Finite Group Databasebeta

← #82109

Subgroups — #82109

Order 384.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#953C211yesmaximal, characteristic, complemented
3C2×#953C211yesmaximal, characteristic, complemented
4C2×#930C211yesmaximal, characteristic, complemented
5C2×C4×#67C211yesmaximal, characteristic, complemented
6C2×#884C211yesmaximal, characteristic, complemented
7#5213C211yesmaximal, characteristic, complemented
8#5210C211yesmaximal, characteristic, complemented
9#4960C211yesmaximal, characteristic, complemented
10#4959C211yesmaximal, characteristic, complemented
11#4959C211yesmaximal, characteristic, complemented
12#4957C211yesmaximal, characteristic, complemented
13#4956C211yesmaximal, characteristic, complemented
14#4951C211yesmaximal, characteristic, complemented
15#4948C211yesmaximal, characteristic, complemented
16#2424—31nomaximal, Sylow
17C25×C3C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic
25C22×#67C2211yescharacteristic
26C22×D12C2211yescharacteristic, complemented
27C22×C4×D3C2211yescharacteristic, complemented
28C2×C3×C22⋊C4—24no—
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33#953—24no—
34#953—24no—
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39C2×#217C2211yescharacteristic
40C2×#217C2211yescharacteristic
41#930—24no—
42C4×#67—24no—
43C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
48C2×C4⋊Dic3C2211yescharacteristic, complemented
49C2×Dic3⋊C4C2211yescharacteristic, complemented
50C2×Dic3⋊C4C2211yescharacteristic, complemented
51C2×Dic3⋊C4C2211yescharacteristic, complemented
52C2×C4×Dic3C2211yescharacteristic, complemented
53#884—24no—
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C3⋊#95C2211yescharacteristic, complemented
57C3⋊#95C2211yescharacteristic, complemented
58C2×#121—31no—
59C2×#121—31no—
60C2×#120—31no—
61C2×#120—31no—
62C2×C4×D4—31no—
63C22×C22⋊C4D311yescharacteristic, complemented
64#396—31no—
65#394—31no—
66#394—31no—
67#392—31no—
68#390—31no—
69#388—31no—
70#386—31no—
71#386—31no—
72#385—31no—
73C24×C3—22no—
74C24×C3—22no—
75C24×C3—22no—
76C24×C3—22no—
77C24×C3—22no—
78C24×C3—41no—
79C24×C3—41no—
80C24×C3C2311yescharacteristic
81C24×C3C2311yescharacteristic
82C24×C3C2311yescharacteristic
83C23×D3C2311yescharacteristic
84C23×D3C2311yescharacteristic
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—22no—
98C2×#67—22no—
99C2×#67—22no—
100C2×#67—22no—
101C2×#67—22no—
102C2×#67—22no—
103C2×#67—22no—
104C2×#67—22no—
105C2×#67—22no—
106C2×#67—22no—
107C2×#67—22no—
108C2×#67—41no—
109C2×#67—41no—
110C2×#67—41no—
111C2×#67—41no—
112C2×#67D414yescomplemented
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C22×Dic3C2311yescharacteristic
118C22×Dic3C2311yescharacteristic
119C2×D12—22no—
120C2×D12—22no—
121C2×D12—41no—
122C2×C4×D3—22no—
123C2×C4×D3—22no—
124C2×C4×D3—41no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—22no—
128C3×C22⋊C4D414yescomplemented
129#217—22no—
130#217—22no—
131#217—22no—
132#217—22no—
133#217—22no—
134#217D414yescomplemented
135(C2×D3)⋊C4—22no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—22no—
141Dic3⋊C4—22no—
142Dic3⋊C4—22no—
143Dic3⋊C4—41no—
144C4×Dic3—22no—
145C25C2×D311yescharacteristic
146C22×D4—31no—
147C22×D4—31no—
148C22×D4—31no—
149C22×D4—31no—
150C22×D4—31no—
151C22×D4—31no—
152C22×D4—31no—
153C23×C4—31no—
154C23×C4C2×D311yescharacteristic
155C23×C4C2×D311yescharacteristic
156#121—64no—
157#121—64no—
158#120—64no—
159#120—64no—
160C4×D4—64no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—31no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4—31no—
170C2×C22⋊C4—31no—
171C2×C22⋊C4—31no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—31no—
176C2×C22⋊C4—31no—
177C2×C22⋊C4C2×D311yescharacteristic
178C2×C22⋊C4C2×D311yescharacteristic
179C2×C22⋊C4C2×D311yescharacteristic
180C2×C22⋊C4C2×D311yescharacteristic
181C2×C42—31no—
182#95—31no—
183#95—31no—
184#95—31no—
185#95—31no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—28no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—44no—
204C23×C3—44no—
205C23×C3—44no—
206C23×C3—44no—
207C23×C3—44no—
208C23×C3—44no—
209C23×C3C2×D412yes—
210C23×C3C2×D412yes—
211C23×C3C2×D412yes—
212C23×C3#4112yes—
213C23×C3#4112yes—
214C23×C3C2411yescharacteristic
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—21no—
219C22×D3—21no—
220C22×D3—21no—
221C22×D3—21no—
222C22×D3—21no—
223C22×D3—21no—
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—21no—
227C22×D3—21no—
228C22×D3C2×D412yes—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—44no—
240C2×C3×C4—44no—
241C2×C3×C4C2×D412yes—
242C2×C3×C4C2×D412yes—
243#67—28no—
244#67—44no—
245#67—44no—
246#67—44no—
247#67—44no—
248#67—44no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3C2×D412yes—
265C2×Dic3C2×D412yes—
266C2×Dic3C2×D412yes—
267D12—44no—
268C4×D3—44no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—31no—
275C24—31no—
276C24—41no—
277C24—41no—
278C24C22×D311yescharacteristic
279C24C22×D311yescharacteristic
280C24C22×D311yescharacteristic
281C2×D4—34no—
282C2×D4—62no—
283C2×D4—62no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—62no—
287C2×D4—62no—
288C2×D4—62no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—62no—
296C2×D4—62no—
297C2×D4—121no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—121no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—121no—
317C22×C4C22×D311yescharacteristic
318C22×C4C22×D311yescharacteristic
319C22×C4C22×D311yescharacteristic
320C22×C4C22×D311yescharacteristic
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—121no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—34no—
329C22⋊C4—62no—
330C22⋊C4—62no—
331C22⋊C4—62no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C22⋊C4—62no—
335C22⋊C4—62no—
336C22⋊C4—62no—
337C22⋊C4—62no—
338C22⋊C4—121no—
339C22⋊C4#6714yescomplemented
340C42—62no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—416no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3#12114yescomplemented
363C22×C3C22×D411yescharacteristic
364C22×C3C22×D411yescharacteristic
365C22×C3C22×D411yescharacteristic
366C22×C3C2×#4111yescharacteristic
367C22×C3C2×#4111yescharacteristic
368C22×C3H(5,𝔽2)11yescharacteristic
369C22×C3H(5,𝔽2)11yescharacteristic
370C2×D3—22no—
371C2×D3—22no—
372C2×D3—22no—
373C2×D3—41no—
374C2×D3—41no—
375C2×D3—41no—
376C2×D3—41no—
377C2×D3—41no—
378C2×D3—41no—
379C2×D3—41no—
380C2×D3—41no—
381C2×D3—41no—
382C2×D3—41no—
383C2×D3—41no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—41no—
387C3×C4—41no—
388Dic3—22no—
389Dic3—22no—
390Dic3—22no—
391Dic3—41no—
392Dic3—41no—
393Dic3—41no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—28no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—32no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—61no—
419C23—61no—
420C23—61no—
421C23—61no—
422C23—61no—
423C23—61no—
424C23—61no—
425C23—61no—
426C23—61no—
427C23—61no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23#23512yes—
432C23#23512yes—
433C23D3×D412yes—
434C23D3×D412yes—
435C23C2×#6712yes—
436C23C23×D311yescharacteristic
437D4—68no—
438D4—124no—
439D4—124no—
440D4—124no—
441D4—124no—
442D4—124no—
443D4—124no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—32no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—124no—
475C2×C4C2×#6712yes—
476C2×C4C2×#6712yes—
477C2×C3—24no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3C2×#12111yescharacteristic
481C2×C3#52511yescharacteristic
482C2×C3#53811yescharacteristic
483C2×C3D4211yescharacteristic
484C2×C3D4211yescharacteristic
485C2×C3#54611yescharacteristic
486C2×C3#54611yescharacteristic
487D3—42no—
488D3—81no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—416no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—62no—
511C22—62no—
512C22—62no—
513C22—121no—
514C22—121no—
515C22—121no—
516C22—121no—
517C22—121no—
518C22—121no—
519C22—121no—
520C22—121no—
521C22—121no—
522C22—121no—
523C22—121no—
524C22#88414yescomplemented
525C22C2×#23511yescharacteristic
526C22C2×#23511yescharacteristic
527C22C2×D3×D411yescharacteristic
528C22C2×D3×D411yescharacteristic
529C22#100411yescharacteristic
530C22#100411yescharacteristic
531C22C22×#6711yescharacteristic
532C4—22no—
533C4—22no—
534C4—41no—
535C4—41no—
536C4—62no—
537C4—62no—
538C4—62no—
539C4—121no—
540C4—121no—
541C4—121no—
542C3#242411yescharacteristic, complemented, Sylow
543C2—24no—
544C2—42no—
545C2—42no—
546C2—122no—
547C2—241no—
548C2C2×#88411yescharacteristic
549C2#549011yescharacteristic
550C2#549011yescharacteristic
551C2#549411yescharacteristic
552C2#579111yescharacteristic
553C2D4×#6711yescharacteristic
554C2D4×#6711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C22⋊C4C2
hypercenterC22×C22⋊C4D3
hypocenterC3#2424
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊C4C2semidirect
2. C2×#953C2semidirect
3. C2×#953C2semidirect
4. C2×#930C2semidirect
5. C2×C4×#67C2semidirect
6. C2×#884C2semidirect
7. #5213C2semidirect
8. #5210C2semidirect
9. #4960C2semidirect
10. #4959C2semidirect
11. #4959C2semidirect
12. #4957C2semidirect
13. #4956C2semidirect
14. #4951C2semidirect
15. #4948C2semidirect
26. C22×D12C22semidirect
27. C22×C4×D3C22semidirect
43. C2×(C2×D3)⋊C4C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×(C2×D3)⋊C4C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×(C2×D3)⋊C4C22semidirect
48. C2×C4⋊Dic3C22semidirect
49. C2×Dic3⋊C4C22semidirect
50. C2×Dic3⋊C4C22semidirect
51. C2×Dic3⋊C4C22semidirect
52. C2×C4×Dic3C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
56. C3⋊#95C22semidirect
57. C3⋊#95C22semidirect
63. C22×C22⋊C4D3semidirect
112. C2×#67D4semidirect
128. C3×C22⋊C4D4semidirect
134. #217D4semidirect
339. C22⋊C4#67semidirect
362. C22×C3#121semidirect
524. C22#884semidirect
542. C3#2424semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊C42
C2×#9532
C2×#9302
C2×C4×#672
C2×#8842
#52132
#52102
#49602
#49592
#49572
#49562
#49512
#49482
#24243