Finite Group Databasebeta

← #10692533

Subgroups — #10692533

Order 768.

496 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 103 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#82209C211yesmaximal, characteristic, complemented
3#71718C212yesmaximal, complemented
4#71572C211yesmaximal, characteristic, complemented
5#67615C212yesmaximal, complemented
6#9389—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C3×#395C411yescharacteristic
10C3×#394C411yescharacteristic
11C3×#393C411yescharacteristic
12C3×#392C411yescharacteristic
13C3×#390—21no—
14C3×#388—21no—
15C3×#388—21no—
16C3×#386—21no—
17C3×#376C2211yescharacteristic
18#5207—21no—
19#5000—24no—
20#4926C2212yes—
21#4924—21no—
22C4×#208C2212yes—
23#4540—24no—
24#4528—21no—
25#4525—21no—
26#2921—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—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95C2×C411yescharacteristic
58C3×#95C2×C411yescharacteristic
59#830—24no—
60#830—22no—
61C4⋊(C3⋊C8)—24no—
62C4⋊(C3⋊C8)—22no—
63#803—22no—
64#803D414yescomplemented
65C4×C3⋊C8—22no—
66C2×#124C2×D311yescharacteristic
67C2×#117C2×D311yescharacteristic
68#433—64no—
69#432—32no—
70#423—61no—
71#407—61no—
72C4×M16—32no—
73#395Dic311yescharacteristic
74#394Dic311yescharacteristic
75#393Dic311yescharacteristic
76#392Dic311yescharacteristic
77#390—21no—
78#388—21no—
79#388—21no—
80#386—21no—
81#376C2×D311yescharacteristic
82#329—64no—
83#324—61no—
84#323—61no—
85C24×C3—21no—
86C24×C3C22⋊C411yescharacteristic
87C2×C3×Q8—22no—
88C2×C3×Q8—81no—
89C2×C3×D4—22no—
90C2×C3×D4—42no—
91C2×C3×D4—42no—
92C2×C3×D4—81no—
93C2×C3×D4—81no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—81no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—42no—
113C3×C22⋊C4—42no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—81no—
118C3×C22⋊C4—81no—
119C3×C42—21no—
120C3×C42—21no—
121C3×C42—42no—
122C3×C42C2×D412yes—
123C3×C42#4112yes—
124#208—42no—
125#208—42no—
126C2×C3⋊C8—24no—
127C2×C3⋊C8—42no—
128C22×Q8C2×Dic311yescharacteristic
129C22×D4—21no—
130C22×D4C2×Dic311yescharacteristic
131C23×C4#6711yescharacteristic
132C2×M16—62no—
133#124—22no—
134#124—22no—
135#117—22no—
136#117—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4C2×Dic311yescharacteristic
139C2×C4⋊C4#6711yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4C2×Dic311yescharacteristic
147C2×C22⋊C4#6711yescharacteristic
148C2×C42—21no—
149C2×C42#6711yescharacteristic
150C2×C42C22×D311yescharacteristic
151C4⋊C8—64no—
152C4⋊C8—62no—
153C22⋊C8—64no—
154C22⋊C8—62no—
155#97—34no—
156#97—62no—
157C4×C8—62no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95C2×Dic311yescharacteristic
162#95C2×Dic311yescharacteristic
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
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×C3—41no—
175C23×C3C2×C22⋊C411yescharacteristic
176C3×Q8—82no—
177C3×D4—82no—
178C3×D4—84no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—22no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
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×C4C4×D412yes—
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12112yes—
204C2×C3×C4#12311yescharacteristic
205C3⋊C8—44no—
206C3⋊C8—82no—
207C24—21no—
208C24#21711yescharacteristic
209C2×Q8—22no—
210C2×Q8—81no—
211C2×D4—22no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—81no—
215C2×D4—81no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4#21711yescharacteristic
224C22×C4#21711yescharacteristic
225C22×C4#21711yescharacteristic
226C22×C4C22×Dic311yescharacteristic
227C22×C4C2×#6711yescharacteristic
228C22×C4C2×#6711yescharacteristic
229M16—122no—
230M16—122no—
231C2×C8—64no—
232C2×C8—122no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—81no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—81no—
244C22⋊C4—81no—
245C42—21no—
246C42—21no—
247C42—42no—
248C42D3×D412yes—
249C42#23712yes—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—42no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—81no—
257C22×C3—81no—
258C22×C3—81no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3—81no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3#38611yescharacteristic
265C22×C3#42111yescharacteristic
266C22×C3#42111yescharacteristic
267C3×C4—22no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—42no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—81no—
276C3×C4—81no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×#21711yescharacteristic
290Q8—82no—
291D4—82no—
292D4—84no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4D4×Dic312yes—
316C2×C4#93511yescharacteristic
317C2×C4#93711yescharacteristic
318C2×C4#93912yes—
319C8—124no—
320C8—242no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—81no—
324C2×C3—161no—
325C2×C3#188411yescharacteristic
326C2×C3#189711yescharacteristic
327C2×C3#191411yescharacteristic
328C22—21no—
329C22—21no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22#522011yescharacteristic
343C22#523611yescharacteristic
344C22#523611yescharacteristic
345C4—21no—
346C4—21no—
347C4—22no—
348C4—22no—
349C4—42no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—81no—
354C4—81no—
355C3#938911yescharacteristic, complemented, Sylow
356C2—21no—
357C2—21no—
358C2—81no—
359C2—161no—
360C2#7171211yescharacteristic
361C2#7181711yescharacteristic
362C2#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#9389
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. #82209C2semidirect
3. #71718C2semidirect
4. #71572C2semidirect
5. #67615C2semidirect
64. #803D4semidirect
355. C3#9389semidirect

Maximal subgroups

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

subgroupindex
C3×#26102
#822092
#717182
#715722
#676152
#93893