Finite Group Databasebeta

← #10705690

Subgroups — #10705690

Order 768.

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

normal subgroups only

This group has 428 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#77810C211yesmaximal, characteristic
2#68989C211yesmaximal, characteristic
3#68126C211yesmaximal, characteristic
4C3×#1346C211yesmaximal, characteristic, complemented
5#67650C211yesmaximal, characteristic, complemented
6#67494C211yesmaximal, characteristic, complemented
7#67457C211yesmaximal, characteristic, complemented
8#11763—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×#892C2211yescharacteristic
11C2×#887C411yescharacteristic
12C3×#398C2211yescharacteristic
13#4992C411yescharacteristic
14#4989—21no—
15#4980—21no—
16#4979C2211yescharacteristic
17#4719—21no—
18#4673C411yescharacteristic
19#4669—21no—
20#4652—21no—
21#4651C411yescharacteristic
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4554—21no—
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4468—21no—
28#2599—31no—
29#1742—31no—
30#1464—31no—
31#1354—31no—
32#1354—31no—
33#1346—31no—
34#1346D311yescharacteristic, complemented
35C22×C4×D3D411yescharacteristic
36C22×Dic6C2×C411yescharacteristic
37C2×C3×M16—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4C2311yescharacteristic
41C2×(C2×D3)⋊C4—21no—
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4C2×C411yescharacteristic
44C2×(C2×D3)⋊C4D411yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×C4×Dic3C2×C411yescharacteristic
50C2×C4×Dic3D411yescharacteristic
51C2×#208—21no—
52D3×M16—24no—
53D3×M16—42no—
54C2×ΓL(1,𝔽25)—41no—
55C2×D3×C8—41no—
56#892—41no—
57#892—41no—
58#887—41no—
59#887—41no—
60C3×C22⋊C8—21no—
61C3×#95—21no—
62C3×#95C2×C411yescharacteristic
63C3⋊#95—21no—
64C3⋊#95—21no—
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×C4×Q8—31no—
74C2×#117—31no—
75#432—61no—
76#398C2×D311yescharacteristic
77#397—31no—
78#395—31no—
79#393—31no—
80#388—61no—
81#387—61no—
82#387—61no—
83#385—61no—
84#383—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—
107C2×Dic6—41no—
108C2×Dic6—42no—
109C3×M16—22no—
110C3×M16—41no—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117(C2×D3)⋊C4—41no—
118(C2×D3)⋊C4—41no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—42no—
121C4⋊Dic3—41no—
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—
136C22×Q8—31no—
137C23×C4—31no—
138C2×M16—21no—
139C2×M16—64no—
140C2×M16—61no—
141C2×M16—121no—
142C2×M16—122no—
143C22×C8—121no—
144C4×Q8—121no—
145C4×Q8—121no—
146#117—121no—
147#117—121no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4C4×D311yescharacteristic
153C2×C4⋊C4C22×D311yescharacteristic
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C42—31no—
159C2×C42—31no—
160C2×C42—61no—
161C4⋊C8—121no—
162C22⋊C8—21no—
163C22⋊C8—61no—
164#97—121no—
165#95—21no—
166#95—31no—
167#95—61no—
168#95—61no—
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—
211Dic6—42no—
212Dic6—42no—
213C3×C8—42no—
214C3×C8—81no—
215C3⋊C8—42no—
216C3⋊C8—81no—
217C24—31no—
218C2×Q8—121no—
219C2×Q8—122no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—62no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4C2×C4×D311yescharacteristic
233C22×C4D3×D411yescharacteristic
234C22×C4D3×D411yescharacteristic
235M16—22no—
236M16—41no—
237M16—62no—
238M16—124no—
239M16—121no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—124no—
243C2×C8—121no—
244C2×C8—121no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—122no—
253C22⋊C4—121no—
254C22⋊C4—121no—
255C22⋊C4—122no—
256C22⋊C4—122no—
257C42—61no—
258C42—61no—
259C42—61no—
260C42—61no—
261C42—122no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3C2×#9911yescharacteristic
265C22×C3#41711yescharacteristic
266C22×C3C2×#10411yescharacteristic
267C2×D3—22no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—42no—
271C2×D3—81no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—81no—
277C3×C4—81no—
278Dic3—21no—
279Dic3—21no—
280Dic3—42no—
281Dic3—41no—
282Dic3—41no—
283Dic3—81no—
284C23—32no—
285C23—61no—
286C23—61no—
287C23—121no—
288C23—121no—
289C23D3×C22⋊C411yescharacteristic
290Q8—122no—
291Q8—122no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—62no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4#91111yescharacteristic
323C2×C4#91511yescharacteristic
324C8—42no—
325C8—81no—
326C8—122no—
327C8—241no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3#150411yescharacteristic
331C2×C3#212011yescharacteristic
332C2×C3#213511yescharacteristic
333D3—42no—
334C22—21no—
335C22—21no—
336C22—62no—
337C22—122no—
338C22—121no—
339C22—121no—
340C22—241no—
341C22D3×#9911yescharacteristic
342C22#477111yescharacteristic
343C22D3×#10411yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—61no—
349C4—61no—
350C4—81no—
351C4—81no—
352C4—122no—
353C4—121no—
354C4—121no—
355C4—241no—
356C3#1176311yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—122no—
360C2#6830211yescharacteristic
361C2#6914411yescharacteristic
362C2#6917311yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1346C2
socleC22×C3C2×#99
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1346D3
hypocenterC3#11763
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1346

Complements

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

NH ≅ G/Ntype
4. C3×#1346C2semidirect
5. #67650C2semidirect
6. #67494C2semidirect
7. #67457C2semidirect
34. #1346D3semidirect
356. C3#11763semidirect

Maximal subgroups

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

subgroupindex
#778102
#689892
#681262
C3×#13462
#676502
#674942
#674572
#117633