Finite Group Databasebeta

← #70205

Subgroups — #70205

Order 384.

444 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 91 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×#938C211yesmaximal, characteristic
2C22×#214C211yesmaximal, characteristic, complemented
3C2×#897C211yesmaximal, characteristic, complemented
4C2×C3×#102C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6C2×#816C211yesmaximal, characteristic, complemented
7#4529C211yesmaximal, characteristic, complemented
8#2000—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C22×#67—21no—
11C22×C4×D3C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C22×C3×C8C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×#217—21no—
16#938—22no—
17#938—22no—
18C2×#214—22no—
19C2×#214—22no—
20C2×#214—22no—
21C2×(C2×D3)⋊C4—21no—
22C2×C4⋊Dic3—21no—
23C2×C4⋊Dic3C2211yescharacteristic
24C22×C3⋊C8C2211yescharacteristic
25#897—22no—
26#897—22no—
27C3×#102—22no—
28(C2×D3)⋊C8—22no—
29#816—22no—
30C22×SD16—31no—
31C2×C22⋊Q8—31no—
32C2×#121—31no—
33C2×#103—31no—
34C2×#102D311yescharacteristic, complemented
35C2×C22⋊C8—31no—
36#340—31no—
37C24×C3—21no—
38C23×D3D411yescharacteristic
39C2×C3×D4—21no—
40C2×C3×D4—21no—
41C2×C3×D4—21no—
42C2×C3×D4—21no—
43C2×C3×D4—21no—
44C2×C3×D4D412yes—
45C22×C3×C4—21no—
46C22×C3×C4C2311yescharacteristic
47C2×#67—24no—
48C2×#67—42no—
49C2×#67—42no—
50C22×Dic3—21no—
51C22×Dic3—21no—
52C22×Dic3D411yescharacteristic
53C2×C4×D3—21no—
54C2×C4×D3—21no—
55C2×C4×D3—21no—
56C2×C4×D3—21no—
57C2×C4×D3—21no—
58C2×C4×D3—21no—
59C2×Dic6—21no—
60C2×Dic6—21no—
61C2×Dic6—21no—
62C2×Dic6—21no—
63C2×Dic6—21no—
64C2×Dic6D412yescomplemented
65C2×C3×C8—21no—
66C2×C3×C8—21no—
67C2×C3×C8—21no—
68C3×C4⋊C4—21no—
69C3×C4⋊C4D412yescomplemented
70#217—24no—
71#214—28no—
72(C2×D3)⋊C4—24no—
73C4⋊Dic3—24no—
74C4⋊Dic3—21no—
75C4⋊Dic3D412yes—
76C2×C3⋊C8—21no—
77C2×C3⋊C8—21no—
78C2×C3⋊C8D412yescomplemented
79C22×Q8—31no—
80C22×D4—61no—
81C22×D4C2×D311yescharacteristic
82C23×C4—31no—
83C2×SD16—62no—
84C2×SD16—62no—
85C2×SD16—62no—
86C22×C8—31no—
87C22×C8C2×D311yescharacteristic
88C22⋊Q8—62no—
89C22⋊Q8—62no—
90#121—62no—
91#121—62no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—61no—
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C22⋊C4—61no—
96C2×C22⋊C4—61no—
97#103—62no—
98#102—22no—
99C22⋊C8—62no—
100C23×C3—22no—
101C23×C3—41no—
102C23×C3—41no—
103C23×C3—41no—
104C23×C3—41no—
105C23×C3—41no—
106C23×C3—41no—
107C23×C3C2×D411yescharacteristic
108C22×D3—21no—
109C22×D3—21no—
110C22×D3—21no—
111C22×D3—21no—
112C22×D3—21no—
113C22×D3—21no—
114C22×D3SD1612yescomplemented
115C3×D4—22no—
116C3×D4—22no—
117C3×D4—41no—
118C3×D4—41no—
119C2×C3×C4—22no—
120C2×C3×C4—41no—
121C2×C3×C4—41no—
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4#4111yescharacteristic
128#67—48no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—41no—
135C2×Dic3—41no—
136C2×Dic3—41no—
137C2×Dic3—41no—
138C4×D3—41no—
139C4×D3—41no—
140C4×D3—41no—
141C4×D3—41no—
142Dic6—22no—
143Dic6—22no—
144Dic6—41no—
145Dic6—41no—
146C3×C8—41no—
147C3⋊C8—22no—
148C24—21no—
149C24—31no—
150C2×Q8—32no—
151C2×Q8—61no—
152C2×Q8—61no—
153C2×Q8—61no—
154C2×Q8—61no—
155C2×Q8—61no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—64no—
162C2×D4—122no—
163C2×D4—122no—
164C2×D4D1212yes—
165C22×C4—21no—
166C22×C4—31no—
167C22×C4—61no—
168C22×C4—61no—
169C22×C4—61no—
170C22×C4—61no—
171C22×C4—61no—
172C22×C4—61no—
173C22×C4—61no—
174C22×C4—61no—
175C22×C4C22×D311yescharacteristic
176SD16—68no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—32no—
181C2×C8—61no—
182C2×C8—61no—
183C4⋊C4—21no—
184C4⋊C4—32no—
185C4⋊C4—64no—
186C4⋊C4—61no—
187C4⋊C4#6712yescomplemented
188C22⋊C4—64no—
189C22⋊C4—64no—
190C22×C3—42no—
191C22×C3—42no—
192C22×C3—42no—
193C22×C3—81no—
194C22×C3—81no—
195C22×C3—81no—
196C22×C3—81no—
197C22×C3#12011yescharacteristic
198C22×C3#12011yescharacteristic
199C22×C3#12111yescharacteristic
200C22×C3C2×SD1611yescharacteristic
201C22×C3#13511yescharacteristic
202C22×C3AGL(1,ℤ/8)11yescharacteristic
203C22×C3#13711yescharacteristic
204C2×D3—22no—
205C2×D3—22no—
206C2×D3—22no—
207C2×D3—41no—
208C2×D3—41no—
209C2×D3—41no—
210C2×D3—41no—
211C3×C4—42no—
212C3×C4#12011yescharacteristic
213C3×C4#12111yescharacteristic
214C3×C4#12111yescharacteristic
215C3×C4#12711yescharacteristic
216Dic3—42no—
217Dic3—41no—
218Dic3—42no—
219C23—22no—
220C23—32no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—61no—
228C23—61no—
229C23—61no—
230C23—61no—
231C23—61no—
232C23—61no—
233C23D3×D411yescharacteristic
234Q8—62no—
235Q8—62no—
236Q8—121no—
237Q8—121no—
238D4—22no—
239D4—22no—
240D4—41no—
241D4—41no—
242D4—128no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—61no—
247C2×C4—62no—
248C2×C4—61no—
249C2×C4—62no—
250C2×C4—61no—
251C2×C4—121no—
252C2×C4—121no—
253C2×C4—121no—
254C2×C4—121no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4—121no—
258C2×C4—121no—
259C2×C4C2×D1211yescharacteristic
260C2×C4#23511yescharacteristic
261C2×C4D3×D411yescharacteristic
262C2×C4D3×D411yescharacteristic
263C2×C4D3×D411yescharacteristic
264C2×C4C2×#6711yescharacteristic
265C8—41no—
266C8—62no—
267C2×C3—82no—
268C2×C3#39211yescharacteristic
269C2×C3#44811yescharacteristic
270C2×C3#44911yescharacteristic
271C2×C3#45011yescharacteristic
272C2×C3#45211yescharacteristic
273C2×C3#46511yescharacteristic
274C2×C3#46811yescharacteristic
275D3—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—62no—
280C22—62no—
281C22—62no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—121no—
287C22—121no—
288C22—121no—
289C22—121no—
290C22#88211yescharacteristic
291C22#88411yescharacteristic
292C22#91111yescharacteristic
293C22#91211yescharacteristic
294C22D3×SD1611yescharacteristic
295C22#91511yescharacteristic
296C22#93711yescharacteristic
297C4—42no—
298C4—121no—
299C4—122no—
300C4—122no—
301C4#88211yescharacteristic
302C4#88411yescharacteristic
303C4#93011yescharacteristic
304C4#94011yescharacteristic
305C3#200011yescharacteristic, complemented, Sylow
306C2—82no—
307C2—122no—
308C2#477611yescharacteristic
309C2#477811yescharacteristic
310C2#478011yescharacteristic
311C2#478311yescharacteristic
312C2#496011yescharacteristic
313C2#515811yescharacteristic
314C2#517011yescharacteristic
subgroupG/H
centerC23D3×D4
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC2×C3×#102C2
socleC23×C3C2×D4
prime radicalC2×#938C2
hypercenterC2×#102D3
hypocenterC3#2000
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C22×#214C2semidirect
3. C2×#897C2semidirect
4. C2×C3×#102C2semidirect
5. C2×(C2×D3)⋊C8C2semidirect
6. C2×#816C2semidirect
7. #4529C2semidirect
34. C2×#102D3semidirect
64. C2×Dic6D4semidirect
69. C3×C4⋊C4D4semidirect
78. C2×C3⋊C8D4semidirect
114. C22×D3SD16semidirect
187. C4⋊C4#67semidirect
305. C3#2000semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C22×#2142
C2×#8972
C2×C3×#1022
C2×(C2×D3)⋊C82
C2×#8162
#45292
#20003