Finite Group Databasebeta

← #10705653

Subgroups — #10705653

Order 768.

586 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2482C211yesmaximal, characteristic
2C2×#4526C211yesmaximal, characteristic
3#71539C211yesmaximal, characteristic
4#67651C212yesmaximal, complemented
5#67649C212yesmaximal, complemented
6#11755—31nomaximal, Sylow
7C2×C3×#117C411yescharacteristic, complemented
8C22×C3×C22⋊C4C2211yescharacteristic
9C22×#217—21no—
10C3×#399C411yescharacteristic, complemented
11C3×#399C2211yescharacteristic
12C3×#397C411yescharacteristic, complemented
13C3×#397C2211yescharacteristic
14C3×#388—21no—
15C3×#388—21no—
16C3×#385—21no—
17C3×#381C411yescharacteristic, complemented
18C3×#379—21no—
19#5207—21no—
20#4526—21no—
21#4526C2212yes—
22#4525—21no—
23#4525C2212yes—
24#2482D311yescharacteristic
25C2×#342—31no—
26#1462—31no—
27#1352—32no—
28#1344—32no—
29C25×C3D411yescharacteristic
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×Dic3—41no—
33C3×#117—41no—
34C3×#117—41no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
38C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—41no—
46C2×C3×C22⋊C4—41no—
47C2×C3×C22⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4D411yescharacteristic
49C2×C3×C22⋊C4D411yescharacteristic
50C2×C3×C22⋊C4C2311yescharacteristic
51C2×C3×C42—21no—
52C2×#217—24no—
53C2×#217—22no—
54C2×#217—41no—
55C2×#217—42no—
56C2×#208—41no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95C2×C411yescharacteristic, complemented
61C3×#95C2×C411yescharacteristic, complemented
62#830—22no—
63#830—41no—
64C2×#117Dic311yescharacteristic, complemented
65C22×C22⋊C4—61no—
66C22×C22⋊C4C2×D311yescharacteristic
67#407—61no—
68#399Dic311yescharacteristic, complemented
69#399C2×D311yescharacteristic
70#397Dic311yescharacteristic, complemented
71#397C2×D311yescharacteristic
72#388—21no—
73#388—21no—
74#385—21no—
75#381Dic311yescharacteristic, complemented
76#379—21no—
77#342—32no—
78#342—61no—
79#323—32no—
80#323—61no—
81C24×C3—21no—
82C24×C3—21no—
83C24×C3—22no—
84C24×C3—21no—
85C24×C3—21no—
86C24×C3—42no—
87C24×C3—81no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C22⋊C411yescharacteristic
90C24×C3C2×D411yescharacteristic
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—42no—
100C22×C3×C4—81no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—44no—
106C22×Dic3—42no—
107C22×Dic3—82no—
108C22×Dic3—82no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—81no—
112C3×C4⋊C4—81no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—42no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—42no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—41no—
123C3×C22⋊C4—81no—
124C3×C22⋊C4—81no—
125C3×C42—42no—
126#217—24no—
127#217—44no—
128#217—42no—
129#208—82no—
130C2×C3⋊C8—42no—
131C25#6711yescharacteristic
132C23×C4—121no—
133C23×C4C2×Dic311yescharacteristic
134C23×C4#6711yescharacteristic
135C2×M16—121no—
136#117—41no—
137#117—41no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4C2×Dic311yescharacteristic, complemented
141C2×C4⋊C4C2×Dic311yescharacteristic, complemented
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—64no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—121no—
153C2×C22⋊C4—122no—
154C2×C22⋊C4C2×Dic311yescharacteristic
155C2×C22⋊C4#6711yescharacteristic
156C2×C22⋊C4#6711yescharacteristic
157C2×C22⋊C4C22×D311yescharacteristic
158C2×C42—21no—
159C22⋊C8—62no—
160C22⋊C8—121no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95C2×Dic311yescharacteristic, complemented
165#95C2×Dic311yescharacteristic, complemented
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—22no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—41no—
178C23×C3—42no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3—42no—
184C23×C3—81no—
185C23×C3—81no—
186C23×C3—84no—
187C23×C3—82no—
188C23×C3—82no—
189C23×C3#9911yescharacteristic
190C23×C3#9911yescharacteristic
191C23×C3#10412yes—
192C23×C3C2×C22⋊C411yescharacteristic
193C23×C3AGL(1,ℤ/8)11yescharacteristic
194C23×C3#13711yescharacteristic
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—81no—
219C2×Dic3—44no—
220C2×Dic3—88no—
221C2×Dic3—82no—
222C2×Dic3—82no—
223C3⋊C8—82no—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—21no—
228C24—21no—
229C24—42no—
230C24—81no—
231C24#21711yescharacteristic
232C24#21711yescharacteristic
233C24C2×#6711yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—81no—
244C22×C4—124no—
245C22×C4—122no—
246C22×C4—242no—
247C22×C4—242no—
248C22×C4#21711yescharacteristic
249C22×C4#21711yescharacteristic
250C22×C4C22×Dic311yescharacteristic
251C22×C4C2×#6711yescharacteristic
252M16—242no—
253C2×C8—122no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—64no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—124no—
272C22⋊C4—122no—
273C42—42no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—42no—
285C22×C3—42no—
286C22×C3—42no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—44no—
290C22×C3—44no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—82no—
298C22×C3—82no—
299C22×C3#35212yes—
300C22×C3C2×#9911yescharacteristic
301C22×C3#41711yescharacteristic
302C22×C3C2×#10411yescharacteristic
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—81no—
308C3×C4—81no—
309C3×C4—81no—
310C3×C4—81no—
311Dic3—84no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—22no—
321C23—21no—
322C23—21no—
323C23—41no—
324C23—42no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—42no—
330C23—81no—
331C23—81no—
332C23—84no—
333C23—82no—
334C23—82no—
335C23#83411yescharacteristic
336C23#83411yescharacteristic
337C23#83712yes—
338C23#94911yescharacteristic
339C23#95111yescharacteristic
340C23C2×#21711yescharacteristic
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—124no—
366C2×C4—248no—
367C2×C4—242no—
368C2×C4—242no—
369C8—242no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—22no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—42no—
376C2×C3—41no—
377C2×C3#150411yescharacteristic
378C2×C3C2×#35211yescharacteristic
379C2×C3#212711yescharacteristic
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—44no—
396C22—44no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—81no—
402C22—81no—
403C22—82no—
404C22—82no—
405C22#453912yes—
406C22C2×#83411yescharacteristic
407C22#523211yescharacteristic
408C22C2×#83711yescharacteristic
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—244no—
418C3#1175511yescharacteristic, complemented, Sylow
419C2—21no—
420C2—21no—
421C2—22no—
422C2—41no—
423C2—41no—
424C2—42no—
425C2—41no—
426C2C2×#453911yescharacteristic
427C2#7170811yescharacteristic
428C2#7186011yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#2482C2
socleC22×C3C2×#99
prime radicalC25×C3D4
cosocleC2×C3×C22⋊C4C23
hypercenter#2482D3
hypocenterC3#11755
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
4. #67651C2semidirect
5. #67649C2semidirect
7. C2×C3×#117C4semidirect
10. C3×#399C4semidirect
12. C3×#397C4semidirect
17. C3×#381C4semidirect
37. C2×C3×C4⋊C4C2×C4semidirect
38. C2×C3×C4⋊C4C2×C4semidirect
60. C3×#95C2×C4semidirect
61. C3×#95C2×C4semidirect
64. C2×#117Dic3semidirect
68. #399Dic3semidirect
70. #397Dic3semidirect
75. #381Dic3semidirect
140. C2×C4⋊C4C2×Dic3semidirect
141. C2×C4⋊C4C2×Dic3semidirect
164. #95C2×Dic3semidirect
165. #95C2×Dic3semidirect
418. C3#11755semidirect

Maximal subgroups

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

subgroupindex
C3×#24822
C2×#45262
#715392
#676512
#676492
#117553