Finite Group Databasebeta

← #75370

Subgroups — #75370

Order 384.

584 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C4×D12C211yesmaximal, characteristic
3C3×#387C211yesmaximal, characteristic, complemented
4#4994C211yesmaximal, characteristic, complemented
5#4989C211yesmaximal, characteristic, complemented
6#4989C211yesmaximal, characteristic, complemented
7#4933C211yesmaximal, characteristic, complemented
8#4676C211yesmaximal, characteristic, complemented
9#4668C211yesmaximal, characteristic, complemented
10#4668C211yesmaximal, characteristic, complemented
11#4667C211yesmaximal, characteristic, complemented
12#4665C211yesmaximal, characteristic
13#4665C211yesmaximal, characteristic
14#4661C211yesmaximal, characteristic, complemented
15#4657C211yesmaximal, characteristic, complemented
16#2579—31nomaximal, Sylow
17C22×D12C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C42C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42D3×C4⋊C4—24no—
43C4×D12—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—
55C22×C4⋊C4—31no—
56#398—31no—
57#398—31no—
58#396—31no—
59#396—31no—
60#387—31no—
61#387D311yescharacteristic, complemented
62#386—31no—
63#386—31no—
64#385—31no—
65#385—31no—
66#385—31no—
67#385—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×D12Q814yes—
87C2×C4×D3—22no—
88C2×C4×D3—22no—
89C2×C4×D3—22no—
90C2×C4×D3—22no—
91C2×C4×D3—22no—
92C2×C4×D3—41no—
93C2×C4×D3—41no—
94C2×C4×D3—41no—
95C2×C4×D3—41no—
96C2×C4×D3—41no—
97C2×C4×D3—41no—
98C2×C4×D3D414yes—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—22no—
101C3×C42—22no—
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—22no—
108(C2×D3)⋊C4—41no—
109(C2×D3)⋊C4—41no—
110C4⋊Dic3—22no—
111C4⋊Dic3—22no—
112C4⋊Dic3—41no—
113Dic3⋊C4—22no—
114Dic3⋊C4—22no—
115Dic3⋊C4—22no—
116Dic3⋊C4—41no—
117Dic3⋊C4—41no—
118Dic3⋊C4—41no—
119C22×D4—31no—
120C23×C4—31no—
121C23×C4—31no—
122C23×C4—31no—
123C23×C4—31no—
124C4×D4—64no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—64no—
135C2×C4⋊C4C2×D311yescharacteristic
136C2×C4⋊C4C2×D311yescharacteristic
137C2×C22⋊C4—31no—
138C2×C22⋊C4—31no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C42C2×D311yescharacteristic
146#95—31no—
147#95—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152#95C2×D311yescharacteristic
153#95C2×D311yescharacteristic
154#95C2×D311yescharacteristic
155#95C2×D311yescharacteristic
156C23×C3C2411yescharacteristic
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×D3—21no—
166C22×D3—21no—
167C22×D3C2×D412yes—
168C22×D3C2×Q812yes—
169C22×D3C2×Q812yes—
170C22×D3#4112yes—
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×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4C2×D412yes—
190C2×C3×C4C2×Q812yes—
191C2×C3×C4#4112yes—
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—21no—
207C2×Dic3—21no—
208C2×Dic3C2×D412yes—
209C2×Dic3#4112yes—
210D12—28no—
211C4×D3—28no—
212C4×D3—44no—
213C4×D3—44no—
214C4×D3—44no—
215C24—31no—
216C24—31no—
217C2×D4—34no—
218C2×D4—62no—
219C2×D4—62no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—34no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—62no—
230C22×C4—62no—
231C22×C4—62no—
232C22×C4—121no—
233C22×C4—121no—
234C22×C4—121no—
235C22×C4—121no—
236C22×C4—121no—
237C22×C4—121no—
238C22×C4C22×D311yescharacteristic
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241C22×C4C22×D311yescharacteristic
242C22×C4C22×D311yescharacteristic
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D311yescharacteristic
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C22⋊C4—62no—
257C22⋊C4—62no—
258C22⋊C4—62no—
259C22⋊C4—62no—
260C22⋊C4—62no—
261C22⋊C4—62no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C42—22no—
265C22×C3C22×D411yescharacteristic
266C22×C3C22×Q811yescharacteristic
267C22×C3C2×#4111yescharacteristic
268C22×C3C2×#4111yescharacteristic
269C22×C3C2×#4111yescharacteristic
270C22×C3H(5,𝔽2)11yescharacteristic
271C22×C3H(5,𝔽2)11yescharacteristic
272C2×D3—22no—
273C2×D3—22no—
274C2×D3—22no—
275C2×D3—22no—
276C2×D3—22no—
277C2×D3—22no—
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—22no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3C22⋊Q814yes—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293Dic3—22no—
294Dic3—22no—
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299C23—32no—
300C23—32no—
301C23—32no—
302C23—32no—
303C23—61no—
304C23—61no—
305C23—61no—
306C23—61no—
307C23—61no—
308C23—61no—
309C23—61no—
310C23—61no—
311C23—61no—
312C23—61no—
313C23C23×D311yescharacteristic
314D4—68no—
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—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—68no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—124no—
353C2×C4—124no—
354C2×C4—124no—
355C2×C4#23512yes—
356C2×C4D3×D412yes—
357C2×C4D3×Q812yes—
358C2×C3C2×C22⋊Q811yescharacteristic
359C2×C3#52511yescharacteristic
360C2×C3#53411yescharacteristic
361C2×C3#55311yescharacteristic
362C2×C3#55311yescharacteristic
363C2×C3#55411yescharacteristic
364C2×C3#55411yescharacteristic
365D3—24no—
366D3—42no—
367C22—34no—
368C22—62no—
369C22—62no—
370C22—62no—
371C22—62no—
372C22—62no—
373C22—62no—
374C22—62no—
375C22—62no—
376C22—62no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22C2×#23511yescharacteristic
382C22C2×D3×D411yescharacteristic
383C22C2×D3×Q811yescharacteristic
384C22D3×#4111yescharacteristic
385C22D3×#4111yescharacteristic
386C22#100911yescharacteristic
387C22#100911yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—62no—
396C4—62no—
397C4—121no—
398C4—121no—
399C4—121no—
400C4—121no—
401C3#257911yescharacteristic, complemented, Sylow
402C2—64no—
403C2—122no—
404C2#553011yescharacteristic
405C2#554711yescharacteristic
406C2#557911yescharacteristic
407C2D3×C22⋊Q811yescharacteristic
408C2#565311yescharacteristic
409C2#569011yescharacteristic
410C2#570911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#387C2
prime radicalC22×D12C22
hypercenter#387D3
hypocenterC3#2579
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊C4C2semidirect
3. C3×#387C2semidirect
4. #4994C2semidirect
5. #4989C2semidirect
6. #4989C2semidirect
7. #4933C2semidirect
8. #4676C2semidirect
9. #4668C2semidirect
10. #4668C2semidirect
11. #4667C2semidirect
14. #4661C2semidirect
15. #4657C2semidirect
61. #387D3semidirect
401. C3#2579semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊C42
C2×C4×D122
C3×#3872
#49942
#49892
#49332
#46762
#46682
#46672
#46652
#46612
#46572
#25793