Finite Group Databasebeta

← #75452

Subgroups — #75452

Order 384.

824 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#893C211yesmaximal, characteristic, complemented
2C2×#884C211yesmaximal, characteristic
3C2×#67⋊C4C211yesmaximal, characteristic, complemented
4C2×#67⋊C4C211yesmaximal, characteristic, complemented
5C2×C42×D3C211yesmaximal, characteristic, complemented
6C3×#388C211yesmaximal, characteristic, complemented
7#4976C211yesmaximal, characteristic, complemented
8#4957C211yesmaximal, characteristic, complemented
9#4956C211yesmaximal, characteristic, complemented
10Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
11#4940C211yesmaximal, characteristic, complemented
12#4667C211yesmaximal, characteristic, complemented
13#4666C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4649C211yesmaximal, characteristic, complemented
16C4×#121—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C42C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×C4×Dic3C2211yescharacteristic
42C2×C4×Dic3C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44C2×C4×Dic3C2211yescharacteristic
45C2×C4×Dic3C2211yescharacteristic
46#893—24no—
47#884C418yescomplemented
48#67⋊C4—24no—
49#67⋊C4—24no—
50C42×D3—24no—
51C3×#95C2211yescharacteristic
52C3×#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C3⋊#95C2211yescharacteristic
57C2×#121—31no—
58C2×C4×D4—31no—
59C2×C4×D4—31no—
60C2×C4×D4—31no—
61C22×C42—31no—
62#390—31no—
63#389—31no—
64#388—31no—
65#388D311yescharacteristic, complemented
66#386—31no—
67#386—31no—
68#380—31no—
69C4×C4⋊C4—31no—
70C4×C22⋊C4—31no—
71C4×C22⋊C4—31no—
72C24×C3C2311yescharacteristic
73C23×D3C2311yescharacteristic
74C23×D3C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—22no—
82C2×#67—22no—
83C2×#67—22no—
84C2×#67—22no—
85C2×#67C2×C414yescomplemented
86C2×#67C2×C414yescomplemented
87C22×Dic3—22no—
88C22×Dic3—22no—
89C22×Dic3—22no—
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C2×D12—22no—
97C2×D12—22no—
98C2×D12C2×C414yescomplemented
99C2×C4×D3—22no—
100C2×C4×D3—22no—
101C2×C4×D3—22no—
102C2×C4×D3—22no—
103C2×C4×D3—22no—
104C2×C4×D3—22no—
105C2×C4×D3—22no—
106C2×C4×D3—22no—
107C2×C4×D3—22no—
108C2×C4×D3—22no—
109C2×C4×D3C2×C414yescomplemented
110C2×C4×D3D414yescomplemented
111C3×C4⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4C2×C414yescomplemented
115C3×C42—22no—
116#217—22no—
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—22no—
120(C2×D3)⋊C4—22no—
121(C2×D3)⋊C4—22no—
122(C2×D3)⋊C4C2×C414yescomplemented
123Dic3⋊C4—22no—
124Dic3⋊C4—22no—
125Dic3⋊C4—22no—
126Dic3⋊C4C2×C414yescomplemented
127C4×Dic3—22no—
128C4×Dic3—22no—
129C4×Dic3—22no—
130C4×Dic3—22no—
131C4×Dic3D414yescomplemented
132C22×D4—31no—
133C22×D4—31no—
134C22×D4—31no—
135C23×C4—31no—
136C23×C4—31no—
137C23×C4—31no—
138C23×C4—31no—
139C23×C4—31no—
140#121—38no—
141C4×D4—64no—
142C4×D4—64no—
143C4×D4—64no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4C2×D311yescharacteristic
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4C2×D311yescharacteristic
157C2×C22⋊C4C2×D311yescharacteristic
158C2×C22⋊C4C2×D311yescharacteristic
159C2×C42—31no—
160C2×C42—31no—
161C2×C42—31no—
162C2×C42—31no—
163C2×C42—31no—
164C2×C42—64no—
165C2×C42C2×D311yescharacteristic
166#95—31no—
167#95—31no—
168#95—31no—
169#95—31no—
170#95C2×D311yescharacteristic
171#95C2×D311yescharacteristic
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3C22×C412yes—
179C23×C3C2411yescharacteristic
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3C22×C412yes—
191C22×D3C22×C412yes—
192C22×D3C2×D412yes—
193C22×D3#4112yes—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4C22×C412yes—
209C2×C3×C4C22×C412yes—
210C2×C3×C4C2×D412yes—
211C2×C3×C4C2×D412yes—
212#67—28no—
213#67—28no—
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—21no—
224C2×Dic3—44no—
225C2×Dic3C22×C412yes—
226C2×Dic3C22×C412yes—
227C2×Dic3C2×D412yes—
228C2×Dic3C2×D412yes—
229C2×Dic3C2×D412yes—
230C2×Dic3#4112yes—
231C2×Dic3#4112yes—
232C2×Dic3#4112yes—
233D12—28no—
234C4×D3—28no—
235C4×D3—28no—
236C4×D3—28no—
237C4×D3—44no—
238C24—31no—
239C24—31no—
240C24C22×D311yescharacteristic
241C2×D4—34no—
242C2×D4—34no—
243C2×D4—34no—
244C2×D4—62no—
245C2×D4—62no—
246C2×D4—62no—
247C2×D4—62no—
248C2×D4—62no—
249C2×D4—62no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—31no—
254C22×C4—34no—
255C22×C4—34no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—62no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4C22×D311yescharacteristic
272C22×C4C22×D311yescharacteristic
273C22×C4C22×D311yescharacteristic
274C22×C4C22×D311yescharacteristic
275C22×C4C22×D311yescharacteristic
276C22×C4C22×D311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—34no—
279C4⋊C4—62no—
280C4⋊C4—62no—
281C4⋊C4—62no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—34no—
285C22⋊C4—62no—
286C22⋊C4—62no—
287C22⋊C4—62no—
288C22⋊C4—62no—
289C22⋊C4—62no—
290C22⋊C4—62no—
291C22⋊C4C4×D314yescomplemented
292C42—22no—
293C42—34no—
294C42—62no—
295C42—62no—
296C42—62no—
297C42—62no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—22no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3C23×C411yescharacteristic
306C22×C3C22×D411yescharacteristic
307C22×C3C22×D411yescharacteristic
308C22×C3C2×#4111yescharacteristic
309C22×C3C2×#4111yescharacteristic
310C22×C3C2×#4111yescharacteristic
311C22×C3C2×#4111yescharacteristic
312C2×D3—22no—
313C2×D3—22no—
314C2×D3—22no—
315C2×D3—22no—
316C2×D3—22no—
317C2×D3—22no—
318C2×D3—22no—
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3C4×D414yescomplemented
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—41no—
331C3×C4—41no—
332Dic3—22no—
333Dic3—22no—
334Dic3—41no—
335Dic3—41no—
336Dic3C4×D414yescomplemented
337Dic3#12114yescomplemented
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—32no—
345C23—32no—
346C23—32no—
347C23—32no—
348C23—61no—
349C23—61no—
350C23—61no—
351C23—61no—
352C23—61no—
353C23—61no—
354C23—61no—
355C23—61no—
356C23—61no—
357C23—61no—
358C23C2×C4×D312yes—
359C23C23×D311yescharacteristic
360D4—68no—
361D4—68no—
362D4—68no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—32no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—32no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—68no—
386C2×C4—68no—
387C2×C4—68no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—124no—
399C2×C4—124no—
400C2×C4C2×C4×D312yes—
401C2×C4C2×C4×D312yes—
402C2×C4D3×D412yes—
403C2×C4D3×D412yes—
404C2×C3—42no—
405C2×C3C2×C4×D411yescharacteristic
406C2×C3C2×C4×D411yescharacteristic
407C2×C3C4×#4111yescharacteristic
408C2×C3C2×#12111yescharacteristic
409C2×C3#52511yescharacteristic
410C2×C3#52911yescharacteristic
411C2×C3#53211yescharacteristic
412D3—24no—
413D3—42no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—34no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—62no—
423C22—62no—
424C22—62no—
425C22—62no—
426C22—62no—
427C22—62no—
428C22—62no—
429C22—62no—
430C22—62no—
431C22—121no—
432C22—121no—
433C22—121no—
434C22—121no—
435C22C22×C4×D311yescharacteristic
436C22C2×D3×D411yescharacteristic
437C22C2×D3×D411yescharacteristic
438C22D3×#4111yescharacteristic
439C22D3×#4111yescharacteristic
440C22D3×#4111yescharacteristic
441C22D3×#4111yescharacteristic
442C4—22no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—34no—
447C4—34no—
448C4—41no—
449C4—41no—
450C4—62no—
451C4—62no—
452C4—121no—
453C4—121no—
454C3C4×#12111yescharacteristic, complemented, Sylow
455C2—42no—
456C2—64no—
457C2—122no—
458C2#552211yescharacteristic
459C2C4×D3×D411yescharacteristic
460C2C4×D3×D411yescharacteristic
461C2D3×#12111yescharacteristic
462C2#565611yescharacteristic
463C2#566811yescharacteristic
464C2#570611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#388C2
socle, cosocleC23×C3C24
prime radicalC2×#884C2
hypercenter#388D3
hypocenterC3C4×#121
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#893C2semidirect
3. C2×#67⋊C4C2semidirect
4. C2×#67⋊C4C2semidirect
5. C2×C42×D3C2semidirect
6. C3×#388C2semidirect
7. #4976C2semidirect
8. #4957C2semidirect
9. #4956C2semidirect
10. Dic3×C22⋊C4C2semidirect
11. #4940C2semidirect
12. #4667C2semidirect
13. #4666C2semidirect
14. #4665C2semidirect
15. #4649C2semidirect
47. #884C4semidirect
65. #388D3semidirect
85. C2×#67C2×C4semidirect
86. C2×#67C2×C4semidirect
98. C2×D12C2×C4semidirect
109. C2×C4×D3C2×C4semidirect
110. C2×C4×D3D4semidirect
114. C3×C22⋊C4C2×C4semidirect
122. (C2×D3)⋊C4C2×C4semidirect
126. Dic3⋊C4C2×C4semidirect
131. C4×Dic3D4semidirect
291. C22⋊C4C4×D3semidirect
325. C2×D3C4×D4semidirect
336. Dic3C4×D4semidirect
337. Dic3#121semidirect
454. C3C4×#121semidirect

Maximal subgroups

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

subgroupindex
C2×#8932
C2×#8842
C2×#67⋊C42
C2×C42×D32
C3×#3882
#49762
#49572
#49562
Dic3×C22⋊C42
#49402
#46672
#46662
#46652
#46492
C4×#1213