Finite Group Databasebeta

← #76744

Subgroups — #76744

Order 384.

630 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 125 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×#940C211yesmaximal, characteristic, complemented
2C2×#879C211yesmaximal, characteristic, complemented
3C3×#392C211yesmaximal, characteristic, complemented
4#5223C212yesmaximal, complemented
5#5223C212yesmaximal, complemented
6#5222C211yesmaximal, characteristic, complemented
7#4961C212yesmaximal, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4959C212yesmaximal, complemented
10#4672C211yesmaximal, characteristic, complemented
11#4651C211yesmaximal, characteristic, complemented
12#2818—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×#67C2212yes—
16C22×#67C2212yes—
17C22×#67C2212yes—
18C22×D12C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2212yes—
24C2×#217C2212yes—
25#940—24no—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33#879—24no—
34C3×#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#127—31no—
39C2×#126—31no—
40#394—31no—
41#394—32no—
42#394—32no—
43#394—31no—
44#392—32no—
45#392—32no—
46#392—31no—
47#392D311yescharacteristic, complemented
48#383—31no—
49C24×C3C2311yescharacteristic
50C24×C3C2312yes—
51C23×D3C2312yes—
52C2×C3×D4—22no—
53C2×C3×D4—42no—
54C2×C3×D4—42no—
55C2×C3×D4—42no—
56C2×C3×D4—42no—
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—42no—
63C2×#67—42no—
64C2×#67—42no—
65C2×#67—42no—
66C2×#67—42no—
67C2×#67—42no—
68C2×#67—42no—
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2311yescharacteristic
73C22×Dic3C2311yescharacteristic
74C2×D12—22no—
75C2×D12—42no—
76C3×C22⋊C4—22no—
77C3×C22⋊C4—42no—
78#217—22no—
79#217—22no—
80#217—42no—
81#217—42no—
82(C2×D3)⋊C4—42no—
83(C2×D3)⋊C4—42no—
84(C2×D3)⋊C4—42no—
85C4⋊Dic3—22no—
86Dic3⋊C4—22no—
87Dic3⋊C4—22no—
88C4×Dic3D414yescomplemented
89C22×D4—32no—
90C22×D4—32no—
91C22×D4—32no—
92C22×D4—31no—
93C22×D4C2×D311yescharacteristic
94C22×D4C2×D312yes—
95#127—64no—
96#126—64no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—31no—
100C2×C22⋊C4—31no—
101C2×C22⋊C4—31no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4C2×D311yescharacteristic
108C2×C22⋊C4C2×D312yes—
109C2×C42—31no—
110#95—32no—
111#95—32no—
112#95—31no—
113#95C2×D311yescharacteristic
114C23×C3—21no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—21no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—21no—
126C23×C3C2411yescharacteristic
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—22no—
134C3×D4—44no—
135C3×D4—48no—
136C2×C3×C4—22no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4C2×D412yes—
143#67—48no—
144#67—48no—
145#67—48no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3C2×D412yes—
158C2×Dic3C2×D412yes—
159D12—44no—
160C24—32no—
161C24C22×D311yescharacteristic
162C24C22×D312yes—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—64no—
169C2×D4—64no—
170C2×D4—62no—
171C2×D4—122no—
172C2×D4—122no—
173C2×D4—122no—
174C2×D4—122no—
175C2×D4—122no—
176C2×D4—122no—
177C2×D4—122no—
178C2×D4—122no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—32no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4C22×D312yes—
185C22×C4C22×D311yescharacteristic
186C22×C4C22×D311yescharacteristic
187C4⋊C4—62no—
188C4⋊C4—62no—
189C4⋊C4—62no—
190C22⋊C4—22no—
191C22⋊C4—42no—
192C22⋊C4—62no—
193C22⋊C4—62no—
194C22⋊C4—122no—
195C22⋊C4—122no—
196C22⋊C4—122no—
197C22⋊C4—122no—
198C22⋊C4—122no—
199C42—34no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3—41no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3C22×D411yescharacteristic
213C22×C3H(5,𝔽2)12yes—
214C22×C3H(5,𝔽2)12yes—
215C22×C3H(5,𝔽2)11yescharacteristic
216C22×C3H(5,𝔽2)11yescharacteristic
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—42no—
221C2×D3—42no—
222C2×D3—42no—
223C2×D3—42no—
224C3×C4—22no—
225C3×C4—41no—
226C3×C4—42no—
227Dic3—22no—
228Dic3—22no—
229Dic3—41no—
230Dic3—41no—
231Dic3—42no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—62no—
245C23—62no—
246C23—62no—
247C23—62no—
248C23—62no—
249C23—62no—
250C23—62no—
251C23C23×D311yescharacteristic
252D4—44no—
253D4—48no—
254D4—124no—
255D4—128no—
256D4—128no—
257D4—128no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—32no—
265C2×C4—32no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—62no—
272C2×C4—61no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—61no—
276C2×C4—62no—
277C2×C4D3×D412yes—
278C2×C3—82no—
279C2×C3—81no—
280C2×C3#53511yescharacteristic
281C2×C3#53511yescharacteristic
282C2×C3#53511yescharacteristic
283C2×C3#56012yes—
284C2×C3#56012yes—
285D3—82no—
286C22—42no—
287C22—41no—
288C22—42no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—122no—
299C22—122no—
300C22—122no—
301C22—122no—
302C22—122no—
303C22—122no—
304C22—122no—
305C22C2×D3×D411yescharacteristic
306C22#100412yes—
307C22#100412yes—
308C22#100411yescharacteristic
309C22#100911yescharacteristic
310C4—22no—
311C4—41no—
312C4—42no—
313C4—62no—
314C4—62no—
315C4—121no—
316C4—121no—
317C4—122no—
318C3#281811yescharacteristic, complemented, Sylow
319C2—82no—
320C2—81no—
321C2—242no—
322C2#559211yescharacteristic
323C2#559512yes—
324C2#560711yescharacteristic
325C2#562012yes—
326C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2818
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#940C2semidirect
2. C2×#879C2semidirect
3. C3×#392C2semidirect
4. #5223C2semidirect
5. #5223C2semidirect
6. #5222C2semidirect
7. #4961C2semidirect
8. #4960C2semidirect
9. #4959C2semidirect
10. #4672C2semidirect
11. #4651C2semidirect
47. #392D3semidirect
88. C4×Dic3D4semidirect
318. C3#2818semidirect

Maximal subgroups

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

subgroupindex
C2×#9402
C2×#8792
C3×#3922
#52232
#52222
#49612
#49602
#49592
#46722
#46512
#28183