Finite Group Databasebeta

← #10713699

Subgroups — #10713699

Order 768.

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

normal subgroups only

This group has 462 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
1C2×#5183C211yesmaximal, characteristic
2C3×#2141C211yesmaximal, characteristic
3C2×#4565C211yesmaximal, characteristic
4C2×#4517C211yesmaximal, characteristic
5C2×#4511C211yesmaximal, characteristic
6#70659C211yesmaximal, characteristic
7#70654C211yesmaximal, characteristic
8#13262—31nomaximal, Sylow
9C22×C3×Q16C2211yescharacteristic
10C2×#944—21no—
11C22×#216—21no—
12C22×Dic12C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C3×#427C2211yescharacteristic
15#5183—22no—
16#5183C414yescomplemented
17C2×#819C2211yescharacteristic
18C2×C8×Dic3C2211yescharacteristic
19C2×#812C2211yescharacteristic
20#4997—21no—
21C3×#358—22no—
22#4565—22no—
23#4532—21no—
24#4517—22no—
25#4511—22no—
26#4492—22no—
27C2×#494—31no—
28C2×#364—31no—
29C2×#362—31no—
30C2×#362—31no—
31#2141—31no—
32#2141D311yescharacteristic
33#1772—31no—
34C22×C3×Q8—21no—
35C22×Dic6—21no—
36C2×C3×Q16—21no—
37C2×C3×Q16—21no—
38C2×C3×Q16—21no—
39C2×C3×Q16—21no—
40C2×C3×Q16—21no—
41C2×C3×Q16C2×C412yes—
42C22×C3×C8C2311yescharacteristic
43C3×#117—21no—
44#944—24no—
45#944—42no—
46C2×#216—24no—
47C2×#216—41no—
48C2×#216—41no—
49C2×#216—42no—
50C2×Dic3⋊C4—42no—
51C2×C4×Dic3D411yescharacteristic
52C2×#208—21no—
53C22×C3⋊C8D411yescharacteristic
54C2×Dic12—21no—
55C2×Dic12—21no—
56C2×Dic12—21no—
57C2×Dic12—21no—
58C2×Dic12—21no—
59C2×Dic12C2×C412yes—
60C3×M32—21no—
61C3×M32—21no—
62C2×C3×C16—21no—
63C3×#107D412yes—
64C3×#106—21no—
65#819—21no—
66#819D412yes—
67C8×Dic3C2×C412yes—
68C8×Dic3D412yes—
69#812—21no—
70#812—21no—
71C2×C3⋊C16—21no—
72#803—41no—
73C4×C3⋊C8—41no—
74C22×Q16—31no—
75C22×Q16—61no—
76C22×Q16C2×D311yescharacteristic
77C2×C4⋊Q8—61no—
78C2×M32—31no—
79C2×M32C2×D311yescharacteristic
80#494—34no—
81#494—62no—
82C2×#108—31no—
83#427C2×D311yescharacteristic
84#405—61no—
85C2×C4×C8—31no—
86#364—62no—
87#362—62no—
88#362—62no—
89#358—22no—
90#358—62no—
91#339—61no—
92C2×C3×Q8—22no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C22×C3×C4—41no—
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—41no—
102C2×Dic6—22no—
103C2×Dic6—41no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C2×Dic6—41no—
107C2×Dic6—41no—
108C3×Q16—22no—
109C3×Q16—22no—
110C3×Q16—41no—
111C3×Q16—41no—
112C2×C3×C8C22×C411yescharacteristic
113C2×C3×C8C2×D411yescharacteristic
114C2×C3×C8C2×D411yescharacteristic
115C2×C3×C8C2×D411yescharacteristic
116C2×C3×C8#4111yescharacteristic
117C2×C3×C8#4111yescharacteristic
118C3×C4⋊C4—22no—
119C3×C22⋊C4—41no—
120C3×C42—41no—
121#216—44no—
122#216—44no—
123Dic3⋊C4—48no—
124C4×Dic3—21no—
125C4×Dic3C22⋊C412yes—
126#208—22no—
127#208—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8C22⋊C412yes—
130C2×C3⋊C8D811yescharacteristic
131C2×C3⋊C8D811yescharacteristic
132C2×C3⋊C8SD1612yes—
133Dic12—22no—
134Dic12—22no—
135Dic12—41no—
136Dic12—41no—
137C3×C16—41no—
138C3⋊C16—41no—
139C22×Q8—21no—
140C22×Q8—61no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—32no—
147C2×Q16—64no—
148C2×Q16—61no—
149C2×Q16—61no—
150C2×Q16—61no—
151C2×Q16—61no—
152C2×Q16—61no—
153C2×Q16—121no—
154C2×Q16—121no—
155C2×Q16—122no—
156C2×Q16C4×D312yes—
157C2×M16—61no—
158C22×C8—31no—
159C22×C8C22×D311yescharacteristic
160C4⋊Q8—64no—
161C4⋊Q8—122no—
162#117—21no—
163C2×C4⋊C4—122no—
164C2×C42—31no—
165M32—21no—
166M32—21no—
167M32—61no—
168M32—61no—
169C2×C16—21no—
170C2×C16—61no—
171#108—32no—
172#108—61no—
173#107#6712yes—
174#106—21no—
175#97—121no—
176C4×C8—32no—
177C4×C8—32no—
178C4×C8—121no—
179C23×C3AGL(1,ℤ/8)11yescharacteristic
180C3×Q8—42no—
181C3×Q8—42no—
182C3×Q8—81no—
183C3×Q8—81no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4C2×C22⋊C411yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4C2×D811yescharacteristic
192C2×C3×C4C2×SD1611yescharacteristic
193C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
194C2×Dic3—24no—
195C2×Dic3—22no—
196C2×Dic3—42no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199Dic6—42no—
200Dic6—42no—
201Dic6—81no—
202Dic6—81no—
203C3×C8C4×D411yescharacteristic
204C3×C8C4×D411yescharacteristic
205C3×C8#12411yescharacteristic
206C3×C8#12711yescharacteristic
207C3⋊C8—42no—
208C3⋊C8#10212yes—
209C3⋊C8#10212yes—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—62no—
217C2×Q8—121no—
218C2×Q8—121no—
219C2×Q8—121no—
220C2×Q8—121no—
221C2×Q8—121no—
222C22×C4—41no—
223C22×C4—61no—
224C22×C4—121no—
225C22×C4D3×D411yescharacteristic
226Q16—22no—
227Q16—22no—
228Q16—41no—
229Q16—41no—
230Q16—62no—
231Q16—62no—
232Q16—124no—
233Q16—124no—
234Q16—121no—
235Q16—121no—
236M16—62no—
237M16—121no—
238C2×C8—32no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8—32no—
242C2×C8—121no—
243C2×C8C2×C4×D311yescharacteristic
244C2×C8#23511yescharacteristic
245C2×C8D3×D411yescharacteristic
246C2×C8D3×D411yescharacteristic
247C2×C8#23711yescharacteristic
248C2×C8C2×#6711yescharacteristic
249C4⋊C4—22no—
250C4⋊C4—128no—
251C22⋊C4—41no—
252C42—32no—
253C42—41no—
254C42—61no—
255C16—41no—
256C16—121no—
257C22×C3—22no—
258C22×C3#41811yescharacteristic
259C22×C3#45911yescharacteristic
260C22×C3#46011yescharacteristic
261C3×C4—82no—
262C3×C4—82no—
263C3×C4#39011yescharacteristic
264C3×C4C2×#10211yescharacteristic
265C3×C4#45911yescharacteristic
266C3×C4#46011yescharacteristic
267Dic3—24no—
268Dic3—82no—
269C23#91111yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—81no—
273Q8—81no—
274Q8—122no—
275Q8—122no—
276Q8—241no—
277Q8—241no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—64no—
281C2×C4—62no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—122no—
285C2×C4—241no—
286C2×C4—241no—
287C2×C4D3×C22⋊C411yescharacteristic
288C2×C4#88411yescharacteristic
289C2×C4D3×D811yescharacteristic
290C2×C4D3×SD1611yescharacteristic
291C2×C4#91411yescharacteristic
292C2×C4#93911yescharacteristic
293C8—32no—
294C8—32no—
295C8—122no—
296C8#67⋊C411yescharacteristic
297C8#88511yescharacteristic
298C8C4×#6711yescharacteristic
299C8#94011yescharacteristic
300C2×C3—21no—
301C2×C3—21no—
302C2×C3#198211yescharacteristic
303C2×C3#221412yes—
304C22—22no—
305C22#476211yescharacteristic
306C22#477011yescharacteristic
307C22#515011yescharacteristic
308C4—64no—
309C4—82no—
310C4—82no—
311C4—242no—
312C4D3×#10211yescharacteristic
313C4#478611yescharacteristic
314C4#495711yescharacteristic
315C4#516311yescharacteristic
316C3#1326211yescharacteristic, complemented, Sylow
317C2—21no—
318C2—21no—
319C2#7016111yescharacteristic
320C2#7152012yes—
subgroupG/H
centerC22#5150
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2141C2
socleC22×C3#459
prime radicalC23×C3AGL(1,ℤ/8)
cosocleC22×C3×C8C23
hypercenter#2141D3
hypocenterC3#13262
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
16. #5183C4semidirect
316. C3#13262semidirect

Maximal subgroups

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

subgroupindex
C2×#51832
C3×#21412
C2×#45652
C2×#45172
C2×#45112
#706592
#706542
#132623