Finite Group Databasebeta

← #10695074

Subgroups — #10695074

Order 768.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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
6#69750C211yesmaximal, characteristic, complemented
7#69735C211yesmaximal, characteristic
8#9846—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C2×#825—22no—
14#5088C411yescharacteristic, complemented
15#5085C411yescharacteristic, complemented
16#5080C411yescharacteristic, complemented
17#5079C411yescharacteristic, complemented
18#5048—22no—
19C4×#215—22no—
20C42×Dic3C2211yescharacteristic
21#4924C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×Q8⋊C8—22no—
24#4550—22no—
25#4532—22no—
26#4480—22no—
27#4459—22no—
28#3167—31no—
29#1994—31no—
30#1962—31no—
31#1744—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—
66#809—22no—
67#808—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4×C3⋊C8C2×C411yescharacteristic
71C4×C3⋊C8C2×C411yescharacteristic
72C4×C3⋊C8D412yes—
73#533C2×D311yescharacteristic
74#532—31no—
75#498—31no—
76#492—31no—
77#487—31no—
78#486—31no—
79#465—62no—
80C4×SD16—62no—
81#423—31no—
82#423C2×D311yescharacteristic
83C2×#104—62no—
84C2×C4×C8—31no—
85C2×C4×C8C2×D311yescharacteristic
86C43—31no—
87#339—62no—
88#336—62no—
89#334—62no—
90Q8⋊C8—22no—
91D4⋊C8—62no—
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C2×#67—41no—
98C22×Dic3—22no—
99#235—42no—
100#235—42no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12—21no—
104C2×C4×D3—41no—
105C2×Dic6—21no—
106C3×M16—42no—
107C2×C3×C8—22no—
108C2×C3×C8—42no—
109C2×C3×C8—41no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C42—41no—
118C3×C42C22×C411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122#215—44no—
123(C2×D3)⋊C4—41no—
124(C2×D3)⋊C4—41no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4×Dic3—22no—
128C4×Dic3—42no—
129C4×Dic3—42no—
130C4×Dic3SD1614yescomplemented
131#208—42no—
132C2×C3⋊C8—42no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8C22⋊C412yes—
135C2×C3⋊C8C22⋊C412yes—
136C2×#41—61no—
137C2×SD16—62no—
138C2×M16—21no—
139C2×M16—61no—
140C22×C8—21no—
141C22×C8—32no—
142C4⋊Q8—31no—
143C4⋊Q8C4×D311yescharacteristic
144C4⋊Q8C4×D311yescharacteristic
145#127—31no—
146#125—21no—
147#124—61no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150#121—61no—
151#121—61no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×D4—61no—
155C4×D4—61no—
156#117—21no—
157C2×C42—32no—
158C2×C42—61no—
159C2×C42—61no—
160C2×C42C22×D311yescharacteristic
161#106—62no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#104—124no—
167#103—62no—
168#102—62no—
169C4×C8—21no—
170C4×C8—31no—
171C4×C8—31no—
172C4×C8—32no—
173C4×C8#6712yes—
174C23×C3#12111yescharacteristic
175C22×D3—41no—
176C3×Q8—42no—
177C3×Q8—42no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4C2×C22⋊C411yescharacteristic
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4C4×D411yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12411yescharacteristic
187C2×C3×C4#12711yescharacteristic
188C2×C3×C4C2×SD1612yes—
189C2×C3×C4#13512yes—
190#67—82no—
191C2×Dic3—24no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—41no—
195D12—42no—
196D12—42no—
197D12—42no—
198C4×D3—81no—
199C4×D3—81no—
200Dic6—42no—
201C3×C8—42no—
202C3×C8—81no—
203C3⋊C8—22no—
204C3⋊C8—22no—
205C3⋊C8—81no—
206#41—122no—
207#41—122no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—61no—
211C2×D4—61no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—121no—
215C22×C4—62no—
216C22×C4—121no—
217C22×C4#23511yescharacteristic
218C22×C4D3×D411yescharacteristic
219C22×C4D3×D411yescharacteristic
220SD16—124no—
221M16—42no—
222M16—122no—
223C2×C8—22no—
224C2×C8—32no—
225C2×C8—32no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8—122no—
229C2×C8—121no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C22⋊C4—41no—
239C22⋊C4—121no—
240C22⋊C4—121no—
241C42—34no—
242C42—41no—
243C42—62no—
244C42—122no—
245C42—122no—
246C42C2×C4×D311yescharacteristic
247C42D3×D411yescharacteristic
248C42#23711yescharacteristic
249C42C2×#6711yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3#39011yescharacteristic
253C22×C3#46512yes—
254C2×D3—81no—
255C2×D3—81no—
256C2×D3—81no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—81no—
260C3×C4—81no—
261C3×C4#42112yes—
262C3×C4C4×SD1612yes—
263C3×C4#44412yes—
264C3×C4#49211yescharacteristic
265C3×C4#49511yescharacteristic
266Dic3—44no—
267Dic3—81no—
268C23—121no—
269C23#88411yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—122no—
273D4—122no—
274D4—122no—
275D4—122no—
276D4—242no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—64no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—121no—
285C2×C4—241no—
286C2×C4—241no—
287C2×C4D3×C22⋊C411yescharacteristic
288C2×C4#67⋊C411yescharacteristic
289C2×C4#88511yescharacteristic
290C2×C4D3×SD1612yes—
291C2×C4#91912yes—
292C2×C4C4×#6711yescharacteristic
293C2×C4#93911yescharacteristic
294C2×C4#94011yescharacteristic
295C8—42no—
296C8—62no—
297C8—62no—
298C8—81no—
299C8—241no—
300C2×C3—22no—
301C2×C3#196311yescharacteristic
302C2×C3#196811yescharacteristic
303C2×C3#197511yescharacteristic
304D3—161no—
305C22—21no—
306C22—21no—
307C22—241no—
308C22—241no—
309C22—241no—
310C22#480712yes—
311C22#495711yescharacteristic
312C4—21no—
313C4—21no—
314C4—81no—
315C4—81no—
316C4—124no—
317C4—241no—
318C4#478812yes—
319C4#482112yes—
320C4#514512yes—
321C4#517511yescharacteristic
322C4#519111yescharacteristic
323C3#984611yescharacteristic, complemented, Sylow
324C2—22no—
325C2—481no—
326C2#7000011yescharacteristic
327C2#7025711yescharacteristic
328C2#7035711yescharacteristic
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#9846
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. #69750C2semidirect
14. #5088C4semidirect
15. #5085C4semidirect
16. #5080C4semidirect
17. #5079C4semidirect
33. #1493D3semidirect
130. C4×Dic3SD16semidirect
323. C3#9846semidirect

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
#697502
#697352
#98463