Finite Group Databasebeta

← #76698

Subgroups — #76698

Order 384.

728 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 143 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×#939C212yesmaximal, complemented
3C2×#877C211yesmaximal, characteristic, complemented
4C3×#392C211yesmaximal, characteristic, complemented
5#5223C212yesmaximal, complemented
6#5222C211yesmaximal, characteristic, complemented
7#4960C211yesmaximal, characteristic, complemented
8#4959C212yesmaximal, complemented
9#4957C212yesmaximal, complemented
10#4672C211yesmaximal, characteristic, complemented
11#4650C211yesmaximal, characteristic, complemented
12#2619—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×#67C2212yes—
16C22×#67C2212yes—
17C22×#67C2212yes—
18C23×Dic3C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217C2212yes—
23C2×#217C2211yescharacteristic
24C2×#217C2212yes—
25#940—24no—
26#939—28no—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35#877—24no—
36C3×#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C2×#127—31no—
40C2×#121—32no—
41C2×#117—31no—
42#394—31no—
43#394—32no—
44#394—31no—
45#392—32no—
46#392—31no—
47#392D311yescharacteristic, complemented
48#390—32no—
49#382—31no—
50C24×C3C2312yes—
51C24×C3C2311yescharacteristic
52C23×D3C2312yes—
53C2×C3×D4—24no—
54C2×C3×D4—22no—
55C2×C3×D4—42no—
56C2×C3×D4—42no—
57C2×C3×D4—42no—
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C2×#67—24no—
62C2×#67—24no—
63C2×#67—24no—
64C2×#67—24no—
65C2×#67—24no—
66C2×#67—42no—
67C2×#67—42no—
68C2×#67—42no—
69C2×#67—42no—
70C22×Dic3—22no—
71C22×Dic3—22no—
72C22×Dic3D414yescomplemented
73C22×Dic3C2312yes—
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2311yescharacteristic
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C2×D12—22no—
79C2×D12—42no—
80C3×C22⋊C4—22no—
81C3×C22⋊C4—42no—
82#217—22no—
83#217—24no—
84#217—42no—
85(C2×D3)⋊C4—24no—
86(C2×D3)⋊C4—42no—
87(C2×D3)⋊C4—42no—
88Dic3⋊C4—22no—
89Dic3⋊C4—22no—
90Dic3⋊C4—24no—
91C4×Dic3—22no—
92C4×Dic3D414yescomplemented
93C22×D4—32no—
94C22×D4—32no—
95C22×D4—32no—
96C22×D4—31no—
97C22×D4C2×D312yes—
98C22×D4C2×D311yescharacteristic
99C23×C4—31no—
100#127—64no—
101#121—68no—
102#117—64no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—31no—
105C2×C4⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4C2×D311yescharacteristic
113C2×C22⋊C4C2×D312yes—
114C2×C42—31no—
115C2×C42—31no—
116#95—32no—
117#95—31no—
118#95C2×D311yescharacteristic
119C23×C3—21no—
120C23×C3—21no—
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×D412yes—
131C23×C3C2411yescharacteristic
132C22×D3—22no—
133C22×D3—22no—
134C22×D3—22no—
135C22×D3—22no—
136C22×D3—22no—
137C22×D3—22no—
138C22×D3—22no—
139C3×D4—44no—
140C3×D4—48no—
141C2×C3×C4—22no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4C2×D412yes—
148#67—48no—
149#67—48no—
150#67—48no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3—28no—
155C2×Dic3—21no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3C2×D412yes—
161C2×Dic3C2×D412yes—
162C2×Dic3C2×D412yes—
163C2×Dic3C2×D412yes—
164C2×Dic3#4112yes—
165D12—44no—
166C24—32no—
167C24C22×D312yes—
168C24C22×D311yescharacteristic
169C2×D4—24no—
170C2×D4—22no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—64no—
175C2×D4—64no—
176C2×D4—64no—
177C2×D4—64no—
178C2×D4—64no—
179C2×D4—62no—
180C2×D4—122no—
181C2×D4—122no—
182C2×D4—122no—
183C2×D4—122no—
184C2×D4—122no—
185C22×C4—31no—
186C22×C4—34no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—32no—
190C22×C4—31no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4C22×D312yes—
194C22×C4C22×D311yescharacteristic
195C22×C4C22×D311yescharacteristic
196C4⋊C4—62no—
197C4⋊C4—64no—
198C4⋊C4—62no—
199C22⋊C4—22no—
200C22⋊C4—42no—
201C22⋊C4—64no—
202C22⋊C4—62no—
203C22⋊C4—64no—
204C22⋊C4—122no—
205C22⋊C4—122no—
206C22⋊C4—122no—
207C42—34no—
208C42—62no—
209C22×C3—22no—
210C22×C3—24no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—42no—
221C22×C3C22×D411yescharacteristic
222C22×C3C22×D411yescharacteristic
223C22×C3C2×#4111yescharacteristic
224C22×C3H(5,𝔽2)12yes—
225C22×C3H(5,𝔽2)12yes—
226C2×D3—42no—
227C2×D3—42no—
228C2×D3—42no—
229C2×D3—42no—
230C2×D3—42no—
231C2×D3—42no—
232C2×D3—42no—
233C3×C4—22no—
234C3×C4—41no—
235C3×C4—42no—
236Dic3—22no—
237Dic3—22no—
238Dic3—41no—
239Dic3—42no—
240Dic3#12114yes—
241C23—21no—
242C23—22no—
243C23—22no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—62no—
253C23—62no—
254C23—62no—
255C23—62no—
256C23—62no—
257C23—62no—
258C23—62no—
259C23D3×D412yes—
260C23C23×D311yescharacteristic
261D4—44no—
262D4—48no—
263D4—124no—
264D4—128no—
265D4—128no—
266D4—128no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—32no—
274C2×C4—32no—
275C2×C4—32no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—68no—
279C2×C4—61no—
280C2×C4—62no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4D3×D412yes—
288C2×C3—42no—
289C2×C3—82no—
290C2×C3C2×#12111yescharacteristic
291C2×C3#53511yescharacteristic
292C2×C3#53511yescharacteristic
293C2×C3#53512yes—
294C2×C3#54012yes—
295D3—82no—
296C22—24no—
297C22—22no—
298C22—41no—
299C22—41no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—122no—
309C22—122no—
310C22—122no—
311C22—122no—
312C22—122no—
313C22—122no—
314C22—122no—
315C22C2×D3×D411yescharacteristic
316C22C2×D3×D411yescharacteristic
317C22#100412yes—
318C22#100412yes—
319C22D3×#4111yescharacteristic
320C4—22no—
321C4—34no—
322C4—41no—
323C4—42no—
324C4—62no—
325C4—62no—
326C4—122no—
327C4—121no—
328C3#261911yescharacteristic, complemented, Sylow
329C2—42no—
330C2—82no—
331C2—242no—
332C2#559211yescharacteristic
333C2#559212yes—
334C2D3×#12111yescharacteristic
335C2#561512yes—
336C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2619
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×#939C2semidirect
3. C2×#877C2semidirect
4. C3×#392C2semidirect
5. #5223C2semidirect
6. #5222C2semidirect
7. #4960C2semidirect
8. #4959C2semidirect
9. #4957C2semidirect
10. #4672C2semidirect
11. #4650C2semidirect
47. #392D3semidirect
72. C22×Dic3D4semidirect
92. C4×Dic3D4semidirect
328. C3#2619semidirect

Maximal subgroups

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

subgroupindex
C2×#9402
C2×#9392
C2×#8772
C3×#3922
#52232
#52222
#49602
#49592
#49572
#46722
#46502
#26193