Finite Group Databasebeta

← #10690229

Subgroups — #10690229

Order 768.

740 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 127 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×#4949C211yesmaximal, characteristic
2#82093C211yesmaximal, characteristic
3C2×C3×#323C211yesmaximal, characteristic
4#69925C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9024—31nomaximal, Sylow
7C22×C3×C22⋊C4C2211yescharacteristic
8C22×#217—21no—
9C22×C4⋊Dic3C2211yescharacteristic
10C22×Dic3⋊C4—21no—
11C2×C3×C22⋊C8C2212yes—
12#5247—21no—
13#5212C2212yes—
14C4×#217—21no—
15C2×C3⋊#95C2211yescharacteristic
16#4951—24no—
17#4949—22no—
18#4949C414yescomplemented
19C3×#323—22no—
20#4551—24no—
21C2×#399—31no—
22#2312—31no—
23#1824—32no—
24C2×#323D311yescharacteristic
25#1294—32no—
26C25×C3D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—21no—
30C23×Dic3—21no—
31C22×C3×C8—22no—
32C2×C3×C22⋊C4—22no—
33C2×C3×C22⋊C4—22no—
34C2×C3×C22⋊C4C2×C412yes—
35C2×C3×C22⋊C4D412yes—
36C2×#217—24no—
37C2×#217—22no—
38C2×#217—22no—
39C2×#217—22no—
40C2×#217—44no—
41C2×C4⋊Dic3—22no—
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3C2×C412yes—
45C2×C4⋊Dic3D414yes—
46C2×Dic3⋊C4—24no—
47C2×Dic3⋊C4—44no—
48C2×C4×Dic3—22no—
49C3×C22⋊C8—22no—
50C3×C22⋊C8—22no—
51C3⋊#95—22no—
52C3⋊#95—21no—
53C3⋊#95—22no—
54C3⋊#95C2×C412yes—
55C22×C4⋊C4—31no—
56C22×C4⋊C4—61no—
57C22×C22⋊C4—61no—
58C22×C22⋊C4C2×D311yescharacteristic
59C2×C22⋊C8C2×D312yes—
60#399—34no—
61#399—62no—
62#388—64no—
63#380—32no—
64#379—61no—
65C4×C22⋊C4—61no—
66C2×#95—31no—
67#340—64no—
68#323—22no—
69C24×C3—21no—
70C24×C3—22no—
71C24×C3—21no—
72C24×C3—22no—
73C24×C3—42no—
74C24×C3—42no—
75C24×C3C22⋊C412yes—
76C24×C3C2×D411yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—42no—
80C22×C3×C4—42no—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D412yes—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4#4112yes—
86C22×Dic3—24no—
87C22×Dic3—22no—
88C22×Dic3—22no—
89C22×Dic3—24no—
90C22×Dic3—42no—
91C22×Dic3—42no—
92C22×Dic3—42no—
93C22×Dic3—42no—
94C22×Dic3—41no—
95C22×Dic3—41no—
96C2×C3×C8—44no—
97C2×C3×C8—42no—
98C3×C22⋊C4—24no—
99C3×C22⋊C4—24no—
100#217—48no—
101#217—42no—
102#217—42no—
103#217—82no—
104#217—82no—
105C4⋊Dic3—24no—
106C4⋊Dic3—42no—
107C4⋊Dic3—44no—
108Dic3⋊C4—48no—
109C4×Dic3—82no—
110C25D1211yescharacteristic
111C23×C4—61no—
112C23×C4—61no—
113C23×C4D1211yescharacteristic
114C23×C4C22×D311yescharacteristic
115C22×C8—22no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—34no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—64no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—124no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—64no—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4—124no—
130C2×C22⋊C4C4×D312yes—
131C2×C22⋊C4#6712yes—
132C2×C42—62no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135#95—32no—
136#95—62no—
137#95—62no—
138#95—61no—
139C23×C3—22no—
140C23×C3—22no—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—22no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3—48no—
147C23×C3—42no—
148C23×C3—42no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—81no—
152C23×C3—81no—
153C23×C3—84no—
154C23×C3C2×C22⋊C411yescharacteristic
155C23×C3#12011yescharacteristic
156C23×C3#12311yescharacteristic
157C23×C3#13512yes—
158C23×C3#13712yes—
159C2×C3×C4—24no—
160C2×C3×C4—22no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—81no—
167C2×C3×C4C4×D412yes—
168C2×C3×C4#12112yes—
169C2×Dic3—24no—
170C2×Dic3—48no—
171C2×Dic3—42no—
172C2×Dic3—42no—
173C2×Dic3—42no—
174C2×Dic3—42no—
175C2×Dic3—42no—
176C2×Dic3—44no—
177C2×Dic3—44no—
178C2×Dic3—44no—
179C2×Dic3—82no—
180C2×Dic3—82no—
181C3×C8—82no—
182C24—21no—
183C24—22no—
184C24—21no—
185C24—22no—
186C24—42no—
187C24—42no—
188C24(C2×D3)⋊C412yes—
189C24C2×D1211yescharacteristic
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—64no—
195C22×C4—62no—
196C22×C4—64no—
197C22×C4—62no—
198C22×C4—122no—
199C22×C4—122no—
200C22×C4—122no—
201C22×C4—122no—
202C22×C4—121no—
203C22×C4—121no—
204C22×C4(C2×D3)⋊C412yes—
205C22×C4C2×C4×D311yescharacteristic
206C22×C4D3×D412yes—
207C22×C4#23712yes—
208C22×C4C2×#6711yescharacteristic
209C2×C8—44no—
210C2×C8—42no—
211C4⋊C4—64no—
212C4⋊C4—124no—
213C4⋊C4—128no—
214C4⋊C4—122no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—128no—
218C22⋊C4—122no—
219C22⋊C4—122no—
220C22⋊C4—242no—
221C22⋊C4—242no—
222C42—242no—
223C22×C3—24no—
224C22×C3—24no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—44no—
230C22×C3—42no—
231C22×C3—42no—
232C22×C3—42no—
233C22×C3—82no—
234C22×C3—84no—
235C22×C3—84no—
236C22×C3#38611yescharacteristic
237C22×C3#41611yescharacteristic
238C22×C3#41911yescharacteristic
239C22×C3#45212yes—
240C22×C3#45312yes—
241C22×C3#45812yes—
242C22×C3#48512yes—
243C3×C4—22no—
244C3×C4—22no—
245C3×C4—42no—
246Dic3—44no—
247Dic3—84no—
248Dic3—82no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—42no—
255C23—42no—
256C23—48no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—81no—
262C23—81no—
263C23—84no—
264C23#88211yescharacteristic
265C23#88611yescharacteristic
266C23#90412yes—
267C23#90912yes—
268C23C2×(C2×D3)⋊C411yescharacteristic
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—64no—
277C2×C4—81no—
278C2×C4—128no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—124no—
285C2×C4—124no—
286C2×C4—124no—
287C2×C4—242no—
288C2×C4—242no—
289C2×C4#67⋊C412yes—
290C2×C4#93912yes—
291C8—82no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—22no—
295C2×C3—82no—
296C2×C3#161814yes—
297C2×C3#187311yescharacteristic
298C2×C3#188711yescharacteristic
299C2×C3#189011yescharacteristic
300C22—24no—
301C22—24no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—44no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—82no—
311C22—84no—
312C22—84no—
313C22#472312yes—
314C22#473312yes—
315C22#474212yes—
316C22#474712yes—
317C22#495811yescharacteristic
318C22#511311yescharacteristic
319C22#512011yescharacteristic
320C4—22no—
321C4—22no—
322C4—42no—
323C4—124no—
324C4—244no—
325C4—242no—
326C3#902411yescharacteristic, complemented, Sylow
327C2—21no—
328C2—21no—
329C2—22no—
330C2—82no—
331C2#6811314yes—
332C2#6993111yescharacteristic
333C2#7002311yescharacteristic
334C2#7007011yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#323C2
socleC23×C3#120
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenterC2×#323D3
hypocenterC3#9024
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
4. #69925C2semidirect
5. #67675C2semidirect
18. #4949C4semidirect
326. C3#9024semidirect

Maximal subgroups

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

subgroupindex
C2×#49492
#820932
C2×C3×#3232
#699252
#676752
#90243