Finite Group Databasebeta

← #73455

Subgroups — #73455

Order 384.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#877C211yesmaximal, characteristic
2C3×C4×C22⋊C4C211yesmaximal, characteristic
3#5210C211yesmaximal, characteristic
4C2×C3⋊#95C211yesmaximal, characteristic
5#4946C211yesmaximal, characteristic
6#4943C211yesmaximal, characteristic
7Dic3×C22⋊C4C211yesmaximal, characteristic
8#4933C211yesmaximal, characteristic, complemented
9#4933C211yesmaximal, characteristic, complemented
10#4932C211yesmaximal, characteristic, complemented
11C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
12#4652C211yesmaximal, characteristic, complemented
13#4652C211yesmaximal, characteristic, complemented
14#4651C211yesmaximal, characteristic, complemented
15#4650C211yesmaximal, characteristic, complemented
16#2343—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C42C2211yescharacteristic
23C2×C3×C42C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36#877C418yescomplemented
37C3×#95C2211yescharacteristic
38C3×#95C2211yescharacteristic
39C3⋊#95C2211yescharacteristic
40C3⋊#95C2211yescharacteristic
41C3⋊#95C2211yescharacteristic
42C3⋊#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C2×#117—31no—
54#387—31no—
55#387—31no—
56#387—31no—
57#387—31no—
58#385—31no—
59#385—31no—
60#383—31no—
61#382—31no—
62#381—31no—
63C4×C4⋊C4—31no—
64C4×C22⋊C4—31no—
65C4×C22⋊C4D311yescharacteristic
66#376—31no—
67C2×#95—31no—
68C24×C3C2311yescharacteristic
69C22×C3×C4—22no—
70C22×C3×C4—22no—
71C22×C3×C4—41no—
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×Dic3—22no—
79C22×Dic3—22no—
80C22×Dic3—22no—
81C22×Dic3—22no—
82C22×Dic3—41no—
83C22×Dic3C2×C414yes—
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C3×C22⋊C4—22no—
93C3×C22⋊C4C2×C414yes—
94C3×C42—22no—
95C3×C42—41no—
96#217—22no—
97#217C2×C414yes—
98Dic3⋊C4—22no—
99Dic3⋊C4—22no—
100Dic3⋊C4—41no—
101Dic3⋊C4—41no—
102Dic3⋊C4C2×C414yescomplemented
103Dic3⋊C4C2×C414yescomplemented
104C4×Dic3—22no—
105C4×Dic3—41no—
106C4×Dic3C2×C414yescomplemented
107C4×Dic3C2×C414yescomplemented
108C23×C4—31no—
109C23×C4—31no—
110C23×C4C2×D311yescharacteristic
111#117—38no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C22⋊C4—31no—
119C2×C22⋊C4—31no—
120C2×C22⋊C4C2×D311yescharacteristic
121C2×C22⋊C4C2×D311yescharacteristic
122C2×C42—31no—
123C2×C42—31no—
124C2×C42—31no—
125C2×C42—31no—
126C2×C42C2×D311yescharacteristic
127C2×C42C2×D311yescharacteristic
128#95—31no—
129#95—31no—
130#95—31no—
131#95—31no—
132#95—31no—
133#95—31no—
134#95—31no—
135#95—31no—
136#95—31no—
137#95—31no—
138#95—31no—
139#95—31no—
140#95—31no—
141#95—31no—
142#95C2×D311yescharacteristic
143#95C2×D311yescharacteristic
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3C22×C412yes—
149C23×C3#4112yes—
150C23×C3#4112yes—
151C23×C3C2411yescharacteristic
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—44no—
168C2×C3×C4C22×C412yes—
169C2×C3×C4C22×C412yes—
170C2×C3×C4#4112yes—
171C2×Dic3—28no—
172C2×Dic3—21no—
173C2×Dic3—21no—
174C2×Dic3—21no—
175C2×Dic3—21no—
176C2×Dic3—21no—
177C2×Dic3—21no—
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—44no—
192C2×Dic3C22×C412yes—
193C2×Dic3C22×C412yes—
194C2×Dic3C22×C412yes—
195C2×Dic3C22×C412yes—
196C2×Dic3#4112yes—
197C24C22×D311yescharacteristic
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—34no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—41no—
210C22×C4—62no—
211C22×C4—62no—
212C22×C4—62no—
213C22×C4—62no—
214C22×C4—121no—
215C22×C4C22×D311yescharacteristic
216C22×C4C22×D311yescharacteristic
217C22×C4C22×D311yescharacteristic
218C22×C4C22×D311yescharacteristic
219C22×C4C22×D311yescharacteristic
220C22×C4C22×D311yescharacteristic
221C4⋊C4—34no—
222C4⋊C4—34no—
223C4⋊C4—62no—
224C4⋊C4—62no—
225C4⋊C4—121no—
226C4⋊C4—121no—
227C22⋊C4—22no—
228C22⋊C4—34no—
229C22⋊C4—62no—
230C22⋊C4C4×D314yes—
231C42—22no—
232C42—34no—
233C42—34no—
234C42—41no—
235C42—62no—
236C42—121no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3#11714yes—
244C22×C3C23×C411yescharacteristic
245C22×C3C2×#4111yescharacteristic
246C22×C3C2×#4111yescharacteristic
247C22×C3C2×#4111yescharacteristic
248C22×C3C2×#4111yescharacteristic
249C22×C3H(5,𝔽2)11yescharacteristic
250C22×C32−1+411yescharacteristic
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—41no—
257Dic3—22no—
258Dic3—22no—
259Dic3—22no—
260Dic3—22no—
261Dic3—22no—
262Dic3—41no—
263Dic3—41no—
264Dic3—41no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23C2×C4×D312yes—
270C23#23512yes—
271C23#23512yes—
272C23C23×D311yescharacteristic
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—32no—
289C2×C4—32no—
290C2×C4—32no—
291C2×C4—32no—
292C2×C4—32no—
293C2×C4—44no—
294C2×C4—68no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—124no—
315C2×C4C2×C4×D312yes—
316C2×C4C2×C4×D312yes—
317C2×C4#23712yes—
318C2×C3—24no—
319C2×C3C2×#11711yescharacteristic
320C2×C3C4×#4111yescharacteristic
321C2×C3#52011yescharacteristic
322C2×C3#55111yescharacteristic
323C2×C3#55111yescharacteristic
324C2×C3#55211yescharacteristic
325C2×C3#55211yescharacteristic
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22#87214yes—
333C22C22×C4×D311yescharacteristic
334C22C2×#23511yescharacteristic
335C22C2×#23511yescharacteristic
336C22C2×#23711yescharacteristic
337C22#100411yescharacteristic
338C22D3×#4111yescharacteristic
339C22#101011yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—62no—
347C4—62no—
348C4—62no—
349C4—62no—
350C4—62no—
351C4—121no—
352C4—121no—
353C4—121no—
354C3#234311yescharacteristic, complemented, Sylow
355C2—24no—
356C2C2×#87211yescharacteristic
357C2#549511yescharacteristic
358C2#553111yescharacteristic
359C2C4×#23711yescharacteristic
360C2#554111yescharacteristic
361C2#554611yescharacteristic
362C2#555611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×C4×C22⋊C4C2
socle, cosocleC23×C3C24
prime radicalC24×C3C23
hypercenterC4×C22⋊C4D3
hypocenterC3#2343
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
8. #4933C2semidirect
9. #4933C2semidirect
10. #4932C2semidirect
11. C4×Dic3⋊C4C2semidirect
12. #4652C2semidirect
13. #4652C2semidirect
14. #4651C2semidirect
15. #4650C2semidirect
36. #877C4semidirect
102. Dic3⋊C4C2×C4semidirect
103. Dic3⋊C4C2×C4semidirect
106. C4×Dic3C2×C4semidirect
107. C4×Dic3C2×C4semidirect
354. C3#2343semidirect

Maximal subgroups

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

subgroupindex
C2×#8772
C3×C4×C22⋊C42
#52102
C2×C3⋊#952
#49462
#49432
Dic3×C22⋊C42
#49332
#49322
C4×Dic3⋊C42
#46522
#46512
#46502
#23433