Finite Group Databasebeta

← #83176

Subgroups — #83176

Order 384.

696 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#127C211yesmaximal, characteristic
2C2×#935C212yesmaximal, complemented
3C2×D4×Dic3C211yesmaximal, characteristic, complemented
4#5222C212yesmaximal, complemented
5#5222C212yesmaximal, complemented
6#5221C211yesmaximal, characteristic, complemented
7#5221C211yesmaximal, characteristic, complemented
8#5221C211yesmaximal, characteristic, complemented
9#5220C212yesmaximal, complemented
10C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2445—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×C3×D4C2212yes—
16C22×C3×D4C2211yescharacteristic
17C23×Dic3C2212yes—
18C3×#127—24no—
19C2×C3×C42C2211yescharacteristic
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26C2×#217C2212yes—
27#935—28no—
28D4×Dic3—24no—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2211yescharacteristic
39C2×#127D311yescharacteristic
40C2×#123—32no—
41C2×C4×D4—31no—
42#394—32no—
43#394—32no—
44#390—31no—
45#390—31no—
46#390—31no—
47#389—31no—
48#386—32no—
49C4×C4⋊C4—31no—
50C24×C3C2312yes—
51C24×C3C2312yes—
52C2×C3×D4—24no—
53C2×C3×D4—24no—
54C2×C3×D4—24no—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—24no—
59C2×C3×D4—22no—
60C2×C3×D4—22no—
61C2×C3×D4—42no—
62C2×C3×D4—42no—
63C2×C3×D4D414yes—
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2311yescharacteristic
67C22×Dic3—24no—
68C22×Dic3—24no—
69C22×Dic3—42no—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2312yes—
73C22×Dic3C2312yes—
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2311yescharacteristic
76C3×C42—22no—
77#217—24no—
78#217—24no—
79#217—24no—
80#217—24no—
81#217—42no—
82#217—42no—
83#217—42no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86Dic3⋊C4—24no—
87Dic3⋊C4—22no—
88Dic3⋊C4D414yescomplemented
89C4×Dic3—22no—
90C4×Dic3—22no—
91C22×D4C2×D311yescharacteristic
92C22×D4C2×D312yes—
93C22×D4C2×D312yes—
94C22×D4C2×D311yescharacteristic
95C23×C4—32no—
96#127—24no—
97#123—68no—
98C4×D4—64no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4—31no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C42—31no—
112C2×C42—31no—
113C2×C42C2×D311yescharacteristic
114#95—31no—
115#95—31no—
116#95—32no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3C2×D414yes—
131C23×C3C2411yescharacteristic
132C3×D4—28no—
133C3×D4—48no—
134C3×D4—48no—
135C3×D4—44no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4C2×D412yes—
141C2×C3×C4C2×D412yes—
142C2×C3×C4#4112yes—
143C2×C3×C4#4112yes—
144C2×C3×C4#4112yes—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156C2×Dic3—21no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—48no—
162C2×Dic3C2×D412yes—
163C2×Dic3C2×D412yes—
164C24C22×D312yes—
165C24C22×D312yes—
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—24no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4#6714yes—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—32no—
183C22×C4—32no—
184C22×C4—64no—
185C22×C4—64no—
186C22×C4—122no—
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C4⋊C4—34no—
191C4⋊C4—64no—
192C4⋊C4—62no—
193C4⋊C4—62no—
194C4⋊C4—62no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—64no—
199C22⋊C4—122no—
200C22⋊C4—122no—
201C22⋊C4—122no—
202C42—22no—
203C42—62no—
204C42—62no—
205C22×C3—24no—
206C22×C3—24no—
207C22×C3—24no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3C22×D411yescharacteristic
220C22×C3C22×D411yescharacteristic
221C22×C3C2×#4111yescharacteristic
222C22×C3C2×#4111yescharacteristic
223C22×C3C2×#4111yescharacteristic
224C22×C3H(5,𝔽2)11yescharacteristic
225C22×C3H(5,𝔽2)11yescharacteristic
226C3×C4—22no—
227C3×C4—22no—
228C3×C4#12314yescomplemented
229Dic3—22no—
230Dic3—22no—
231Dic3—41no—
232Dic3—42no—
233Dic3—41no—
234Dic3—42no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23C2×#6714yes—
249C23C23×D311yescharacteristic
250D4—28no—
251D4—48no—
252D4—48no—
253D4—44no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—32no—
259C2×C4—32no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—61no—
269C2×C4—62no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—128no—
277C2×C4D3×D412yes—
278C2×C4#23712yes—
279C2×C4#23712yes—
280C2×C4#23712yes—
281C2×C4C2×#6712yes—
282C2×C3—44no—
283C2×C3—82no—
284C2×C3C2×#12311yescharacteristic
285C2×C3#53211yescharacteristic
286C2×C3#54011yescharacteristic
287C2×C3D4211yescharacteristic
288C2×C3#54611yescharacteristic
289C2×C3#55011yescharacteristic
290C2×C3#55911yescharacteristic
291C22—24no—
292C22—24no—
293C22—24no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22C2×D3×D411yescharacteristic
306C22C2×#23711yescharacteristic
307C22C2×#23711yescharacteristic
308C22C2×#23711yescharacteristic
309C22#100411yescharacteristic
310C22#100411yescharacteristic
311C22C22×#6711yescharacteristic
312C4—22no—
313C4—22no—
314C4—62no—
315C4—62no—
316C4—122no—
317C4—121no—
318C4—122no—
319C4—121no—
320C4#93514yescomplemented
321C3#244511yescharacteristic, complemented, Sylow
322C2—44no—
323C2—82no—
324C2#571311yescharacteristic
325C2#571611yescharacteristic
326C2#571811yescharacteristic
327C2#571911yescharacteristic
328C2C2×#93511yescharacteristic
329C2D4×#6711yescharacteristic
330C2#580611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC2×C3×#127C2
hypercenterC2×#127D3
hypocenterC3#2445
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
2. C2×#935C2semidirect
3. C2×D4×Dic3C2semidirect
4. #5222C2semidirect
5. #5222C2semidirect
6. #5221C2semidirect
7. #5221C2semidirect
8. #5221C2semidirect
9. #5220C2semidirect
10. C4×Dic3⋊C4C2semidirect
11. #4928C2semidirect
88. Dic3⋊C4D4semidirect
228. C3×C4#123semidirect
320. C4#935semidirect
321. C3#2445semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1272
C2×#9352
C2×D4×Dic32
#52222
#52212
#52202
C4×Dic3⋊C42
#49282
#24453