Finite Group Databasebeta

← #10714472

Subgroups — #10714472

Order 768.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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×C3×#470C211yesmaximal, characteristic
2C2×#4565C211yesmaximal, characteristic, complemented
3C2×#4565C211yesmaximal, characteristic, complemented
4#72000C211yesmaximal, characteristic
5C2×#4553C211yesmaximal, characteristic
6C2×#4487C211yesmaximal, characteristic, complemented
7#67720C211yesmaximal, characteristic, complemented
8#13399—31nomaximal, Sylow
9C22×C3×Q16C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×C22⋊Q8—21no—
12C3×#470—22no—
13C3×#470—22no—
14C2×C3×#106C2211yescharacteristic
15C2×C3×#103—21no—
16#5207—21no—
17C2×#819C2211yescharacteristic
18C2×#819C2211yescharacteristic
19C2×#812C2211yescharacteristic
20C2×#812C2211yescharacteristic
21#4565—22no—
22#4565—22no—
23#4553—22no—
24#4552—21no—
25#4487—22no—
26C2×#470D311yescharacteristic
27C2×#365—31no—
28C2×#362—31no—
29C2×#362—31no—
30C2×#349—31no—
31#1811—31no—
32#1383—31no—
33C22×C3×Q8—21no—
34C23×C3×C4D411yescharacteristic
35C2×C3×Q16—21no—
36C2×C3×Q16—21no—
37C2×C3×Q16—21no—
38C2×C3×Q16—21no—
39C2×C3×Q16—21no—
40C2×C3×Q16D412yes—
41C2×C3×M16—21no—
42C2×C3×M16—21no—
43C2×C3×M16—21no—
44C2×C3×M16—21no—
45C2×C3×M16—21no—
46C2×C3×M16—21no—
47C22×C3×C8D411yescharacteristic
48C22×C3×C8C2311yescharacteristic
49C3×C22⋊Q8—24no—
50C3×C22⋊Q8—42no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—41no—
53C2×C3×C22⋊C4—41no—
54C2×#208—21no—
55C2×#208—21no—
56C3×#106—21no—
57C3×#106D412yes—
58C3×#103—24no—
59#830—42no—
60#819—21no—
61#819—21no—
62#819—21no—
63#819—21no—
64#812—21no—
65#812—21no—
66#812—21no—
67#812D412yes—
68C2×C3⋊C16—21no—
69C2×C3⋊C16—21no—
70C22×Q16C2×D311yescharacteristic
71C22×M16C2×D311yescharacteristic
72C2×C22⋊Q8—21no—
73C2×M32—31no—
74C2×M32—31no—
75#470—22no—
76#470—22no—
77C2×#108—31no—
78C2×#108—31no—
79C2×#106C2×D311yescharacteristic
80C2×#103—21no—
81#407—61no—
82#365—62no—
83#362—62no—
84#362—62no—
85#349—62no—
86#341—61no—
87C24×C3SD1611yescharacteristic
88C2×C3×Q8—22no—
89C2×C3×Q8—41no—
90C2×C3×Q8—41no—
91C2×C3×Q8—41no—
92C2×C3×Q8—41no—
93C2×C3×Q8—41no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×C3×C4SD1611yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C3×Q16—22no—
105C3×Q16—22no—
106C3×Q16—41no—
107C3×Q16—41no—
108C3×M16—41no—
109C3×M16—41no—
110C3×M16—41no—
111C3×M16—41no—
112C2×C3×C8—21no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8C2×D411yescharacteristic
116C2×C3×C8C2×D411yescharacteristic
117C2×C3×C8C2×D411yescharacteristic
118C2×C3×C8#4111yescharacteristic
119C2×C3×C8#4111yescharacteristic
120C2×C3×C8#4111yescharacteristic
121C3×C4⋊C4—22no—
122C3×C4⋊C4—44no—
123C3×C4⋊C4—41no—
124C3×C22⋊C4—44no—
125#208—41no—
126#208—41no—
127#208—41no—
128#208—41no—
129C2×C3⋊C8—41no—
130C2×C3⋊C8—41no—
131C3⋊C16—22no—
132C3⋊C16—41no—
133C22×Q8—21no—
134C23×C4#6711yescharacteristic
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16#6712yes—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—61no—
148C2×M16—61no—
149C22×C8#6711yescharacteristic
150C22×C8C22×D311yescharacteristic
151C22⋊Q8—24no—
152C22⋊Q8—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—41no—
155C2×C22⋊C4—41no—
156M32—32no—
157M32—61no—
158M32—61no—
159M32—61no—
160C2×C16—61no—
161C2×C16—61no—
162#108—61no—
163#108—61no—
164#108—61no—
165#108—61no—
166#106—21no—
167#106#6712yes—
168#103—24no—
169C22⋊C8—122no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—42no—
174C23×C3C2×SD1611yescharacteristic
175C3×Q8—42no—
176C3×Q8—42no—
177C3×Q8—81no—
178C3×Q8—81no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—82no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12311yescharacteristic
194C2×C3×C4#13511yescharacteristic
195C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
196C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
197C3×C8—41no—
198C3×C8#12111yescharacteristic
199C3×C8#12111yescharacteristic
200C3×C8#12311yescharacteristic
201C3×C8#12411yescharacteristic
202C3⋊C8—81no—
203C3⋊C8—81no—
204C24#21411yescharacteristic
205C2×Q8—22no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4#21411yescharacteristic
220C22×C4C2×#6711yescharacteristic
221Q16—22no—
222Q16—22no—
223Q16—41no—
224Q16—41no—
225M16—41no—
226M16—41no—
227M16—41no—
228M16—41no—
229M16—121no—
230M16—121no—
231M16—121no—
232M16—121no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—121no—
237C2×C8—121no—
238C2×C8D3×D411yescharacteristic
239C2×C8#23711yescharacteristic
240C2×C8#23711yescharacteristic
241C2×C8#23711yescharacteristic
242C2×C8C2×#6711yescharacteristic
243C2×C8C2×#6711yescharacteristic
244C4⋊C4—22no—
245C4⋊C4—44no—
246C4⋊C4—41no—
247C22⋊C4—44no—
248C16—62no—
249C16—121no—
250C22×C3—22no—
251C22×C3—42no—
252C22×C3—42no—
253C22×C3—41no—
254C22×C3—81no—
255C22×C3#45011yescharacteristic
256C22×C3#46511yescharacteristic
257C22×C3#48111yescharacteristic
258C3×C4—41no—
259C3×C4—82no—
260C3×C4—82no—
261C3×C4#39411yescharacteristic
262C3×C4#44911yescharacteristic
263C3×C4#46911yescharacteristic
264C3×C4#48211yescharacteristic
265C23—21no—
266C23—21no—
267C23—21no—
268C23—42no—
269C23C2×#21411yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—81no—
273Q8—81no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4#93211yescharacteristic
287C2×C4#93511yescharacteristic
288C2×C4#93911yescharacteristic
289C2×C4#94911yescharacteristic
290C2×C4#95011yescharacteristic
291C2×C4#95311yescharacteristic
292C8—41no—
293C8—241no—
294C8—241no—
295C8#93511yescharacteristic
296C8#93611yescharacteristic
297C8#93811yescharacteristic
298C8#93911yescharacteristic
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—81no—
302C2×C3#203411yescharacteristic
303C2×C3#223912yes—
304C22—22no—
305C22—42no—
306C22—41no—
307C22—42no—
308C22—81no—
309C22#503411yescharacteristic
310C22#504111yescharacteristic
311C22#521811yescharacteristic
312C4—41no—
313C4—82no—
314C4—82no—
315C4#503511yescharacteristic
316C4#503911yescharacteristic
317C4#522211yescharacteristic
318C4#523811yescharacteristic
319C3#1339911yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2—81no—
323C2#7109012yes—
324C2#7168011yescharacteristic
subgroupG/H
centerC22#5034
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC2×C3×#470C2
socleC22×C3#481
prime radicalC24×C3SD16
hypercenterC2×#470D3
hypocenterC3#13399
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ C2×#470

Complements

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

NH ≅ G/Ntype
2. C2×#4565C2semidirect
3. C2×#4565C2semidirect
6. C2×#4487C2semidirect
7. #67720C2semidirect
319. C3#13399semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4702
C2×#45652
#720002
C2×#45532
C2×#44872
#677202
#133993