Finite Group Databasebeta

← #82351

Subgroups — #82351

Order 384.

604 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic
2C2×#926C211yesmaximal, characteristic, complemented
3C2×#924C211yesmaximal, characteristic, complemented
4C2×#924C211yesmaximal, characteristic, complemented
5#5222C212yesmaximal
6#5212C211yesmaximal, characteristic, complemented
7#4976C212yesmaximal, complemented
8#4952C212yesmaximal, complemented
9#4945C212yesmaximal, complemented
10#4929C211yesmaximal, characteristic, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2415—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C3×C4×D4—24no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C42C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26#926—24no—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30#924—24no—
31#924—24no—
32C2×Dic3⋊C4C2212yes—
33C2×Dic3⋊C4C2212yes—
34C2×Dic3⋊C4C2212yes—
35C2×C4×Dic3C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×C22⋊Q8—31no—
41C2×C22⋊Q8—31no—
42C2×C4×D4D311yescharacteristic
43C2×#117—31no—
44#397—32no—
45#394—32no—
46#391—31no—
47#389—32no—
48#389—31no—
49#388—32no—
50#380—31no—
51C24×C3C2311yescharacteristic
52C24×C3C2311yescharacteristic
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4D414yes—
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2311yescharacteristic
67C22×Dic3C2312yes—
68C22×Dic3C2312yes—
69C22×Dic3C2312yes—
70C22×Dic3C2312yes—
71C2×Dic6—22no—
72C2×Dic6—22no—
73C2×Dic6—22no—
74C3×C4⋊C4—22no—
75C3×C22⋊C4—22no—
76C3×C22⋊C4—22no—
77C3×C42—22no—
78#217—24no—
79#217—24no—
80#217—24no—
81#217—24no—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3D414yescomplemented
85Dic3⋊C4—24no—
86Dic3⋊C4—24no—
87Dic3⋊C4—24no—
88C4×Dic3—24no—
89C22×Q8—31no—
90C22×D4C2×D311yescharacteristic
91C23×C4C2×D311yescharacteristic
92C23×C4C2×D311yescharacteristic
93C22⋊Q8—64no—
94C22⋊Q8—64no—
95C4×D4—24no—
96#117—64no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4C2×D311yescharacteristic
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4C2×D311yescharacteristic
109C2×C22⋊C4C2×D311yescharacteristic
110C2×C42—32no—
111C2×C42C2×D311yescharacteristic
112#95—32no—
113#95—32no—
114#95—32no—
115#95—32no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3C2×D412yes—
129C23×C3#4112yes—
130C23×C3C2411yescharacteristic
131C3×D4—28no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—28no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—44no—
144C2×C3×C4C2×D412yes—
145C2×C3×C4C2×D412yes—
146C2×C3×C4C2×D412yes—
147C2×C3×C4#4112yes—
148C2×C3×C4#4112yes—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3C2×D414yes—
161Dic6—44no—
162C24C22×D311yescharacteristic
163C24C22×D311yescharacteristic
164C2×Q8—62no—
165C2×Q8—62no—
166C2×Q8—62no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—32no—
176C22×C4—32no—
177C22×C4—32no—
178C22×C4—32no—
179C22×C4#6714yes—
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182C22×C4C22×D311yescharacteristic
183C22×C4C22×D311yescharacteristic
184C22×C4C22×D311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—34no—
187C4⋊C4—64no—
188C4⋊C4—64no—
189C4⋊C4—62no—
190C4⋊C4—64no—
191C4⋊C4—62no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—64no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C42—22no—
199C42—64no—
200C22×C3—22no—
201C22×C3—22no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3C22×D411yescharacteristic
215C22×C3C22×D411yescharacteristic
216C22×C3C2×#4111yescharacteristic
217C22×C3C2×#4111yescharacteristic
218C22×C3C2×#4111yescharacteristic
219C22×C32−1+411yescharacteristic
220C22×C32−1+411yescharacteristic
221C3×C4—22no—
222C3×C4—22no—
223C3×C4—41no—
224C3×C4—41no—
225C3×C4#12114yes—
226Dic3—24no—
227Dic3—42no—
228Dic3—42no—
229Dic3—42no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23#23512yes—
243C23C2×#6712yes—
244C23C23×D311yescharacteristic
245Q8—124no—
246D4—28no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—28no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—34no—
259C2×C4—44no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4#23512yes—
272C2×C4D3×D412yes—
273C2×C4#23712yes—
274C2×C4C2×#6712yes—
275C2×C4C2×#6712yes—
276C2×C3—42no—
277C2×C3—42no—
278C2×C3C2×#12111yescharacteristic
279C2×C3#52511yescharacteristic
280C2×C3#53611yescharacteristic
281C2×C3#54711yescharacteristic
282C2×C3#54811yescharacteristic
283C2×C3#55211yescharacteristic
284C2×C3#55611yescharacteristic
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22C2×#23511yescharacteristic
300C22C2×#23511yescharacteristic
301C22C2×D3×D411yescharacteristic
302C22C2×#23711yescharacteristic
303C22#101011yescharacteristic
304C22#101011yescharacteristic
305C22C22×#6711yescharacteristic
306C4—22no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—64no—
311C4—122no—
312C4—122no—
313C4—122no—
314C4#93914yes—
315C3#241511yescharacteristic, complemented, Sylow
316C2—42no—
317C2—42no—
318C2#554111yescharacteristic
319C2#554211yescharacteristic
320C2#555111yescharacteristic
321C2#555211yescharacteristic
322C2#579111yescharacteristic
323C2C2×#93911yescharacteristic
324C2#582511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2415
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#926C2semidirect
3. C2×#924C2semidirect
4. C2×#924C2semidirect
6. #5212C2semidirect
7. #4976C2semidirect
8. #4952C2semidirect
9. #4945C2semidirect
10. #4929C2semidirect
11. #4928C2semidirect
84. C4⋊Dic3D4semidirect
315. C3#2415semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9262
C2×#9242
#52222
#52122
#49762
#49522
#49452
#49292
#49282
#24153