Finite Group Databasebeta

← #71706

Subgroups — #71706

Order 384.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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×#123C211yesmaximal, characteristic, complemented
2#5804C211yesmaximal, characteristic
3C2×#929C211yesmaximal, characteristic, complemented
4C2×#834C211yesmaximal, characteristic, complemented
5#5211C211yesmaximal, characteristic, complemented
6C2×#806C211yesmaximal, characteristic, complemented
7#4527C211yesmaximal, characteristic, complemented
8#2022—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C23×C3×C4C2211yescharacteristic
11C22×#67C2211yescharacteristic
12C3×#123—22no—
13C3×#123—22no—
14C2×C3×C4⋊C4—21no—
15C2×C3×C22⋊C4—21no—
16C2×C3×C22⋊C4C2211yescharacteristic
17#953—21no—
18C2×#217C2211yescharacteristic
19C2×#217C2211yescharacteristic
20C2×#217C2211yescharacteristic
21#939—22no—
22#937—21no—
23#935—21no—
24#929—21no—
25#929—22no—
26#929—22no—
27C2×(C2×D3)⋊C4—21no—
28C2×Dic3⋊C4—21no—
29#834—22no—
30C3⋊#95—22no—
31#806—22no—
32#534—31no—
33C2×#123—31no—
34C2×#123D311yescharacteristic, complemented
35C2×#99—31no—
36C2×#99—31no—
37#381—31no—
38#342—31no—
39C24×C3D411yescharacteristic
40C24×C3C2311yescharacteristic
41C23×D3D411yescharacteristic
42C2×C3×D4—21no—
43C2×C3×D4—21no—
44C2×C3×D4—21no—
45C2×C3×D4—21no—
46C2×C3×D4—21no—
47C2×C3×D4—21no—
48C22×C3×C4—21no—
49C22×C3×C4—21no—
50C22×C3×C4—21no—
51C22×C3×C4—21no—
52C22×C3×C4—21no—
53C22×C3×C4—21no—
54C22×C3×C4—21no—
55C22×C3×C4D411yescharacteristic
56C22×C3×C4D412yescomplemented
57C2×#67—21no—
58C2×#67—21no—
59C2×#67—21no—
60C2×#67—21no—
61C2×#67—21no—
62C2×#67—21no—
63C2×#67—22no—
64C22×Dic3—21no—
65C22×Dic3—21no—
66C22×Dic3D411yescharacteristic
67C22×Dic3D412yes—
68C3×C4⋊C4—24no—
69C3×C22⋊C4—24no—
70C3×C22⋊C4—21no—
71C3×C22⋊C4D412yescomplemented
72#217—22no—
73#217—21no—
74#217—21no—
75#217—21no—
76#217—21no—
77#217—21no—
78#217D412yescomplemented
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—22no—
81Dic3⋊C4—24no—
82Dic3⋊C4—22no—
83C22×D4—31no—
84C22×D4C2×D311yescharacteristic
85C23×C4C2×D311yescharacteristic
86#123—22no—
87#123—22no—
88#123—62no—
89#123—61no—
90#123—61no—
91#123—62no—
92#121—62no—
93#120—61no—
94#120—61no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—61no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—31no—
101C2×C22⋊C4—61no—
102C2×C22⋊C4C2×D311yescharacteristic
103#99—62no—
104#99—62no—
105#95—62no—
106C23×C3—21no—
107C23×C3—21no—
108C23×C3—21no—
109C23×C3—42no—
110C23×C3—42no—
111C23×C3C2×D411yescharacteristic
112C23×C3C2×D411yescharacteristic
113C23×C3C2×D411yescharacteristic
114C23×C3C2×D411yescharacteristic
115C23×C3C2×D411yescharacteristic
116C23×C3C2×D411yescharacteristic
117C23×C3#4111yescharacteristic
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—42no—
122C3×D4—41no—
123C3×D4—41no—
124C3×D4—41no—
125C3×D4—41no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—42no—
134C2×C3×C4—41no—
135C2×C3×C4—41no—
136C2×C3×C4—41no—
137C2×C3×C4—41no—
138#67—41no—
139#67—41no—
140#67—41no—
141#67—41no—
142#67—82no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—21no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—42no—
151C2×Dic3—41no—
152C24—31no—
153C24#6711yescharacteristic
154C24C22×D311yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—61no—
162C2×D4—61no—
163C2×D4—61no—
164C2×D4—61no—
165C2×D4—61no—
166C2×D4—61no—
167C2×D4—62no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—31no—
176C22×C4—32no—
177C22×C4—61no—
178C22×C4—61no—
179C22×C4#6711yescharacteristic
180C22×C4#6712yescomplemented
181C4⋊C4—24no—
182C4⋊C4—64no—
183C4⋊C4—62no—
184C22⋊C4—24no—
185C22⋊C4—21no—
186C22⋊C4—32no—
187C22⋊C4—64no—
188C22⋊C4—62no—
189C22⋊C4—62no—
190C22⋊C4—61no—
191C22⋊C4—61no—
192C22⋊C4—61no—
193C22⋊C4—61no—
194C22⋊C4—61no—
195C22⋊C4#6712yescomplemented
196C22×C3—22no—
197C22×C3—22no—
198C22×C3—22no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3—42no—
204C22×C3—41no—
205C22×C3—42no—
206C22×C3—81no—
207C22×C3#12011yescharacteristic
208C22×C3#12011yescharacteristic
209C22×C3#12011yescharacteristic
210C22×C3#12111yescharacteristic
211C22×C3#12111yescharacteristic
212C22×C3#12111yescharacteristic
213C22×C3#12711yescharacteristic
214C2×D3—42no—
215C2×D3—41no—
216C2×D3—42no—
217C2×D3—81no—
218C3×C4—42no—
219C3×C4—42no—
220C3×C4—41no—
221Dic3—42no—
222Dic3—41no—
223Dic3—42no—
224Dic3—81no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—42no—
229C23—42no—
230C23—61no—
231C23—61no—
232C23—61no—
233C23—122no—
234C23D3×D411yescharacteristic
235C23D3×D411yescharacteristic
236C23D3×D411yescharacteristic
237C23#23711yescharacteristic
238C23C2×#6711yescharacteristic
239C23C2×#6711yescharacteristic
240C23C2×#6711yescharacteristic
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—121no—
246D4—121no—
247D4—121no—
248D4—121no—
249D4—242no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—62no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—122no—
268C2×C4—122no—
269C2×C4—122no—
270C2×C4—121no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—81no—
278C2×C3#39211yescharacteristic
279C2×C3#45812yes—
280D3—81no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22—41no—
290C22—42no—
291C22—81no—
292C22—122no—
293C22—121no—
294C22—122no—
295C22—241no—
296C22#93711yescharacteristic
297C22#93711yescharacteristic
298C22#93811yescharacteristic
299C22#93911yescharacteristic
300C22#93911yescharacteristic
301C22#94011yescharacteristic
302C22#95311yescharacteristic
303C4—42no—
304C4—42no—
305C4—41no—
306C4—122no—
307C4—121no—
308C4—122no—
309C4—241no—
310C3#202211yescharacteristic, complemented, Sylow
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—241no—
319C2#505312yes—
320C2#522311yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC24×C3C23
FrattiniC24C22×D3
FittingC2×C3×#123C2
socleC22×C3#120
prime radical#5804C2
hypercenterC2×#123D3
hypocenterC3#2022
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
3. C2×#929C2semidirect
4. C2×#834C2semidirect
5. #5211C2semidirect
6. C2×#806C2semidirect
7. #4527C2semidirect
34. C2×#123D3semidirect
56. C22×C3×C4D4semidirect
71. C3×C22⋊C4D4semidirect
78. #217D4semidirect
180. C22×C4#67semidirect
195. C22⋊C4#67semidirect
310. C3#2022semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
#58042
C2×#9292
C2×#8342
#52112
C2×#8062
#45272
#20223