Finite Group Databasebeta

← #10690192

Subgroups — #10690192

Order 768.

836 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×C3×#393C211yesmaximal, characteristic
2#82093C211yesmaximal, characteristic
3#72132C212yesmaximal, complemented
4C2×#4525C211yesmaximal, characteristic
5#67612C212yesmaximal, complemented
6#9016—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C22×C3×C22⋊C4—21no—
9C22×C3×C22⋊C4C2211yescharacteristic
10C3×#393—22no—
11C3×#393C414yescomplemented
12C2×C3×#95—21no—
13C2×C3×#95C2211yescharacteristic
14#5247—21no—
15C2×#835—24no—
16#5212C2212yes—
17C4×#217—21no—
18C2×#830C2212yes—
19#4951—24no—
20#4525—22no—
21C2×#393D311yescharacteristic
22#2312—31no—
23#1787—32no—
24C2×#323—31no—
25#1294—32no—
26C25×C3D411yescharacteristic
27C22×C3×Q8—21no—
28C22×C3×Q8—22no—
29C22×C3×Q8—22no—
30C22×C3×Q8—44no—
31C22×C3×Q8C2×C412yes—
32C23×C3×C4—21no—
33C23×C3×C4—21no—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—24no—
39C2×C3×C22⋊C4—44no—
40C2×C3×C22⋊C4C2×C412yes—
41C2×C3×C22⋊C4D412yes—
42C2×#217—22no—
43C2×#217—22no—
44C2×#217—22no—
45C2×C4⋊Dic3D414yes—
46C2×C4×Dic3—22no—
47C22×C3⋊C8—22no—
48C3×#95—24no—
49C3×#95—21no—
50C3×#95C2×C412yes—
51#835—48no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54#830—22no—
55#830—22no—
56C23×Q8C2×D311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4C2×D311yescharacteristic
59C2×#103—64no—
60C2×C22⋊C8—32no—
61#393—22no—
62#393Dic314yescomplemented
63#388—64no—
64#380—32no—
65#379—61no—
66C4×C22⋊C4—61no—
67C2×#95—21no—
68C2×#95C2×D311yescharacteristic
69#323—62no—
70C24×C3—21no—
71C24×C3—22no—
72C24×C3—21no—
73C24×C3—22no—
74C24×C3—42no—
75C24×C3—42no—
76C24×C3C22⋊C412yes—
77C24×C3C2×D411yescharacteristic
78C2×C3×Q8—28no—
79C2×C3×Q8—24no—
80C2×C3×Q8—44no—
81C2×C3×Q8—44no—
82C2×C3×Q8—44no—
83C2×C3×Q8—48no—
84C2×C3×Q8—42no—
85C22×C3×C4—24no—
86C22×C3×C4—22no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—22no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—41no—
93C22×C3×C4—41no—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4—42no—
97C22×C3×C4—42no—
98C22×C3×C4C22⋊C412yes—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D412yes—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4112yes—
103C22×Dic3—24no—
104C3×C22⋊C4—24no—
105C3×C22⋊C4—24no—
106C3×C22⋊C4—48no—
107#217—42no—
108#217—42no—
109#217—82no—
110#217—82no—
111C4⋊Dic3—44no—
112C4×Dic3—82no—
113C2×C3⋊C8—44no—
114C2×C3⋊C8—42no—
115C25#6711yescharacteristic
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—44no—
120C22×Q8C2×Dic312yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4#6711yescharacteristic
124C23×C4C22×D311yescharacteristic
125C22×C8—62no—
126C2×C4⋊C4—34no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—44no—
131C2×C22⋊C4—62no—
132C2×C22⋊C4—62no—
133C2×C22⋊C4—62no—
134C2×C22⋊C4C2×Dic312yes—
135C2×C22⋊C4#6712yes—
136C2×C42—62no—
137#103—128no—
138C22⋊C8—62no—
139C22⋊C8—62no—
140#95—24no—
141#95—21no—
142#95—62no—
143#95—62no—
144#95C2×Dic312yes—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—22no—
148C23×C3—22no—
149C23×C3—22no—
150C23×C3—42no—
151C23×C3—42no—
152C23×C3—48no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3—42no—
156C23×C3—42no—
157C23×C3—81no—
158C23×C3—81no—
159C23×C3—84no—
160C23×C3C2×C22⋊C411yescharacteristic
161C23×C3#12011yescharacteristic
162C23×C3#12311yescharacteristic
163C23×C3#13512yes—
164C23×C3#13712yes—
165C3×Q8—28no—
166C3×Q8—48no—
167C3×Q8—44no—
168C2×C3×C4—24no—
169C2×C3×C4—24no—
170C2×C3×C4—22no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—44no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—44no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—81no—
184C2×C3×C4—84no—
185C2×C3×C4C4×D412yes—
186C2×C3×C4#12112yes—
187C2×Dic3—44no—
188C2×Dic3—44no—
189C2×Dic3—44no—
190C3⋊C8—82no—
191C24—21no—
192C24—22no—
193C24—21no—
194C24—22no—
195C24—42no—
196C24—42no—
197C24#21712yes—
198C24C2×#6711yescharacteristic
199C2×Q8—28no—
200C2×Q8—24no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—48no—
205C2×Q8—42no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—64no—
220C22×C4#21712yes—
221C22×C4D3×D412yes—
222C22×C4#23712yes—
223C22×C4C22×Dic311yescharacteristic
224C22×C4C2×#6711yescharacteristic
225C2×C8—124no—
226C2×C8—122no—
227C4⋊C4—124no—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—48no—
231C22⋊C4—122no—
232C22⋊C4—122no—
233C22⋊C4—242no—
234C22⋊C4—242no—
235C42—242no—
236C22×C3—24no—
237C22×C3—24no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—44no—
243C22×C3—42no—
244C22×C3—42no—
245C22×C3—42no—
246C22×C3—82no—
247C22×C3—84no—
248C22×C3—84no—
249C22×C3#38611yescharacteristic
250C22×C3#41611yescharacteristic
251C22×C3#41911yescharacteristic
252C22×C3#45212yes—
253C22×C3#45512yes—
254C22×C3B(4,𝔽2)12yes—
255C22×C3#48512yes—
256C3×C4—22no—
257C3×C4—22no—
258C3×C4—42no—
259C3×C4—44no—
260C3×C4—82no—
261Dic3—84no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—48no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23—81no—
276C23—84no—
277C23#93511yescharacteristic
278C23#93711yescharacteristic
279C23#94212yes—
280C23#95012yes—
281C23C2×#21711yescharacteristic
282Q8—28no—
283Q8—48no—
284Q8—44no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—84no—
302C2×C4—124no—
303C2×C4—124no—
304C2×C4—124no—
305C2×C4D4×Dic312yes—
306C2×C4#93912yes—
307C8—242no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—22no—
311C2×C3—82no—
312C2×C3#160614yes—
313C2×C3#187311yescharacteristic
314C2×C3#188511yescharacteristic
315C2×C3#188911yescharacteristic
316C22—21no—
317C22—21no—
318C22—24no—
319C22—24no—
320C22—21no—
321C22—21no—
322C22—44no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—82no—
327C22—84no—
328C22—84no—
329C22#503212yes—
330C22#504512yes—
331C22#504712yes—
332C22#506112yes—
333C22#522011yescharacteristic
334C22#522511yescharacteristic
335C22#523311yescharacteristic
336C4—22no—
337C4—22no—
338C4—42no—
339C4—44no—
340C4—82no—
341C4—244no—
342C3#901611yescharacteristic, complemented, Sylow
343C2—21no—
344C2—21no—
345C2—22no—
346C2—82no—
347C2#6970014yes—
348C2#7166911yescharacteristic
349C2#7169311yescharacteristic
350C2#7172311yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#393C2
socleC23×C3#120
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenterC2×#393D3
hypocenterC3#9016
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
3. #72132C2semidirect
5. #67612C2semidirect
11. C3×#393C4semidirect
62. #393Dic3semidirect
342. C3#9016semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3932
#820932
#721322
C2×#45252
#676122
#90163