Finite Group Databasebeta

← #82668

Subgroups — #82668

Order 384.

760 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic, complemented
2C2×#938C211yesmaximal, characteristic, complemented
3C2×#936C211yesmaximal, characteristic, complemented
4C2×C4×#67C211yesmaximal, characteristic, complemented
5C2×#924C211yesmaximal, characteristic, complemented
6C2×#897C211yesmaximal, characteristic, complemented
7#5223C212yesmaximal, complemented
8#5221C212yesmaximal, complemented
9Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
10#4957C212yesmaximal, complemented
11#4949C212yesmaximal, complemented
12#2540—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C23×C3×C4C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C22×#67C2212yes—
18C22×C4×D3C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C3×#121—24no—
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26C2×#217C2211yescharacteristic
27C2×#217C2212yes—
28#938—24no—
29#936—24no—
30C4×#67—24no—
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2212yes—
33C2×C4⋊Dic3C2212yes—
34C2×C4⋊Dic3C2211yescharacteristic
35#924—24no—
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2212yes—
38C2×C4×Dic3C2212yes—
39C2×C4×Dic3C2211yescharacteristic
40#897—24no—
41C3⋊#95C2212yes—
42C2×#124—31no—
43C2×C22⋊Q8—31no—
44C2×C22⋊Q8—31no—
45C2×#121—31no—
46C2×#121D311yescharacteristic, complemented
47C2×C4×D4—31no—
48#399—32no—
49#392—32no—
50#390—32no—
51#390—32no—
52C4×C4⋊C4—31no—
53C24×C3C2312yes—
54C24×C3C2311yescharacteristic
55C23×D3C2311yescharacteristic
56C2×C3×D4—24no—
57C2×C3×D4—22no—
58C2×C3×D4—22no—
59C2×C3×D4—24no—
60C2×C3×D4—22no—
61C2×C3×D4—42no—
62C22×C3×C4—22no—
63C22×C3×C4—22no—
64C22×C3×C4—22no—
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2312yes—
67C22×C3×C4C2311yescharacteristic
68C2×#67—22no—
69C2×#67—22no—
70C2×#67—24no—
71C2×#67—22no—
72C2×#67—24no—
73C2×#67—42no—
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2312yes—
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2312yes—
78C22×Dic3C2311yescharacteristic
79C2×C4×D3—22no—
80C2×C4×D3—22no—
81C2×C4×D3—22no—
82C2×Dic6—22no—
83C2×Dic6—22no—
84C2×Dic6—22no—
85C3×C4⋊C4D414yescomplemented
86C3×C22⋊C4—24no—
87#217—24no—
88#217—22no—
89#217—24no—
90#217—24no—
91#217—24no—
92(C2×D3)⋊C4—24no—
93(C2×D3)⋊C4—22no—
94C4⋊Dic3—24no—
95C4⋊Dic3D414yescomplemented
96Dic3⋊C4—24no—
97Dic3⋊C4—22no—
98C4×Dic3—42no—
99C4×Dic3D414yescomplemented
100C22×Q8—31no—
101C22×D4—31no—
102C22×D4—32no—
103C22×D4C2×D311yescharacteristic
104C22×D4C2×D312yes—
105C23×C4—31no—
106C23×C4C2×D311yescharacteristic
107#124—64no—
108C22⋊Q8—64no—
109C22⋊Q8—64no—
110#121—24no—
111#121—64no—
112C4×D4—64no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4C2×D311yescharacteristic
118C2×C22⋊C4—31no—
119C2×C22⋊C4—32no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4—32no—
122C2×C22⋊C4—32no—
123C2×C22⋊C4—31no—
124C2×C22⋊C4—32no—
125C2×C22⋊C4C2×D312yes—
126C2×C42—31no—
127C2×C42—32no—
128#95—32no—
129C23×C3—21no—
130C23×C3—21no—
131C23×C3—21no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—21no—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—22no—
141C23×C3—22no—
142C23×C3#4112yes—
143C23×C3C2411yescharacteristic
144C22×D3—21no—
145C22×D3—21no—
146C22×D3—21no—
147C22×D3—21no—
148C22×D3—21no—
149C22×D3—21no—
150C22×D3#4112yes—
151C3×D4—48no—
152C3×D4—44no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—44no—
159C2×C3×C4C2×D412yes—
160C2×C3×C4C2×D412yes—
161C2×C3×C4C2×D414yes—
162C2×C3×C4C2×D412yes—
163#67—28no—
164#67—48no—
165C2×Dic3—21no—
166C2×Dic3—21no—
167C2×Dic3—22no—
168C2×Dic3—21no—
169C2×Dic3—22no—
170C2×Dic3—21no—
171C2×Dic3—22no—
172C2×Dic3—21no—
173C2×Dic3—21no—
174C2×Dic3—22no—
175C2×Dic3—22no—
176C2×Dic3—21no—
177C2×Dic3C2×D412yes—
178C2×Dic3C2×D412yes—
179C2×Dic3C2×D414yes—
180C4×D3—44no—
181Dic6—44no—
182C24—31no—
183C24C22×D312yes—
184C24C22×D311yescharacteristic
185C2×Q8—62no—
186C2×Q8—62no—
187C2×Q8—62no—
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—24no—
192C2×D4—22no—
193C2×D4—42no—
194C2×D4—62no—
195C2×D4—64no—
196C2×D4—62no—
197C2×D4—62no—
198C2×D4—64no—
199C2×D4—122no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—31no—
204C22×C4—32no—
205C22×C4—31no—
206C22×C4—32no—
207C22×C4—31no—
208C22×C4—62no—
209C22×C4—62no—
210C22×C4—62no—
211C22×C4C22×D311yescharacteristic
212C22×C4C22×D312yes—
213C22×C4C22×D311yescharacteristic
214C4⋊C4—34no—
215C4⋊C4—62no—
216C4⋊C4—64no—
217C4⋊C4—64no—
218C4⋊C4#6714yescomplemented
219C22⋊C4—24no—
220C22⋊C4—62no—
221C22⋊C4—62no—
222C22⋊C4—64no—
223C22⋊C4—64no—
224C22⋊C4—64no—
225C22⋊C4—64no—
226C22⋊C4—64no—
227C42—34no—
228C42—122no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3—42no—
237C22×C3—42no—
238C22×C3—42no—
239C22×C3—42no—
240C22×C3—42no—
241C22×C3—42no—
242C22×C3—42no—
243C22×C3C22×D411yescharacteristic
244C22×C3C22×D411yescharacteristic
245C22×C3C22×D411yescharacteristic
246C22×C3C2×#4111yescharacteristic
247C22×C3C2×#4111yescharacteristic
248C22×C3H(5,𝔽2)11yescharacteristic
249C22×C32−1+411yescharacteristic
250C2×D3—22no—
251C2×D3—22no—
252C2×D3—22no—
253C2×D3—41no—
254C2×D3—41no—
255C2×D3—41no—
256C2×D3—41no—
257C3×C4—24no—
258C3×C4—41no—
259C3×C4#12714yes—
260Dic3—22no—
261Dic3—22no—
262Dic3—24no—
263Dic3—42no—
264Dic3—41no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—32no—
279C23—61no—
280C23—61no—
281C23—61no—
282C23—61no—
283C23—61no—
284C23—61no—
285C23#23712yes—
286C23C23×D311yescharacteristic
287Q8—124no—
288D4—48no—
289D4—44no—
290D4—68no—
291D4—128no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—32no—
298C2×C4—32no—
299C2×C4—34no—
300C2×C4—44no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—62no—
304C2×C4—61no—
305C2×C4—62no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—62no—
309C2×C4—61no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—61no—
313C2×C4—124no—
314C2×C4D3×D412yes—
315C2×C4D3×D412yes—
316C2×C4C2×#6714yes—
317C2×C4C2×#6712yes—
318C2×C3—42no—
319C2×C3—82no—
320C2×C3#52511yescharacteristic
321C2×C3C2×#12711yescharacteristic
322C2×C3#53711yescharacteristic
323C2×C3#54611yescharacteristic
324C2×C3#54611yescharacteristic
325C2×C3#54811yescharacteristic
326C2×C3#54811yescharacteristic
327D3—42no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—62no—
343C22—62no—
344C22—62no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22—121no—
349C22C2×D3×D411yescharacteristic
350C22C2×D3×D411yescharacteristic
351C22C2×#23711yescharacteristic
352C22#100411yescharacteristic
353C22D3×#4111yescharacteristic
354C22#101011yescharacteristic
355C22C22×#6711yescharacteristic
356C4—24no—
357C4—41no—
358C4—62no—
359C4—62no—
360C4—64no—
361C4—121no—
362C4—122no—
363C4#94014yes—
364C3#254011yescharacteristic, complemented, Sylow
365C2—42no—
366C2—82no—
367C2—122no—
368C2#559811yescharacteristic
369C2#560611yescharacteristic
370C2#560811yescharacteristic
371C2#561211yescharacteristic
372C2C2×#94011yescharacteristic
373C2#580611yescharacteristic
374C2#583111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2540
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#121C2semidirect
2. C2×#938C2semidirect
3. C2×#936C2semidirect
4. C2×C4×#67C2semidirect
5. C2×#924C2semidirect
6. C2×#897C2semidirect
7. #5223C2semidirect
8. #5221C2semidirect
9. Dic3×C4⋊C4C2semidirect
10. #4957C2semidirect
11. #4949C2semidirect
46. C2×#121D3semidirect
85. C3×C4⋊C4D4semidirect
95. C4⋊Dic3D4semidirect
99. C4×Dic3D4semidirect
218. C4⋊C4#67semidirect
364. C3#2540semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#9382
C2×#9362
C2×C4×#672
C2×#9242
C2×#8972
#52232
#52212
Dic3×C4⋊C42
#49572
#49492
#25403