Finite Group Databasebeta

← #82106

Subgroups — #82106

Order 384.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 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×C3×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#953C211yesmaximal, characteristic
3C2×#929C211yesmaximal, characteristic, complemented
4C2×#926C211yesmaximal, characteristic, complemented
5#5247C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#5210C211yesmaximal, characteristic, complemented
8#4961C212yesmaximal, complemented
9#4957C212yesmaximal, complemented
10#4948C212yesmaximal, complemented
11#4945C212yesmaximal, complemented
12#2411—31nomaximal, Sylow
13C25×C3C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C22×#67C2212yes—
18C2×C3×C22⋊C4—24no—
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C22⋊C4C2212yes—
21#953—24no—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2212yes—
27C2×#217C2211yescharacteristic
28#929—24no—
29C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
30C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
31C2×(C2×D3)⋊C4C2212yescomplemented
32#926—24no—
33C2×C4⋊Dic3C2212yescomplemented
34C2×Dic3⋊C4C2211yescharacteristic, complemented
35C2×Dic3⋊C4C2211yescharacteristic, complemented
36C2×Dic3⋊C4C2212yescomplemented
37C2×C4×Dic3C2212yescomplemented
38C3⋊#95C2212yescomplemented
39C3⋊#95C2212yescomplemented
40C2×#123—31no—
41C2×#120—31no—
42C2×#117—31no—
43C22×C22⋊C4D311yescharacteristic, complemented
44#396—32no—
45#394—32no—
46#390—32no—
47#388—32no—
48#386—31no—
49#385—31no—
50#379—31no—
51C24×C3—22no—
52C24×C3—24no—
53C24×C3—22no—
54C24×C3—22no—
55C24×C3—42no—
56C24×C3C2312yes—
57C24×C3C2311yescharacteristic
58C23×D3C2311yescharacteristic
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4—22no—
62C22×C3×C4—42no—
63C22×C3×C4D414yescomplemented
64C22×C3×C4C2312yes—
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2311yescharacteristic
67C2×#67—22no—
68C2×#67—22no—
69C2×#67—24no—
70C2×#67—22no—
71C2×#67—24no—
72C2×#67—42no—
73C22×Dic3C2311yescharacteristic
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2312yes—
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2312yes—
78C3×C22⋊C4—24no—
79C3×C22⋊C4—24no—
80#217—22no—
81#217—22no—
82#217—24no—
83#217—24no—
84#217—42no—
85#217D414yes—
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—42no—
89C4⋊Dic3—24no—
90Dic3⋊C4—22no—
91Dic3⋊C4—22no—
92Dic3⋊C4—42no—
93C4×Dic3—24no—
94C25C2×D311yescharacteristic
95C22×D4—31no—
96C22×D4—32no—
97C23×C4C2×D311yescharacteristic
98C23×C4C2×D311yescharacteristic
99#123—64no—
100#120—64no—
101#117—64no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—31no—
105C2×C4⋊C4—32no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—31no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4—31no—
114C2×C22⋊C4—32no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4C2×D312yes—
117C2×C22⋊C4C2×D312yes—
118C2×C42—32no—
119#95—32no—
120#95—32no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—28no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—48no—
131C23×C3—48no—
132C23×C3—44no—
133C23×C3—44no—
134C23×C3C2×D412yes—
135C23×C3C2×D412yes—
136C23×C3#4114yes—
137C23×C3#4112yes—
138C23×C3C2411yescharacteristic
139C22×D3—21no—
140C22×D3—21no—
141C22×D3—22no—
142C22×D3—21no—
143C22×D3—22no—
144C2×C3×C4—28no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—44no—
153C2×C3×C4C2×D412yes—
154C2×C3×C4C2×D412yes—
155#67—48no—
156#67—44no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—21no—
161C2×Dic3—21no—
162C2×Dic3—21no—
163C2×Dic3—21no—
164C2×Dic3—22no—
165C2×Dic3—22no—
166C2×Dic3—21no—
167C2×Dic3—22no—
168C2×Dic3—22no—
169C2×Dic3C2×D414yes—
170C24—24no—
171C24—22no—
172C24—22no—
173C24—22no—
174C24—31no—
175C24—42no—
176C24C22×D312yes—
177C24C22×D311yescharacteristic
178C2×D4—64no—
179C2×D4—64no—
180C2×D4—62no—
181C2×D4—62no—
182C2×D4—62no—
183C2×D4—122no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—32no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—32no—
191C22×C4—31no—
192C22×C4—42no—
193C22×C4#6714yescomplemented
194C22×C4C22×D312yes—
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D311yescharacteristic
197C4⋊C4—64no—
198C4⋊C4—62no—
199C4⋊C4—62no—
200C4⋊C4—122no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—34no—
204C22⋊C4—64no—
205C22⋊C4—64no—
206C22⋊C4—62no—
207C22⋊C4—62no—
208C22⋊C4—62no—
209C22⋊C4—62no—
210C22⋊C4—122no—
211C22⋊C4—122no—
212C42—64no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—24no—
216C22×C3—24no—
217C22×C3—24no—
218C22×C3—24no—
219C22×C3—24no—
220C22×C3—416no—
221C22×C3—42no—
222C22×C3—42no—
223C22×C3—42no—
224C22×C3—42no—
225C22×C3#12114yes—
226C22×C3C22×D411yescharacteristic
227C22×C3C22×D411yescharacteristic
228C22×C3C2×#4112yes—
229C22×C3C2×#4111yescharacteristic
230C22×C3H(5,𝔽2)12yes—
231C2×D3—41no—
232C2×D3—41no—
233C2×D3—41no—
234C2×D3—42no—
235C2×D3—42no—
236C3×C4—22no—
237C3×C4—22no—
238C3×C4—42no—
239Dic3—24no—
240Dic3—41no—
241Dic3—41no—
242Dic3—41no—
243Dic3—42no—
244C23—22no—
245C23—22no—
246C23—28no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—48no—
254C23—48no—
255C23—44no—
256C23—44no—
257C23—62no—
258C23—61no—
259C23—61no—
260C23—62no—
261C23—61no—
262C23#23514yes—
263C23D3×D412yes—
264C23#23712yes—
265C23C2×#6712yes—
266C23C23×D311yescharacteristic
267D4—128no—
268D4—124no—
269C2×C4—28no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—34no—
278C2×C4—44no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4C2×#6712yes—
292C2×C4C2×#6712yes—
293C2×C3—24no—
294C2×C3—44no—
295C2×C3C2×#12111yescharacteristic
296C2×C3#52511yescharacteristic
297C2×C3#53511yescharacteristic
298C2×C3#54612yes—
299C2×C3#55112yes—
300D3—81no—
301C22—24no—
302C22—24no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—416no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—122no—
314C22—122no—
315C22—121no—
316C22—121no—
317C22—121no—
318C22#93914yes—
319C22C2×#23512yes—
320C22C2×D3×D411yescharacteristic
321C22C2×#23711yescharacteristic
322C22#100412yes—
323C22C22×#6711yescharacteristic
324C4—22no—
325C4—22no—
326C4—42no—
327C4—64no—
328C4—122no—
329C4—121no—
330C4—121no—
331C4—121no—
332C3#241111yescharacteristic, complemented, Sylow
333C2—24no—
334C2—44no—
335C2—241no—
336C2#549012yes—
337C2#549512yes—
338C2#579111yescharacteristic
339C2C2×#93911yescharacteristic
340C2#580511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C22⋊C4C2
prime radicalC2×#953C2
hypercenterC22×C22⋊C4D3
hypocenterC3#2411
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊C4C2semidirect
3. C2×#929C2semidirect
4. C2×#926C2semidirect
5. #5247C2semidirect
6. #5213C2semidirect
7. #5210C2semidirect
8. #4961C2semidirect
9. #4957C2semidirect
10. #4948C2semidirect
11. #4945C2semidirect
29. C2×(C2×D3)⋊C4C22semidirect
30. C2×(C2×D3)⋊C4C22semidirect
31. C2×(C2×D3)⋊C4C22semidirect
33. C2×C4⋊Dic3C22semidirect
34. C2×Dic3⋊C4C22semidirect
35. C2×Dic3⋊C4C22semidirect
36. C2×Dic3⋊C4C22semidirect
37. C2×C4×Dic3C22semidirect
38. C3⋊#95C22semidirect
39. C3⋊#95C22semidirect
43. C22×C22⋊C4D3semidirect
63. C22×C3×C4D4semidirect
193. C22×C4#67semidirect
332. C3#2411semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊C42
C2×#9532
C2×#9292
C2×#9262
#52472
#52132
#52102
#49612
#49572
#49482
#49452
#24113