Finite Group Databasebeta

← #78036

Subgroups — #78036

Order 384.

760 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 157 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×#896C212yesmaximal, complemented
2C2×#893C211yesmaximal, characteristic, complemented
3C2×#885C211yesmaximal, characteristic, complemented
4C2×#884C212yesmaximal, complemented
5C3×#399C211yesmaximal, characteristic, complemented
6#4993C211yesmaximal, characteristic, complemented
7#4988C212yesmaximal, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4957C212yesmaximal, complemented
10#4671C211yesmaximal, characteristic, complemented
11#4649C211yesmaximal, characteristic, complemented
12#2540—31nomaximal, Sylow
13C22×#67C2211yescharacteristic
14C22×#67C2212yes—
15C22×D12C2211yescharacteristic
16C22×D12C2212yes—
17C22×C4×D3C2212yes—
18C22×Dic6C2211yescharacteristic
19C2×C3×C4⋊C4C2212yes—
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2212yes—
33C2×C4×Dic3C2211yescharacteristic
34#896—28no—
35#893—24no—
36#885—24no—
37#884—28no—
38C3×#95C2211yescharacteristic
39C3⋊#95C2211yescharacteristic
40C2×#124—31no—
41C2×C22⋊Q8—32no—
42C2×#121—32no—
43C2×C4×D4—31no—
44#399—31no—
45#399D311yescharacteristic, complemented
46#392—31no—
47#392—31no—
48#390—32no—
49#390—32no—
50C4×C4⋊C4—31no—
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C23×D3C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2312yes—
58C2×#67—24no—
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—22no—
62C2×#67—42no—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C2×D12—24no—
68C2×D12—22no—
69C2×D12—24no—
70C2×D12—24no—
71C2×D12—42no—
72C2×C4×D3—24no—
73C2×C4×D3—24no—
74C2×C4×D3—24no—
75C2×Dic6—22no—
76C2×Dic6—24no—
77C3×C4⋊C4—22no—
78C3×C4⋊C4—24no—
79C3×C22⋊C4—24no—
80C3×C22⋊C4—22no—
81#217—22no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—24no—
87(C2×D3)⋊C4—24no—
88Dic3⋊C4—24no—
89Dic3⋊C4D418yescomplemented
90C4×Dic3—42no—
91C4×Dic3D414yescomplemented
92C22×Q8—31no—
93C22×D4—31no—
94C22×D4—32no—
95C22×D4—31no—
96C22×D4—32no—
97C23×C4—32no—
98#124—64no—
99C22⋊Q8—68no—
100#121—68no—
101C4×D4—64no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4C2×D311yescharacteristic
105C2×C4⋊C4C2×D312yes—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4C2×D312yes—
114C2×C22⋊C4C2×D311yescharacteristic
115C2×C42—31no—
116C2×C42—32no—
117#95—31no—
118#95C2×D311yescharacteristic
119C23×C3—21no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3C2411yescharacteristic
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—21no—
132C22×D3—22no—
133C22×D3—22no—
134C22×D3—21no—
135C22×D3—21no—
136C22×D3#4114yes—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—21no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4C2×D412yes—
146C2×C3×C4C2×D414yes—
147#67—48no—
148#67—44no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3C2×D412yes—
156C2×Dic3C2×D414yes—
157C2×Dic3C2×D412yes—
158C2×Dic3C2×D414yes—
159D12—28no—
160D12—48no—
161C4×D3—48no—
162Dic6—44no—
163C24—32no—
164C24—31no—
165C24C22×D311yescharacteristic
166C2×Q8—64no—
167C2×Q8—62no—
168C2×D4—64no—
169C2×D4—64no—
170C2×D4—64no—
171C2×D4—64no—
172C2×D4—62no—
173C2×D4—64no—
174C2×D4—62no—
175C2×D4—64no—
176C2×D4—122no—
177C2×D4—122no—
178C22×C4—31no—
179C22×C4—32no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—64no—
183C22×C4—64no—
184C22×C4—64no—
185C22×C4C22×D311yescharacteristic
186C22×C4C22×D312yes—
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D312yes—
189C4⋊C4—22no—
190C4⋊C4—24no—
191C4⋊C4—38no—
192C4⋊C4—64no—
193C22⋊C4—24no—
194C22⋊C4—22no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—62no—
199C22⋊C4—62no—
200C22⋊C4—64no—
201C22⋊C4—62no—
202C42—34no—
203C42—122no—
204C22×C3—41no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—42no—
208C22×C3—42no—
209C22×C3C22×D411yescharacteristic
210C22×C3C22×D412yes—
211C22×C3C2×#4112yes—
212C22×C3H(5,𝔽2)11yescharacteristic
213C22×C32−1+411yescharacteristic
214C2×D3—24no—
215C2×D3—24no—
216C2×D3—24no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—42no—
221C2×D3—41no—
222C2×D3—41no—
223C2×D3—41no—
224C2×D3—42no—
225C2×D3—42no—
226C3×C4—22no—
227C3×C4—24no—
228C3×C4—41no—
229C3×C4—42no—
230Dic3—22no—
231Dic3—24no—
232Dic3—41no—
233Dic3#12714yes—
234C23—21no—
235C23—22no—
236C23—21no—
237C23—22no—
238C23—21no—
239C23—34no—
240C23—62no—
241C23—62no—
242C23—62no—
243C23—62no—
244C23—62no—
245C23—62no—
246C23—61no—
247C23—62no—
248C23—61no—
249C23—62no—
250C23—61no—
251C23C23×D311yescharacteristic
252Q8—124no—
253D4—68no—
254D4—128no—
255D4—124no—
256D4—128no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—32no—
266C2×C4—34no—
267C2×C4—32no—
268C2×C4—34no—
269C2×C4—61no—
270C2×C4—62no—
271C2×C4—61no—
272C2×C4—62no—
273C2×C4—61no—
274C2×C4—62no—
275C2×C4—128no—
276C2×C4D3×D412yes—
277C2×C4D3×D414yes—
278C2×C3—81no—
279C2×C3#52511yescharacteristic
280C2×C3C2×#12711yescharacteristic
281C2×C3#53711yescharacteristic
282C2×C3#54612yes—
283C2×C3#54812yes—
284D3—44no—
285D3—81no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—64no—
292C22—64no—
293C22—64no—
294C22—122no—
295C22—122no—
296C22—122no—
297C22—122no—
298C22—122no—
299C22—122no—
300C22—121no—
301C22—121no—
302C22—121no—
303C22C2×D3×D412yes—
304C22C2×D3×D411yescharacteristic
305C22#100711yescharacteristic
306C22D3×#4112yes—
307C22#100911yescharacteristic
308C4—24no—
309C4—22no—
310C4—34no—
311C4—42no—
312C4—41no—
313C4—64no—
314C4—62no—
315C4—121no—
316C3#254011yescharacteristic, complemented, Sylow
317C2—81no—
318C2—124no—
319C2—241no—
320C2#562911yescharacteristic
321C2#563312yes—
322C2#565412yes—
323C2#565611yescharacteristic
324C2D3×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
hypercenter#399D3
hypocenterC3#2540
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
2. C2×#893C2semidirect
3. C2×#885C2semidirect
4. C2×#884C2semidirect
5. C3×#399C2semidirect
6. #4993C2semidirect
7. #4988C2semidirect
8. #4960C2semidirect
9. #4957C2semidirect
10. #4671C2semidirect
11. #4649C2semidirect
45. #399D3semidirect
89. Dic3⋊C4D4semidirect
91. C4×Dic3D4semidirect
316. C3#2540semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#8932
C2×#8852
C2×#8842
C3×#3992
#49932
#49882
#49602
#49572
#46712
#46492
#25403