Finite Group Databasebeta

← #76673

Subgroups — #76673

Order 384.

1348 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×D3×D4C211yesmaximal, characteristic, complemented
2C2×#940C212yesmaximal, complemented
3C2×#937C212yesmaximal, complemented
4C2×#884C211yesmaximal, characteristic, complemented
5C2×#882C212yesmaximal, complemented
6C3×#392C211yesmaximal, characteristic, complemented
7#5223C211yesmaximal, characteristic, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4957C212yesmaximal, complemented
10#4671C211yesmaximal, characteristic, complemented
11#4666C211yesmaximal, characteristic, complemented
12#2433—31nomaximal, Sylow
13C24×D3C2212yes—
14C22×C3×D4C2211yescharacteristic
15C22×C3×D4C2212yescomplemented
16C22×#67C2211yescharacteristic
17C22×#67C2212yes—
18C22×#67C2211yescharacteristic, complemented
19C22×#67C2212yescomplemented
20C22×#67C2212yescomplemented
21C2×D3×D4—28no—
22C2×D3×D4—24no—
23C22×D12C2211yescharacteristic
24C22×D12C2211yescharacteristic
25C22×D12C2212yescomplemented
26C22×C4×D3C2211yescharacteristic
27C2×C3×C22⋊C4C2212yescomplemented
28C2×C3×C22⋊C4C2211yescharacteristic, complemented
29C2×#217C2212yescomplemented
30#940—28no—
31#937—28no—
32C2×(C2×D3)⋊C4C2212yes—
33C2×(C2×D3)⋊C4C2212yescomplemented
34C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
35C2×(C2×D3)⋊C4C2212yescomplemented
36C2×Dic3⋊C4C2211yescharacteristic, complemented
37C2×C4×Dic3C2212yescomplemented
38#884—24no—
39#882—28no—
40C3×#95C2211yescharacteristic, complemented
41C3⋊#95C2211yescharacteristic, complemented
42C23×D4—31no—
43C2×#127—32no—
44C2×#121—31no—
45C2×#120—32no—
46C2×#120—32no—
47#392—31no—
48#392—31no—
49#392—31no—
50#392D311yescharacteristic, complemented
51#390—32no—
52C4×C22⋊C4—31no—
53C24×C3C2311yescharacteristic
54C24×C3C2312yes—
55C23×D3—24no—
56C23×D3—24no—
57C23×D3—24no—
58C23×D3—24no—
59C23×D3—24no—
60C23×D3—42no—
61C23×D3—42no—
62C23×D3C2311yescharacteristic
63C23×D3C2312yes—
64C23×D3C2311yescharacteristic
65C2×C3×D4—24no—
66C2×C3×D4—24no—
67C2×C3×D4—22no—
68C2×C3×D4—24no—
69C2×C3×D4—42no—
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2312yes—
73C2×#67—24no—
74C2×#67—22no—
75C2×#67—24no—
76C2×#67—24no—
77C2×#67—24no—
78C2×#67—24no—
79C2×#67—22no—
80C2×#67—24no—
81C2×#67—24no—
82C2×#67—24no—
83C2×#67—42no—
84C2×#67—42no—
85C2×#67D418yescomplemented
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2312yes—
88C22×Dic3C2311yescharacteristic
89D3×D4—232no—
90C2×D12—24no—
91C2×D12—22no—
92C2×D12—24no—
93C2×D12—22no—
94C2×D12—24no—
95C2×D12—24no—
96C2×D12—42no—
97C2×C4×D3—24no—
98C2×C4×D3D414yescomplemented
99C3×C22⋊C4—22no—
100C3×C22⋊C4—24no—
101#217—24no—
102(C2×D3)⋊C4—22no—
103(C2×D3)⋊C4—24no—
104(C2×D3)⋊C4—24no—
105(C2×D3)⋊C4D418yescomplemented
106Dic3⋊C4—22no—
107C4×Dic3—24no—
108C25—32no—
109C22×D4—32no—
110C22×D4—31no—
111C22×D4—31no—
112C22×D4—32no—
113C22×D4—32no—
114C22×D4—31no—
115C22×D4—31no—
116C22×D4—32no—
117C22×D4—64no—
118C22×D4—68no—
119C22×D4C2×D311yescharacteristic
120C22×D4C2×D312yescomplemented
121C23×C4—31no—
122#127—68no—
123#121—64no—
124#120—68no—
125#120—68no—
126C2×C4⋊C4—31no—
127C2×C22⋊C4—32no—
128C2×C22⋊C4—32no—
129C2×C22⋊C4—32no—
130C2×C22⋊C4—32no—
131C2×C22⋊C4—31no—
132C2×C22⋊C4C2×D312yescomplemented
133C2×C22⋊C4C2×D311yescharacteristic, complemented
134C2×C42—32no—
135#95—31no—
136#95C2×D311yescharacteristic, complemented
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—22no—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—21no—
144C23×C3—22no—
145C23×C3—22no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3C2×D414yes—
149C23×C3C2411yescharacteristic
150C22×D3—22no—
151C22×D3—21no—
152C22×D3—21no—
153C22×D3—216no—
154C22×D3—22no—
155C22×D3—22no—
156C22×D3—22no—
157C22×D3—22no—
158C22×D3—22no—
159C22×D3—22no—
160C22×D3—21no—
161C22×D3—22no—
162C22×D3—22no—
163C22×D3—21no—
164C22×D3—21no—
165C22×D3—48no—
166C22×D3—48no—
167C22×D3—48no—
168C22×D3—48no—
169C22×D3—48no—
170C22×D3—48no—
171C22×D3C2×D414yes—
172C22×D3C2×D414yes—
173C22×D3C2×D412yes—
174C3×D4—28no—
175C3×D4—48no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—21no—
180C2×C3×C4—22no—
181C2×C3×C4C2×D412yes—
182C2×C3×C4C2×D414yescomplemented
183#67—216no—
184#67—48no—
185#67—48no—
186#67—44no—
187#67—44no—
188C2×Dic3—22no—
189C2×Dic3—22no—
190C2×Dic3—21no—
191C2×Dic3—22no—
192C2×Dic3C2×D414yes—
193C2×Dic3C2×D412yes—
194C2×Dic3C2×D414yescomplemented
195D12—28no—
196D12—48no—
197D12—44no—
198C4×D3—28no—
199C24—31no—
200C24—32no—
201C24—31no—
202C24—64no—
203C24—64no—
204C24—64no—
205C24—64no—
206C24—64no—
207C24—122no—
208C24—122no—
209C24C22×D311yescharacteristic
210C24C22×D312yes—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—24no—
215C2×D4—38no—
216C2×D4—42no—
217C2×D4—64no—
218C2×D4—64no—
219C2×D4—62no—
220C2×D4—64no—
221C2×D4—64no—
222C2×D4—64no—
223C2×D4—64no—
224C2×D4—64no—
225C2×D4—62no—
226C2×D4—64no—
227C2×D4—632no—
228C2×D4—64no—
229C2×D4—62no—
230C2×D4—64no—
231C2×D4—64no—
232C2×D4—62no—
233C2×D4—64no—
234C2×D4—122no—
235C2×D4—122no—
236C2×D4—122no—
237C22×C4—31no—
238C22×C4—32no—
239C22×C4—34no—
240C22×C4—31no—
241C22×C4—64no—
242C22×C4C22×D311yescharacteristic
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D312yes—
245C4⋊C4—62no—
246C22⋊C4—22no—
247C22⋊C4—24no—
248C22⋊C4—38no—
249C22⋊C4—64no—
250C22⋊C4—64no—
251C22⋊C4—62no—
252C22⋊C4—64no—
253C42—64no—
254C22×C3—24no—
255C22×C3—24no—
256C22×C3—24no—
257C22×C3—42no—
258C22×C3—42no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3C22×D411yescharacteristic
267C22×C3C22×D412yes—
268C22×C3C22×D412yes—
269C22×C3H(5,𝔽2)11yescharacteristic
270C22×C3H(5,𝔽2)11yescharacteristic
271C2×D3—22no—
272C2×D3—22no—
273C2×D3—24no—
274C2×D3—24no—
275C2×D3—24no—
276C2×D3—24no—
277C2×D3—24no—
278C2×D3—432no—
279C2×D3—42no—
280C2×D3—42no—
281C2×D3—42no—
282C2×D3—42no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—42no—
287C2×D3—42no—
288C2×D3#12714yescomplemented
289C3×C4—22no—
290C3×C4—24no—
291C3×C4—41no—
292Dic3—24no—
293Dic3—41no—
294Dic3#12014yescomplemented
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23—21no—
305C23—21no—
306C23—34no—
307C23—32no—
308C23—34no—
309C23—616no—
310C23—62no—
311C23—61no—
312C23—61no—
313C23—62no—
314C23—62no—
315C23—62no—
316C23—62no—
317C23—62no—
318C23—62no—
319C23—61no—
320C23—62no—
321C23—62no—
322C23—61no—
323C23—61no—
324C23—128no—
325C23—128no—
326C23—128no—
327C23—128no—
328C23—128no—
329C23—128no—
330C23D3×D414yes—
331C23C23×D311yescharacteristic
332D4—28no—
333D4—48no—
334D4—68no—
335D4—616no—
336D4—128no—
337D4—124no—
338D4—124no—
339D4—128no—
340D4—128no—
341D4—124no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—34no—
348C2×C4—32no—
349C2×C4—34no—
350C2×C4—68no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—61no—
355C2×C4D3×D412yes—
356C2×C4D3×D414yescomplemented
357C2×C3—44no—
358C2×C3—81no—
359C2×C3C2×#12011yescharacteristic
360C2×C3C2×#12711yescharacteristic
361C2×C3#53511yescharacteristic
362C2×C3D4212yes—
363C2×C3D4212yes—
364D3—24no—
365D3—44no—
366D3—81no—
367C22—24no—
368C22—24no—
369C22—24no—
370C22—34no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—42no—
378C22—41no—
379C22—42no—
380C22—62no—
381C22—64no—
382C22—62no—
383C22—64no—
384C22—64no—
385C22—64no—
386C22—64no—
387C22—1232no—
388C22—122no—
389C22—122no—
390C22—122no—
391C22—122no—
392C22—121no—
393C22—121no—
394C22—122no—
395C22—121no—
396C22—122no—
397C22C2×D3×D411yescharacteristic
398C22C2×D3×D412yes—
399C22C2×D3×D412yes—
400C22#100411yescharacteristic
401C22#100911yescharacteristic
402C4—22no—
403C4—24no—
404C4—34no—
405C4—41no—
406C4—64no—
407C4—121no—
408C3#243311yescharacteristic, complemented, Sylow
409C2—44no—
410C2—64no—
411C2—81no—
412C2—124no—
413C2—241no—
414C2D3×#12011yescharacteristic
415C2#559012yes—
416C2#560711yescharacteristic
417C2#560912yes—
418C2D3×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2433
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D3×D4C2semidirect
2. C2×#940C2semidirect
3. C2×#937C2semidirect
4. C2×#884C2semidirect
5. C2×#882C2semidirect
6. C3×#392C2semidirect
7. #5223C2semidirect
8. #4960C2semidirect
9. #4957C2semidirect
10. #4671C2semidirect
11. #4666C2semidirect
15. C22×C3×D4C22semidirect
18. C22×#67C22semidirect
19. C22×#67C22semidirect
20. C22×#67C22semidirect
25. C22×D12C22semidirect
27. C2×C3×C22⋊C4C22semidirect
28. C2×C3×C22⋊C4C22semidirect
29. C2×#217C22semidirect
33. C2×(C2×D3)⋊C4C22semidirect
34. C2×(C2×D3)⋊C4C22semidirect
35. C2×(C2×D3)⋊C4C22semidirect
36. C2×Dic3⋊C4C22semidirect
37. C2×C4×Dic3C22semidirect
40. C3×#95C22semidirect
41. C3⋊#95C22semidirect
50. #392D3semidirect
85. C2×#67D4semidirect
98. C2×C4×D3D4semidirect
105. (C2×D3)⋊C4D4semidirect
120. C22×D4C2×D3semidirect
132. C2×C22⋊C4C2×D3semidirect
133. C2×C22⋊C4C2×D3semidirect
136. #95C2×D3semidirect
182. C2×C3×C4C2×D4semidirect
194. C2×Dic3C2×D4semidirect
288. C2×D3#127semidirect
294. Dic3#120semidirect
356. C2×C4D3×D4semidirect
408. C3#2433semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×#9402
C2×#9372
C2×#8842
C2×#8822
C3×#3922
#52232
#49602
#49572
#46712
#46662
#24333