Finite Group Databasebeta

← #82683

Subgroups — #82683

Order 384.

734 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic, complemented
2C2×#940C211yesmaximal, characteristic, complemented
3C2×#939C211yesmaximal, characteristic, complemented
4C2×#929C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5223C211yesmaximal, characteristic, complemented
8#5220C211yesmaximal, characteristic, complemented
9#5213C211yesmaximal, characteristic, complemented
10#4991C211yesmaximal, characteristic, complemented
11#4978C211yesmaximal, characteristic, complemented
12#4960C211yesmaximal, characteristic, complemented
13#4959C211yesmaximal, characteristic, complemented
14#4958C211yesmaximal, characteristic, complemented
15#4944C211yesmaximal, characteristic, complemented
16#2673—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic
25C22×#67C2211yescharacteristic
26C22×#67C2211yescharacteristic
27C23×Dic3C2211yescharacteristic
28C22×D12C2211yescharacteristic
29C3×#121—24no—
30C2×C3×C4⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38#940—24no—
39#939—24no—
40#929—24no—
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×(C2×D3)⋊C4C2211yescharacteristic
46C2×(C2×D3)⋊C4C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49C2×C4×Dic3C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#127—31no—
57C2×#123—31no—
58C2×#121—31no—
59C2×#121D311yescharacteristic, complemented
60#394—31no—
61#394—31no—
62#392—31no—
63#392—31no—
64#392—31no—
65#392—31no—
66#388—31no—
67#387—31no—
68#386—31no—
69#386—31no—
70#386—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C24×C3C2311yescharacteristic
74C23×D3C2311yescharacteristic
75C23×D3C2311yescharacteristic
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4—41no—
83C2×C3×D4—41no—
84C2×C3×D4D414yescomplemented
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—41no—
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—22no—
98C2×#67—22no—
99C2×#67—22no—
100C2×#67—22no—
101C2×#67—41no—
102C2×#67—41no—
103C2×#67—41no—
104C2×#67—41no—
105C2×#67—41no—
106C2×#67—41no—
107C2×#67—41no—
108C2×#67—41no—
109C2×#67—41no—
110C22×Dic3—22no—
111C22×Dic3—41no—
112C22×Dic3—41no—
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C22×Dic3C2311yescharacteristic
118C22×Dic3C2311yescharacteristic
119C2×D12—22no—
120C2×D12—41no—
121C2×D12—41no—
122C3×C4⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125#217—22no—
126#217—22no—
127#217—22no—
128#217—41no—
129#217—41no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136Dic3⋊C4—22no—
137Dic3⋊C4D414yescomplemented
138C4×Dic3—22no—
139C22×D4—31no—
140C22×D4—31no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—31no—
145C22×D4—31no—
146C22×D4C2×D311yescharacteristic
147C22×D4C2×D311yescharacteristic
148C22×D4C2×D311yescharacteristic
149C23×C4—31no—
150C23×C4C2×D311yescharacteristic
151#127—64no—
152#123—64no—
153#121—24no—
154#121—64no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4C2×D311yescharacteristic
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—31no—
162C2×C22⋊C4—31no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4—31no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4C2×D311yescharacteristic
170C2×C22⋊C4C2×D311yescharacteristic
171C2×C42—31no—
172#95—31no—
173#95—31no—
174#95—31no—
175#95—31no—
176#95—31no—
177#95—31no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3C2×D412yes—
198C23×C3C2×D412yes—
199C23×C3C2411yescharacteristic
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—21no—
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—21no—
213C22×D3—21no—
214C3×D4—44no—
215C3×D4—44no—
216C3×D4—44no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—44no—
228C2×C3×C4C2×D412yes—
229C2×C3×C4C2×D412yes—
230#67—44no—
231#67—44no—
232#67—44no—
233#67—44no—
234#67—44no—
235#67—44no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—44no—
252C2×Dic3C2×D412yes—
253C2×Dic3C2×D412yes—
254C2×Dic3#4112yes—
255D12—44no—
256C24—31no—
257C24—31no—
258C24C22×D311yescharacteristic
259C24C22×D311yescharacteristic
260C24C22×D311yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—62no—
270C2×D4—62no—
271C2×D4—62no—
272C2×D4—62no—
273C2×D4—62no—
274C2×D4—62no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4#6714yescomplemented
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—41no—
300C22×C4—62no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4C22×D311yescharacteristic
304C22×C4C22×D311yescharacteristic
305C22×C4C22×D311yescharacteristic
306C22×C4C22×D311yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—34no—
309C4⋊C4—62no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C42—62no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3C22×D411yescharacteristic
346C22×C3C22×D411yescharacteristic
347C22×C3C2×#4111yescharacteristic
348C22×C3H(5,𝔽2)11yescharacteristic
349C22×C3H(5,𝔽2)11yescharacteristic
350C22×C3H(5,𝔽2)11yescharacteristic
351C22×C3H(5,𝔽2)11yescharacteristic
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C2×D3—41no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—41no—
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—41no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—41no—
369C3×C4—41no—
370Dic3—22no—
371Dic3—22no—
372Dic3—22no—
373Dic3—41no—
374Dic3—41no—
375Dic3—41no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—61no—
399C23—61no—
400C23—61no—
401C23—61no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23—61no—
407C23—61no—
408C23—61no—
409C23C2×#6712yes—
410C23C2×#6712yes—
411C23C23×D311yescharacteristic
412D4—44no—
413D4—44no—
414D4—44no—
415D4—124no—
416D4—124no—
417D4—124no—
418D4—124no—
419D4—124no—
420D4—124no—
421D4—124no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—32no—
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—44no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—124no—
452C2×C4D3×D412yes—
453C2×C4C2×#6712yes—
454C2×C3—42no—
455C2×C3—42no—
456C2×C3—81no—
457C2×C3#53411yescharacteristic
458C2×C3#53511yescharacteristic
459C2×C3#54011yescharacteristic
460C2×C3D4211yescharacteristic
461C2×C3#54611yescharacteristic
462C2×C3#55011yescharacteristic
463C2×C3#56011yescharacteristic
464D3—81no—
465D3—81no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22—121no—
491C22—121no—
492C22—121no—
493C22—121no—
494C22—121no—
495C22—121no—
496C22—121no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22—121no—
501C22C2×D3×D411yescharacteristic
502C22#100411yescharacteristic
503C22#100411yescharacteristic
504C22#100411yescharacteristic
505C22D3×#4111yescharacteristic
506C22#100911yescharacteristic
507C22C22×#6711yescharacteristic
508C4—22no—
509C4—22no—
510C4—41no—
511C4—41no—
512C4—62no—
513C4—62no—
514C4—62no—
515C4—121no—
516C4—121no—
517C4—121no—
518C3#267311yescharacteristic, complemented, Sylow
519C2—42no—
520C2—42no—
521C2—81no—
522C2—241no—
523C2—241no—
524C2#559911yescharacteristic
525C2#560711yescharacteristic
526C2#561511yescharacteristic
527C2#562011yescharacteristic
528C2D4×#6711yescharacteristic
529C2#580411yescharacteristic
530C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2673
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#121C2semidirect
2. C2×#940C2semidirect
3. C2×#939C2semidirect
4. C2×#929C2semidirect
5. #5223C2semidirect
6. #5223C2semidirect
7. #5223C2semidirect
8. #5220C2semidirect
9. #5213C2semidirect
10. #4991C2semidirect
11. #4978C2semidirect
12. #4960C2semidirect
13. #4959C2semidirect
14. #4958C2semidirect
15. #4944C2semidirect
59. C2×#121D3semidirect
84. C2×C3×D4D4semidirect
137. Dic3⋊C4D4semidirect
290. C2×D4#67semidirect
518. C3#2673semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#9402
C2×#9392
C2×#9292
#52232
#52202
#52132
#49912
#49782
#49602
#49592
#49582
#49442
#26733