Finite Group Databasebeta

← #10693009

Subgroups — #10693009

Order 768.

726 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#82192C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic, complemented
3C3×#1463C211yesmaximal, characteristic, complemented
4#69991C212yesmaximal
5#67682C212yesmaximal, complemented
6#9480—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#929—21no—
10C3×#407C2212yes—
11#5213—21no—
12#5212—21no—
13#5211—21no—
14#5211C2211yescharacteristic
15C4×#217C2212yes—
16C2×#825—24no—
17#4991C412yescomplemented
18#4989—22no—
19#4980—24no—
20#4979C412yescomplemented
21C3×#324—22no—
22#2406—31no—
23#2313—31no—
24#1789—32no—
25#1463D311yescharacteristic, complemented
26#1284—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C2×#235—22no—
32C2×#235—22no—
33C2×#235—44no—
34C22×D12C2×C411yescharacteristic
35C22×C4×D3—21no—
36C22×Dic6C2×C411yescharacteristic
37C2×C3×M16—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4C2×C412yes—
41C2×C3×C4⋊C4D412yes—
42C2×#217—22no—
43C2×#217—21no—
44#929—44no—
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—22no—
47C2×C4⋊Dic3—22no—
48C2×Dic3⋊C4—22no—
49C2×C4×Dic3D414yes—
50C3×C22⋊C8—22no—
51C3×C22⋊C8—22no—
52C3⋊#95—24no—
53C3⋊#95—22no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95—21no—
57C3⋊#95C2×C412yes—
58#825—48no—
59C22×#41—31no—
60C2×#123—61no—
61C22×C4⋊C4C2×D311yescharacteristic
62C2×#104—64no—
63#407C2×D312yes—
64#395—32no—
65#394—32no—
66#387—64no—
67#386—61no—
68#385—62no—
69#381—31no—
70#381—61no—
71#380—61no—
72C4×C22⋊C4—32no—
73#324—22no—
74C24×C3C2×D411yescharacteristic
75C23×D3—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—42no—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22⋊C412yes—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4#4112yes—
89C2×#67—24no—
90C2×#67—42no—
91C2×#67—41no—
92C2×#67—41no—
93C22×Dic3—21no—
94C22×Dic3—22no—
95C22×Dic3—24no—
96#235—28no—
97#235—44no—
98#235—44no—
99#235—44no—
100C2×D12—22no—
101C2×D12—42no—
102C2×C4×D3—42no—
103C2×C4×D3—42no—
104C2×C4×D3—42no—
105C2×Dic6—22no—
106C2×Dic6—42no—
107C3×M16—44no—
108C2×C3×C8—42no—
109C3×C4⋊C4—24no—
110C3×C4⋊C4—24no—
111#217—42no—
112#217—44no—
113(C2×D3)⋊C4—44no—
114(C2×D3)⋊C4—44no—
115C4⋊Dic3—82no—
116Dic3⋊C4—44no—
117C4×Dic3—44no—
118C2×#41—62no—
119C2×#41—62no—
120C2×#41—124no—
121C22×Q8—31no—
122C22×D4—31no—
123C22×D4—61no—
124C23×C4—61no—
125C23×C4D1211yescharacteristic
126C23×C4D1211yescharacteristic
127C23×C4C22×D311yescharacteristic
128C2×M16—22no—
129#123—124no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4C4×D312yes—
135C2×C4⋊C4#6712yes—
136C2×C22⋊C4—61no—
137C2×C22⋊C4—62no—
138C2×C22⋊C4—62no—
139C2×C22⋊C4—62no—
140C2×C42—34no—
141#104—128no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144#95—32no—
145#95—62no—
146#95—64no—
147#95—62no—
148#95—62no—
149#95—61no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—41no—
153C23×C3C2×C22⋊C411yescharacteristic
154C23×C3#12011yescharacteristic
155C23×C3#12311yescharacteristic
156C22×D3—22no—
157C22×D3—41no—
158C22×D3—42no—
159C22×D3—41no—
160C22×D3—42no—
161C2×C3×C4—24no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—81no—
174C2×C3×C4C4×D412yes—
175C2×C3×C4#12112yes—
176#67—44no—
177#67—44no—
178C2×Dic3—22no—
179C2×Dic3—41no—
180C2×Dic3—41no—
181C2×Dic3—42no—
182C2×Dic3—44no—
183C2×Dic3—44no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—44no—
187D12—24no—
188D12—42no—
189C4×D3—48no—
190Dic6—24no—
191Dic6—42no—
192C3×C8—82no—
193C24—61no—
194C24C2×D1211yescharacteristic
195#41—68no—
196#41—124no—
197#41—124no—
198#41—124no—
199C2×Q8—62no—
200C2×Q8—122no—
201C2×D4—64no—
202C2×D4—62no—
203C2×D4—122no—
204C2×D4—122no—
205C2×D4—121no—
206C2×D4—121no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—61no—
215C22×C4—64no—
216C22×C4—62no—
217C22×C4—122no—
218C22×C4—122no—
219C22×C4—122no—
220C22×C4(C2×D3)⋊C412yes—
221C22×C4(C2×D3)⋊C412yes—
222C22×C4C2×C4×D311yescharacteristic
223C22×C4D3×D412yes—
224C22×C4#23712yes—
225C22×C4C2×#6711yescharacteristic
226M16—44no—
227C2×C8—42no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—124no—
231C4⋊C4—242no—
232C22⋊C4—124no—
233C22⋊C4—124no—
234C22⋊C4—122no—
235C22⋊C4—124no—
236C42—124no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—44no—
245C22×C3#38611yescharacteristic
246C22×C3#42112yes—
247C22×C3B(4,𝔽2)12yes—
248C2×D3—44no—
249C2×D3—42no—
250C2×D3—82no—
251C2×D3—81no—
252C2×D3—81no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—42no—
256C3×C4—42no—
257Dic3—42no—
258Dic3—84no—
259Dic3—82no—
260C23—22no—
261C23—22no—
262C23—41no—
263C23—62no—
264C23—121no—
265C23—122no—
266C23—122no—
267C23—121no—
268C23#88211yescharacteristic
269C23#88611yescharacteristic
270C23C2×(C2×D3)⋊C411yescharacteristic
271Q8—64no—
272Q8—122no—
273D4—64no—
274D4—122no—
275D4—124no—
276D4—124no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—62no—
290C2×C4—81no—
291C2×C4—128no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—122no—
295C2×C4—124no—
296C2×C4—124no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—124no—
300C2×C4#67⋊C412yes—
301C2×C4#93912yes—
302C8—82no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—41no—
306C2×C3#188911yescharacteristic
307C2×C3#189711yescharacteristic
308C2×C3#191211yescharacteristic
309D3—82no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—44no—
318C22—124no—
319C22—122no—
320C22—242no—
321C22—241no—
322C22—241no—
323C22#474112yes—
324C22#495811yescharacteristic
325C22#513912yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—42no—
330C4—122no—
331C4—244no—
332C4—242no—
333C3#948011yescharacteristic, complemented, Sylow
334C2—22no—
335C2—22no—
336C2—41no—
337C2—242no—
338C2#6996611yescharacteristic
339C2#7002211yescharacteristic
340C2#7010511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1463C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1463D3
hypocenterC3#9480
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
3. C3×#1463C2semidirect
5. #67682C2semidirect
17. #4991C4semidirect
20. #4979C4semidirect
25. #1463D3semidirect
333. C3#9480semidirect

Maximal subgroups

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

subgroupindex
#821922
#821552
C3×#14632
#699912
#676822
#94803