Finite Group Databasebeta

← #82342

Subgroups — #82342

Order 384.

716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic
3C2×#67⋊C4C211yesmaximal, characteristic, complemented
4#5220C211yesmaximal, characteristic, complemented
5#5213C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7C4×#217C211yesmaximal, characteristic, complemented
8C2×C3⋊#95C211yesmaximal, characteristic, complemented
9#4990C211yesmaximal, characteristic, complemented
10#4978C211yesmaximal, characteristic, complemented
11#4956C211yesmaximal, characteristic, complemented
12#4945C211yesmaximal, characteristic, complemented
13Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
14C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
15#4933C211yesmaximal, characteristic, complemented
16#2364—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C23×Dic3C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C3×C4×D4—24no—
26C2×C3×C4⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C42C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34#939C418yescomplemented
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×C4×Dic3C2211yescharacteristic
42C2×C4×Dic3C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44#67⋊C4—24no—
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C2×#121—31no—
56C2×C4×D4—31no—
57C2×C4×D4D311yescharacteristic, complemented
58#388—31no—
59#388—31no—
60#387—31no—
61#387—31no—
62#386—31no—
63#386—31no—
64#386—31no—
65#386—31no—
66C4×C22⋊C4—31no—
67C4×C22⋊C4—31no—
68C4×C22⋊C4—31no—
69C2×#95—31no—
70C24×C3C2311yescharacteristic
71C24×C3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4C2×C414yescomplemented
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—41no—
81C22×C3×C4—41no—
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C2×#67—22no—
88C2×#67—22no—
89C2×#67—41no—
90C2×#67—41no—
91C2×#67C2×C414yescomplemented
92C2×#67C2×C414yescomplemented
93C22×Dic3—22no—
94C22×Dic3—22no—
95C22×Dic3—41no—
96C22×Dic3—41no—
97C22×Dic3—41no—
98C22×Dic3C2×C414yescomplemented
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C2×C4×D3—22no—
107C2×C4×D3—22no—
108C2×C4×D3—41no—
109C3×C4⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4D414yescomplemented
112C3×C42—22no—
113#217—22no—
114#217—22no—
115#217—41no—
116#217C2×C414yescomplemented
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4C2×C414yescomplemented
121Dic3⋊C4—22no—
122Dic3⋊C4C2×C414yescomplemented
123C4×Dic3—22no—
124C4×Dic3—41no—
125C4×Dic3—41no—
126C22×D4—31no—
127C22×D4—31no—
128C22×D4C2×D311yescharacteristic
129C23×C4—31no—
130C23×C4—31no—
131C23×C4—31no—
132C23×C4C2×D311yescharacteristic
133C23×C4C2×D311yescharacteristic
134#121—38no—
135C4×D4—24no—
136C4×D4—64no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4C2×D311yescharacteristic
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊C4—31no—
146C2×C22⋊C4—31no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4C2×D311yescharacteristic
149C2×C22⋊C4C2×D311yescharacteristic
150C2×C42—31no—
151C2×C42—31no—
152C2×C42—31no—
153C2×C42C2×D311yescharacteristic
154#95—31no—
155#95—31no—
156#95—31no—
157#95—31no—
158#95—31no—
159#95—31no—
160#95—31no—
161#95—31no—
162#95—31no—
163#95—31no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3C22×C412yes—
175C23×C3C22×C412yes—
176C23×C3C2×D412yes—
177C23×C3#4112yes—
178C23×C3C2411yescharacteristic
179C22×D3—21no—
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3C22×C412yes—
186C3×D4—28no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—44no—
199C2×C3×C4—44no—
200C2×C3×C4C22×C412yes—
201C2×C3×C4C2×D412yes—
202C2×C3×C4C2×D412yes—
203C2×C3×C4#4112yes—
204#67—28no—
205#67—44no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—44no—
224C2×Dic3—44no—
225C2×Dic3C22×C412yes—
226C2×Dic3C22×C412yes—
227C2×Dic3C22×C412yes—
228C2×Dic3#4112yes—
229C4×D3—44no—
230C24—31no—
231C24C22×D311yescharacteristic
232C24C22×D311yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—34no—
236C2×D4—34no—
237C2×D4—62no—
238C2×D4—62no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4C4×D314yescomplemented
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—34no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—31no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—62no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4—62no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4—121no—
263C22×C4—121no—
264C22×C4C22×D311yescharacteristic
265C22×C4C22×D311yescharacteristic
266C22×C4C22×D311yescharacteristic
267C22×C4C22×D311yescharacteristic
268C22×C4C22×D311yescharacteristic
269C4⋊C4—22no—
270C4⋊C4—34no—
271C4⋊C4—62no—
272C22⋊C4—22no—
273C22⋊C4—34no—
274C22⋊C4—34no—
275C22⋊C4—62no—
276C22⋊C4—62no—
277C22⋊C4—62no—
278C22⋊C4—62no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4#6714yescomplemented
282C42—22no—
283C42—62no—
284C42—121no—
285C42—121no—
286C22×C3—22no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3C4×D414yes—
300C22×C3C23×C411yescharacteristic
301C22×C3C22×D411yescharacteristic
302C22×C3C2×#4111yescharacteristic
303C22×C3C2×#4111yescharacteristic
304C22×C3C2×#4111yescharacteristic
305C22×C3H(5,𝔽2)11yescharacteristic
306C22×C3H(5,𝔽2)11yescharacteristic
307C2×D3—22no—
308C2×D3—22no—
309C2×D3—22no—
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—41no—
319Dic3—22no—
320Dic3—22no—
321Dic3—22no—
322Dic3—22no—
323Dic3—41no—
324Dic3—41no—
325Dic3—41no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—32no—
337C23—61no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23—61no—
342C23—61no—
343C23C2×C4×D312yes—
344C23C2×C4×D312yes—
345C23#23512yes—
346C23C2×#6712yes—
347C23C23×D311yescharacteristic
348D4—28no—
349D4—68no—
350D4—124no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—32no—
363C2×C4—32no—
364C2×C4—32no—
365C2×C4—32no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4—124no—
388C2×C4C2×C4×D312yes—
389C2×C4#23712yes—
390C2×C4C2×#6712yes—
391C2×C4C2×#6712yes—
392C2×C3—24no—
393C2×C3—42no—
394C2×C3C2×C4×D411yescharacteristic
395C2×C3C4×#4111yescharacteristic
396C2×C3#51811yescharacteristic
397C2×C3#54611yescharacteristic
398C2×C3#54611yescharacteristic
399C2×C3#55311yescharacteristic
400C2×C3#55311yescharacteristic
401D3—42no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—62no—
416C22—62no—
417C22—62no—
418C22—121no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22C4×#6714yes—
423C22C22×C4×D311yescharacteristic
424C22C2×#23511yescharacteristic
425C22C2×#23711yescharacteristic
426C22#100411yescharacteristic
427C22D3×#4111yescharacteristic
428C22#100911yescharacteristic
429C22C22×#6711yescharacteristic
430C4—22no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—62no—
436C4—62no—
437C4—62no—
438C4—62no—
439C4—121no—
440C4—121no—
441C4—121no—
442C3#236411yescharacteristic, complemented, Sylow
443C2—24no—
444C2—42no—
445C2—122no—
446C2C4×#23711yescharacteristic
447C2#554011yescharacteristic
448C2#554511yescharacteristic
449C2#556511yescharacteristic
450C2C2×C4×#6711yescharacteristic
451C2#580611yescharacteristic
452C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC2×C3×C4×D4C2
socle, cosocleC23×C3C24
prime radicalC2×#939C2
hypercenterC2×C4×D4D3
hypocenterC3#2364
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
3. C2×#67⋊C4C2semidirect
4. #5220C2semidirect
5. #5213C2semidirect
6. #5213C2semidirect
7. C4×#217C2semidirect
8. C2×C3⋊#95C2semidirect
9. #4990C2semidirect
10. #4978C2semidirect
11. #4956C2semidirect
12. #4945C2semidirect
13. Dic3×C22⋊C4C2semidirect
14. C4×(C2×D3)⋊C4C2semidirect
15. #4933C2semidirect
34. #939C4semidirect
57. C2×C4×D4D3semidirect
75. C2×C3×D4C2×C4semidirect
91. C2×#67C2×C4semidirect
92. C2×#67C2×C4semidirect
98. C22×Dic3C2×C4semidirect
111. C3×C22⋊C4D4semidirect
116. #217C2×C4semidirect
120. (C2×D3)⋊C4C2×C4semidirect
122. Dic3⋊C4C2×C4semidirect
241. C2×D4C4×D3semidirect
281. C22⋊C4#67semidirect
442. C3#2364semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9392
C2×#67⋊C42
#52202
#52132
C4×#2172
C2×C3⋊#952
#49902
#49782
#49562
#49452
Dic3×C22⋊C42
C4×(C2×D3)⋊C42
#49332
#23643