Finite Group Databasebeta

← #74687

Subgroups — #74687

Order 384.

864 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#237C211yesmaximal, characteristic
2C2×#935C211yesmaximal, characteristic, complemented
3C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C2×#886C211yesmaximal, characteristic, complemented
5C3×#386C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4961C212yesmaximal, complemented
8#4950C212yesmaximal, complemented
9#4943C211yesmaximal, characteristic, complemented
10#4668C212yesmaximal, complemented
11#4656C212yesmaximal, complemented
12#2406—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C2×#237—24no—
21C2×#237—24no—
22C2×#237—24no—
23C22×C4×D3C2211yescharacteristic
24C22×Dic6C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2212yes—
27C2×#217C2211yescharacteristic
28C2×#217C2212yes—
29#935—24no—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2212yes—
32C2×C4⋊Dic3—24no—
33C2×C4⋊Dic3C2212yes—
34C2×C4⋊Dic3C2212yes—
35C2×C4⋊Dic3C2212yes—
36C2×Dic3⋊C4C2212yes—
37#886—24no—
38C3×#95C2212yes—
39C3⋊#95C2212yes—
40C3⋊#95C2212yes—
41C3⋊#95C2212yes—
42C22×#41—31no—
43C2×#123—31no—
44C2×#123—31no—
45C22×C4⋊C4—31no—
46#395—32no—
47#394—32no—
48#386—31no—
49#386D311yescharacteristic, complemented
50#385—32no—
51#385—32no—
52#381—31no—
53C24×C3C2311yescharacteristic
54C24×C3C2311yescharacteristic
55C23×D3C2311yescharacteristic
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4D414yes—
59C22×C3×C4—22no—
60C22×C3×C4—24no—
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2312yes—
64C2×#67—22no—
65C2×#67—22no—
66C2×#67—22no—
67C2×#67—22no—
68C2×#67—22no—
69C2×#67D414yes—
70C22×Dic3—22no—
71C22×Dic3—22no—
72C22×Dic3—22no—
73C22×Dic3—22no—
74C22×Dic3—22no—
75C22×Dic3—22no—
76C22×Dic3—22no—
77C22×Dic3—22no—
78C22×Dic3D414yes—
79C22×Dic3C2312yes—
80C22×Dic3C2312yes—
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84#237—232no—
85C2×C4×D3—22no—
86C2×C4×D3—22no—
87C2×C4×D3—22no—
88C2×Dic6—22no—
89C2×Dic6—22no—
90C2×Dic6—22no—
91C3×C22⋊C4—22no—
92C3×C22⋊C4—24no—
93#217—22no—
94#217—24no—
95(C2×D3)⋊C4—24no—
96(C2×D3)⋊C4—24no—
97C4⋊Dic3—24no—
98C4⋊Dic3—24no—
99C4⋊Dic3—24no—
100Dic3⋊C4—24no—
101C2×#41—64no—
102C2×#41—64no—
103C2×#41—64no—
104C22×Q8—31no—
105C22×D4—31no—
106C22×D4—31no—
107C22×D4C2×D311yescharacteristic
108C23×C4—31no—
109C23×C4—31no—
110C23×C4—31no—
111C23×C4—31no—
112C23×C4C2×D311yescharacteristic
113#123—64no—
114#123—64no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—32no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—64no—
120C2×C22⋊C4—31no—
121C2×C22⋊C4—32no—
122C2×C22⋊C4—32no—
123C2×C22⋊C4—32no—
124C2×C22⋊C4C2×D311yescharacteristic
125C2×C22⋊C4C2×D312yes—
126#95—32no—
127#95—32no—
128#95—32no—
129#95C2×D312yes—
130C23×C3—22no—
131C23×C3—22no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3C2×D412yes—
137C23×C3C2×D412yes—
138C23×C3C2×D412yes—
139C23×C3C2×D412yes—
140C23×C3C2411yescharacteristic
141C22×D3—21no—
142C22×D3—21no—
143C22×D3—22no—
144C22×D3—22no—
145C22×D3C2×D412yes—
146C3×D4—28no—
147C2×C3×C4—28no—
148C2×C3×C4—22no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4C2×D412yes—
156C2×C3×C4#4114yescomplemented
157#67—28no—
158#67—28no—
159C2×Dic3—28no—
160C2×Dic3—28no—
161C2×Dic3—21no—
162C2×Dic3—21no—
163C2×Dic3—21no—
164C2×Dic3—21no—
165C2×Dic3—21no—
166C2×Dic3—21no—
167C2×Dic3—22no—
168C2×Dic3—22no—
169C2×Dic3—22no—
170C2×Dic3—22no—
171C2×Dic3—22no—
172C2×Dic3—22no—
173C2×Dic3—44no—
174C2×Dic3C2×D412yes—
175C2×Dic3C2×D412yes—
176C2×Dic3C2×D412yes—
177C4×D3—28no—
178Dic6—28no—
179C24—31no—
180C24C22×D311yescharacteristic
181C24C22×D311yescharacteristic
182#41—632no—
183C2×Q8—62no—
184C2×Q8—62no—
185C2×Q8—62no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—34no—
189C2×D4—62no—
190C2×D4—62no—
191C2×D4—62no—
192C2×D4—62no—
193C2×D4—62no—
194C2×D4D1214yes—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—34no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—32no—
202C22×C4—32no—
203C22×C4—62no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—62no—
208C22×C4—62no—
209C22×C4—62no—
210C22×C4—62no—
211C22×C4—62no—
212C22×C4—62no—
213C22×C4—62no—
214C22×C4C22×D312yes—
215C22×C4C22×D311yescharacteristic
216C22×C4C22×D312yes—
217C4⋊C4—64no—
218C4⋊C4—64no—
219C4⋊C4—64no—
220C4⋊C4—64no—
221C22⋊C4—22no—
222C22⋊C4—24no—
223C22⋊C4—64no—
224C22⋊C4—62no—
225C22⋊C4—64no—
226C22⋊C4—64no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—24no—
230C22×C3—24no—
231C22×C3—22no—
232C22×C3—24no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—42no—
236C22×C3#12014yes—
237C22×C3C22×D411yescharacteristic
238C22×C3C22×D411yescharacteristic
239C22×C3C22×D411yescharacteristic
240C22×C3C2×#4112yes—
241C22×C32−1+412yes—
242C2×D3—22no—
243C2×D3—24no—
244C2×D3—41no—
245C2×D3—41no—
246C2×D3—42no—
247C3×C4—22no—
248C3×C4—24no—
249C3×C4—42no—
250Dic3—22no—
251Dic3—22no—
252Dic3—22no—
253Dic3—42no—
254Dic3—42no—
255C23—22no—
256C23—22no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—32no—
262C23—61no—
263C23—61no—
264C23—62no—
265C23—62no—
266C23C2×D1212yes—
267C23C2×D1212yes—
268C23D3×D412yes—
269C23D3×D412yes—
270C23C23×D311yescharacteristic
271Q8—68no—
272D4—28no—
273D4—68no—
274D4—68no—
275C2×C4—28no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—32no—
284C2×C4—32no—
285C2×C4—32no—
286C2×C4—68no—
287C2×C4—68no—
288C2×C4—68no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4—124no—
302C2×C4C2×D1212yes—
303C2×C4#23714yescomplemented
304C2×C3—24no—
305C2×C3—42no—
306C2×C3C2×#12011yescharacteristic
307C2×C3#52511yescharacteristic
308C2×C3#53611yescharacteristic
309C2×C3#54712yes—
310C2×C3#54712yes—
311D3—42no—
312C22—22no—
313C22—24no—
314C22—22no—
315C22—24no—
316C22—22no—
317C22—24no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22—62no—
322C22—64no—
323C22—121no—
324C22—122no—
325C22—121no—
326C22#88214yes—
327C22C22×D1211yescharacteristic
328C22C2×D3×D411yescharacteristic
329C22C2×D3×D411yescharacteristic
330C22C2×#23712yes—
331C22#101012yes—
332C4—22no—
333C4—24no—
334C4—42no—
335C4—62no—
336C4—62no—
337C4—62no—
338C4—122no—
339C4—122no—
340C3#240611yescharacteristic, complemented, Sylow
341C2—24no—
342C2—42no—
343C2—122no—
344C2C2×#88211yescharacteristic
345C2#555512yes—
346C2#558611yescharacteristic
347C2#561112yes—
348C2#564811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC22×#237C2
hypercenter#386D3
hypocenterC3#2406
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#935C2semidirect
3. C22×C4⋊Dic3C2semidirect
4. C2×#886C2semidirect
5. C3×#386C2semidirect
6. #5213C2semidirect
7. #4961C2semidirect
8. #4950C2semidirect
9. #4943C2semidirect
10. #4668C2semidirect
11. #4656C2semidirect
49. #386D3semidirect
156. C2×C3×C4#41semidirect
303. C2×C4#237semidirect
340. C3#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×#9352
C22×C4⋊Dic32
C2×#8862
C3×#3862
#52132
#49612
#49502
#49432
#46682
#46562
#24063