Finite Group Databasebeta

← #10692852

Subgroups — #10692852

Order 768.

486 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2545C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3#71719C212yesmaximal
4#71625C211yesmaximal, characteristic, complemented
5#67621C212yesmaximal, complemented
6#9448—31nomaximal, Sylow
7C2×C3×C4⋊Q8C411yescharacteristic
8C2×C3×#124C411yescharacteristic
9C2×C3×#124C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C3×#396—21no—
12C3×#396—21no—
13C3×#391C411yescharacteristic
14C3×#390C411yescharacteristic
15C3×#388—21no—
16C3×#388—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5024—24no—
19#4926—21no—
20#4926C2212yes—
21#4926C2212yes—
22#4925—21no—
23#4540—24no—
24#4528—21no—
25#4528—21no—
26#2983—31no—
27#2545D311yescharacteristic
28#1533—32no—
29#1467—31no—
30#1285—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×Q8C2×C411yescharacteristic
33C22×C3×D4C2×C411yescharacteristic
34C3×C4⋊Q8—42no—
35C3×#124—22no—
36C3×#124—22no—
37C3×#124—42no—
38C3×C4×Q8—22no—
39C3×C4×Q8—22no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2×C411yescharacteristic
53C2×C3×C42C2×C411yescharacteristic
54C2×C3×C42D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60#830—22no—
61#830—22no—
62C4⋊(C3⋊C8)—22no—
63C4⋊(C3⋊C8)—22no—
64C4⋊(C3⋊C8)—22no—
65C4⋊(C3⋊C8)—22no—
66#803—22no—
67#803—22no—
68#803D414yes—
69C4×C3⋊C8—22no—
70C2×C4⋊Q8Dic311yescharacteristic
71C2×#124Dic311yescharacteristic
72C2×#124C2×D311yescharacteristic
73C2×C4×Q8C2×D311yescharacteristic
74#446—64no—
75#432—32no—
76#432—32no—
77#432—61no—
78#431—61no—
79#396—21no—
80#396—21no—
81#391Dic311yescharacteristic
82#390Dic311yescharacteristic
83#388—21no—
84#388—21no—
85C4×C4⋊C4C2×D311yescharacteristic
86#329—64no—
87#324—61no—
88#324—61no—
89C24×C3—21no—
90C2×C3×Q8—22no—
91C2×C3×Q8—22no—
92C2×C3×Q8—22no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×D4—22no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4C22⋊C411yescharacteristic
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—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—42no—
113C3×C4⋊C4—42no—
114C3×C4⋊C4—42no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—42no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—42no—
121C3×C22⋊C4—41no—
122C3×C42—21no—
123C3×C42—21no—
124C3×C42—21no—
125C3×C42—21no—
126C3×C42—41no—
127C3×C42—41no—
128C3×C42C2×D412yes—
129C3×C42#4112yes—
130C2×C3⋊C8—24no—
131C2×C3⋊C8—42no—
132C22×Q8C2×Dic311yescharacteristic
133C22×Q8C2×Dic311yescharacteristic
134C22×D4C2×Dic311yescharacteristic
135C4⋊Q8—42no—
136#124—22no—
137#124—22no—
138#124—42no—
139C4×Q8—22no—
140C4×Q8—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4C2×Dic311yescharacteristic
145C2×C4⋊C4#6711yescharacteristic
146C2×C4⋊C4#6711yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C42C2×Dic311yescharacteristic
154C2×C42C2×Dic311yescharacteristic
155C2×C42#6711yescharacteristic
156C2×C42#6711yescharacteristic
157C2×C42C22×D311yescharacteristic
158C4⋊C8—62no—
159C4⋊C8—62no—
160C4⋊C8—62no—
161C4⋊C8—62no—
162C22⋊C8—62no—
163C22⋊C8—62no—
164#97—34no—
165#97—62no—
166#97—62no—
167C4×C8—62no—
168#95—21no—
169#95—21no—
170#95—21no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3C2×C22⋊C411yescharacteristic
179C3×Q8—44no—
180C3×Q8—82no—
181C3×D4—82no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—41no—
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#10012yes—
205C2×C3×C4C4×D412yes—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12112yes—
208C2×C3×C4#12311yescharacteristic
209C3⋊C8—44no—
210C3⋊C8—82no—
211C24—21no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×D4—22no—
219C2×D4—41no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4#21711yescharacteristic
225C22×C4#21711yescharacteristic
226C22×C4#21711yescharacteristic
227C22×C4#21711yescharacteristic
228C22×C4C22×Dic311yescharacteristic
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×#6711yescharacteristic
231C2×C8—64no—
232C2×C8—122no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—41no—
251C42—41no—
252C42D3×D412yes—
253C42#23712yes—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—81no—
257C22×C3—81no—
258C22×C3—81no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3—81no—
262C22×C3—81no—
263C22×C3#38611yescharacteristic
264C22×C3C2×#10011yescharacteristic
265C22×C3#41311yescharacteristic
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—42no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—81no—
276C3×C4—81no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×#21711yescharacteristic
285Q8—44no—
286Q8—82no—
287D4—82no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
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—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4#83312yes—
311C2×C4D4×Dic312yes—
312C2×C4#93511yescharacteristic
313C2×C4#93711yescharacteristic
314C2×C4#93912yes—
315C8—124no—
316C8—242no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—161no—
320C2×C3#188411yescharacteristic
321C2×C3#190011yescharacteristic
322C2×C3#191211yescharacteristic
323C22—21no—
324C22—21no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C2×#83311yescharacteristic
333C22#522011yescharacteristic
334C22#523711yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C3#944811yescharacteristic, complemented, Sylow
347C2—21no—
348C2—21no—
349C2—161no—
350C2#7168311yescharacteristic
351C2#7181711yescharacteristic
352C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2545C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#2545D3
hypocenterC3#9448
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2545

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
4. #71625C2semidirect
5. #67621C2semidirect
346. C3#9448semidirect

Maximal subgroups

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

subgroupindex
C3×#25452
#824362
#717192
#716252
#676212
#94483