Finite Group Databasebeta

← #82099

Subgroups — #82099

Order 384.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 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
3C22×#217C211yesmaximal, characteristic, complemented
4C2×C4×#67C211yesmaximal, characteristic, complemented
5C2×#67⋊C4C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#5210C211yesmaximal, characteristic, complemented
8C4×#217C211yesmaximal, characteristic, complemented
9#4958C211yesmaximal, characteristic, complemented
10#4957C211yesmaximal, characteristic, complemented
11#4956C211yesmaximal, characteristic, complemented
12#4946C211yesmaximal, characteristic, complemented
13#4945C211yesmaximal, characteristic, complemented
14#4944C211yesmaximal, characteristic, complemented
15Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
16#2358—31nomaximal, Sylow
17C25×C3C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C23×Dic3C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic, complemented
26C2×C3×C22⋊C4—24no—
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31#953C418yescomplemented
32C2×#217—24no—
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39C2×#217C2211yescharacteristic
40C4×#67—24no—
41C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C2×Dic3⋊C4C2211yescharacteristic, complemented
45C2×Dic3⋊C4C2211yescharacteristic, complemented
46C2×Dic3⋊C4C2211yescharacteristic, complemented
47C2×C4×Dic3C2211yescharacteristic, complemented
48C2×C4×Dic3C2211yescharacteristic, complemented
49C2×C4×Dic3C2211yescharacteristic, complemented
50#67⋊C4—24no—
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C3⋊#95C2211yescharacteristic, complemented
57C2×#120—31no—
58C2×C4×D4—31no—
59C2×C4×D4—31no—
60C22×C22⋊C4—31no—
61C22×C22⋊C4D311yescharacteristic, complemented
62#390—31no—
63#388—31no—
64#388—31no—
65#386—31no—
66#386—31no—
67#386—31no—
68#385—31no—
69#385—31no—
70C4×C22⋊C4—31no—
71C4×C22⋊C4—31no—
72C24×C3—22no—
73C24×C3—22no—
74C24×C3—22no—
75C24×C3—22no—
76C24×C3—41no—
77C24×C3—41no—
78C24×C3C2×C414yescomplemented
79C24×C3C2311yescharacteristic
80C24×C3C2311yescharacteristic
81C24×C3C2311yescharacteristic
82C23×D3C2311yescharacteristic
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—41no—
88C22×C3×C4—41no—
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—41no—
98C2×#67—41no—
99C2×#67C2×C414yescomplemented
100C2×#67C2×C414yescomplemented
101C2×#67C2×C414yescomplemented
102C22×Dic3—22no—
103C22×Dic3—22no—
104C22×Dic3—22no—
105C22×Dic3—22no—
106C22×Dic3—41no—
107C22×Dic3—41no—
108C22×Dic3C2311yescharacteristic
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C2×C4×D3—22no—
116C2×C4×D3—22no—
117C2×C4×D3—41no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—22no—
120C3×C22⋊C4—22no—
121C3×C22⋊C4D414yescomplemented
122#217—22no—
123#217—22no—
124#217—22no—
125#217C2×C414yescomplemented
126#217C2×C414yescomplemented
127#217C2×C414yescomplemented
128#217D414yescomplemented
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—41no—
132Dic3⋊C4—22no—
133Dic3⋊C4—22no—
134Dic3⋊C4—41no—
135C4×Dic3—22no—
136C4×Dic3—22no—
137C4×Dic3—41no—
138C25C2×D311yescharacteristic
139C22×D4—31no—
140C22×D4—31no—
141C22×D4—31no—
142C23×C4—31no—
143C23×C4—31no—
144C23×C4—31no—
145C23×C4C2×D311yescharacteristic
146C23×C4C2×D311yescharacteristic
147#120—38no—
148C4×D4—64no—
149C4×D4—64no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—31no—
157C2×C22⋊C4—31no—
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—64no—
165C2×C22⋊C4C2×D311yescharacteristic
166C2×C22⋊C4C2×D311yescharacteristic
167C2×C22⋊C4C2×D311yescharacteristic
168C2×C22⋊C4C2×D311yescharacteristic
169C2×C42—31no—
170C2×C42—31no—
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—28no—
187C23×C3—28no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—44no—
195C23×C3—44no—
196C23×C3—44no—
197C23×C3—44no—
198C23×C3—44no—
199C23×C3C22×C412yes—
200C23×C3C22×C412yes—
201C23×C3C22×C412yes—
202C23×C3C2×D412yes—
203C23×C3C2×D412yes—
204C23×C3#4112yes—
205C23×C3#4112yes—
206C23×C3C2411yescharacteristic
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—21no—
213C22×D3C22×C412yes—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
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—44no—
225C2×C3×C4—44no—
226C2×C3×C4C2×D412yes—
227C2×C3×C4C2×D412yes—
228#67—28no—
229#67—28no—
230#67—44no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
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—44no—
248C2×Dic3—44no—
249C2×Dic3C22×C412yes—
250C2×Dic3C22×C412yes—
251C2×Dic3C22×C412yes—
252C2×Dic3C2×D412yes—
253C2×Dic3C2×D412yes—
254C4×D3—44no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—31no—
260C24—41no—
261C24—41no—
262C24C4×D314yescomplemented
263C24C22×D311yescharacteristic
264C24C22×D311yescharacteristic
265C24C22×D311yescharacteristic
266C2×D4—34no—
267C2×D4—34no—
268C2×D4—34no—
269C2×D4—62no—
270C2×D4—62no—
271C2×D4—62no—
272C2×D4—62no—
273C2×D4—121no—
274C2×D4—121no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—62no—
293C22×C4—62no—
294C22×C4—121no—
295C22×C4—121no—
296C22×C4—121no—
297C22×C4C22×D311yescharacteristic
298C22×C4C22×D311yescharacteristic
299C22×C4C22×D311yescharacteristic
300C22×C4C22×D311yescharacteristic
301C4⋊C4—62no—
302C4⋊C4—62no—
303C4⋊C4—121no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—34no—
308C22⋊C4—34no—
309C22⋊C4—34no—
310C22⋊C4—34no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—121no—
317C22⋊C4#6714yescomplemented
318C42—62no—
319C42—62no—
320C42—121no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—416no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3C4×D414yescomplemented
342C22×C3C4×D414yescomplemented
343C22×C3C23×C411yescharacteristic
344C22×C3C22×D411yescharacteristic
345C22×C3C22×D411yescharacteristic
346C22×C3C2×#4111yescharacteristic
347C22×C3C2×#4111yescharacteristic
348C22×C3H(5,𝔽2)11yescharacteristic
349C22×C3H(5,𝔽2)11yescharacteristic
350C2×D3—22no—
351C2×D3—22no—
352C2×D3—22no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—41no—
360C3×C4—41no—
361Dic3—22no—
362Dic3—22no—
363Dic3—22no—
364Dic3—22no—
365Dic3—22no—
366Dic3—41no—
367Dic3—41no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—28no—
377C23—28no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—32no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—61no—
391C23—61no—
392C23—61no—
393C23—61no—
394C23—61no—
395C23—61no—
396C23C2×C4×D312yes—
397C23C2×C4×D312yes—
398C23C2×C4×D312yes—
399C23#23512yes—
400C23D3×D412yes—
401C23#23712yes—
402C23C2×#6712yes—
403C23C23×D311yescharacteristic
404D4—68no—
405D4—68no—
406D4—124no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—32no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—32no—
421C2×C4—32no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—124no—
441C2×C4—124no—
442C2×C4—124no—
443C2×C4C2×#6712yes—
444C2×C4C2×#6712yes—
445C2×C3—24no—
446C2×C3—24no—
447C2×C3—42no—
448C2×C3C2×C4×D411yescharacteristic
449C2×C3C2×C4×D411yescharacteristic
450C2×C3#51811yescharacteristic
451C2×C3D4211yescharacteristic
452C2×C3#54611yescharacteristic
453C2×C3#54611yescharacteristic
454C2×C3#55111yescharacteristic
455D3—42no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—416no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—62no—
477C22—62no—
478C22—62no—
479C22—121no—
480C22—121no—
481C22—121no—
482C22—121no—
483C22#67⋊C414yescomplemented
484C22C4×#6714yescomplemented
485C22C22×C4×D311yescharacteristic
486C22C2×#23511yescharacteristic
487C22C2×D3×D411yescharacteristic
488C22C2×#23711yescharacteristic
489C22#100411yescharacteristic
490C22#100411yescharacteristic
491C22C22×#6711yescharacteristic
492C4—22no—
493C4—22no—
494C4—41no—
495C4—41no—
496C4—62no—
497C4—62no—
498C4—62no—
499C4—62no—
500C4—62no—
501C4—121no—
502C4—121no—
503C3#235811yescharacteristic, complemented, Sylow
504C2—24no—
505C2—24no—
506C2—42no—
507C2—122no—
508C2C2×#67⋊C411yescharacteristic
509C2#548611yescharacteristic
510C2#549011yescharacteristic
511C2#549511yescharacteristic
512C2C2×C4×#6711yescharacteristic
513C2D4×#6711yescharacteristic
514C2#580611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC22×C3×C22⋊C4C2
socle, cosocleC23×C3C24
prime radicalC2×#953C2
hypercenterC22×C22⋊C4D3
hypocenterC3#2358
LCSG ▷ C22×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
3. C22×#217C2semidirect
4. C2×C4×#67C2semidirect
5. C2×#67⋊C4C2semidirect
6. #5213C2semidirect
7. #5210C2semidirect
8. C4×#217C2semidirect
9. #4958C2semidirect
10. #4957C2semidirect
11. #4956C2semidirect
12. #4946C2semidirect
13. #4945C2semidirect
14. #4944C2semidirect
15. Dic3×C22⋊C4C2semidirect
25. C22×C4×D3C22semidirect
31. #953C4semidirect
41. C2×(C2×D3)⋊C4C22semidirect
42. C2×(C2×D3)⋊C4C22semidirect
43. C2×(C2×D3)⋊C4C22semidirect
44. C2×Dic3⋊C4C22semidirect
45. C2×Dic3⋊C4C22semidirect
46. C2×Dic3⋊C4C22semidirect
47. C2×C4×Dic3C22semidirect
48. C2×C4×Dic3C22semidirect
49. C2×C4×Dic3C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
56. C3⋊#95C22semidirect
61. C22×C22⋊C4D3semidirect
78. C24×C3C2×C4semidirect
99. C2×#67C2×C4semidirect
100. C2×#67C2×C4semidirect
101. C2×#67C2×C4semidirect
121. C3×C22⋊C4D4semidirect
125. #217C2×C4semidirect
126. #217C2×C4semidirect
127. #217C2×C4semidirect
128. #217D4semidirect
262. C24C4×D3semidirect
317. C22⋊C4#67semidirect
341. C22×C3C4×D4semidirect
342. C22×C3C4×D4semidirect
483. C22#67⋊C4semidirect
484. C22C4×#67semidirect
503. C3#2358semidirect

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
C22×#2172
C2×C4×#672
C2×#67⋊C42
#52132
#52102
C4×#2172
#49582
#49572
#49562
#49462
#49452
#49442
Dic3×C22⋊C42
#23583