Finite Group Databasebeta

← #10705648

Subgroups — #10705648

Order 768.

500 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C3×#2598C211yesmaximal, characteristic
2#71592C211yesmaximal, characteristic, complemented
3#71572C211yesmaximal, characteristic, complemented
4#67651C212yesmaximal, complemented
5#67645C212yesmaximal
6#11754—31nomaximal, Sylow
7C2×C3×C4×D4C411yescharacteristic, complemented
8C2×C3×#117C2211yescharacteristic
9C2×#926—21no—
10C3×#400C411yescharacteristic, complemented
11C3×#399C2211yescharacteristic
12C3×#394C411yescharacteristic, complemented
13C3×#392C2211yescharacteristic
14C3×#388—21no—
15C3×#386—21no—
16C3×#386—21no—
17C3×#385—21no—
18C3×#383C411yescharacteristic, complemented
19#4926—21no—
20#4528—21no—
21#4526—21no—
22#4526C2212yes—
23#4525C2212yes—
24#2598D311yescharacteristic
25#1740—31no—
26#1464—31no—
27#1352—32no—
28#1343—32no—
29C22×C3×D4—21no—
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4—21no—
32C23×C3×C4D411yescharacteristic
33C3×C4×D4—41no—
34C3×C4×D4—41no—
35C3×#117—41no—
36C3×#117—41no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4D411yescharacteristic
47C2×C3×C22⋊C4C2311yescharacteristic
48C2×C3×C42C2×C411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×#217—22no—
51#926—24no—
52#926—42no—
53C2×C4⋊Dic3—41no—
54C2×C4×Dic3—41no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95C2×C411yescharacteristic
60C3×#95C2×C411yescharacteristic
61#830—22no—
62C4⋊(C3⋊C8)—41no—
63#803—41no—
64C2×C4×D4Dic311yescharacteristic, complemented
65C2×#117—61no—
66C2×#117C2×D311yescharacteristic
67#432—61no—
68#400Dic311yescharacteristic, complemented
69#399C2×D311yescharacteristic
70#394Dic311yescharacteristic, complemented
71#392C2×D311yescharacteristic
72#388—21no—
73#386—21no—
74#386—21no—
75#385—21no—
76#383Dic311yescharacteristic, complemented
77#342—32no—
78#342—61no—
79#324—61no—
80#323—32no—
81C24×C3—21no—
82C24×C3C2×D411yescharacteristic
83C2×C3×D4—41no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—41no—
87C2×C3×D4—41no—
88C2×C3×D4—81no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—42no—
98C22×C3×C4—81no—
99C22×C3×C4—81no—
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—42no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—81no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—41no—
113C3×C22⋊C4—42no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—81no—
118C3×C22⋊C4—81no—
119C3×C42—21no—
120C3×C42—21no—
121C3×C42—21no—
122C3×C42—21no—
123#217—24no—
124#217—42no—
125C4⋊Dic3—44no—
126C4×Dic3—44no—
127C2×C3⋊C8—42no—
128C22×D4—21no—
129C22×D4C2×Dic311yescharacteristic
130C23×C4—21no—
131C23×C4#6711yescharacteristic
132C4×D4—41no—
133C4×D4—41no—
134#117—41no—
135#117—41no—
136#117—64no—
137#117—122no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—121no—
140C2×C4⋊C4C2×Dic311yescharacteristic
141C2×C4⋊C4C2×Dic311yescharacteristic
142C2×C4⋊C4#6711yescharacteristic
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—62no—
149C2×C22⋊C4#6711yescharacteristic
150C2×C22⋊C4C22×D311yescharacteristic
151C2×C42—121no—
152C2×C42C2×Dic311yescharacteristic
153C2×C42#6711yescharacteristic
154C4⋊C8—121no—
155C22⋊C8—62no—
156#97—121no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95C2×Dic311yescharacteristic
162#95C2×Dic311yescharacteristic
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—22no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3C2×C22⋊C411yescharacteristic
175C23×C3AGL(1,ℤ/8)11yescharacteristic
176C23×C3#13711yescharacteristic
177C3×D4—42no—
178C3×D4—42no—
179C3×D4—82no—
180C3×D4—82no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—84no—
207C2×C3×C4#9911yescharacteristic
208C2×C3×C4#9911yescharacteristic
209C2×C3×C4#10412yes—
210C2×Dic3—44no—
211C2×Dic3—82no—
212C2×Dic3—82no—
213C3⋊C8—82no—
214C24—21no—
215C24C2×#6711yescharacteristic
216C2×D4—41no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—81no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—122no—
234C22×C4#21711yescharacteristic
235C22×C4#21711yescharacteristic
236C22×C4#21711yescharacteristic
237C22×C4#21711yescharacteristic
238C22×C4C22×Dic311yescharacteristic
239C22×C4C2×#6711yescharacteristic
240C2×C8—122no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—81no—
245C4⋊C4—124no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—42no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—42no—
252C22⋊C4—64no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C22⋊C4—122no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—124no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—42no—
268C22×C3—42no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—42no—
272C22×C3—81no—
273C22×C3—81no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3C2×#9911yescharacteristic
278C22×C3#41711yescharacteristic
279C22×C3C2×#10411yescharacteristic
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—22no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4—81no—
290Dic3—84no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23#94911yescharacteristic
303C23#95111yescharacteristic
304C23C2×#21711yescharacteristic
305D4—42no—
306D4—42no—
307D4—82no—
308D4—82no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—84no—
335C2×C4—124no—
336C2×C4—242no—
337C2×C4—242no—
338C2×C4#83411yescharacteristic
339C2×C4#83411yescharacteristic
340C2×C4#83712yes—
341C8—242no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—82no—
347C2×C3#150411yescharacteristic
348C2×C3#212011yescharacteristic
349C2×C3#212811yescharacteristic
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22C2×#83411yescharacteristic
367C22#523211yescharacteristic
368C22C2×#83711yescharacteristic
369C4—21no—
370C4—21no—
371C4—22no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—244no—
380C3#1175411yescharacteristic, complemented, Sylow
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—82no—
386C2#7170811yescharacteristic
387C2#7171611yescharacteristic
388C2#7186111yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#2598C2
socleC22×C3C2×#99
prime radicalC3×#392C22
cosocleC2×C3×C22⋊C4C23
hypercenter#2598D3
hypocenterC3#11754
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2598

Complements

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

NH ≅ G/Ntype
2. #71592C2semidirect
3. #71572C2semidirect
4. #67651C2semidirect
7. C2×C3×C4×D4C4semidirect
10. C3×#400C4semidirect
12. C3×#394C4semidirect
18. C3×#383C4semidirect
64. C2×C4×D4Dic3semidirect
68. #400Dic3semidirect
70. #394Dic3semidirect
76. #383Dic3semidirect
380. C3#11754semidirect

Maximal subgroups

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

subgroupindex
C3×#25982
#715922
#715722
#676512
#676452
#117543