4—122no—
264D4—122no—
265D4—242no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—121no—
271C2×C4—241no—
272C2×C4—241no—
273C2×C4D3×C22⋊C411yescharacteristic
274C2×C4#67⋊C411yescharacteristic
275C2×C4#88411yescharacteristic
276C2×C4#91612yes—
277C2×C4#91912yes—
278C2×C4C4×#6711yescharacteristic
279C2×C4#93911yescharacteristic
280C2×C4#94011yescharacteristic
281C8—42no—
282C8—62no—
283C8—62no—
284C8—81no—
285C8—122no—
286C8—241no—
287C2×C3—22no—
288C2×C3#195811yescharacteristic
289C2×C3#197511yescharacteristic
290C2×C3#197511yescharacteristic
291D3—161no—
292C22—21no—
293C22—21no—
294C22—241no—
295C22—241no—
296C22—241no—
297C22#480012yes—
298C22#495711yescharacteristic
299C4—21no—
300C4—21no—
301C4—81no—
302C4—81no—
303C4—241no—
304C4#480412yes—
305C4#482312yes—
306C4#514512yes—
307C4#516811yescharacteristic
308C4#519111yescharacteristic
309C3#986311yescharacteristic, complemented, Sylow
310C2—22no—
311C2—481no—
312C2#7000011yescharacteristic
313C2#7022611yescharacteristic
314C2#7037411yescharacteristic
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#9863
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. #71826C2semidirect
4. #70516C2semidirect
7. C2×#4454C2semidirect
31. #1493D3semidirect
309. C3#9863semidirect

Maximal subgroups

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

subgroupindex
#833342
C3×#14932
#718262
#705162
#704982
#697352
C2×#44542
#98633