Finite Group Databasebeta

← #75021

Subgroups — #75021

Order 384.

540 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×#893C211yesmaximal, characteristic
2C2×#872C211yesmaximal, characteristic, complemented
3C3×#387C211yesmaximal, characteristic, complemented
4#4991C211yesmaximal, characteristic
5#4978C211yesmaximal, characteristic, complemented
6#4976C211yesmaximal, characteristic, complemented
7#4940C211yesmaximal, characteristic, complemented
8#4675C211yesmaximal, characteristic, complemented
9#4674C211yesmaximal, characteristic, complemented
10#4672C211yesmaximal, characteristic
11#4670C211yesmaximal, characteristic, complemented
12#4667C211yesmaximal, characteristic, complemented
13#4667C211yesmaximal, characteristic, complemented
14#4667C211yesmaximal, characteristic, complemented
15#4647C211yesmaximal, characteristic, complemented
16#2485—31nomaximal, Sylow
17C22×D12C2211yescharacteristic
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×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×C4×Dic3C2211yescharacteristic
41C2×C4×Dic3C2211yescharacteristic
42#893—24no—
43#872—24no—
44C3×#95C2211yescharacteristic
45C3×#95C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3×#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×C4×D4—31no—
55C2×#117—31no—
56#397—31no—
57#397—31no—
58#394—31no—
59#394—31no—
60#389—31no—
61#389—31no—
62#388—31no—
63#388—31no—
64#387—31no—
65#387D311yescharacteristic, complemented
66#380—31no—
67#380—31no—
68#380—31no—
69C23×D3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C2×D12—22no—
85C2×D12—22no—
86C2×D12—22no—
87C2×C4×D3—22no—
88C2×C4×D3—22no—
89C2×C4×D3—22no—
90C2×C4×D3—22no—
91C2×C4×D3—22no—
92C2×C4×D3D414yes—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C42—22no—
96(C2×D3)⋊C4—22no—
97(C2×D3)⋊C4—22no—
98(C2×D3)⋊C4—22no—
99(C2×D3)⋊C4—22no—
100(C2×D3)⋊C4—22no—
101(C2×D3)⋊C4—22no—
102(C2×D3)⋊C4—41no—
103(C2×D3)⋊C4—41no—
104(C2×D3)⋊C4—41no—
105(C2×D3)⋊C4—41no—
106C4⋊Dic3—41no—
107Dic3⋊C4—22no—
108Dic3⋊C4—22no—
109Dic3⋊C4—22no—
110Dic3⋊C4—22no—
111Dic3⋊C4—22no—
112Dic3⋊C4—41no—
113C4×Dic3—22no—
114C4×Dic3—22no—
115C22×D4—31no—
116C23×C4—31no—
117C23×C4—31no—
118C4×D4—64no—
119#117—64no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4C2×D311yescharacteristic
128C2×C4⋊C4C2×D311yescharacteristic
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⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C22⋊C4—31no—
139C2×C42—31no—
140C2×C42—31no—
141C2×C42C2×D311yescharacteristic
142#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146#95—31no—
147#95—31no—
148#95C2×D311yescharacteristic
149#95C2×D311yescharacteristic
150#95C2×D311yescharacteristic
151#95C2×D311yescharacteristic
152C23×C3C2411yescharacteristic
153C22×D3—21no—
154C22×D3—21no—
155C22×D3—21no—
156C22×D3—21no—
157C22×D3—21no—
158C22×D3—21no—
159C22×D3—21no—
160C22×D3—21no—
161C22×D3—21no—
162C22×D3—21no—
163C22×D3—21no—
164C22×D3—21no—
165C22×D3C2×D412yes—
166C22×D3#4112yes—
167C2×C3×C4—21no—
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×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4C2×D412yes—
187C2×C3×C4#4112yes—
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×Dic3C2×D412yes—
204C2×Dic3#4112yes—
205C2×Dic3#4112yes—
206D12—28no—
207C4×D3—28no—
208C4×D3—44no—
209C24—31no—
210C24—31no—
211C2×D4—62no—
212C2×D4—62no—
213C2×D4—62no—
214C22×C4—31no—
215C22×C4—34no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—62no—
222C22×C4—62no—
223C22×C4—62no—
224C22×C4—62no—
225C22×C4—62no—
226C22×C4C22×D311yescharacteristic
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—62no—
239C4⋊C4—62no—
240C4⋊C4—121no—
241C4⋊C4—121no—
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—121no—
251C22⋊C4—121no—
252C42—22no—
253C42—62no—
254C42—62no—
255C22×C3C22×D411yescharacteristic
256C22×C3C2×#4111yescharacteristic
257C22×C3C2×#4111yescharacteristic
258C22×C3C2×#4111yescharacteristic
259C22×C3C2×#4111yescharacteristic
260C22×C3H(5,𝔽2)11yescharacteristic
261C22×C32−1+411yescharacteristic
262C2×D3—22no—
263C2×D3—22no—
264C2×D3—22no—
265C2×D3—22no—
266C2×D3—22no—
267C2×D3—22no—
268C2×D3—41no—
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—41no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—41no—
283Dic3—41no—
284Dic3—41no—
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288Dic3#12314yes—
289C23—32no—
290C23—32no—
291C23—61no—
292C23—61no—
293C23—61no—
294C23—61no—
295C23—61no—
296C23—61no—
297C23—61no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23—61no—
302C23—61no—
303C23C23×D311yescharacteristic
304D4—68no—
305C2×C4—21no—
306C2×C4—21no—
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—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—32no—
327C2×C4—68no—
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—61no—
343C2×C4—124no—
344C2×C4#23512yes—
345C2×C4D3×D412yes—
346C2×C3C2×#12311yescharacteristic
347C2×C3#52511yescharacteristic
348C2×C3#53711yescharacteristic
349C2×C3#55211yescharacteristic
350C2×C3#55311yescharacteristic
351C2×C3#55511yescharacteristic
352C2×C3#55611yescharacteristic
353D3—42no—
354D3—42no—
355C22—62no—
356C22—62no—
357C22—62no—
358C22—62no—
359C22—62no—
360C22—62no—
361C22—121no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22C2×#23511yescharacteristic
370C22C2×D3×D411yescharacteristic
371C22#100711yescharacteristic
372C22D3×#4111yescharacteristic
373C22D3×#4111yescharacteristic
374C22D3×#4111yescharacteristic
375C22#100911yescharacteristic
376C4—22no—
377C4—22no—
378C4—34no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—121no—
385C4—121no—
386C4—121no—
387C4—121no—
388C4—121no—
389C3#248511yescharacteristic, complemented, Sylow
390C2—122no—
391C2—122no—
392C2#553211yescharacteristic
393C2#554711yescharacteristic
394C2#556811yescharacteristic
395C2#562911yescharacteristic
396C2D3×#12311yescharacteristic
397C2#569311yescharacteristic
398C2#570911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#387C2
prime radicalC22×D12C22
hypercenter#387D3
hypocenterC3#2485
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×#872C2semidirect
3. C3×#387C2semidirect
5. #4978C2semidirect
6. #4976C2semidirect
7. #4940C2semidirect
8. #4675C2semidirect
9. #4674C2semidirect
11. #4670C2semidirect
12. #4667C2semidirect
13. #4667C2semidirect
14. #4667C2semidirect
15. #4647C2semidirect
65. #387D3semidirect
389. C3#2485semidirect

Maximal subgroups

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

subgroupindex
C2×#8932
C2×#8722
C3×#3872
#49912
#49782
#49762
#49402
#46752
#46742
#46722
#46702
#46672
#46472
#24853