Finite Group Databasebeta

← #82128

Subgroups — #82128

Order 384.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 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
3C2×C4×#67C211yesmaximal, characteristic, complemented
4C2×#883C211yesmaximal, characteristic, complemented
5#5247C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#5212C211yesmaximal, characteristic, complemented
8#5210C211yesmaximal, characteristic, complemented
9#4961C211yesmaximal, characteristic, complemented
10#4959C211yesmaximal, characteristic, complemented
11#4956C211yesmaximal, characteristic, complemented
12#4952C211yesmaximal, characteristic, complemented
13#4951C211yesmaximal, characteristic, complemented
14#4949C211yesmaximal, characteristic, complemented
15#4945C211yesmaximal, characteristic, complemented
16#2418—31nomaximal, Sylow
17C25×C3C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic, complemented
24C2×C3×C22⋊C4—24no—
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29#953—24no—
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39C4×#67—24no—
40C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C2×C4⋊Dic3C2211yescharacteristic, complemented
44C2×C4⋊Dic3C2211yescharacteristic, complemented
45C2×Dic3⋊C4C2211yescharacteristic, complemented
46C2×Dic3⋊C4C2211yescharacteristic, complemented
47C2×Dic3⋊C4C2211yescharacteristic, complemented
48C2×C4×Dic3C2211yescharacteristic, complemented
49#883—24no—
50C3⋊#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#123—31no—
57C2×#120—31no—
58C2×C4×D4—31no—
59C22×C22⋊C4D311yescharacteristic, complemented
60#399—31no—
61#397—31no—
62#394—31no—
63#394—31no—
64#388—31no—
65#388—31no—
66#386—31no—
67#386—31no—
68#385—31no—
69#380—31no—
70#379—31no—
71C24×C3—22no—
72C24×C3—22no—
73C24×C3—22no—
74C24×C3—22no—
75C24×C3—22no—
76C24×C3—41no—
77C24×C3—41no—
78C24×C3C2311yescharacteristic
79C24×C3C2311yescharacteristic
80C24×C3C2311yescharacteristic
81C23×D3C2311yescharacteristic
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—41no—
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—41no—
99C2×#67—41no—
100C2×#67D414yes—
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C2×C4×D3—22no—
109C2×C4×D3—22no—
110C2×C4×D3—41no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4D414yescomplemented
115#217—22no—
116#217—22no—
117#217—22no—
118#217—22no—
119#217—22no—
120#217—22no—
121#217—22no—
122#217—41no—
123#217—41no—
124(C2×D3)⋊C4—22no—
125(C2×D3)⋊C4—22no—
126(C2×D3)⋊C4—41no—
127C4⋊Dic3—22no—
128C4⋊Dic3—41no—
129Dic3⋊C4—22no—
130Dic3⋊C4—22no—
131Dic3⋊C4—41no—
132C4×Dic3—22no—
133C25C2×D311yescharacteristic
134C22×D4—31no—
135C22×D4—31no—
136C22×D4—31no—
137C23×C4—31no—
138C23×C4C2×D311yescharacteristic
139C23×C4C2×D311yescharacteristic
140#123—64no—
141#120—64no—
142C4×D4—64no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C22⋊C4—24no—
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⋊C4—31no—
157C2×C22⋊C4—31no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4C2×D311yescharacteristic
162C2×C22⋊C4C2×D311yescharacteristic
163C2×C22⋊C4C2×D311yescharacteristic
164C2×C22⋊C4C2×D311yescharacteristic
165C2×C42—31no—
166#95—31no—
167#95—31no—
168#95—31no—
169#95—31no—
170#95—31no—
171#95—31no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—28no—
177C23×C3—21no—
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—44no—
190C23×C3—44no—
191C23×C3—44no—
192C23×C3—44no—
193C23×C3—44no—
194C23×C3—44no—
195C23×C3C2×D412yes—
196C23×C3C2×D412yes—
197C23×C3#4112yes—
198C23×C3#4112yes—
199C23×C3#4112yes—
200C23×C3C2411yescharacteristic
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—21no—
207C22×D3C2×D412yes—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—44no—
219C2×C3×C4—44no—
220C2×C3×C4C2×D412yes—
221C2×C3×C4C2×D412yes—
222#67—28no—
223#67—44no—
224#67—44no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
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×Dic3C2×D412yes—
246C4×D3—44no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—31no—
253C24—41no—
254C24—41no—
255C24C22×D311yescharacteristic
256C24C22×D311yescharacteristic
257C24C22×D311yescharacteristic
258C2×D4—34no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—121no—
266C2×D4—121no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—62no—
281C22×C4—62no—
282C22×C4—121no—
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D311yescharacteristic
285C22×C4C22×D311yescharacteristic
286C22×C4C22×D311yescharacteristic
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—62no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C22⋊C4—62no—
303C22⋊C4—62no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C22⋊C4—121no—
307C22⋊C4#6714yescomplemented
308C42—62no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—416no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3#12314yescomplemented
331C22×C3C22×D411yescharacteristic
332C22×C3C22×D411yescharacteristic
333C22×C3C2×#4111yescharacteristic
334C22×C3C2×#4111yescharacteristic
335C22×C3C2×#4111yescharacteristic
336C22×C3H(5,𝔽2)11yescharacteristic
337C22×C3H(5,𝔽2)11yescharacteristic
338C2×D3—22no—
339C2×D3—22no—
340C2×D3—22no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—41no—
344C2×D3—41no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—41no—
348C3×C4—41no—
349Dic3—22no—
350Dic3—41no—
351Dic3—41no—
352Dic3—41no—
353Dic3—41no—
354Dic3—41no—
355Dic3—41no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—28no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—32no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23—61no—
384C23—61no—
385C23—61no—
386C23#23512yes—
387C23#23512yes—
388C23D3×D412yes—
389C23#23712yes—
390C23C2×#6712yes—
391C23C23×D311yescharacteristic
392D4—68no—
393D4—124no—
394D4—124no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—32no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—124no—
429C2×C4C2×#6712yes—
430C2×C4C2×#6712yes—
431C2×C3—24no—
432C2×C3—42no—
433C2×C3—42no—
434C2×C3C2×#12311yescharacteristic
435C2×C3#52511yescharacteristic
436C2×C3#53811yescharacteristic
437C2×C3D4211yescharacteristic
438C2×C3#54611yescharacteristic
439C2×C3#54611yescharacteristic
440C2×C3#55111yescharacteristic
441D3—42no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—416no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—62no—
464C22—62no—
465C22—62no—
466C22—121no—
467C22—121no—
468C22—121no—
469C22—121no—
470C22#88314yescomplemented
471C22C2×#23511yescharacteristic
472C22C2×#23511yescharacteristic
473C22C2×D3×D411yescharacteristic
474C22C2×#23711yescharacteristic
475C22#100411yescharacteristic
476C22#100411yescharacteristic
477C22C22×#6711yescharacteristic
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—62no—
483C4—121no—
484C4—121no—
485C4—121no—
486C4—121no—
487C4—121no—
488C4—121no—
489C3#241811yescharacteristic, complemented, Sylow
490C2—24no—
491C2—42no—
492C2—42no—
493C2—122no—
494C2C2×#88311yescharacteristic
495C2#549011yescharacteristic
496C2#549411yescharacteristic
497C2#549511yescharacteristic
498C2#579111yescharacteristic
499C2D4×#6711yescharacteristic
500C2#580611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C22⋊C4C2
prime radicalC2×#953C2
hypercenterC22×C22⋊C4D3
hypocenterC3#2418
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
3. C2×C4×#67C2semidirect
4. C2×#883C2semidirect
5. #5247C2semidirect
6. #5213C2semidirect
7. #5212C2semidirect
8. #5210C2semidirect
9. #4961C2semidirect
10. #4959C2semidirect
11. #4956C2semidirect
12. #4952C2semidirect
13. #4951C2semidirect
14. #4949C2semidirect
15. #4945C2semidirect
23. C22×C4×D3C22semidirect
40. C2×(C2×D3)⋊C4C22semidirect
41. C2×(C2×D3)⋊C4C22semidirect
42. C2×(C2×D3)⋊C4C22semidirect
43. C2×C4⋊Dic3C22semidirect
44. C2×C4⋊Dic3C22semidirect
45. C2×Dic3⋊C4C22semidirect
46. C2×Dic3⋊C4C22semidirect
47. C2×Dic3⋊C4C22semidirect
48. C2×C4×Dic3C22semidirect
50. C3⋊#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
59. C22×C22⋊C4D3semidirect
114. C3×C22⋊C4D4semidirect
307. C22⋊C4#67semidirect
330. C22×C3#123semidirect
470. C22#883semidirect
489. C3#2418semidirect

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×C4×#672
C2×#8832
#52472
#52132
#52122
#52102
#49612
#49592
#49562
#49522
#49512
#49492
#49452
#24183