Finite Group Databasebeta

← #82251

Subgroups — #82251

Order 384.

476 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#117C211yesmaximal, characteristic
2C2×#889C211yesmaximal, characteristic, complemented
3#5210C211yesmaximal, characteristic
4#5210C211yesmaximal, characteristic
5#4985C211yesmaximal, characteristic, complemented
6#4983C211yesmaximal, characteristic, complemented
7#4980C211yesmaximal, characteristic, complemented
8#4953C211yesmaximal, characteristic
9#4948C211yesmaximal, characteristic
10#4945C211yesmaximal, characteristic
11#4944C211yesmaximal, characteristic
12#4933C211yesmaximal, characteristic, complemented
13#4933C211yesmaximal, characteristic, complemented
14C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
15#4928C211yesmaximal, characteristic, complemented
16#2475—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C3×#117—24no—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C42C2211yescharacteristic
24C2×C3×C42C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
40C2×C4×Dic3C2211yescharacteristic
41#889—24no—
42C3⋊#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#125—31no—
55C2×#117D311yescharacteristic
56#400—31no—
57#398—31no—
58#397—31no—
59#396—31no—
60#389—31no—
61#388—31no—
62#388—31no—
63#387—31no—
64#387—31no—
65#387—31no—
66#385—31no—
67#385—31no—
68C4×C4⋊C4—31no—
69C24×C3C2311yescharacteristic
70C22×C3×C4—22no—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C3×C4⋊C4—22no—
88C3×C4⋊C4D414yes—
89C3×C22⋊C4—22no—
90C3×C22⋊C4—22no—
91C3×C42—22no—
92C3×C42—22no—
93#217—22no—
94#217—22no—
95#217—22no—
96#217—22no—
97C4⋊Dic3—22no—
98C4⋊Dic3—22no—
99Dic3⋊C4—22no—
100Dic3⋊C4—22no—
101Dic3⋊C4—22no—
102Dic3⋊C4—22no—
103Dic3⋊C4—22no—
104Dic3⋊C4—22no—
105Dic3⋊C4—22no—
106Dic3⋊C4—22no—
107C4×Dic3—22no—
108C4×Dic3—22no—
109C23×C4C2×D311yescharacteristic
110#125—64no—
111#117—24no—
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×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4C2×D311yescharacteristic
123C2×C4⋊C4C2×D311yescharacteristic
124C2×C22⋊C4—31no—
125C2×C22⋊C4—31no—
126C2×C22⋊C4—31no—
127C2×C22⋊C4—31no—
128C2×C22⋊C4C2×D311yescharacteristic
129C2×C22⋊C4C2×D311yescharacteristic
130C2×C42—31no—
131C2×C42—31no—
132C2×C42C2×D311yescharacteristic
133C2×C42C2×D311yescharacteristic
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#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3#4112yes—
153C23×C3C2411yescharacteristic
154C2×C3×C4—28no—
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—21no—
168C2×C3×C4—21no—
169C2×C3×C4C2×D412yes—
170C2×C3×C4C2×D412yes—
171C2×C3×C4C2×D412yes—
172C2×C3×C4#4112yes—
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—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3#4112yes—
196C2×Dic3#4112yes—
197C24C22×D311yescharacteristic
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
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×C4C22×D311yescharacteristic
210C22×C4C22×D311yescharacteristic
211C22×C4C22×D311yescharacteristic
212C22×C4C22×D311yescharacteristic
213C22×C4C22×D311yescharacteristic
214C22×C4C22×D311yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—62no—
217C4⋊C4—62no—
218C4⋊C4—62no—
219C4⋊C4—62no—
220C4⋊C4—62no—
221C4⋊C4—62no—
222C4⋊C4—62no—
223C4⋊C4—62no—
224C4⋊C4—62no—
225C4⋊C4—62no—
226C4⋊C4#6714yes—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—62no—
230C22⋊C4—62no—
231C22⋊C4—62no—
232C22⋊C4—62no—
233C42—22no—
234C42—22no—
235C42—62no—
236C42—62no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3C22×D411yescharacteristic
245C22×C3C2×#4111yescharacteristic
246C22×C3C2×#4111yescharacteristic
247C22×C3C2×#4111yescharacteristic
248C22×C3C2×#4111yescharacteristic
249C22×C32−1+411yescharacteristic
250C22×C32−1+411yescharacteristic
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—41no—
256C3×C4—41no—
257Dic3—22no—
258Dic3—22no—
259Dic3—41no—
260Dic3—41no—
261Dic3—41no—
262Dic3—41no—
263Dic3—41no—
264Dic3—41no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23#23512yes—
272C23C23×D311yescharacteristic
273C2×C4—28no—
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—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
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#23512yes—
313C2×C4C2×#6712yes—
314C2×C4C2×#6712yes—
315C2×C4C2×#6712yes—
316C2×C3—42no—
317C2×C3#52511yescharacteristic
318C2×C3#52911yescharacteristic
319C2×C3#54111yescharacteristic
320C2×C3#54711yescharacteristic
321C2×C3#54811yescharacteristic
322C2×C3#55211yescharacteristic
323C2×C3#55211yescharacteristic
324C22—22no—
325C22—22no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22C2×#23511yescharacteristic
332C22C2×#23511yescharacteristic
333C22D3×#4111yescharacteristic
334C22D3×#4111yescharacteristic
335C22#101011yescharacteristic
336C22#101011yescharacteristic
337C22C22×#6711yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—62no—
345C4—62no—
346C4—121no—
347C4—121no—
348C4—121no—
349C4—121no—
350C4—121no—
351C4—121no—
352C3#247511yescharacteristic, complemented, Sylow
353C2—42no—
354C2#551811yescharacteristic
355C2#552811yescharacteristic
356C2#553111yescharacteristic
357C2#553111yescharacteristic
358C2#579111yescharacteristic
359C2#582411yescharacteristic
360C2#583111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#117C2
prime radicalC24×C3C23
hypercenterC2×#117D3
hypocenterC3#2475
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#889C2semidirect
5. #4985C2semidirect
6. #4983C2semidirect
7. #4980C2semidirect
12. #4933C2semidirect
13. #4933C2semidirect
14. C4×Dic3⋊C4C2semidirect
15. #4928C2semidirect
352. C3#2475semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1172
C2×#8892
#52102
#49852
#49832
#49802
#49532
#49482
#49452
#49442
#49332
C4×Dic3⋊C42
#49282
#24753