Finite Group Databasebeta

← #10713691

Subgroups — #10713691

Order 768.

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

normal subgroups only

This group has 482 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×#5191C211yesmaximal, characteristic
2C3×#2141C211yesmaximal, characteristic, complemented
3C2×#4565C211yesmaximal, characteristic, complemented
4C2×#4516C211yesmaximal, characteristic
5C2×#4511C211yesmaximal, characteristic, complemented
6#70659C211yesmaximal, characteristic, complemented
7#70652C211yesmaximal, characteristic
8#13260—31nomaximal, Sylow
9C22×C3×Q16C2211yescharacteristic
10C2×#947—21no—
11C22×#215—21no—
12C22×D24C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C3×#427C2211yescharacteristic
15#5191—22no—
16#5191C414yescomplemented
17C2×#819C2211yescharacteristic
18C2×C8×Dic3C2211yescharacteristic
19C2×#812C2211yescharacteristic
20#4997—21no—
21C3×#358—22no—
22#4565—22no—
23#4532—21no—
24#4516—22no—
25#4511—22no—
26#4491—22no—
27C2×#495—31no—
28C2×#364—31no—
29C2×#362—31no—
30C2×#361—31no—
31#2141D311yescharacteristic, complemented
32#2140—31no—
33#1772—31no—
34C22×C3×Q8—21no—
35C22×D12—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#947—24no—
45#947—42no—
46C2×#215—24no—
47C2×#215—41no—
48C2×#215—41no—
49C2×#215—42no—
50C2×(C2×D3)⋊C4—42no—
51C2×C4×Dic3D411yescharacteristic
52C2×#208—21no—
53C22×C3⋊C8D411yescharacteristic
54C2×D24—21no—
55C2×D24—21no—
56C2×D24—21no—
57C2×D24—21no—
58C2×D24—21no—
59C2×D24C2×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×Q16C2×D311yescharacteristic
75C22×SD16—61no—
76C22×D8—31no—
77C2×#124—61no—
78C2×M32—31no—
79C2×M32C2×D311yescharacteristic
80#495—34no—
81#495—62no—
82C2×#108—31no—
83#427C2×D311yescharacteristic
84#405—61no—
85C2×C4×C8—31no—
86#364—62no—
87#362—62no—
88#361—62no—
89#358—22no—
90#357—62no—
91#339—61no—
92C23×D3—41no—
93C2×C3×Q8—22no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C22×C3×C4—41no—
100C22×C3×C4C2×D411yescharacteristic
101C22×Dic3—21no—
102C2×D12—22no—
103C2×D12—41no—
104C2×D12—41no—
105C2×D12—41no—
106C2×D12—41no—
107C2×D12—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#215—44no—
122#215—44no—
123(C2×D3)⋊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—
133D24—22no—
134D24—22no—
135D24—41no—
136D24—41no—
137C3×C16—41no—
138C3⋊C16—41no—
139C22×Q8—21no—
140C22×D4—61no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16C4×D312yes—
147C2×SD16—64no—
148C2×SD16—121no—
149C2×SD16—121no—
150C2×SD16—122no—
151C2×D8—32no—
152C2×D8—61no—
153C2×D8—61no—
154C2×D8—61no—
155C2×D8—61no—
156C2×D8—61no—
157C2×M16—61no—
158C22×C8—31no—
159C22×C8C22×D311yescharacteristic
160#124—64no—
161#124—122no—
162#117—21no—
163C2×C22⋊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
180C22×D3—42no—
181C22×D3—84no—
182C22×D3—81no—
183C22×D3—81no—
184C3×Q8—42no—
185C3×Q8—42no—
186C3×Q8—81no—
187C3×Q8—81no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×C3×C4C2×C22⋊C411yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4C2×D811yescharacteristic
196C2×C3×C4C2×SD1611yescharacteristic
197C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
198C2×Dic3—22no—
199C2×Dic3—42no—
200D12—42no—
201D12—42no—
202D12—81no—
203D12—81no—
204C3×C8C4×D411yescharacteristic
205C3×C8C4×D411yescharacteristic
206C3×C8#12411yescharacteristic
207C3×C8#12711yescharacteristic
208C3⋊C8—42no—
209C3⋊C8#10212yes—
210C3⋊C8#10212yes—
211C24—121no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×D4—62no—
219C2×D4—121no—
220C2×D4—121no—
221C2×D4—121no—
222C2×D4—121no—
223C2×D4—121no—
224C22×C4—41no—
225C22×C4—61no—
226C22×C4D3×D411yescharacteristic
227Q16—22no—
228Q16—22no—
229Q16—41no—
230Q16—41no—
231SD16—124no—
232SD16—124no—
233D8—62no—
234D8—62no—
235D8—121no—
236D8—121no—
237M16—62no—
238M16—121no—
239C2×C8—32no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8—32no—
243C2×C8—121no—
244C2×C8C2×C4×D311yescharacteristic
245C2×C8#23511yescharacteristic
246C2×C8D3×D411yescharacteristic
247C2×C8D3×D411yescharacteristic
248C2×C8#23711yescharacteristic
249C2×C8C2×#6711yescharacteristic
250C4⋊C4—22no—
251C22⋊C4—41no—
252C22⋊C4—128no—
253C42—32no—
254C42—41no—
255C42—61no—
256C16—41no—
257C16—121no—
258C22×C3—22no—
259C22×C3#41811yescharacteristic
260C22×C3#45911yescharacteristic
261C22×C3#46011yescharacteristic
262C2×D3—84no—
263C2×D3—82no—
264C2×D3—162no—
265C2×D3—162no—
266C3×C4—82no—
267C3×C4—82no—
268C3×C4#39011yescharacteristic
269C3×C4C2×#10211yescharacteristic
270C3×C4#45911yescharacteristic
271C3×C4#46011yescharacteristic
272Dic3—42no—
273C23—122no—
274C23—244no—
275C23—241no—
276C23—241no—
277C23#91111yescharacteristic
278Q8—42no—
279Q8—42no—
280Q8—81no—
281Q8—81no—
282D4—122no—
283D4—122no—
284D4—241no—
285D4—241no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—62no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—122no—
292C2×C4D3×C22⋊C411yescharacteristic
293C2×C4#88411yescharacteristic
294C2×C4D3×D811yescharacteristic
295C2×C4D3×SD1611yescharacteristic
296C2×C4#91411yescharacteristic
297C2×C4#93911yescharacteristic
298C8—32no—
299C8—32no—
300C8—122no—
301C8#67⋊C411yescharacteristic
302C8#88511yescharacteristic
303C8C4×#6711yescharacteristic
304C8#94011yescharacteristic
305C2×C3—21no—
306C2×C3—21no—
307C2×C3#198211yescharacteristic
308C2×C3#221212yes—
309D3—162no—
310C22—22no—
311C22—244no—
312C22—242no—
313C22—482no—
314C22—482no—
315C22#476211yescharacteristic
316C22#477011yescharacteristic
317C22#515011yescharacteristic
318C4—82no—
319C4—82no—
320C4—122no—
321C4D3×#10211yescharacteristic
322C4#478611yescharacteristic
323C4#495711yescharacteristic
324C4#516311yescharacteristic
325C3#1326011yescharacteristic, complemented, Sylow
326C2—21no—
327C2—21no—
328C2—482no—
329C2#7016111yescharacteristic
330C2#7151912yes—
subgroupG/H
centerC22#5150
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2141C2
socleC22×C3#459
prime radicalC22×D24C22
cosocleC22×C3×C8C23
hypercenter#2141D3
hypocenterC3#13260
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
2. C3×#2141C2semidirect
3. C2×#4565C2semidirect
5. C2×#4511C2semidirect
6. #70659C2semidirect
16. #5191C4semidirect
31. #2141D3semidirect
325. C3#13260semidirect

Maximal subgroups

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

subgroupindex
C2×#51912
C3×#21412
C2×#45652
C2×#45162
C2×#45112
#706592
#706522
#132603