Finite Group Databasebeta

← #10692842

Subgroups — #10692842

Order 768.

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

normal subgroups only

This group has 516 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×#2500C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic, complemented
3#71674C212yesmaximal
4#71594C211yesmaximal, characteristic, complemented
5#67616C212yesmaximal, complemented
6#9446—31nomaximal, Sylow
7C2×C3×#121C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C3×#394—21no—
10C3×#394—21no—
11C3×#390C411yescharacteristic, complemented
12C3×#390C411yescharacteristic, complemented
13C3×#388—21no—
14C3×#388—21no—
15C3×#384C411yescharacteristic, complemented
16C3×#382C411yescharacteristic, complemented
17C3×C4×C22⋊C4C2211yescharacteristic
18#5211C2212yes—
19#5207C2212yes—
20#5000—24no—
21#4924—21no—
22C4×#208—21no—
23#4537—24no—
24#4526—21no—
25#4526—21no—
26#2921—31no—
27#2500D311yescharacteristic
28#1820—31no—
29#1510—32no—
30#1284—32no—
31C22×C3×D4—21no—
32C22×C3×D4C2×C411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C3×#121—22no—
35C3×#121—22no—
36C3×#117—22no—
37C3×#117—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2×C411yescharacteristic
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×C22⋊C4D411yescharacteristic
50C2×C3×C22⋊C4D411yescharacteristic
51C2×C3×C42C2×C411yescharacteristic
52C2×C3×C42C2×C411yescharacteristic
53C2×C3×C42D411yescharacteristic
54C2×C3×C42D411yescharacteristic
55C2×#217—22no—
56C2×#208—22no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3⋊#95—22no—
61C3⋊#95—22no—
62#830—22no—
63#830D414yes—
64C4⋊(C3⋊C8)—24no—
65#803—22no—
66C4×C3⋊C8—22no—
67C2×#121C2×D311yescharacteristic
68C2×#117C2×D311yescharacteristic
69#433—64no—
70#423—61no—
71#407—32no—
72C4×M16—61no—
73#394—21no—
74#394—21no—
75#390Dic311yescharacteristic, complemented
76#390Dic311yescharacteristic, complemented
77#388—21no—
78#388—21no—
79#384Dic311yescharacteristic, complemented
80#382Dic311yescharacteristic, complemented
81#381—32no—
82C4×C22⋊C4C2×D311yescharacteristic
83#342—61no—
84#342—61no—
85#327—64no—
86C24×C3—21no—
87C24×C3C2×D411yescharacteristic
88C2×C3×D4—22no—
89C2×C3×D4—41no—
90C2×C3×D4—41no—
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—41no—
98C22×C3×C4—41no—
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D412yes—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4112yes—
107C22×Dic3—42no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—81no—
112C3×C4⋊C4—81no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—42no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—81no—
121C3×C22⋊C4—81no—
122C3×C42—21no—
123C3×C42—21no—
124C3×C42—21no—
125C3×C42—21no—
126C3×C42—41no—
127C3×C42—41no—
128#217—42no—
129#217—42no—
130#208—42no—
131#208—42no—
132C2×C3⋊C8—24no—
133C22×D4—21no—
134C22×D4C2×Dic311yescharacteristic
135C23×C4C22×D311yescharacteristic
136C2×M16—62no—
137#121—22no—
138#121—22no—
139#117—22no—
140#117—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4C2×Dic311yescharacteristic
144C2×C4⋊C4C2×Dic311yescharacteristic
145C2×C4⋊C4C2×Dic311yescharacteristic
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—62no—
153C2×C22⋊C4#6711yescharacteristic
154C2×C22⋊C4#6711yescharacteristic
155C2×C42C2×Dic311yescharacteristic
156C2×C42C2×Dic311yescharacteristic
157C2×C42#6711yescharacteristic
158C2×C42#6711yescharacteristic
159C4⋊C8—64no—
160C22⋊C8—34no—
161C22⋊C8—62no—
162#97—62no—
163C4×C8—62no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—62no—
168#95—62no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3C2×C22⋊C411yescharacteristic
181C23×C3#12112yes—
182C3×D4—82no—
183C3×D4—84no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—22no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
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×C4C4×D412yes—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12311yescharacteristic
210C2×Dic3—82no—
211C2×Dic3—82no—
212C2×Dic3—82no—
213C3⋊C8—44no—
214C24—21no—
215C24C2×#6711yescharacteristic
216C2×D4—22no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—42no—
220C2×D4—81no—
221C2×D4—81no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—122no—
228C22×C4#21711yescharacteristic
229C22×C4#21711yescharacteristic
230C22×C4#21711yescharacteristic
231C22×C4#21711yescharacteristic
232C22×C4D3×D412yes—
233C22×C4#23712yes—
234C22×C4C22×Dic311yescharacteristic
235C22×C4C2×#6711yescharacteristic
236M16—122no—
237M16—122no—
238C2×C8—64no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—81no—
252C22⋊C4—81no—
253C22⋊C4—122no—
254C22⋊C4—122no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—41no—
260C42—41no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—81no—
267C22×C3—81no—
268C22×C3—81no—
269C22×C3—81no—
270C22×C3—81no—
271C22×C3—81no—
272C22×C3—81no—
273C22×C3—81no—
274C22×C3—81no—
275C22×C3#38611yescharacteristic
276C22×C3#41011yescharacteristic
277C22×C3#41011yescharacteristic
278C3×C4—21no—
279C3×C4—22no—
280C3×C4—21no—
281C3×C4—42no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—81no—
287C3×C4—81no—
288Dic3—162no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23#93912yes—
301C23C2×#21711yescharacteristic
302D4—82no—
303D4—84no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
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—242no—
328C2×C4—242no—
329C2×C4—242no—
330C2×C4D4×Dic312yes—
331C2×C4#93511yescharacteristic
332C2×C4#93711yescharacteristic
333C8—124no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—42no—
337C2×C3—161no—
338C2×C3#188411yescharacteristic
339C2×C3#189911yescharacteristic
340C2×C3#191011yescharacteristic
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22#522011yescharacteristic
356C22#524011yescharacteristic
357C22#524011yescharacteristic
358C4—21no—
359C4—22no—
360C4—21no—
361C4—42no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C4—482no—
369C3#944611yescharacteristic, complemented, Sylow
370C2—21no—
371C2—21no—
372C2—42no—
373C2—161no—
374C2#7170511yescharacteristic
375C2#7181711yescharacteristic
376C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2500C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2500D3
hypocenterC3#9446
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2500

Complements

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

NH ≅ G/Ntype
2. #82209C2semidirect
4. #71594C2semidirect
5. #67616C2semidirect
11. C3×#390C4semidirect
12. C3×#390C4semidirect
15. C3×#384C4semidirect
16. C3×#382C4semidirect
75. #390Dic3semidirect
76. #390Dic3semidirect
79. #384Dic3semidirect
80. #382Dic3semidirect
369. C3#9446semidirect

Maximal subgroups

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

subgroupindex
C3×#25002
#822092
#716742
#715942
#676162
#94463