Finite Group Databasebeta

← #76697

Subgroups — #76697

Order 384.

932 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 153 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×#237C211yesmaximal, characteristic, complemented
2C2×#939C212yesmaximal, complemented
3C2×#884C211yesmaximal, characteristic, complemented
4C3×#392C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5223C212yesmaximal, complemented
7#4960C212yesmaximal, complemented
8#4947C211yesmaximal, characteristic, complemented
9#4942C212yesmaximal, complemented
10#4673C211yesmaximal, characteristic, complemented
11#4667C211yesmaximal, characteristic, complemented
12#2615—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×#67C2212yes—
16C22×#67C2211yescharacteristic
17C22×#67C2212yes—
18C22×#67C2211yescharacteristic
19C23×Dic3C2212yes—
20C2×#237—28no—
21C2×#237—24no—
22C22×D12C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C22×Dic6C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2212yes—
27C2×#217C2212yes—
28C2×#217C2212yes—
29#939—28no—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2212yes—
33C2×Dic3⋊C4C2212yes—
34C2×Dic3⋊C4C2211yescharacteristic
35#884—24no—
36C3×#95C2211yescharacteristic
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C22×#41—31no—
41C2×#121—32no—
42C2×#121—31no—
43#393—31no—
44#393—31no—
45#392—32no—
46#392—31no—
47#392—32no—
48#392D311yescharacteristic, complemented
49#380—32no—
50#380—31no—
51C24×C3C2311yescharacteristic
52C24×C3C2312yes—
53C23×D3C2311yescharacteristic
54C23×D3C2311yescharacteristic
55C2×C3×D4—24no—
56C2×C3×D4—24no—
57C2×C3×D4—22no—
58C2×C3×D4—42no—
59C2×C3×D4—42no—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C2×#67—24no—
64C2×#67—24no—
65C2×#67—24no—
66C2×#67—22no—
67C2×#67—24no—
68C2×#67—22no—
69C2×#67—24no—
70C2×#67—42no—
71C2×#67—42no—
72C2×#67—42no—
73C22×Dic3—24no—
74C22×Dic3—24no—
75C22×Dic3D418yescomplemented
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2312yes—
78C22×Dic3C2312yes—
79C22×Dic3C2311yescharacteristic
80#237—232no—
81C2×D12—22no—
82C2×D12—42no—
83C2×C4×D3—24no—
84C2×C4×D3D414yescomplemented
85C2×Dic6—22no—
86C2×Dic6—24no—
87C3×C22⋊C4—22no—
88C3×C22⋊C4—42no—
89#217—24no—
90#217—42no—
91(C2×D3)⋊C4—22no—
92(C2×D3)⋊C4—24no—
93(C2×D3)⋊C4—42no—
94Dic3⋊C4—22no—
95Dic3⋊C4—24no—
96C2×#41—68no—
97C2×#41—64no—
98C22×Q8—31no—
99C22×D4—31no—
100C22×D4—32no—
101C22×D4—31no—
102C22×D4—32no—
103C22×D4—31no—
104C22×D4C2×D311yescharacteristic
105C22×D4C2×D312yes—
106C23×C4—32no—
107C23×C4—31no—
108#121—64no—
109#121—68no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—32no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—31no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4—32no—
117C2×C22⋊C4C2×D311yescharacteristic
118C2×C22⋊C4C2×D312yes—
119#95—32no—
120#95—32no—
121#95—31no—
122#95C2×D311yescharacteristic
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—21no—
131C23×C3—22no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3C2×D414yes—
135C23×C3C2411yescharacteristic
136C22×D3—22no—
137C22×D3—21no—
138C22×D3—22no—
139C22×D3—21no—
140C22×D3—21no—
141C22×D3—22no—
142C22×D3—22no—
143C22×D3—21no—
144C22×D3—21no—
145C22×D3C2×D412yes—
146C3×D4—28no—
147C3×D4—48no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4C2×D412yes—
155#67—216no—
156#67—44no—
157#67—48no—
158#67—44no—
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—216no—
162C2×Dic3—21no—
163C2×Dic3—22no—
164C2×Dic3—22no—
165C2×Dic3—22no—
166C2×Dic3—22no—
167C2×Dic3C2×D414yes—
168C2×Dic3C2×D414yes—
169C2×Dic3C2×D412yes—
170D12—44no—
171C4×D3—28no—
172Dic6—28no—
173C24—31no—
174C24—31no—
175C24C22×D311yescharacteristic
176C24C22×D312yes—
177#41—632no—
178C2×Q8—64no—
179C2×Q8—62no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—62no—
186C2×D4—64no—
187C2×D4—62no—
188C2×D4—64no—
189C2×D4—64no—
190C2×D4—64no—
191C2×D4—62no—
192C2×D4—64no—
193C2×D4—122no—
194C2×D4—122no—
195C2×D4—122no—
196C2×D4—122no—
197C22×C4—31no—
198C22×C4—38no—
199C22×C4—32no—
200C22×C4—34no—
201C22×C4—32no—
202C22×C4—31no—
203C22×C4—64no—
204C22×C4—64no—
205C22×C4—64no—
206C22×C4C22×D311yescharacteristic
207C22×C4C22×D312yes—
208C22×C4C22×D311yescharacteristic
209C4⋊C4—62no—
210C4⋊C4—64no—
211C22⋊C4—22no—
212C22⋊C4—42no—
213C22⋊C4—64no—
214C22⋊C4—64no—
215C22⋊C4—62no—
216C22⋊C4—122no—
217C22⋊C4—122no—
218C22×C3—24no—
219C22×C3—24no—
220C22×C3—24no—
221C22×C3—42no—
222C22×C3—42no—
223C22×C3—42no—
224C22×C3—42no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—42no—
228C22×C3—42no—
229C22×C3—41no—
230C22×C3C22×D411yescharacteristic
231C22×C3C22×D412yes—
232C22×C3H(5,𝔽2)12yes—
233C22×C3H(5,𝔽2)11yescharacteristic
234C22×C32−1+411yescharacteristic
235C2×D3—22no—
236C2×D3—22no—
237C2×D3—22no—
238C2×D3—42no—
239C2×D3—42no—
240C2×D3—41no—
241C2×D3—41no—
242C2×D3—42no—
243C2×D3—42no—
244C2×D3—41no—
245C3×C4—22no—
246C3×C4—41no—
247C3×C4—42no—
248Dic3—24no—
249Dic3—41no—
250Dic3—42no—
251Dic3#12014yes—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—22no—
260C23—21no—
261C23—22no—
262C23—21no—
263C23—32no—
264C23—62no—
265C23—61no—
266C23—62no—
267C23—61no—
268C23—61no—
269C23—62no—
270C23—62no—
271C23—61no—
272C23—61no—
273C23D3×D414yes—
274C23C23×D311yescharacteristic
275Q8—68no—
276D4—28no—
277D4—48no—
278D4—616no—
279D4—124no—
280D4—124no—
281D4—128no—
282D4—124no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—32no—
290C2×C4—34no—
291C2×C4—34no—
292C2×C4—68no—
293C2×C4—616no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—61no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4D3×D412yes—
302C2×C3—44no—
303C2×C3—81no—
304C2×C3C2×#12011yescharacteristic
305C2×C3#53511yescharacteristic
306C2×C3#53512yes—
307C2×C3#53712yes—
308C2×C3#53711yescharacteristic
309D3—42no—
310D3—81no—
311C22—24no—
312C22—24no—
313C22—24no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—62no—
324C22—62no—
325C22—62no—
326C22—122no—
327C22—122no—
328C22—121no—
329C22—121no—
330C22—122no—
331C22—122no—
332C22—121no—
333C22C2×D3×D411yescharacteristic
334C22C2×D3×D412yes—
335C22#100412yes—
336C22#100411yescharacteristic
337C22#101011yescharacteristic
338C4—22no—
339C4—34no—
340C4—41no—
341C4—42no—
342C4—64no—
343C4—121no—
344C4—122no—
345C3#261511yescharacteristic, complemented, Sylow
346C2—44no—
347C2—81no—
348C2—122no—
349C2—241no—
350C2D3×#12011yescharacteristic
351C2#559212yes—
352C2#559712yes—
353C2#560811yescharacteristic
354C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2615
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C2×#939C2semidirect
3. C2×#884C2semidirect
4. C3×#392C2semidirect
5. #5223C2semidirect
6. #5223C2semidirect
7. #4960C2semidirect
8. #4947C2semidirect
9. #4942C2semidirect
10. #4673C2semidirect
11. #4667C2semidirect
48. #392D3semidirect
75. C22×Dic3D4semidirect
84. C2×C4×D3D4semidirect
345. C3#2615semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×#9392
C2×#8842
C3×#3922
#52232
#49602
#49472
#49422
#46732
#46672
#26153