Finite Group Databasebeta

← #10705712

Subgroups — #10705712

Order 768.

420 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 71 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#78206C211yesmaximal, characteristic
2#68989C211yesmaximal, characteristic
3#68126C211yesmaximal, characteristic
4C3×#1353C211yesmaximal, characteristic, complemented
5#67652C211yesmaximal, characteristic, complemented
6#67495C211yesmaximal, characteristic, complemented
7#67458C211yesmaximal, characteristic, complemented
8#11767—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×#892C2211yescharacteristic
11C2×#890C411yescharacteristic
12C3×#400C2211yescharacteristic
13#4993C411yescharacteristic
14#4981C2211yescharacteristic
15#4719—21no—
16#4674C411yescharacteristic
17#4669—21no—
18#4668—21no—
19#4655—21no—
20#4648C411yescharacteristic
21#4647—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4554—21no—
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4468—21no—
28#2611—31no—
29#1742—31no—
30#1464—31no—
31#1353—31no—
32#1353D311yescharacteristic, complemented
33#1346—31no—
34#1346—31no—
35C22×C4×D3D411yescharacteristic
36C2×C3×M16—21no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×(C2×D3)⋊C4—21no—
40C2×(C2×D3)⋊C4—21no—
41C2×(C2×D3)⋊C4C2×C411yescharacteristic
42C2×(C2×D3)⋊C4D411yescharacteristic
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3C2×C411yescharacteristic
51C2×C4×Dic3D411yescharacteristic
52C2×#208—21no—
53D3×M16—24no—
54D3×M16—42no—
55C2×ΓL(1,𝔽25)—41no—
56C2×D3×C8—41no—
57#892—41no—
58#892—41no—
59#890—41no—
60#890—41no—
61C3×C22⋊C8—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95C2×C411yescharacteristic
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95C2×C411yescharacteristic
69#830—21no—
70#815—41no—
71Dic3⋊C8—41no—
72C22×M16—61no—
73C2×#125—31no—
74C2×#117—31no—
75#432—61no—
76#400C2×D311yescharacteristic
77#399—31no—
78#398—31no—
79#397—31no—
80#389—61no—
81#388—61no—
82#387—61no—
83#385—61no—
84#376—31no—
85#343—31no—
86#343—61no—
87#343C2×D311yescharacteristic
88#324—31no—
89#324C2×D311yescharacteristic
90#323—61no—
91C23×D3C22⋊C411yescharacteristic
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3C22⋊C411yescharacteristic
100C22×Dic3C22⋊C411yescharacteristic
101C22×Dic3C22⋊C411yescharacteristic
102C2×C4×D3—22no—
103C2×C4×D3—21no—
104C2×C4×D3—21no—
105C2×C4×D3—21no—
106C2×C4×D3—21no—
107C3×M16—22no—
108C3×M16—41no—
109C2×C3×C8—41no—
110C2×C3×C8—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—42no—
116(C2×D3)⋊C4—42no—
117C4⋊Dic3—41no—
118C4⋊Dic3—41no—
119C4⋊Dic3—41no—
120C4⋊Dic3—41no—
121Dic3⋊C4—42no—
122Dic3⋊C4—42no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125C4×Dic3—21no—
126C4×Dic3—21no—
127C4×Dic3—21no—
128C4×Dic3—21no—
129C4×Dic3—42no—
130#208—22no—
131#208—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134ΓL(1,𝔽25)—44no—
135D3×C8—44no—
136C23×C4—31no—
137C2×M16—21no—
138C2×M16—64no—
139C2×M16—61no—
140C2×M16—121no—
141C2×M16—122no—
142C22×C8—121no—
143#125—121no—
144#125—121no—
145#117—121no—
146#117—121no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4C4×D311yescharacteristic
154C2×C4⋊C4C22×D311yescharacteristic
155C2×C22⋊C4—31no—
156C2×C22⋊C4—31no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C42—31no—
160C2×C42—31no—
161C2×C42—61no—
162C4⋊C8—121no—
163C22⋊C8—21no—
164C22⋊C8—61no—
165#97—121no—
166#95—21no—
167#95—21no—
168#95—31no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95C4×D311yescharacteristic
173C23×C3C2×C22⋊C411yescharacteristic
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—41no—
177C22×D3—41no—
178C22×D3#10412yes—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
192C2×C3×C4#13711yescharacteristic
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—22no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3#9911yescharacteristic
206C2×Dic3#9911yescharacteristic
207C4×D3—22no—
208C4×D3—22no—
209C4×D3—22no—
210C4×D3—22no—
211C3×C8—42no—
212C3×C8—81no—
213C3⋊C8—42no—
214C3⋊C8—81no—
215C24—31no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—62no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C2×C4×D311yescharacteristic
229C22×C4D3×D411yescharacteristic
230C22×C4D3×D411yescharacteristic
231M16—22no—
232M16—41no—
233M16—62no—
234M16—124no—
235M16—121no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—124no—
239C2×C8—121no—
240C2×C8—121no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C22⋊C4—121no—
252C22⋊C4—121no—
253C22⋊C4—122no—
254C22⋊C4—122no—
255C42—61no—
256C42—61no—
257C42—61no—
258C42—61no—
259C42—122no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3C2×#9911yescharacteristic
263C22×C3#41711yescharacteristic
264C22×C3C2×#10411yescharacteristic
265C2×D3—22no—
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—42no—
269C2×D3—81no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—81no—
275C3×C4—81no—
276Dic3—21no—
277Dic3—21no—
278Dic3—42no—
279Dic3—41no—
280Dic3—41no—
281Dic3—81no—
282C23—32no—
283C23—61no—
284C23—61no—
285C23—121no—
286C23—121no—
287C23D3×C22⋊C411yescharacteristic
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—62no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—62no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4#91411yescharacteristic
319C2×C4#91811yescharacteristic
320C8—42no—
321C8—81no—
322C8—122no—
323C8—241no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3#150411yescharacteristic
327C2×C3#212311yescharacteristic
328C2×C3#213211yescharacteristic
329D3—42no—
330C22—21no—
331C22—21no—
332C22—62no—
333C22—122no—
334C22—121no—
335C22—121no—
336C22—241no—
337C22D3×#9911yescharacteristic
338C22#480311yescharacteristic
339C22D3×#10411yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—61no—
345C4—61no—
346C4—81no—
347C4—81no—
348C4—122no—
349C4—121no—
350C4—121no—
351C4—241no—
352C3#1176711yescharacteristic, complemented, Sylow
353C2—21no—
354C2—21no—
355C2—122no—
356C2#6833411yescharacteristic
357C2#6914511yescharacteristic
358C2#6916511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1353C2
socleC22×C3C2×#99
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1353D3
hypocenterC3#11767
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1353

Complements

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

NH ≅ G/Ntype
4. C3×#1353C2semidirect
5. #67652C2semidirect
6. #67495C2semidirect
7. #67458C2semidirect
32. #1353D3semidirect
352. C3#11767semidirect

Maximal subgroups

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

subgroupindex
#782062
#689892
#681262
C3×#13532
#676522
#674952
#674582
#117673