Finite Group Databasebeta

← #76971

Subgroups — #76971

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×#892C211yesmaximal, characteristic, complemented
3C2×#884C212yesmaximal, complemented
4C3×#394C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4961C212yesmaximal, complemented
8#4960C212yesmaximal, complemented
9#4957C212yesmaximal, complemented
10#4671C211yesmaximal, characteristic, complemented
11#4650C211yesmaximal, characteristic, complemented
12#2619—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×#67C2212yes—
16C22×#67C2212yes—
17C22×D12C2212yes—
18C22×D12C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×C3×C22⋊C4C2212yes—
23C2×#217C2212yes—
24#940—24no—
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34#892—24no—
35#884—28no—
36C3×#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C2×#127—31no—
40C2×#121—32no—
41C2×#117—31no—
42#394—31no—
43#394—32no—
44#394D311yescharacteristic, complemented
45#392—31no—
46#392—32no—
47#392—31no—
48#390—32no—
49#382—31no—
50C24×C3C2312yes—
51C23×D3C2312yes—
52C23×D3C2311yescharacteristic
53C2×C3×D4—22no—
54C2×C3×D4—42no—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—24no—
63C2×#67—24no—
64C2×#67—42no—
65C2×#67—42no—
66C2×#67—42no—
67C2×#67—42no—
68C22×Dic3C2312yes—
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2311yescharacteristic
72C2×D12—24no—
73C2×D12—22no—
74C2×D12—42no—
75C2×D12—42no—
76C2×D12—42no—
77C2×C4×D3—22no—
78C2×C4×D3—22no—
79C2×C4×D3D414yescomplemented
80C3×C4⋊C4—22no—
81C3×C22⋊C4—24no—
82C3×C22⋊C4—42no—
83#217—42no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—42no—
88(C2×D3)⋊C4—42no—
89C4⋊Dic3—22no—
90Dic3⋊C4—24no—
91C4×Dic3—22no—
92C4×Dic3D414yescomplemented
93C22×D4—32no—
94C22×D4—32no—
95C22×D4—32no—
96C22×D4—32no—
97C22×D4—31no—
98C22×D4C2×D311yescharacteristic
99C23×C4—31no—
100#127—64no—
101#121—68no—
102#117—64no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4C2×D311yescharacteristic
106C2×C22⋊C4—31no—
107C2×C22⋊C4—31no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4C2×D312yes—
113C2×C22⋊C4C2×D312yes—
114C2×C42—31no—
115C2×C42—31no—
116#95—32no—
117#95—31no—
118#95C2×D311yescharacteristic
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3C2411yescharacteristic
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—22no—
134C22×D3—22no—
135C22×D3—22no—
136C22×D3—22no—
137C22×D3—22no—
138C22×D3C2×D412yes—
139C3×D4—44no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—22no—
143C2×C3×C4—21no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4C2×D412yes—
149C2×C3×C4C2×D412yes—
150#67—48no—
151#67—48no—
152#67—48no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3C2×D412yes—
160C2×Dic3C2×D412yes—
161C2×Dic3C2×D412yes—
162C2×Dic3#4112yes—
163D12—44no—
164D12—48no—
165C4×D3—28no—
166C24—31no—
167C24—32no—
168C24C22×D312yes—
169C2×D4—22no—
170C2×D4—42no—
171C2×D4—64no—
172C2×D4—64no—
173C2×D4—64no—
174C2×D4—64no—
175C2×D4—64no—
176C2×D4—64no—
177C2×D4—62no—
178C2×D4—122no—
179C2×D4—122no—
180C2×D4—122no—
181C2×D4—122no—
182C2×D4—122no—
183C2×D4—122no—
184C2×D4—122no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—34no—
188C22×C4—32no—
189C22×C4—31no—
190C22×C4—62no—
191C22×C4—62no—
192C22×C4C22×D312yes—
193C22×C4C22×D311yescharacteristic
194C22×C4C22×D311yescharacteristic
195C22×C4C22×D311yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—62no—
198C4⋊C4—64no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C22⋊C4—62no—
202C22⋊C4—62no—
203C22⋊C4—64no—
204C22⋊C4—122no—
205C22⋊C4—122no—
206C22⋊C4—122no—
207C42—34no—
208C42—62no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3C22×D411yescharacteristic
217C22×C3C22×D411yescharacteristic
218C22×C3C2×#4111yescharacteristic
219C22×C3H(5,𝔽2)12yes—
220C22×C3H(5,𝔽2)12yes—
221C2×D3—22no—
222C2×D3—24no—
223C2×D3—41no—
224C2×D3—41no—
225C2×D3—42no—
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—22no—
235C3×C4—42no—
236C3×C4—41no—
237Dic3—22no—
238Dic3—41no—
239Dic3—42no—
240Dic3#12114yes—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—32no—
249C23—61no—
250C23—62no—
251C23—61no—
252C23—62no—
253C23—62no—
254C23—62no—
255C23—62no—
256C23—62no—
257C23—62no—
258C23—62no—
259C23—62no—
260C23C23×D311yescharacteristic
261D4—44no—
262D4—124no—
263D4—128no—
264D4—128no—
265D4—128no—
266D4—128no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—32no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—68no—
280C2×C4—62no—
281C2×C4—61no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4D3×D412yes—
287C2×C4D3×D412yes—
288C2×C3—82no—
289C2×C3C2×#12111yescharacteristic
290C2×C3#53511yescharacteristic
291C2×C3#53511yescharacteristic
292C2×C3#53512yes—
293C2×C3#54012yes—
294D3—42no—
295D3—82no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—62no—
304C22—64no—
305C22—122no—
306C22—121no—
307C22—121no—
308C22—122no—
309C22—122no—
310C22—122no—
311C22—122no—
312C22—122no—
313C22—122no—
314C22—122no—
315C22C2×D3×D411yescharacteristic
316C22C2×D3×D411yescharacteristic
317C22#100412yes—
318C22D3×#4111yescharacteristic
319C22#100912yes—
320C4—22no—
321C4—22no—
322C4—34no—
323C4—41no—
324C4—42no—
325C4—62no—
326C4—121no—
327C4—122no—
328C3#261911yescharacteristic, complemented, Sylow
329C2—82no—
330C2—122no—
331C2—242no—
332C2#559211yescharacteristic
333C2D3×#12111yescharacteristic
334C2#560712yes—
335C2#566312yes—
336C2#567411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2619
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#940C2semidirect
2. C2×#892C2semidirect
3. C2×#884C2semidirect
4. C3×#394C2semidirect
5. #5223C2semidirect
6. #4991C2semidirect
7. #4961C2semidirect
8. #4960C2semidirect
9. #4957C2semidirect
10. #4671C2semidirect
11. #4650C2semidirect
44. #394D3semidirect
79. C2×C4×D3D4semidirect
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×#8922
C2×#8842
C3×#3942
#52232
#49912
#49612
#49602
#49572
#46712
#46502
#26193