Finite Group Databasebeta

← #78107

Subgroups — #78107

Order 384.

900 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 159 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×#895C211yesmaximal, characteristic, complemented
2C2×#894C212yesmaximal, complemented
3C2×#882C212yesmaximal, complemented
4C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#399C211yesmaximal, characteristic, complemented
6#4993C211yesmaximal, characteristic, complemented
7#4988C212yesmaximal, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4957C212yesmaximal, complemented
10#4671C211yesmaximal, characteristic, complemented
11#4666C211yesmaximal, characteristic, complemented
12#2539—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C22×#67C2212yes—
15C22×D12C2212yes—
16C22×D12C2212yes—
17C22×D12C2211yescharacteristic, complemented
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×C4⋊C4C2212yescomplemented
21C2×C3×C4⋊C4C2211yescharacteristic, complemented
22C2×C3×C22⋊C4C2212yes—
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yescomplemented
30C2×(C2×D3)⋊C4C2212yescomplemented
31C2×(C2×D3)⋊C4C2212yescomplemented
32C2×Dic3⋊C4C2212yescomplemented
33C2×C4×Dic3C2212yescomplemented
34#895—24no—
35#894—28no—
36#882—28no—
37D3×C22⋊C4—24no—
38C3×#95C2211yescharacteristic, complemented
39C3⋊#95C2211yescharacteristic, complemented
40C2×#123—32no—
41C2×#121—31no—
42C2×#120—32no—
43C22×C22⋊C4—31no—
44#399—31no—
45#399D311yescharacteristic, complemented
46#392—31no—
47#392—31no—
48#390—32no—
49#390—32no—
50C4×C22⋊C4—31no—
51C24×C3C2311yescharacteristic
52C23×D3—22no—
53C23×D3—22no—
54C23×D3—22no—
55C23×D3—24no—
56C23×D3—42no—
57C23×D3C2312yes—
58C23×D3C2311yescharacteristic
59C23×D3C2311yescharacteristic
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2312yes—
64C2×#67—24no—
65C2×#67—24no—
66C2×#67—42no—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C22×Dic3C2311yescharacteristic
70C2×D12—24no—
71C2×D12—24no—
72C2×D12—24no—
73C2×D12—24no—
74C2×D12—24no—
75C2×D12—22no—
76C2×D12—42no—
77C2×D12—42no—
78C2×C4×D3—22no—
79C2×C4×D3—24no—
80C2×C4×D3—42no—
81C2×C4×D3D414yescomplemented
82C3×C4⋊C4—24no—
83C3×C4⋊C4—22no—
84C3×C22⋊C4—22no—
85C3×C22⋊C4—24no—
86#217—22no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—24no—
89(C2×D3)⋊C4—22no—
90(C2×D3)⋊C4—24no—
91(C2×D3)⋊C4—24no—
92(C2×D3)⋊C4—24no—
93(C2×D3)⋊C4D418yescomplemented
94Dic3⋊C4—24no—
95C4×Dic3—42no—
96C25—31no—
97C22×D4—32no—
98C22×D4—32no—
99C22×D4—32no—
100C22×D4—31no—
101C23×C4—31no—
102C23×C4—31no—
103#123—68no—
104#121—64no—
105#120—68no—
106C2×C4⋊C4—32no—
107C2×C4⋊C4C2×D311yescharacteristic, complemented
108C2×C4⋊C4C2×D312yescomplemented
109C2×C22⋊C4—32no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—32no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4—32no—
117C2×C22⋊C4—64no—
118C2×C22⋊C4C2×D311yescharacteristic
119C2×C22⋊C4C2×D312yes—
120C2×C42—32no—
121#95—31no—
122#95C2×D311yescharacteristic, complemented
123C23×C3—21no—
124C23×C3—22no—
125C23×C3—21no—
126C23×C3—22no—
127C23×C3#4112yes—
128C23×C3C2411yescharacteristic
129C22×D3—21no—
130C22×D3—21no—
131C22×D3—22no—
132C22×D3—28no—
133C22×D3—21no—
134C22×D3—22no—
135C22×D3—21no—
136C22×D3—22no—
137C22×D3—22no—
138C22×D3—22no—
139C22×D3—22no—
140C22×D3—22no—
141C22×D3—22no—
142C22×D3—22no—
143C22×D3—22no—
144C22×D3—48no—
145C22×D3—48no—
146C22×D3—44no—
147C22×D3—44no—
148C22×D3C2×D414yes—
149C22×D3C2×D412yes—
150C22×D3#4112yes—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—21no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4C2×D414yes—
160C2×C3×C4C2×D412yes—
161#67—48no—
162C2×Dic3—22no—
163C2×Dic3—22no—
164C2×Dic3—22no—
165C2×Dic3—22no—
166C2×Dic3—21no—
167C2×Dic3C2×D414yes—
168C2×Dic3C2×D412yes—
169D12—48no—
170D12—44no—
171D12—48no—
172C4×D3—44no—
173C4×D3—44no—
174C24—31no—
175C24—31no—
176C24—32no—
177C24—62no—
178C24—62no—
179C24—64no—
180C24—62no—
181C24—122no—
182C24C22×D311yescharacteristic
183C2×D4—64no—
184C2×D4—64no—
185C2×D4—64no—
186C2×D4—64no—
187C2×D4—64no—
188C2×D4—64no—
189C2×D4—64no—
190C2×D4—62no—
191C2×D4—122no—
192C2×D4—122no—
193C2×D4—122no—
194C22×C4—32no—
195C22×C4—31no—
196C22×C4—34no—
197C22×C4—31no—
198C22×C4—62no—
199C22×C4—64no—
200C22×C4—122no—
201C22×C4C22×D311yescharacteristic
202C22×C4C22×D312yes—
203C22×C4C22×D311yescharacteristic
204C22×C4C22×D312yes—
205C4⋊C4—22no—
206C4⋊C4—24no—
207C4⋊C4—64no—
208C22⋊C4—22no—
209C22⋊C4—24no—
210C22⋊C4—38no—
211C22⋊C4—62no—
212C22⋊C4—64no—
213C22⋊C4—64no—
214C22⋊C4—64no—
215C22⋊C4—62no—
216C22⋊C4—62no—
217C22⋊C4—64no—
218C42—122no—
219C22×C3—22no—
220C22×C3—24no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—42no—
224C22×C3C22×D411yescharacteristic
225C22×C3C22×D412yes—
226C22×C3C2×#4111yescharacteristic
227C22×C3C2×#4111yescharacteristic
228C22×C3H(5,𝔽2)12yes—
229C2×D3—22no—
230C2×D3—22no—
231C2×D3—24no—
232C2×D3—24no—
233C2×D3—22no—
234C2×D3—24no—
235C2×D3—416no—
236C2×D3—41no—
237C2×D3—41no—
238C2×D3—42no—
239C2×D3—42no—
240C2×D3—42no—
241C2×D3—42no—
242C2×D3—42no—
243C2×D3—42no—
244C2×D3—42no—
245C2×D3—42no—
246C2×D3#12714yes—
247C3×C4—22no—
248C3×C4—24no—
249C3×C4—41no—
250C3×C4—42no—
251Dic3—22no—
252Dic3—24no—
253Dic3—41no—
254C23—21no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—34no—
259C23—32no—
260C23—32no—
261C23—68no—
262C23—61no—
263C23—61no—
264C23—62no—
265C23—61no—
266C23—62no—
267C23—61no—
268C23—62no—
269C23—62no—
270C23—62no—
271C23—62no—
272C23—62no—
273C23—62no—
274C23—62no—
275C23—62no—
276C23—128no—
277C23—124no—
278C23—128no—
279C23—124no—
280C23#23912yes—
281C23C23×D311yescharacteristic
282D4—128no—
283D4—124no—
284D4—128no—
285D4—128no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—34no—
295C2×C4—32no—
296C2×C4—62no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—61no—
300C2×C4—62no—
301C2×C4—124no—
302C2×C4—124no—
303C2×C4D3×D414yes—
304C2×C4D3×D412yes—
305C2×C3—42no—
306C2×C3#52511yescharacteristic
307C2×C3C2×#12711yescharacteristic
308C2×C3#53411yescharacteristic
309C2×C3#54612yes—
310C2×C3#54612yes—
311D3—24no—
312D3—42no—
313D3—82no—
314C22—22no—
315C22—24no—
316C22—34no—
317C22—41no—
318C22—42no—
319C22—41no—
320C22—62no—
321C22—64no—
322C22—62no—
323C22—64no—
324C22—64no—
325C22—62no—
326C22—1216no—
327C22—121no—
328C22—121no—
329C22—122no—
330C22—122no—
331C22—122no—
332C22—122no—
333C22—122no—
334C22—122no—
335C22—122no—
336C22—122no—
337C22C2×D3×D411yescharacteristic
338C22C2×D3×D412yes—
339C22C2×#23911yescharacteristic
340C22D3×#4111yescharacteristic
341C22#100912yes—
342C4—22no—
343C4—24no—
344C4—41no—
345C4—42no—
346C4—62no—
347C4—64no—
348C4—121no—
349C3#253911yescharacteristic, complemented, Sylow
350C2—42no—
351C2—64no—
352C2—122no—
353C2—242no—
354C2#562711yescharacteristic
355C2#563012yes—
356C2#565311yescharacteristic
357C2#565412yes—
358C2D3×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
hypercenter#399D3
hypocenterC3#2539
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#895C2semidirect
2. C2×#894C2semidirect
3. C2×#882C2semidirect
4. C2×D3×C22⋊C4C2semidirect
5. C3×#399C2semidirect
6. #4993C2semidirect
7. #4988C2semidirect
8. #4960C2semidirect
9. #4957C2semidirect
10. #4671C2semidirect
11. #4666C2semidirect
17. C22×D12C22semidirect
20. C2×C3×C4⋊C4C22semidirect
21. C2×C3×C4⋊C4C22semidirect
29. C2×(C2×D3)⋊C4C22semidirect
30. C2×(C2×D3)⋊C4C22semidirect
31. C2×(C2×D3)⋊C4C22semidirect
32. C2×Dic3⋊C4C22semidirect
33. C2×C4×Dic3C22semidirect
38. C3×#95C22semidirect
39. C3⋊#95C22semidirect
45. #399D3semidirect
81. C2×C4×D3D4semidirect
93. (C2×D3)⋊C4D4semidirect
107. C2×C4⋊C4C2×D3semidirect
108. C2×C4⋊C4C2×D3semidirect
122. #95C2×D3semidirect
349. C3#2539semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×#8942
C2×#8822
C2×D3×C22⋊C42
C3×#3992
#49932
#49882
#49602
#49572
#46712
#46662
#25393