Finite Group Databasebeta

← #75296

Subgroups — #75296

Order 384.

512 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×D3×C4⋊C4C211yesmaximal, characteristic
2C3×#387C211yesmaximal, characteristic, complemented
3#4987C211yesmaximal, characteristic
4#4980C211yesmaximal, characteristic, complemented
5#4978C211yesmaximal, characteristic, complemented
6C4×(C2×D3)⋊C4C211yesmaximal, characteristic
7#4933C211yesmaximal, characteristic, complemented
8#4667C211yesmaximal, characteristic
9#4666C211yesmaximal, characteristic
10#4664C211yesmaximal, characteristic
11#4664C211yesmaximal, characteristic
12#4655C211yesmaximal, characteristic, complemented
13#4654C211yesmaximal, characteristic, complemented
14#4653C211yesmaximal, characteristic, complemented
15#4652C211yesmaximal, characteristic, complemented
16#2313—31nomaximal, Sylow
17C22×C4×D3C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C42C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36D3×C4⋊C4C418yescomplemented
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
53C22×C4⋊C4—31no—
54#387—31no—
55#387—31no—
56#387—31no—
57#387—31no—
58#387—31no—
59#387—31no—
60#387—31no—
61#387D311yescharacteristic, complemented
62#381—31no—
63#381—31no—
64#380—31no—
65#380—31no—
66C4×C22⋊C4—31no—
67C4×C22⋊C4—31no—
68C23×D3C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C2×C4×D3—22no—
84C2×C4×D3—22no—
85C2×C4×D3—41no—
86C2×C4×D3—41no—
87C2×C4×D3—41no—
88C2×C4×D3—41no—
89C2×C4×D3C2×C414yes—
90C2×C4×D3C2×C414yes—
91C2×C4×D3C2×C414yes—
92C3×C4⋊C4—41no—
93C3×C4⋊C4C2×C414yes—
94C3×C42—41no—
95(C2×D3)⋊C4—22no—
96(C2×D3)⋊C4—22no—
97(C2×D3)⋊C4—22no—
98(C2×D3)⋊C4—22no—
99C4⋊Dic3—41no—
100C4⋊Dic3C2×C414yes—
101Dic3⋊C4—41no—
102Dic3⋊C4—41no—
103Dic3⋊C4C2×C414yes—
104Dic3⋊C4C2×C414yes—
105C4×Dic3—41no—
106C4×Dic3—41no—
107C4×Dic3—41no—
108C23×C4—31no—
109C23×C4—31no—
110C23×C4—31no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—38no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4C2×D311yescharacteristic
119C2×C4⋊C4C2×D311yescharacteristic
120C2×C22⋊C4—31no—
121C2×C22⋊C4—31no—
122C2×C22⋊C4—31no—
123C2×C22⋊C4—31no—
124C2×C42—31no—
125C2×C42—31no—
126C2×C42—31no—
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#95C2×D311yescharacteristic
141#95C2×D311yescharacteristic
142#95C2×D311yescharacteristic
143#95C2×D311yescharacteristic
144C23×C3C2411yescharacteristic
145C22×D3—21no—
146C22×D3—21no—
147C22×D3—21no—
148C22×D3—21no—
149C22×D3C22×C412yes—
150C22×D3#4112yes—
151C22×D3#4112yes—
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—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4C22×C412yes—
171C2×C3×C4C22×C412yes—
172C2×C3×C4C22×C412yes—
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×Dic3C22×C412yes—
192C2×Dic3C22×C412yes—
193C2×Dic3C22×C412yes—
194C4×D3—28no—
195C4×D3—44no—
196C4×D3—44no—
197C24—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—34no—
202C22×C4—34no—
203C22×C4—34no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—62no—
209C22×C4—62no—
210C22×C4—121no—
211C22×C4—121no—
212C22×C4—121no—
213C22×C4—121no—
214C22×C4C22×D311yescharacteristic
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—34no—
224C4⋊C4—41no—
225C4⋊C4—121no—
226C4⋊C4—121no—
227C4⋊C4—121no—
228C4⋊C4C4×D314yes—
229C22⋊C4—62no—
230C22⋊C4—62no—
231C22⋊C4—62no—
232C22⋊C4—62no—
233C42—41no—
234C42—121no—
235C42—121no—
236C42—121no—
237C22×C3C23×C411yescharacteristic
238C22×C3C2×#4111yescharacteristic
239C22×C3C2×#4111yescharacteristic
240C22×C3H(5,𝔽2)11yescharacteristic
241C22×C3H(5,𝔽2)11yescharacteristic
242C22×C32−1+411yescharacteristic
243C22×C32−1+411yescharacteristic
244C2×D3—22no—
245C2×D3—22no—
246C2×D3—22no—
247C2×D3—22no—
248C2×D3—22no—
249C2×D3—22no—
250C2×D3#11714yes—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—41no—
258Dic3—22no—
259Dic3—22no—
260Dic3—22no—
261Dic3—41no—
262Dic3—41no—
263Dic3—41no—
264Dic3—41no—
265C23—32no—
266C23—32no—
267C23—32no—
268C23—61no—
269C23—61no—
270C23—61no—
271C23—61no—
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—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—32no—
292C2×C4—32no—
293C2×C4—32no—
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—124no—
314C2×C4—124no—
315C2×C4C2×C4×D312yes—
316C2×C4C2×C4×D312yes—
317C2×C4C2×C4×D312yes—
318C2×C3C2×#11711yescharacteristic
319C2×C3#51811yescharacteristic
320C2×C3#51911yescharacteristic
321C2×C3#53911yescharacteristic
322C2×C3#53911yescharacteristic
323C2×C3#53911yescharacteristic
324C2×C3#53911yescharacteristic
325D3—24no—
326C22—34no—
327C22—62no—
328C22—62no—
329C22—62no—
330C22—62no—
331C22—62no—
332C22—62no—
333C22C22×C4×D311yescharacteristic
334C22#100411yescharacteristic
335C22#100711yescharacteristic
336C22D3×#4111yescharacteristic
337C22D3×#4111yescharacteristic
338C22#100911yescharacteristic
339C22#101011yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—62no—
348C4—62no—
349C4—62no—
350C4—121no—
351C4—121no—
352C4—121no—
353C4—121no—
354C3#231311yescharacteristic, complemented, Sylow
355C2—64no—
356C2D3×#11711yescharacteristic
357C2#554511yescharacteristic
358C2#557211yescharacteristic
359C2#563811yescharacteristic
360C2#566011yescharacteristic
361C2#569211yescharacteristic
362C2#570711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#387C2
socle, cosocleC23×C3C24
prime radicalC23×D3C23
hypercenter#387D3
hypocenterC3#2313
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C3×#387C2semidirect
4. #4980C2semidirect
5. #4978C2semidirect
7. #4933C2semidirect
12. #4655C2semidirect
13. #4654C2semidirect
14. #4653C2semidirect
15. #4652C2semidirect
36. D3×C4⋊C4C4semidirect
61. #387D3semidirect
354. C3#2313semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊C42
C3×#3872
#49872
#49802
#49782
C4×(C2×D3)⋊C42
#49332
#46672
#46662
#46642
#46552
#46542
#46532
#46522
#23133