Finite Group Databasebeta

← #76742

Subgroups — #76742

Order 384.

644 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 125 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
2C2×#884C211yesmaximal, characteristic, complemented
3C3×#392C211yesmaximal, characteristic, complemented
4#5223C212yesmaximal, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5223C212yesmaximal, complemented
7#4960C212yesmaximal, complemented
8#4953C211yesmaximal, characteristic, complemented
9#4952C212yesmaximal, complemented
10#4674C211yesmaximal, characteristic, complemented
11#4664C211yesmaximal, characteristic, complemented
12#2815—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×#67C2212yes—
16C22×#67C2211yescharacteristic
17C22×#67C2211yescharacteristic
18C22×#67C2212yes—
19C22×D12C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26#938—24no—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32#884—24no—
33C3×#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#121—31no—
39C2×#121—31no—
40#397—32no—
41#397—31no—
42#397—31no—
43#392—32no—
44#392—31no—
45#392—32no—
46#392—32no—
47#392D311yescharacteristic, complemented
48#381—31no—
49C24×C3C2312yes—
50C24×C3C2311yescharacteristic
51C23×D3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C2×C3×D4—22no—
54C2×C3×D4—42no—
55C2×C3×D4—42no—
56C2×C3×D4—42no—
57C2×C3×D4—42no—
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C2×#67—22no—
62C2×#67—24no—
63C2×#67—22no—
64C2×#67—42no—
65C2×#67—42no—
66C2×#67—42no—
67C2×#67—42no—
68C2×#67—42no—
69C2×#67—42no—
70C2×#67—42no—
71C22×Dic3C2312yes—
72C22×Dic3C2311yescharacteristic
73C22×Dic3C2312yes—
74C22×Dic3C2311yescharacteristic
75C2×D12—22no—
76C2×D12—42no—
77C2×C4×D3—42no—
78C2×C4×D3D414yescomplemented
79C3×C22⋊C4—22no—
80C3×C22⋊C4—42no—
81#217—24no—
82#217—42no—
83#217—42no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—42no—
86(C2×D3)⋊C4—42no—
87C4⋊Dic3—22no—
88Dic3⋊C4—22no—
89C22×D4—31no—
90C22×D4—32no—
91C22×D4—31no—
92C22×D4—32no—
93C22×D4—31no—
94C22×D4C2×D311yescharacteristic
95C22×D4C2×D312yes—
96C23×C4—31no—
97#121—64no—
98#121—64no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—31no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—31no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4C2×D311yescharacteristic
108C2×C22⋊C4C2×D312yes—
109#95—32no—
110#95—31no—
111#95—32no—
112#95—32no—
113#95C2×D311yescharacteristic
114C23×C3—22no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—21no—
126C23×C3C2411yescharacteristic
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—21no—
130C22×D3—21no—
131C22×D3—21no—
132C22×D3—21no—
133C22×D3—22no—
134C22×D3—22no—
135C22×D3—21no—
136C22×D3C2×D412yes—
137C3×D4—44no—
138C3×D4—48no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4C2×D412yes—
146#67—44no—
147#67—44no—
148#67—48no—
149#67—48no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3C2×D412yes—
160D12—44no—
161C4×D3—44no—
162C24—31no—
163C24—31no—
164C24C22×D311yescharacteristic
165C24C22×D312yes—
166C2×D4—22no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—62no—
172C2×D4—62no—
173C2×D4—62no—
174C2×D4—64no—
175C2×D4—122no—
176C2×D4—122no—
177C2×D4—122no—
178C2×D4—122no—
179C2×D4—122no—
180C2×D4—122no—
181C2×D4—122no—
182C2×D4—122no—
183C22×C4—31no—
184C22×C4—34no—
185C22×C4—32no—
186C22×C4—31no—
187C22×C4—32no—
188C22×C4—122no—
189C22×C4C22×D312yes—
190C22×C4C22×D311yescharacteristic
191C22×C4C22×D311yescharacteristic
192C4⋊C4—62no—
193C4⋊C4—62no—
194C22⋊C4—22no—
195C22⋊C4—42no—
196C22⋊C4—64no—
197C22⋊C4—62no—
198C22⋊C4—122no—
199C22⋊C4—122no—
200C22⋊C4—122no—
201C22⋊C4—122no—
202C22×C3—41no—
203C22×C3—42no—
204C22×C3—41no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3—41no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3C22×D411yescharacteristic
215C22×C3H(5,𝔽2)12yes—
216C22×C3H(5,𝔽2)12yes—
217C22×C3H(5,𝔽2)11yescharacteristic
218C22×C3H(5,𝔽2)11yescharacteristic
219C2×D3—22no—
220C2×D3—22no—
221C2×D3—22no—
222C2×D3—42no—
223C2×D3—42no—
224C2×D3—41no—
225C2×D3—41no—
226C2×D3—41no—
227C2×D3—42no—
228C2×D3—42no—
229C3×C4—22no—
230C3×C4—41no—
231C3×C4—42no—
232Dic3—22no—
233Dic3—41no—
234Dic3—42no—
235Dic3—42no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23—22no—
248C23—32no—
249C23—62no—
250C23—62no—
251C23—61no—
252C23—61no—
253C23—61no—
254C23—61no—
255C23—62no—
256C23—62no—
257C23—61no—
258C23C23×D311yescharacteristic
259D4—44no—
260D4—48no—
261D4—124no—
262D4—124no—
263D4—124no—
264D4—128no—
265D4—128no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—32no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—124no—
283C2×C4D3×D412yes—
284C2×C3—81no—
285C2×C3—82no—
286C2×C3#53411yescharacteristic
287C2×C3#53511yescharacteristic
288C2×C3#53511yescharacteristic
289C2×C3#56012yes—
290C2×C3#56012yes—
291D3—42no—
292D3—81no—
293C22—41no—
294C22—42no—
295C22—42no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—62no—
306C22—62no—
307C22—62no—
308C22—122no—
309C22—122no—
310C22—121no—
311C22—121no—
312C22—122no—
313C22—121no—
314C22—122no—
315C22C2×D3×D411yescharacteristic
316C22#100412yes—
317C22#100412yes—
318C22#100411yescharacteristic
319C22#100911yescharacteristic
320C4—22no—
321C4—41no—
322C4—42no—
323C4—62no—
324C4—121no—
325C4—122no—
326C4—122no—
327C3#281511yescharacteristic, complemented, Sylow
328C2—81no—
329C2—82no—
330C2—122no—
331C2—241no—
332C2#558911yescharacteristic
333C2#559512yes—
334C2#560711yescharacteristic
335C2#562012yes—
336C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2815
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#938C2semidirect
2. C2×#884C2semidirect
3. C3×#392C2semidirect
4. #5223C2semidirect
5. #5223C2semidirect
6. #5223C2semidirect
7. #4960C2semidirect
8. #4953C2semidirect
9. #4952C2semidirect
10. #4674C2semidirect
11. #4664C2semidirect
47. #392D3semidirect
78. C2×C4×D3D4semidirect
327. C3#2815semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C2×#8842
C3×#3922
#52232
#49602
#49532
#49522
#46742
#46642
#28153