Finite Group Databasebeta

← #74189

Subgroups — #74189

Order 384.

436 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#382C211yesmaximal, characteristic, complemented
2#4990C211yesmaximal, characteristic
3#4990C211yesmaximal, characteristic
4#4982C211yesmaximal, characteristic, complemented
5#4979C211yesmaximal, characteristic, complemented
6C4×(C2×D3)⋊C4C211yesmaximal, characteristic
7#4933C211yesmaximal, characteristic, complemented
8#4676C211yesmaximal, characteristic
9#4673C211yesmaximal, characteristic
10#4668C211yesmaximal, characteristic
11#4667C211yesmaximal, characteristic
12#4653C211yesmaximal, characteristic, complemented
13#4652C211yesmaximal, characteristic, complemented
14#4649C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2574—31nomaximal, Sylow
17C22×C4×D3C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C42C2211yescharacteristic
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36C2×C4×Dic3C2211yescharacteristic
37C2×C4×Dic3C2211yescharacteristic
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
52#398—31no—
53#396—31no—
54#395—31no—
55#393—31no—
56#391—31no—
57#388—31no—
58#388—31no—
59#387—31no—
60#387—31no—
61#387—31no—
62#385—31no—
63#382D311yescharacteristic, complemented
64#380—31no—
65C4×C4⋊C4—31no—
66C4×C22⋊C4—31no—
67C23×D3C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×Dic3C2311yescharacteristic
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C2×C4×D3—22no—
83C2×C4×D3—22no—
84C2×C4×D3—41no—
85C2×Dic6—22no—
86C2×Dic6—41no—
87C2×Dic6—41no—
88C3×C4⋊C4—41no—
89C3×C4⋊C4—41no—
90C3×C42—22no—
91(C2×D3)⋊C4—22no—
92(C2×D3)⋊C4—22no—
93(C2×D3)⋊C4—41no—
94(C2×D3)⋊C4—41no—
95(C2×D3)⋊C4—41no—
96(C2×D3)⋊C4Q814yescomplemented
97C4⋊Dic3—22no—
98Dic3⋊C4—22no—
99Dic3⋊C4—22no—
100Dic3⋊C4—22no—
101Dic3⋊C4—22no—
102Dic3⋊C4—41no—
103Dic3⋊C4—41no—
104C4×Dic3—22no—
105C4×Dic3—41no—
106C4×Dic3—41no—
107C22×Q8—31no—
108C23×C4—31no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4C2×D311yescharacteristic
117C2×C4⋊C4C2×D311yescharacteristic
118C2×C22⋊C4—31no—
119C2×C22⋊C4—31no—
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#95C2×D311yescharacteristic
139#95C2×D311yescharacteristic
140#95C2×D311yescharacteristic
141#95C2×D311yescharacteristic
142C23×C3C2411yescharacteristic
143C22×D3—21no—
144C22×D3—21no—
145C22×D3—21no—
146C22×D3—21no—
147C22×D3—21no—
148C22×D3—21no—
149C22×D3C2×Q812yes—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
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×C4C2×Q812yes—
170C2×C3×C4#4112yes—
171C2×Dic3—21no—
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×Dic3C2×Q812yes—
189C2×Dic3#4112yes—
190C2×Dic3#4112yes—
191C2×Dic3#4112yes—
192C4×D3—44no—
193Dic6—44no—
194C24—31no—
195C2×Q8—62no—
196C2×Q8—121no—
197C2×Q8—121no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—121no—
208C22×C4C22×D311yescharacteristic
209C22×C4C22×D311yescharacteristic
210C22×C4C22×D311yescharacteristic
211C22×C4C22×D311yescharacteristic
212C22×C4C22×D311yescharacteristic
213C22×C4C22×D311yescharacteristic
214C22×C4C22×D311yescharacteristic
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—62no—
218C4⋊C4—62no—
219C4⋊C4—62no—
220C4⋊C4—62no—
221C4⋊C4—62no—
222C4⋊C4—121no—
223C4⋊C4—121no—
224C22⋊C4—34no—
225C22⋊C4—62no—
226C22⋊C4—62no—
227C22⋊C4—121no—
228C22⋊C4—121no—
229C22⋊C4—121no—
230C42—22no—
231C42—62no—
232C42—121no—
233C42—121no—
234C22×C3C22×Q811yescharacteristic
235C22×C3C2×#4111yescharacteristic
236C22×C3C2×#4111yescharacteristic
237C22×C3C2×#4111yescharacteristic
238C22×C3C2×#4111yescharacteristic
239C22×C3H(5,𝔽2)11yescharacteristic
240C22×C3H(5,𝔽2)11yescharacteristic
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—41no—
245C2×D3—41no—
246C2×D3—41no—
247C2×D3—41no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255Dic3—22no—
256Dic3—22no—
257Dic3—22no—
258Dic3—22no—
259Dic3—41no—
260Dic3—41no—
261Dic3—41no—
262C23—32no—
263C23—61no—
264C23—61no—
265C23—61no—
266C23—61no—
267C23—61no—
268C23—61no—
269C23C23×D311yescharacteristic
270Q8—124no—
271C2×C4—21no—
272C2×C4—21no—
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—32no—
291C2×C4—32no—
292C2×C4—32no—
293C2×C4—32no—
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—124no—
312C2×C4#23512yes—
313C2×C4D3×Q812yes—
314C2×C3#52911yescharacteristic
315C2×C3#53311yescharacteristic
316C2×C3#53811yescharacteristic
317C2×C3#55111yescharacteristic
318C2×C3#55411yescharacteristic
319C2×C3#55411yescharacteristic
320C2×C3#55511yescharacteristic
321D3—42no—
322C22—62no—
323C22—62no—
324C22—62no—
325C22—121no—
326C22—121no—
327C22—121no—
328C22—121no—
329C22C2×#23511yescharacteristic
330C22#100411yescharacteristic
331C22C2×D3×Q811yescharacteristic
332C22D3×#4111yescharacteristic
333C22D3×#4111yescharacteristic
334C22D3×#4111yescharacteristic
335C22#100911yescharacteristic
336C4—22no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—62no—
344C4—62no—
345C4—62no—
346C4—62no—
347C4—121no—
348C4—121no—
349C4—121no—
350C3#257411yescharacteristic, complemented, Sylow
351C2—122no—
352C2#551111yescharacteristic
353C2#552711yescharacteristic
354C2#552811yescharacteristic
355C2#567011yescharacteristic
356C2#567811yescharacteristic
357C2#568811yescharacteristic
358C2#569011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#382C2
prime radicalC23×D3C23
hypercenter#382D3
hypocenterC3#2574
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C3×#382C2semidirect
4. #4982C2semidirect
5. #4979C2semidirect
7. #4933C2semidirect
12. #4653C2semidirect
13. #4652C2semidirect
14. #4649C2semidirect
15. #4646C2semidirect
63. #382D3semidirect
96. (C2×D3)⋊C4Q8semidirect
350. C3#2574semidirect

Maximal subgroups

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

subgroupindex
C3×#3822
#49902
#49822
#49792
C4×(C2×D3)⋊C42
#49332
#46762
#46732
#46682
#46672
#46532
#46522
#46492
#46462
#25743