Finite Group Databasebeta

← #10705708

Subgroups — #10705708

Order 768.

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

normal subgroups only

This group has 488 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×#2591C211yesmaximal, characteristic
2C2×#4530C211yesmaximal, characteristic, complemented
3#71552C211yesmaximal, characteristic, complemented
4#67652C212yesmaximal, complemented
5#67647C212yesmaximal, complemented
6#11766—31nomaximal, Sylow
7C2×C3×#121C411yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C22×#208—21no—
10C3×#400C411yescharacteristic
11C3×#400C2211yescharacteristic
12C3×#394C411yescharacteristic
13C3×#394C2211yescharacteristic
14C3×#388—21no—
15C3×#388—21no—
16C3×#386—21no—
17C3×#386—21no—
18C3×C4×C22⋊C4C411yescharacteristic
19#5207—21no—
20#4530—21no—
21#4530C2212yes—
22#4528—21no—
23#4528C2212yes—
24#2591D311yescharacteristic
25C2×#343—31no—
26#1463—31no—
27#1353—32no—
28#1345—32no—
29C22×C3×D4—21no—
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C3×#121—41no—
35C3×#121—41no—
36C2×C3×C4⋊C4—41no—
37C2×C3×C4⋊C4—41no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42—21no—
50C2×#208—24no—
51C2×#208—22no—
52C2×#208—41no—
53C2×#208—42no—
54C2×#208—41no—
55C22×C3⋊C8—41no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95C2×C411yescharacteristic
61C3×#95C2×C411yescharacteristic
62#830—22no—
63#830—41no—
64C22×M16—61no—
65C2×#121Dic311yescharacteristic
66C22×C4⋊C4C2×D311yescharacteristic
67#407—61no—
68#400Dic311yescharacteristic
69#400C2×D311yescharacteristic
70#394Dic311yescharacteristic
71#394C2×D311yescharacteristic
72#388—21no—
73#388—21no—
74#386—21no—
75#386—21no—
76C4×C22⋊C4Dic311yescharacteristic
77#343—32no—
78#343—61no—
79#324—32no—
80#324—61no—
81C24×C3—21no—
82C24×C3C22⋊C411yescharacteristic
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—22no—
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—21no—
100C22×C3×C4—21no—
101C22×C3×C4—81no—
102C22×C3×C4—81no—
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—81no—
120C3×C22⋊C4—81no—
121C3×C42—81no—
122#208—24no—
123#208—44no—
124#208—42no—
125#208—82no—
126C2×C3⋊C8—44no—
127C2×C3⋊C8—42no—
128C2×C3⋊C8—42no—
129C22×D4—21no—
130C22×D4C2×Dic311yescharacteristic
131C23×C4C2×Dic311yescharacteristic
132C23×C4#6711yescharacteristic
133C23×C4#6711yescharacteristic
134C2×M16—64no—
135C2×M16—62no—
136C2×M16—121no—
137C2×M16—122no—
138C2×M16—121no—
139C22×C8—121no—
140#121—41no—
141#121—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4C2×Dic311yescharacteristic
145C2×C4⋊C4C2×Dic311yescharacteristic
146C2×C4⋊C4#6711yescharacteristic
147C2×C4⋊C4#6711yescharacteristic
148C2×C4⋊C4C22×D311yescharacteristic
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C42—21no—
156C22⋊C8—62no—
157C22⋊C8—121no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95C2×Dic311yescharacteristic
163#95C2×Dic311yescharacteristic
164C23×C3—22no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3#9911yescharacteristic
176C23×C3#9911yescharacteristic
177C23×C3C2×C22⋊C411yescharacteristic
178C3×D4—82no—
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—42no—
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—41no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4#10412yes—
211C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
212C2×C3×C4#13711yescharacteristic
213C3⋊C8—44no—
214C3⋊C8—82no—
215C24—21no—
216C24#21711yescharacteristic
217C2×D4—41no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—81no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4#21711yescharacteristic
238C22×C4#21711yescharacteristic
239C22×C4#21711yescharacteristic
240C22×C4C22×Dic311yescharacteristic
241C22×C4C2×#6711yescharacteristic
242C22×C4C2×#6711yescharacteristic
243M16—64no—
244M16—124no—
245M16—122no—
246M16—242no—
247C2×C8—124no—
248C2×C8—122no—
249C2×C8—122no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C42—81no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3—81no—
273C22×C3—81no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—81no—
279C22×C3#35512yes—
280C22×C3C2×#9911yescharacteristic
281C22×C3#41711yescharacteristic
282C22×C3C2×#10411yescharacteristic
283C3×C4—21no—
284C3×C4—22no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4—81no—
293C23—21no—
294C23—21no—
295C23—22no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#83411yescharacteristic
305C23#83411yescharacteristic
306C23C2×#21711yescharacteristic
307D4—82no—
308D4—82no—
309D4—82no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
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—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4#83712yes—
340C2×C4#94911yescharacteristic
341C2×C4#95111yescharacteristic
342C8—124no—
343C8—242no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—22no—
347C2×C3—41no—
348C2×C3—161no—
349C2×C3#150411yescharacteristic
350C2×C3#212111yescharacteristic
351C2×C3C2×#35511yescharacteristic
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22#454212yes—
369C22C2×#83411yescharacteristic
370C22#523211yescharacteristic
371C22C2×#83711yescharacteristic
372C4—21no—
373C4—22no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C3#1176611yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—22no—
386C2—41no—
387C2—161no—
388C2#7170811yescharacteristic
389C2C2×#454211yescharacteristic
390C2#7184711yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2591C2
socleC22×C3C2×#99
prime radicalC2×C3×#121C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2591D3
hypocenterC3#11766
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
2. C2×#4530C2semidirect
3. #71552C2semidirect
4. #67652C2semidirect
5. #67647C2semidirect
382. C3#11766semidirect

Maximal subgroups

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

subgroupindex
C3×#25912
C2×#45302
#715522
#676522
#676472
#117663