Finite Group Databasebeta

← #80006

Subgroups — #80006

Order 384.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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×#938C211yesmaximal, characteristic, complemented
2D3×#121C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic, complemented
5#5157C212yesmaximal, complemented
6#5156C212yesmaximal, complemented
7#5033C211yesmaximal, characteristic, complemented
8#4775C212yesmaximal, complemented
9#4774C212yesmaximal, complemented
10#4736C211yesmaximal, characteristic, complemented
11D3×C22⋊C8C211yesmaximal, characteristic, complemented
12#3186—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic, complemented
14C22×#67—21no—
15C2×D3×D4—21no—
16C2×D3×D4C2211yescharacteristic, complemented
17C22×C4×D3C2211yescharacteristic
18C2×C3×D8C2212yescomplemented
19C3×#121C2211yescharacteristic
20C2×#217—21no—
21#939—21no—
22#938—21no—
23#938—21no—
24#938C2212yes—
25#938C2211yescharacteristic, complemented
26#938C2212yescomplemented
27C2×#213C2212yescomplemented
28#930C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic, complemented
30C2×D3×C8C2212yescomplemented
31#895C2211yescharacteristic, complemented
32D3×C4⋊C4C2211yescharacteristic, complemented
33#884—21no—
34D3×C22⋊C4—21no—
35C3×#102C2212yescomplemented
36C3×C22⋊C8C2211yescharacteristic, complemented
37#832C2212yescomplemented
38#832C2212yescomplemented
39#830C2211yescharacteristic, complemented
40#821C2212yescomplemented
41(C2×D3)⋊C8C2212yescomplemented
42#818C2212yescomplemented
43#807C2212yescomplemented
44C2×#121—31no—
45C2×#121—31no—
46#480—31no—
47#480—32no—
48#480—31no—
49#466—32no—
50#466—32no—
51#447—31no—
52#447—32no—
53#447D311yescharacteristic, complemented
54C2×C22⋊C8—31no—
55C24×C3—21no—
56C23×D3—21no—
57C23×D3D411yescharacteristic
58C2×C3×D4—21no—
59C2×C3×D4—21no—
60C2×C3×D4—42no—
61C2×C3×D4C2311yescharacteristic
62C2×C3×D4C2312yes—
63C22×C3×C4C2311yescharacteristic
64C2×#67—21no—
65C2×#67—21no—
66C2×#67—22no—
67C2×#67—22no—
68C2×#67—21no—
69C2×#67—44no—
70C22×Dic3—21no—
71C22×Dic3D411yescharacteristic
72D3×D4—42no—
73D3×D4—44no—
74C2×D12—21no—
75C2×D12C2311yescharacteristic
76C2×C4×D3—21no—
77C2×C4×D3—42no—
78C2×C4×D3D411yescharacteristic
79C2×C4×D3D411yescharacteristic
80C2×C4×D3C2311yescharacteristic
81C2×C4×D3C2311yescharacteristic
82C3×D8—42no—
83C2×C3×C8C2312yes—
84C3×C4⋊C4C2311yescharacteristic
85C3×C22⋊C4—21no—
86#217—21no—
87#217—22no—
88#217—22no—
89#213—42no—
90(C2×D3)⋊C4—21no—
91(C2×D3)⋊C4—21no—
92C4⋊Dic3—21no—
93C4⋊Dic3C2312yes—
94C4⋊Dic3C2311yescharacteristic
95Dic3⋊C4—21no—
96C2×C3⋊C8C2312yes—
97D3×C8—42no—
98C22×D4—31no—
99C22×D4—61no—
100C22×D4—61no—
101C22×D4C2×D311yescharacteristic, complemented
102C23×C4—31no—
103C2×D8—32no—
104C2×D8C2×D312yescomplemented
105C22×C8—32no—
106#121—32no—
107#121—31no—
108#121—32no—
109#121—31no—
110#121—31no—
111#121—61no—
112#121—61no—
113#121—61no—
114#121—61no—
115#121C2×D311yescharacteristic
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C22⋊C4—61no—
119C2×C22⋊C4—61no—
120#107—32no—
121#107—32no—
122#102—32no—
123#102—32no—
124#102—32no—
125#102C2×D312yescomplemented
126C22⋊C8—31no—
127C22⋊C8—32no—
128C22⋊C8C2×D311yescharacteristic, complemented
129C23×C3—22no—
130C23×C3—21no—
131C23×C3—42no—
132C23×C3—42no—
133C23×C3—42no—
134C23×C3C2×D411yescharacteristic
135C22×D3—21no—
136C22×D3—41no—
137C22×D3—41no—
138C22×D3—42no—
139C22×D3—42no—
140C22×D3—42no—
141C22×D3D814yescomplemented
142C22×D3C2×D411yescharacteristic
143C22×D3C2×D411yescharacteristic
144C3×D4—22no—
145C3×D4—22no—
146C3×D4—22no—
147C3×D4—42no—
148C3×D4—42no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4C2×D411yescharacteristic
152C2×C3×C4C2411yescharacteristic
153#67—42no—
154#67—42no—
155#67—42no—
156#67—44no—
157#67—44no—
158C2×Dic3—21no—
159C2×Dic3—22no—
160C2×Dic3—41no—
161C2×Dic3—42no—
162C2×Dic3C2×D411yescharacteristic
163C2×Dic3C2×D411yescharacteristic
164D12—22no—
165D12—42no—
166C4×D3—22no—
167C4×D3—22no—
168C4×D3—41no—
169C4×D3—41no—
170C4×D3—42no—
171C3×C8—22no—
172C3⋊C8—22no—
173C24—21no—
174C24—31no—
175C24—61no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—31no—
179C2×D4—42no—
180C2×D4—61no—
181C2×D4—61no—
182C2×D4—61no—
183C2×D4—62no—
184C2×D4—62no—
185C2×D4—61no—
186C2×D4—122no—
187C2×D4—124no—
188C2×D4—124no—
189C2×D4C22×D311yescharacteristic
190C2×D4C22×D312yes—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—122no—
199C22×C4C22×D311yescharacteristic
200D8—42no—
201D8—122no—
202C2×C8—32no—
203C2×C8—122no—
204C2×C8C22×D312yes—
205C4⋊C4—31no—
206C4⋊C4—32no—
207C4⋊C4—61no—
208C4⋊C4—61no—
209C4⋊C4C22×D311yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—61no—
212C22⋊C4—61no—
213C22⋊C4—62no—
214C22⋊C4—62no—
215C22⋊C4—61no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—41no—
219C22×C3—42no—
220C22×C3—42no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—42no—
224C22×C3—84no—
225C22×C3C2×D812yes—
226C22×C3C22×D411yescharacteristic
227C2×D3—21no—
228C2×D3—21no—
229C2×D3—21no—
230C2×D3—21no—
231C2×D3—44no—
232C2×D3—41no—
233C2×D3—41no—
234C2×D3—41no—
235C2×D3—84no—
236C2×D3C2×D812yes—
237C2×D3C2×D812yes—
238C3×C4—21no—
239C3×C4—41no—
240C3×C4H(5,𝔽2)12yes—
241Dic3—21no—
242Dic3—21no—
243Dic3—41no—
244Dic3—42no—
245C23—21no—
246C23—22no—
247C23—31no—
248C23—31no—
249C23—34no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—61no—
254C23—121no—
255C23—121no—
256C23—122no—
257C23—122no—
258C23—122no—
259C23D3×D411yescharacteristic
260D4—22no—
261D4—22no—
262D4—22no—
263D4—42no—
264D4—42no—
265D4—62no—
266D4—122no—
267D4—122no—
268D4—124no—
269D4—122no—
270D4—124no—
271D4—122no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—61no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—122no—
283C2×C4—121no—
284C2×C4—122no—
285C2×C4D3×D411yescharacteristic
286C2×C4C23×D311yescharacteristic
287C8—22no—
288C8—62no—
289C2×C3—22no—
290C2×C3—81no—
291C2×C3—82no—
292C2×C3#53411yescharacteristic
293C2×C3C22×D811yescharacteristic
294C2×C3#57511yescharacteristic
295D3—22no—
296D3—22no—
297D3—81no—
298C22—21no—
299C22—21no—
300C22—32no—
301C22—32no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—42no—
307C22—41no—
308C22—61no—
309C22—61no—
310C22—61no—
311C22—61no—
312C22—84no—
313C22—124no—
314C22—121no—
315C22—121no—
316C22—121no—
317C22—244no—
318C22D3×D812yes—
319C22C2×D3×D411yescharacteristic
320C4—21no—
321C4—41no—
322C4—61no—
323C4—61no—
324C4—121no—
325C4—122no—
326C4#100412yes—
327C3#318611yescharacteristic, complemented, Sylow
328C2—22no—
329C2—62no—
330C2—62no—
331C2—82no—
332C2—81no—
333C2—241no—
334C2#558911yescharacteristic
335C2C2×D3×D811yescharacteristic
336C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3186
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#938C2semidirect
2. D3×#121C2semidirect
3. C3×#447C2semidirect
4. #5217C2semidirect
5. #5157C2semidirect
6. #5156C2semidirect
7. #5033C2semidirect
8. #4775C2semidirect
9. #4774C2semidirect
10. #4736C2semidirect
11. D3×C22⋊C8C2semidirect
13. C22×C3×D4C22semidirect
16. C2×D3×D4C22semidirect
18. C2×C3×D8C22semidirect
25. #938C22semidirect
26. #938C22semidirect
27. C2×#213C22semidirect
29. C2×C4⋊Dic3C22semidirect
30. C2×D3×C8C22semidirect
31. #895C22semidirect
32. D3×C4⋊C4C22semidirect
35. C3×#102C22semidirect
36. C3×C22⋊C8C22semidirect
37. #832C22semidirect
38. #832C22semidirect
39. #830C22semidirect
40. #821C22semidirect
41. (C2×D3)⋊C8C22semidirect
42. #818C22semidirect
43. #807C22semidirect
53. #447D3semidirect
101. C22×D4C2×D3semidirect
104. C2×D8C2×D3semidirect
125. #102C2×D3semidirect
128. C22⋊C8C2×D3semidirect
141. C22×D3D8semidirect
327. C3#3186semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
D3×#1212
C3×#4472
#52172
#51572
#51562
#50332
#47752
#47742
#47362
D3×C22⋊C82
#31863