Finite Group Databasebeta

← #10692523

Subgroups — #10692523

Order 768.

582 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2610C211yesmaximal, characteristic
2#82202C211yesmaximal, characteristic, complemented
3#71718C212yesmaximal, complemented
4#71572C211yesmaximal, characteristic, complemented
5#67615C212yesmaximal, complemented
6#9387—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C22×#208—21no—
10C3×#395C411yescharacteristic
11C3×#394C411yescharacteristic
12C3×#393C411yescharacteristic
13C3×#392C411yescharacteristic
14C3×#390—21no—
15C3×#388—21no—
16C3×#388—21no—
17C3×#386—21no—
18C3×#376C2211yescharacteristic
19#4997—24no—
20#4926C2212yes—
21C4×#208—21no—
22C4×#208C2212yes—
23#4540—24no—
24#4528—21no—
25#4525—21no—
26#2873—31no—
27#2610D311yescharacteristic
28#1527—32no—
29#1464—31no—
30#1300—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×D4—21no—
33C22×C3×D4C2×C411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C3×#124—22no—
36C3×#124—22no—
37C3×#117—22no—
38C3×#117—22no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—21no—
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⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4D411yescharacteristic
50C2×C3×C42—21no—
51C2×C3×C42D411yescharacteristic
52C2×C3×C42C2311yescharacteristic
53C2×#208—22no—
54C2×#208—24no—
55C2×#208—24no—
56C2×#208—22no—
57C22×C3⋊C8—22no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95C2×C411yescharacteristic
62C3×#95C2×C411yescharacteristic
63#830—22no—
64C4⋊(C3⋊C8)—22no—
65#803—22no—
66#803—22no—
67#803D414yescomplemented
68C4×C3⋊C8—22no—
69C4×C3⋊C8—22no—
70C22×M16—61no—
71C2×#124C2×D311yescharacteristic
72C2×#117C2×D311yescharacteristic
73#432—32no—
74#405—64no—
75C4×M16—32no—
76C4×M16—61no—
77#395Dic311yescharacteristic
78#394Dic311yescharacteristic
79#393Dic311yescharacteristic
80#392Dic311yescharacteristic
81#390—21no—
82#388—21no—
83#388—21no—
84#386—21no—
85#376C2×D311yescharacteristic
86#329—64no—
87#324—61no—
88#323—61no—
89C24×C3—21no—
90C24×C3C22⋊C411yescharacteristic
91C2×C3×Q8—22no—
92C2×C3×Q8—81no—
93C2×C3×D4—22no—
94C2×C3×D4—42no—
95C2×C3×D4—42no—
96C2×C3×D4—81no—
97C2×C3×D4—81no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22⋊C411yescharacteristic
107C22×C3×C4C22⋊C411yescharacteristic
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C3×C4⋊C4—22no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—81no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4—22no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—42no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—81no—
122C3×C22⋊C4—81no—
123C3×C42—21no—
124C3×C42—21no—
125C3×C42—42no—
126C3×C42C2×D412yes—
127C3×C42#4112yes—
128#208—28no—
129#208—24no—
130#208—24no—
131C2×C3⋊C8—28no—
132C2×C3⋊C8—24no—
133C2×C3⋊C8—42no—
134C22×Q8C2×Dic311yescharacteristic
135C22×D4—21no—
136C22×D4C2×Dic311yescharacteristic
137C23×C4#6711yescharacteristic
138C2×M16—62no—
139C2×M16—64no—
140C2×M16—64no—
141C2×M16—62no—
142C22×C8—62no—
143#124—22no—
144#124—22no—
145#117—22no—
146#117—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4C2×Dic311yescharacteristic
149C2×C4⋊C4#6711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4C2×Dic311yescharacteristic
157C2×C22⋊C4#6711yescharacteristic
158C2×C42—21no—
159C2×C42#6711yescharacteristic
160C2×C42C22×D311yescharacteristic
161C4⋊C8—62no—
162C22⋊C8—62no—
163#97—34no—
164#97—62no—
165#97—62no—
166C4×C8—62no—
167C4×C8—62no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95C2×Dic311yescharacteristic
172#95C2×Dic311yescharacteristic
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3—41no—
184C23×C3#10412yes—
185C23×C3C2×C22⋊C411yescharacteristic
186C3×Q8—82no—
187C3×D4—82no—
188C3×D4—84no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—22no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
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—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4#10412yes—
212C2×C3×C4C4×D412yes—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12112yes—
215C2×C3×C4#12311yescharacteristic
216C3⋊C8—28no—
217C3⋊C8—82no—
218C24—21no—
219C24#21711yescharacteristic
220C2×Q8—22no—
221C2×Q8—81no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—81no—
226C2×D4—81no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4#21711yescharacteristic
235C22×C4#21711yescharacteristic
236C22×C4#21711yescharacteristic
237C22×C4C22×Dic311yescharacteristic
238C22×C4C2×#6711yescharacteristic
239C22×C4C2×#6711yescharacteristic
240M16—68no—
241M16—64no—
242M16—64no—
243C2×C8—68no—
244C2×C8—64no—
245C2×C8—122no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—81no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C42—21no—
259C42—21no—
260C42—42no—
261C42D3×D412yes—
262C42#23712yes—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—22no—
266C22×C3—42no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—81no—
270C22×C3—81no—
271C22×C3—81no—
272C22×C3—81no—
273C22×C3—81no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3#38611yescharacteristic
278C22×C3C2×#10411yescharacteristic
279C22×C3C2×#10411yescharacteristic
280C3×C4—22no—
281C3×C4—21no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—22no—
285C3×C4—42no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—81no—
289C3×C4—81no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23#83712yes—
302C23C2×#21711yescharacteristic
303Q8—82no—
304D4—82no—
305D4—84no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4#83712yes—
329C2×C4D4×Dic312yes—
330C2×C4#93511yescharacteristic
331C2×C4#93711yescharacteristic
332C2×C4#93912yes—
333C8—68no—
334C8—242no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—42no—
338C2×C3—161no—
339C2×C3#188411yescharacteristic
340C2×C3#189611yescharacteristic
341C2×C3#191411yescharacteristic
342C22—21no—
343C22—21no—
344C22—22no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#522011yescharacteristic
357C22C2×#83711yescharacteristic
358C22C2×#83711yescharacteristic
359C4—21no—
360C4—22no—
361C4—21no—
362C4—22no—
363C4—22no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C3#938711yescharacteristic, complemented, Sylow
370C2—21no—
371C2—21no—
372C2—42no—
373C2—161no—
374C2#7171111yescharacteristic
375C2#7181711yescharacteristic
376C2#7185511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2610C2
socleC22×C3#386
prime radicalC3×#392C4
cosocleC2×C3×C42C23
hypercenter#2610D3
hypocenterC3#9387
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2610

Complements

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

NH ≅ G/Ntype
2. #82202C2semidirect
3. #71718C2semidirect
4. #71572C2semidirect
5. #67615C2semidirect
67. #803D4semidirect
369. C3#9387semidirect

Maximal subgroups

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

subgroupindex
C3×#26102
#822022
#717182
#715722
#676152
#93873