Finite Group Databasebeta

← #10692537

Subgroups — #10692537

Order 768.

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

normal subgroups only

This group has 446 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
1C3×#2921C211yesmaximal, characteristic
2#82261C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic
4#69863C212yesmaximal, complemented
5#67665C212yesmaximal, complemented
6#9390—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#870C2211yescharacteristic
9C3×#433—24no—
10C3×#423—21no—
11C3×#407—21no—
12C3×C4×M16C2212yes—
13#5210—21no—
14#4986C411yescharacteristic
15#4981C411yescharacteristic
16#4976—21no—
17#4953C411yescharacteristic
18#4949C411yescharacteristic
19#4945—21no—
20#4933—21no—
21#4932C2211yescharacteristic
22#4926C2212yes—
23#4528—21no—
24#4525—21no—
25#4465—24no—
26#2921D311yescharacteristic
27#2611—31no—
28#1528—32no—
29#1464—31no—
30#1300—32no—
31C23×C3×C4D411yescharacteristic
32C2×C3×M16—22no—
33C3×#117—22no—
34C3×#117—22no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C22⋊C4C2×C411yescharacteristic
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×#217—21no—
42C2×#217—21no—
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—21no—
50#870—22no—
51#870—22no—
52C3×C4⋊C8—24no—
53C3×C22⋊C8—24no—
54C3×#97D414yes—
55C3×C4×C8—22no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95C2×C411yescharacteristic
62C3⋊#95C2×C411yescharacteristic
63#830—22no—
64C4⋊(C3⋊C8)—22no—
65#803—22no—
66C2×#125—31no—
67C2×#117C2×D311yescharacteristic
68#433—24no—
69#432—32no—
70#423—21no—
71#407—21no—
72C4×M16C2×D312yes—
73#400—31no—
74#399—31no—
75#398—31no—
76#397—31no—
77#389—61no—
78#388—61no—
79#387—61no—
80#385—61no—
81#376—31no—
82#330—64no—
83#324—61no—
84#323—61no—
85C24×C3C22⋊C411yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—41no—
89C22×C3×C4—41no—
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C3×M16—42no—
101C3×M16—42no—
102C2×C3×C8—24no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C22⋊C4—22no—
107C3×C42—21no—
108C3×C42—21no—
109C3×C42C2×D412yes—
110C3×C42#4112yes—
111#217—42no—
112#217—42no—
113C4⋊Dic3—22no—
114C4⋊Dic3—22no—
115Dic3⋊C4—42no—
116Dic3⋊C4—42no—
117Dic3⋊C4—41no—
118Dic3⋊C4—42no—
119Dic3⋊C4—41no—
120C4×Dic3—42no—
121C2×C3⋊C8—42no—
122C23×C4#6711yescharacteristic
123C2×M16—22no—
124#125—62no—
125#125—62no—
126#117—22no—
127#117—22no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4C4×D311yescharacteristic
135C2×C4⋊C4D1211yescharacteristic
136C2×C22⋊C4—61no—
137C2×C22⋊C4—61no—
138C2×C22⋊C4C4×D311yescharacteristic
139C2×C22⋊C4D1211yescharacteristic
140C2×C42—61no—
141C2×C42#6711yescharacteristic
142C2×C42C22×D311yescharacteristic
143C4⋊C8—24no—
144C4⋊C8—62no—
145C22⋊C8—24no—
146C22⋊C8—62no—
147#97—62no—
148#97#6714yes—
149C4×C8—22no—
150#95—31no—
151#95—31no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3C2×C22⋊C411yescharacteristic
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4C4×D412yes—
177C2×C3×C4#12011yescharacteristic
178C2×C3×C4#12112yes—
179C2×C3×C4#12311yescharacteristic
180C2×Dic3—22no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C3×C8—44no—
193C3⋊C8—82no—
194C24(C2×D3)⋊C411yescharacteristic
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—61no—
200C22×C4—61no—
201C22×C4—61no—
202C22×C4—61no—
203C22×C4(C2×D3)⋊C411yescharacteristic
204C22×C4(C2×D3)⋊C411yescharacteristic
205C22×C4(C2×D3)⋊C411yescharacteristic
206C22×C4C2×C4×D311yescharacteristic
207C22×C4C2×D1211yescharacteristic
208C22×C4C2×#6711yescharacteristic
209M16—42no—
210M16—42no—
211C2×C8—24no—
212C2×C8—122no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—62no—
216C4⋊C4—62no—
217C4⋊C4—122no—
218C4⋊C4—122no—
219C4⋊C4—121no—
220C4⋊C4—122no—
221C4⋊C4—121no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—122no—
225C22⋊C4—122no—
226C42—21no—
227C42—21no—
228C42—122no—
229C42#23512yes—
230C42C2×#6712yes—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—42no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3—41no—
237C22×C3—81no—
238C22×C3#38611yescharacteristic
239C22×C3#42111yescharacteristic
240C22×C3#42111yescharacteristic
241C3×C4—22no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—22no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—41no—
248Dic3—42no—
249Dic3—81no—
250Dic3—81no—
251Dic3—81no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×(C2×D3)⋊C411yescharacteristic
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—62no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4C4×#6712yes—
284C2×C4#92911yescharacteristic
285C2×C4#93012yes—
286C2×C4#95311yescharacteristic
287C8—44no—
288C8—242no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—81no—
292C2×C3#188411yescharacteristic
293C2×C3#189711yescharacteristic
294C2×C3#191511yescharacteristic
295C22—21no—
296C22—21no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22#500511yescharacteristic
303C22#500511yescharacteristic
304C22#521311yescharacteristic
305C4—22no—
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—122no—
313C4—241no—
314C4—241no—
315C4—241no—
316C3#939011yescharacteristic, complemented, Sylow
317C2—21no—
318C2—21no—
319C2—81no—
320C2#7159111yescharacteristic
321C2#7202111yescharacteristic
322C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2921C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2921D3
hypocenterC3#9390
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
4. #69863C2semidirect
5. #67665C2semidirect
316. C3#9390semidirect

Maximal subgroups

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

subgroupindex
C3×#29212
#822612
#715722
#698632
#676652
#93903