Finite Group Databasebeta

← #10692967

Subgroups — #10692967

Order 768.

716 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 123 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×#2409C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3#72132C212yesmaximal, complemented
4#71619C211yesmaximal, characteristic
5#67625C212yesmaximal, complemented
6#9471—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C2×C3×C22⋊Q8—21no—
9C2×C3×C4×D4C2211yescharacteristic
10C3×#395C412yescomplemented
11C3×#393C412yescomplemented
12C3×#391—21no—
13C3×#386—21no—
14C3×#385—21no—
15C3×#380C2211yescharacteristic
16C2×#835—24no—
17C2×#834—21no—
18#5212C2212yes—
19#5207—21no—
20C2×#830C2212yes—
21#5011—24no—
22#4926—21no—
23#4526—21no—
24#2973—31no—
25#2409D311yescharacteristic
26#1799—31no—
27#1787—32no—
28#1289—32no—
29C22×C3×Q8—21no—
30C22×C3×Q8—22no—
31C22×C3×Q8—22no—
32C22×C3×Q8—44no—
33C22×C3×Q8C2×C412yes—
34C22×C3×D4D411yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4D411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C3×C22⋊Q8—44no—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×#217—22no—
50C2×C4⋊Dic3—22no—
51C2×#208—22no—
52C22×C3⋊C8D414yes—
53C3×#95—22no—
54C3×#95—22no—
55C3×#95C2×C412yes—
56#835—48no—
57#834—42no—
58C3⋊#95—22no—
59#830—22no—
60#830—22no—
61#830—22no—
62C4⋊(C3⋊C8)—22no—
63#803—22no—
64C23×Q8C2×D311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×C4×D4C2×D311yescharacteristic
67#435—64no—
68#432—61no—
69C2×#103—64no—
70C2×#99—61no—
71#407—61no—
72C2×C22⋊C8—32no—
73#395Dic312yescomplemented
74#393Dic312yescomplemented
75#391—21no—
76#386—21no—
77#385—21no—
78#380—32no—
79#380C2×D311yescharacteristic
80#342—61no—
81C24×C3C22⋊C411yescharacteristic
82C24×C3C2×D411yescharacteristic
83C2×C3×Q8—24no—
84C2×C3×Q8—28no—
85C2×C3×Q8—24no—
86C2×C3×Q8—44no—
87C2×C3×Q8—44no—
88C2×C3×Q8—42no—
89C2×C3×Q8—48no—
90C2×C3×Q8—41no—
91C2×C3×Q8—41no—
92C2×C3×D4—22no—
93C2×C3×D4—21no—
94C2×C3×D4—22no—
95C2×C3×D4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—22no—
100C22×C3×C4—42no—
101C22×C3×C4—44no—
102C22×C3×C4—42no—
103C22×C3×C4—42no—
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22⋊C411yescharacteristic
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C2×D412yes—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4112yes—
111C22×Dic3—42no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—42no—
114C3×C4⋊C4—44no—
115C3×C22⋊C4—22no—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—44no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C42—21no—
121C3×C42—21no—
122#217—42no—
123#217—42no—
124C4⋊Dic3—44no—
125#208—82no—
126C2×C3⋊C8—24no—
127C2×C3⋊C8—44no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—44no—
132C22×Q8C2×Dic312yes—
133C22×D4#6711yescharacteristic
134C23×C4—21no—
135C23×C4#6711yescharacteristic
136C23×C4C22×D311yescharacteristic
137C2×M16—62no—
138C22×C8—34no—
139C22⋊Q8—44no—
140C4×D4—22no—
141C4×D4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—62no—
145C2×C4⋊C4C2×Dic311yescharacteristic
146C2×C22⋊C4—22no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—62no—
149C2×C22⋊C4C2×Dic311yescharacteristic
150C2×C22⋊C4#6711yescharacteristic
151C2×C42#6711yescharacteristic
152C4⋊C8—62no—
153#103—128no—
154#99—122no—
155C22⋊C8—62no—
156C22⋊C8—62no—
157C22⋊C8—62no—
158#97—62no—
159#95—22no—
160#95—22no—
161#95—62no—
162#95C2×Dic312yes—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3C2×C22⋊C411yescharacteristic
171C23×C3#12112yes—
172C23×C3AGL(1,ℤ/8)11yescharacteristic
173C23×C3#13712yes—
174C23×C3#13711yescharacteristic
175C3×Q8—28no—
176C3×Q8—44no—
177C3×Q8—44no—
178C3×Q8—44no—
179C3×D4—42no—
180C3×D4—42no—
181C2×C3×C4—24no—
182C2×C3×C4—22no—
183C2×C3×C4—24no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—41no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—48no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—81no—
201C2×C3×C4C4×D412yes—
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12311yescharacteristic
204C2×Dic3—82no—
205C2×Dic3—82no—
206C2×Dic3—82no—
207C3⋊C8—44no—
208C24#21711yescharacteristic
209C24C2×#6711yescharacteristic
210C2×Q8—24no—
211C2×Q8—28no—
212C2×Q8—24no—
213C2×Q8—44no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—48no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—122no—
232C22×C4#21711yescharacteristic
233C22×C4#21711yescharacteristic
234C22×C4#21711yescharacteristic
235C22×C4D3×D412yes—
236C22×C4#23712yes—
237C22×C4C22×Dic311yescharacteristic
238C22×C4C2×#6711yescharacteristic
239M16—242no—
240C2×C8—64no—
241C2×C8—124no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—124no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—44no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—122no—
252C22⋊C4—122no—
253C42—21no—
254C42—21no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—41no—
260C22×C3—42no—
261C22×C3—41no—
262C22×C3—42no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—81no—
267C22×C3#38611yescharacteristic
268C22×C3#41711yescharacteristic
269C22×C3#41911yescharacteristic
270C22×C3#46812yes—
271C22×C3#47012yes—
272C3×C4—21no—
273C3×C4—22no—
274C3×C4—44no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—82no—
279C3×C4B(4,𝔽2)12yes—
280Dic3—162no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#93912yes—
289C23#94212yes—
290C23#94911yescharacteristic
291C23#95111yescharacteristic
292C23C2×#21711yescharacteristic
293Q8—28no—
294Q8—44no—
295Q8—44no—
296Q8—44no—
297D4—42no—
298D4—42no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—48no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—242no—
320C2×C4—242no—
321C2×C4—242no—
322C2×C4D4×Dic312yes—
323C2×C4#93511yescharacteristic
324C2×C4#93711yescharacteristic
325C8—124no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—81no—
331C2×C3#188911yescharacteristic
332C2×C3#189411yescharacteristic
333C2×C3#190911yescharacteristic
334C22—21no—
335C22—22no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22#504912yes—
347C22#505212yes—
348C22#522011yescharacteristic
349C22#522511yescharacteristic
350C22#523211yescharacteristic
351C4—21no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—44no—
356C4—41no—
357C4—82no—
358C4—482no—
359C4#503212yes—
360C3#947111yescharacteristic, complemented, Sylow
361C2—21no—
362C2—22no—
363C2—21no—
364C2—22no—
365C2—81no—
366C2#7166911yescharacteristic
367C2#7169711yescharacteristic
368C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2409C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2409D3
hypocenterC3#9471
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #72132C2semidirect
5. #67625C2semidirect
10. C3×#395C4semidirect
11. C3×#393C4semidirect
73. #395Dic3semidirect
74. #393Dic3semidirect
360. C3#9471semidirect

Maximal subgroups

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

subgroupindex
C3×#24092
#823022
#721322
#716192
#676252
#94713