Finite Group Databasebeta

← #10691183

Subgroups — #10691183

Order 768.

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

normal subgroups only

This group has 436 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#82486C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3C3×C4×#101C211yesmaximal, characteristic
4#69863C212yesmaximal
5#67666C212yesmaximal
6#9190—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#870C2211yescharacteristic
9C2×C3×#101—21no—
10C3×C4×M16C2212yes—
11#5230—21no—
12#5024—24no—
13#4986C411yescharacteristic
14#4986C411yescharacteristic
15#4985—21no—
16#4981C411yescharacteristic
17#4981C411yescharacteristic
18#4937C2211yescharacteristic
19#4933—21no—
20#4933—21no—
21#4926—21no—
22#4926C2212yes—
23#4925—21no—
24C3×#343—21no—
25#4465—24no—
26#2983—31no—
27#2634—31no—
28C4×#101D311yescharacteristic
29#1528—32no—
30#1288—32no—
31C22×C3×Q8D411yescharacteristic
32C2×C3×M16—22no—
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×C4⋊Dic3C2×C411yescharacteristic
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47#870—22no—
48#870—22no—
49C3×#101—24no—
50C3×#97—22no—
51C3×C4×C8—22no—
52C3⋊#95—21no—
53C3⋊#95—21no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95C2×C411yescharacteristic
61C3⋊#95C2×C411yescharacteristic
62#830—22no—
63C4⋊(C3⋊C8)—22no—
64C4⋊(C3⋊C8)—22no—
65C4⋊(C3⋊C8)—22no—
66#803—22no—
67#803D414yes—
68C4×C3⋊C8—22no—
69C2×#125—31no—
70C2×C4×Q8C2×D311yescharacteristic
71#446—64no—
72#432—32no—
73#432—61no—
74#431—61no—
75C2×#101—21no—
76C4×M16C2×D312yes—
77#400—31no—
78#400—31no—
79#400—61no—
80#398—31no—
81#398—31no—
82#395—61no—
83#387—61no—
84#387—61no—
85#383—31no—
86#343—21no—
87#330—64no—
88C2×C3×Q8—22no—
89C2×C3×Q8—22no—
90C2×C3×Q8—41no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C3×M16—24no—
103C3×M16—42no—
104C2×C3×C8—42no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C42—21no—
109C3×C42—21no—
110C3×C42—21no—
111C3×C42—21no—
112C3×C42C2×D412yes—
113C3×C42#4112yes—
114C4⋊Dic3—22no—
115C4⋊Dic3—22no—
116Dic3⋊C4—42no—
117Dic3⋊C4—42no—
118Dic3⋊C4—41no—
119Dic3⋊C4—42no—
120Dic3⋊C4—41no—
121C2×C3⋊C8—24no—
122C22×Q8#6711yescharacteristic
123C2×M16—22no—
124#125—62no—
125#125—62no—
126C4×Q8—22no—
127C4×Q8—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⋊C4C4×D311yescharacteristic
136C2×C4⋊C4#6711yescharacteristic
137C2×C42D1211yescharacteristic
138C2×C42D1211yescharacteristic
139C2×C42C22×D311yescharacteristic
140C4⋊C8—62no—
141C4⋊C8—62no—
142C4⋊C8—62no—
143#101—24no—
144C22⋊C8—62no—
145#97—22no—
146#97—34no—
147#97—62no—
148C4×C8—22no—
149C4×C8—62no—
150#95—31no—
151#95—31no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160C23×C3C2×C22⋊C411yescharacteristic
161C3×Q8—44no—
162C2×C3×C4—21no—
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—21no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4#10412yes—
176C2×C3×C4C4×D412yes—
177C2×C3×C4#12011yescharacteristic
178C2×C3×C4#12112yes—
179C2×C3×C4#12311yescharacteristic
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—42no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C3×C8—44no—
192C3⋊C8—44no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—41no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4—61no—
200C22×C4(C2×D3)⋊C411yescharacteristic
201C22×C4(C2×D3)⋊C411yescharacteristic
202C22×C4(C2×D3)⋊C411yescharacteristic
203C22×C4(C2×D3)⋊C411yescharacteristic
204C22×C4C2×C4×D311yescharacteristic
205C22×C4C2×D1211yescharacteristic
206C22×C4C2×#6711yescharacteristic
207M16—24no—
208M16—42no—
209C2×C8—42no—
210C2×C8—64no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—62no—
215C4⋊C4—62no—
216C4⋊C4—122no—
217C4⋊C4—122no—
218C4⋊C4—121no—
219C4⋊C4—121no—
220C4⋊C4—122no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42D3×D412yes—
226C42#23712yes—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3#38611yescharacteristic
230C22×C3C2×#10411yescharacteristic
231C22×C3#42111yescharacteristic
232C3×C4—22no—
233C3×C4—22no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—22no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—41no—
240Dic3—81no—
241Dic3—81no—
242Dic3—81no—
243Dic3—81no—
244C23C2×(C2×D3)⋊C411yescharacteristic
245Q8—44no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—121no—
260C2×C4—121no—
261C2×C4—121no—
262C2×C4—121no—
263C2×C4—122no—
264C2×C4—121no—
265C2×C4—121no—
266C2×C4—121no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—121no—
270C2×C4#80512yes—
271C2×C4#67⋊C412yes—
272C2×C4#88211yescharacteristic
273C2×C4#88611yescharacteristic
274C2×C4#93912yes—
275C8—44no—
276C8—124no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3#187111yescharacteristic
280C2×C3#191211yescharacteristic
281C2×C3#191511yescharacteristic
282C22—21no—
283C22—21no—
284C22C2×#80511yescharacteristic
285C22#495811yescharacteristic
286C22#500511yescharacteristic
287C4—22no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—241no—
296C4—241no—
297C4—241no—
298C4—241no—
299C3#919011yescharacteristic, complemented, Sylow
300C2—21no—
301C2—21no—
302C2#6945311yescharacteristic
303C2#7010511yescharacteristic
304C2#7013811yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#101C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenterC4×#101D3
hypocenterC3#9190
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
299. C3#9190semidirect

Maximal subgroups

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

subgroupindex
#824862
#824362
C3×C4×#1012
#698632
#676662
#91903