Finite Group Databasebeta

← #10705697

Subgroups — #10705697

Order 768.

446 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 73 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#78345C211yesmaximal, characteristic
2D3×#343C211yesmaximal, characteristic
3#68144C211yesmaximal, characteristic
4C3×#1353C211yesmaximal, characteristic, complemented
5#67652C211yesmaximal, characteristic, complemented
6#67495C211yesmaximal, characteristic, complemented
7#67458C211yesmaximal, characteristic, complemented
8#11764—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×#892C411yescharacteristic
11C2×D3×C4⋊C4C2211yescharacteristic
12C3×#400C2211yescharacteristic
13#4985C411yescharacteristic
14#4981C2211yescharacteristic
15#4726—21no—
16#4668—21no—
17#4664—21no—
18#4664C411yescharacteristic
19#4661C411yescharacteristic
20#4655—21no—
21#4654—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4554—21no—
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4469—21no—
28#2484—31no—
29C2×#343—31no—
30#1463—31no—
31#1353—31no—
32#1353D311yescharacteristic, complemented
33#1346—31no—
34#1346—31no—
35C22×C4×D3C2×C411yescharacteristic
36C22×C4×D3D411yescharacteristic
37C22×C4×D3D411yescharacteristic
38C2×C3×M16—21no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4C2311yescharacteristic
41C2×(C2×D3)⋊C4—21no—
42C2×(C2×D3)⋊C4—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4C2×C411yescharacteristic
48C2×Dic3⋊C4D411yescharacteristic
49C2×C4×Dic3—21no—
50C2×#208—21no—
51D3×M16—24no—
52D3×M16—42no—
53C2×ΓL(1,𝔽25)—41no—
54C2×ΓL(1,𝔽25)—41no—
55C2×D3×C8—41no—
56#892—41no—
57#892—41no—
58D3×C4⋊C4—41no—
59D3×C4⋊C4—41no—
60C3×C22⋊C8—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95C2×C411yescharacteristic
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95C2×C411yescharacteristic
70#830—21no—
71(C2×D3)⋊C8—41no—
72C22×M16—61no—
73C2×#117—31no—
74C22×C4⋊C4—31no—
75#407—61no—
76#400—31no—
77#400C2×D311yescharacteristic
78#398—31no—
79#398—31no—
80#387—61no—
81#387—61no—
82#385—61no—
83#381—31no—
84#381—61no—
85#343—31no—
86#343—61no—
87#343C2×D311yescharacteristic
88#324—31no—
89#324—61no—
90#324C2×D311yescharacteristic
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—21no—
103C2×C4×D3—21no—
104C2×C4×D3—21no—
105C2×C4×D3—21no—
106C2×C4×D3—22no—
107C2×C4×D3—21no—
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—42no—
112C2×C4×D3—42no—
113C3×M16—22no—
114C3×M16—41no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122C4⋊Dic3—41no—
123C4⋊Dic3—42no—
124C4⋊Dic3—41no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129C4×Dic3—42no—
130#208—22no—
131#208—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134ΓL(1,𝔽25)—44no—
135ΓL(1,𝔽25)—82no—
136D3×C8—44no—
137C23×C4—31no—
138C23×C4—31no—
139C23×C4—31no—
140C2×M16—21no—
141C2×M16—64no—
142C2×M16—61no—
143C2×M16—121no—
144C2×M16—121no—
145C2×M16—122no—
146C22×C8—121no—
147#117—121no—
148#117—121no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—121no—
156C2×C4⋊C4—121no—
157C2×C4⋊C4C4×D311yescharacteristic
158C2×C4⋊C4C22×D311yescharacteristic
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C42—61no—
162C22⋊C8—21no—
163C22⋊C8—61no—
164C22⋊C8—121no—
165#95—21no—
166#95—21no—
167#95—31no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95C4×D311yescharacteristic
174C23×C3C2×C22⋊C411yescharacteristic
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—22no—
178C22×D3—41no—
179C22×D3#9911yescharacteristic
180C22×D3#9911yescharacteristic
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
193C2×C3×C4#13711yescharacteristic
194C2×Dic3—21no—
195C2×Dic3—21no—
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—41no—
206C2×Dic3#10412yes—
207C4×D3—22no—
208C4×D3—22no—
209C4×D3—22no—
210C4×D3—22no—
211C4×D3—42no—
212C4×D3—42no—
213C4×D3—42no—
214C4×D3—81no—
215C3×C8—42no—
216C3×C8—81no—
217C3⋊C8—42no—
218C3⋊C8—81no—
219C24—31no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—62no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—122no—
237C22×C4—122no—
238C22×C4C2×C4×D311yescharacteristic
239C22×C4D3×D411yescharacteristic
240C22×C4D3×D411yescharacteristic
241M16—22no—
242M16—41no—
243M16—62no—
244M16—124no—
245M16—121no—
246M16—242no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—124no—
250C2×C8—121no—
251C2×C8—121no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C4⋊C4—122no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4—122no—
264C42—122no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3C2×#9911yescharacteristic
268C22×C3#41711yescharacteristic
269C22×C3C2×#10411yescharacteristic
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—21no—
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—42no—
276C2×D3—42no—
277C2×D3#35212yes—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—81no—
283C3×C4—81no—
284Dic3—22no—
285Dic3—41no—
286Dic3—41no—
287Dic3—81no—
288Dic3—81no—
289C23—31no—
290C23—31no—
291C23—62no—
292C23—61no—
293C23—61no—
294C23—121no—
295C23D3×C22⋊C411yescharacteristic
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—32no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—241no—
328C2×C4#91411yescharacteristic
329C2×C4#91811yescharacteristic
330C8—42no—
331C8—81no—
332C8—122no—
333C8—241no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3#150411yescharacteristic
337C2×C3C2×#35211yescharacteristic
338C2×C3#213311yescharacteristic
339D3—22no—
340D3—41no—
341C22—21no—
342C22—21no—
343C22—32no—
344C22—61no—
345C22—61no—
346C22—61no—
347C22—61no—
348C22—61no—
349C22—122no—
350C22—122no—
351C22D3×#9911yescharacteristic
352C22#480311yescharacteristic
353C22D3×#10411yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—62no—
359C4—81no—
360C4—81no—
361C4—121no—
362C4—121no—
363C4—241no—
364C4—241no—
365C3#1176411yescharacteristic, complemented, Sylow
366C2—21no—
367C2—21no—
368C2—62no—
369C2—121no—
370C2#6833411yescharacteristic
371C2D3×#35211yescharacteristic
372C2#6916411yescharacteristic
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#11764
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
365. C3#11764semidirect

Maximal subgroups

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

subgroupindex
#783452
D3×#3432
#681442
C3×#13532
#676522
#674952
#674582
#117643