Finite Group Databasebeta

← #82250

Subgroups — #82250

Order 384.

526 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#117C211yesmaximal, characteristic, complemented
2C2×#885C211yesmaximal, characteristic
3#5213C211yesmaximal, characteristic
4#5210C211yesmaximal, characteristic, complemented
5#4994C211yesmaximal, characteristic, complemented
6#4990C211yesmaximal, characteristic, complemented
7#4979C211yesmaximal, characteristic, complemented
8#4978C211yesmaximal, characteristic, complemented
9#4959C211yesmaximal, characteristic
10#4952C211yesmaximal, characteristic, complemented
11#4951C211yesmaximal, characteristic, complemented
12#4940C211yesmaximal, characteristic, complemented
13#4940C211yesmaximal, characteristic, complemented
14C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
15#4929C211yesmaximal, characteristic, complemented
16#2474—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C3×#117—24no—
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C42C2211yescharacteristic
26C2×C3×C42C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×C4×Dic3C2211yescharacteristic
42C2×C4×Dic3C2211yescharacteristic
43#885—24no—
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#124—31no—
55C2×#117D311yescharacteristic, complemented
56#397—31no—
57#396—31no—
58#395—31no—
59#394—31no—
60#391—31no—
61#388—31no—
62#388—31no—
63#388—31no—
64#388—31no—
65#387—31no—
66#386—31no—
67#385—31no—
68C4×C4⋊C4—31no—
69C24×C3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—22no—
81C2×#67—22no—
82C2×#67—41no—
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C2×Dic6—22no—
91C2×Dic6—22no—
92C2×Dic6—41no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C22⋊C4—22no—
96C3×C22⋊C4D414yes—
97C3×C42—22no—
98C3×C42—22no—
99#217—22no—
100#217—22no—
101#217—41no—
102(C2×D3)⋊C4—22no—
103(C2×D3)⋊C4—22no—
104(C2×D3)⋊C4—22no—
105(C2×D3)⋊C4—22no—
106(C2×D3)⋊C4—22no—
107(C2×D3)⋊C4—41no—
108C4⋊Dic3—41no—
109Dic3⋊C4—22no—
110Dic3⋊C4—22no—
111Dic3⋊C4—22no—
112Dic3⋊C4—41no—
113C4×Dic3—22no—
114C4×Dic3—22no—
115C22×Q8—31no—
116C22×D4—31no—
117C23×C4C2×D311yescharacteristic
118#124—64no—
119#117—24no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4C2×D311yescharacteristic
126C2×C4⋊C4C2×D311yescharacteristic
127C2×C22⋊C4—31no—
128C2×C22⋊C4—31no—
129C2×C22⋊C4—31no—
130C2×C22⋊C4—31no—
131C2×C22⋊C4—31no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—31no—
134C2×C22⋊C4—31no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4C2×D311yescharacteristic
137C2×C22⋊C4C2×D311yescharacteristic
138C2×C42—31no—
139C2×C42—31no—
140C2×C42C2×D311yescharacteristic
141C2×C42C2×D311yescharacteristic
142#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146#95—31no—
147#95—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3C2×D412yes—
159C23×C3C2411yescharacteristic
160C22×D3—21no—
161C22×D3—21no—
162C22×D3—21no—
163C22×D3—21no—
164C22×D3—21no—
165C22×D3—21no—
166C22×D3—21no—
167C2×C3×C4—28no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4C2×D412yes—
183C2×C3×C4C2×D412yes—
184C2×C3×C4#4112yes—
185C2×C3×C4#4112yes—
186#67—44no—
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—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3#4112yes—
207C2×Dic3#4112yes—
208Dic6—44no—
209C24—31no—
210C24C22×D311yescharacteristic
211C2×Q8—62no—
212C2×Q8—62no—
213C2×Q8—121no—
214C2×D4—62no—
215C2×D4—62no—
216C2×D4—121no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4C22×D311yescharacteristic
228C22×C4C22×D311yescharacteristic
229C22×C4C22×D311yescharacteristic
230C22×C4C22×D311yescharacteristic
231C22×C4C22×D311yescharacteristic
232C22×C4C22×D311yescharacteristic
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—62no—
236C4⋊C4—62no—
237C4⋊C4—62no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C22⋊C4—22no—
241C22⋊C4—62no—
242C22⋊C4—62no—
243C22⋊C4—62no—
244C22⋊C4—62no—
245C22⋊C4—62no—
246C22⋊C4—62no—
247C22⋊C4—62no—
248C22⋊C4—121no—
249C22⋊C4—121no—
250C22⋊C4#6714yes—
251C42—22no—
252C42—22no—
253C42—62no—
254C42—62no—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3C22×D411yescharacteristic
263C22×C3C2×#4111yescharacteristic
264C22×C3C2×#4111yescharacteristic
265C22×C3C2×#4111yescharacteristic
266C22×C3C2×#4111yescharacteristic
267C22×C3H(5,𝔽2)11yescharacteristic
268C22×C32−1+411yescharacteristic
269C2×D3—41no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C3×C4—22no—
277C3×C4—22no—
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—41no—
281C3×C4—41no—
282Dic3—22no—
283Dic3—22no—
284Dic3—41no—
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288Dic3—41no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—61no—
296C23—61no—
297C23—61no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23—61no—
302C23C2×#6712yes—
303C23C23×D311yescharacteristic
304Q8—124no—
305D4—124no—
306C2×C4—28no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—32no—
322C2×C4—32no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4#23512yes—
343C2×C4#23512yes—
344C2×C4C2×#6712yes—
345C2×C4C2×#6712yes—
346C2×C3—42no—
347C2×C3#52511yescharacteristic
348C2×C3#52911yescharacteristic
349C2×C3#54211yescharacteristic
350C2×C3#54611yescharacteristic
351C2×C3#54711yescharacteristic
352C2×C3#55511yescharacteristic
353C2×C3#55611yescharacteristic
354D3—81no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22C2×#23511yescharacteristic
370C22C2×#23511yescharacteristic
371C22D3×#4111yescharacteristic
372C22D3×#4111yescharacteristic
373C22#100911yescharacteristic
374C22#101011yescharacteristic
375C22C22×#6711yescharacteristic
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—62no—
383C4—62no—
384C4—121no—
385C4—121no—
386C4—121no—
387C4—121no—
388C4—121no—
389C3#247411yescharacteristic, complemented, Sylow
390C2—42no—
391C2—241no—
392C2#552811yescharacteristic
393C2#553211yescharacteristic
394C2#553311yescharacteristic
395C2#553411yescharacteristic
396C2#579111yescharacteristic
397C2#582411yescharacteristic
398C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#117C2
prime radicalC22×#67C22
hypercenterC2×#117D3
hypocenterC3#2474
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C3×#117C2semidirect
4. #5210C2semidirect
5. #4994C2semidirect
6. #4990C2semidirect
7. #4979C2semidirect
8. #4978C2semidirect
10. #4952C2semidirect
11. #4951C2semidirect
12. #4940C2semidirect
13. #4940C2semidirect
14. C4×Dic3⋊C4C2semidirect
15. #4929C2semidirect
55. C2×#117D3semidirect
389. C3#2474semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1172
C2×#8852
#52132
#52102
#49942
#49902
#49792
#49782
#49592
#49522
#49512
#49402
C4×Dic3⋊C42
#49292
#24743