Finite Group Databasebeta

← #82769

Subgroups — #82769

Order 384.

786 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
1C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#947C211yesmaximal, characteristic, complemented
3C22×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
4C2×#882C211yesmaximal, characteristic
5#5231C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4992C211yesmaximal, characteristic, complemented
8#4991C211yesmaximal, characteristic, complemented
9#4991C211yesmaximal, characteristic, complemented
10#4990C211yesmaximal, characteristic, complemented
11#4990C211yesmaximal, characteristic, complemented
12#4988C211yesmaximal, characteristic, complemented
13#4958C211yesmaximal, characteristic, complemented
14#4955C211yesmaximal, characteristic, complemented
15Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
16#2472—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic, complemented
19C23×C3×C4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C23×Dic3C2211yescharacteristic
22C22×D12C2211yescharacteristic
23C22×D12C2211yescharacteristic
24C3×C22⋊Q8—24no—
25C2×C3×C4⋊C4C2211yescharacteristic, complemented
26C2×C3×C4⋊C4C2211yescharacteristic, complemented
27C2×C3×C4⋊C4C2211yescharacteristic, complemented
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×#217C2211yescharacteristic
31#947—24no—
32C2×(C2×D3)⋊C4—24no—
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C2×C4×Dic3C2211yescharacteristic, complemented
48C2×C4×Dic3C2211yescharacteristic, complemented
49#882—24no—
50C3⋊#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#124—31no—
57C2×C22⋊Q8D311yescharacteristic, complemented
58C2×#120—31no—
59C22×C22⋊C4—31no—
60#397—31no—
61#394—31no—
62#394—31no—
63#393—31no—
64#390—31no—
65#388—31no—
66#388—31no—
67#386—31no—
68#386—31no—
69#379—31no—
70C4×C22⋊C4—31no—
71C24×C3C2311yescharacteristic
72C23×D3—22no—
73C23×D3—22no—
74C23×D3—22no—
75C23×D3—22no—
76C23×D3—22no—
77C23×D3—41no—
78C23×D3—41no—
79C23×D3C2311yescharacteristic
80C23×D3C2311yescharacteristic
81C23×D3C2311yescharacteristic
82C2×C3×Q8—22no—
83C2×C3×Q8—22no—
84C2×C3×Q8—41no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—41no—
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C2×#67—22no—
95C2×#67—22no—
96C2×#67—22no—
97C22×Dic3—22no—
98C22×Dic3—22no—
99C22×Dic3—41no—
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C2×D12—22no—
106C2×D12—22no—
107C2×D12—22no—
108C2×D12—22no—
109C2×D12—41no—
110C2×D12—41no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4D414yes—
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)⋊C4—22no—
123(C2×D3)⋊C4—22no—
124(C2×D3)⋊C4—22no—
125(C2×D3)⋊C4—22no—
126(C2×D3)⋊C4—41no—
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4D414yescomplemented
131C4×Dic3—22no—
132C4×Dic3—41no—
133C25—31no—
134C22×Q8C2×D311yescharacteristic, complemented
135C22×D4—31no—
136C22×D4—31no—
137C22×D4—31no—
138C23×C4—31no—
139C23×C4C2×D311yescharacteristic
140#124—64no—
141C22⋊Q8—24no—
142#120—64no—
143C2×C4⋊C4C2×D311yescharacteristic, complemented
144C2×C4⋊C4C2×D311yescharacteristic, complemented
145C2×C4⋊C4C2×D311yescharacteristic, complemented
146C2×C22⋊C4—31no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
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⋊C4—64no—
162C2×C22⋊C4C2×D311yescharacteristic
163C2×C22⋊C4C2×D311yescharacteristic
164C2×C42—31no—
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×C3C2×D412yes—
177C23×C3#4112yes—
178C23×C3#4112yes—
179C23×C3C2411yescharacteristic
180C22×D3—28no—
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×D3—21no—
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
199C22×D3—21no—
200C22×D3—44no—
201C22×D3—44no—
202C22×D3—44no—
203C22×D3—44no—
204C22×D3—44no—
205C22×D3—44no—
206C22×D3C2×D412yes—
207C22×D3#4112yes—
208C3×Q8—44no—
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—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×C4C2×D412yes—
226C2×C3×C4C2×D412yes—
227C2×C3×C4C2×D412yes—
228#67—44no—
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—44no—
244C2×Dic3C2×D412yes—
245D12—44no—
246D12—44no—
247C24—31no—
248C24—31no—
249C24—31no—
250C24—62no—
251C24—62no—
252C24—62no—
253C24—62no—
254C24—62no—
255C24—121no—
256C24—121no—
257C24C22×D311yescharacteristic
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—121no—
269C2×D4—121no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—31no—
277C22×C4—41no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—121no—
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D311yescharacteristic
285C22×C4C22×D311yescharacteristic
286C22×C4C22×D311yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—34no—
292C22⋊C4—62no—
293C22⋊C4—62no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C22⋊C4#6714yes—
307C42—62no—
308C42—121no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3#12414yescomplemented
316C22×C3C22×D411yescharacteristic
317C22×C3C22×D411yescharacteristic
318C22×C3C2×#4111yescharacteristic
319C22×C3C2×#4111yescharacteristic
320C22×C3C2×#4111yescharacteristic
321C22×C3H(5,𝔽2)11yescharacteristic
322C22×C3H(5,𝔽2)11yescharacteristic
323C2×D3—22no—
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—22no—
327C2×D3—22no—
328C2×D3—22no—
329C2×D3—416no—
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—41no—
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—41no—
344C2×D3—41no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—41no—
351Dic3—22no—
352Dic3—41no—
353Dic3—41no—
354Dic3—41no—
355Dic3—41no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—32no—
361C23—32no—
362C23—68no—
363C23—61no—
364C23—61no—
365C23—61no—
366C23—61no—
367C23—61no—
368C23—61no—
369C23—61no—
370C23—61no—
371C23—61no—
372C23—61no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—124no—
383C23—124no—
384C23—124no—
385C23—124no—
386C23—124no—
387C23—124no—
388C23#23912yes—
389C23#23912yes—
390C23C2×#6712yes—
391C23C23×D311yescharacteristic
392Q8—44no—
393D4—124no—
394D4—124no—
395D4—124no—
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—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—32no—
412C2×C4—44no—
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—124no—
428C2×C4D3×D412yes—
429C2×C4C2×#6712yes—
430C2×C4C2×#6712yes—
431C2×C3—24no—
432C2×C3#52511yescharacteristic
433C2×C3C2×#12411yescharacteristic
434C2×C3#53811yescharacteristic
435C2×C3D4211yescharacteristic
436C2×C3#54611yescharacteristic
437C2×C3#54611yescharacteristic
438C2×C3#55111yescharacteristic
439D3—42no—
440D3—42no—
441D3—81no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—62no—
449C22—62no—
450C22—62no—
451C22—62no—
452C22—62no—
453C22—62no—
454C22—1216no—
455C22—121no—
456C22—121no—
457C22—121no—
458C22—121no—
459C22—121no—
460C22—121no—
461C22—121no—
462C22—121no—
463C22—121no—
464C22—121no—
465C22—121no—
466C22—121no—
467C22—121no—
468C22—121no—
469C22—121no—
470C22#94714yescomplemented
471C22C2×D3×D411yescharacteristic
472C22#100411yescharacteristic
473C22C2×#23911yescharacteristic
474C22C2×#23911yescharacteristic
475C22D3×#4111yescharacteristic
476C22#100911yescharacteristic
477C22C22×#6711yescharacteristic
478C4—22no—
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—62no—
485C4—121no—
486C4—121no—
487C4—121no—
488C4—121no—
489C3#247211yescharacteristic, complemented, Sylow
490C2—24no—
491C2—122no—
492C2—122no—
493C2—241no—
494C2#562711yescharacteristic
495C2#563011yescharacteristic
496C2#563711yescharacteristic
497C2#564411yescharacteristic
498C2D4×#6711yescharacteristic
499C2C2×#94711yescharacteristic
500C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC2×#882C2
hypercenterC2×C22⋊Q8D3
hypocenterC3#2472
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊Q8C2semidirect
2. C2×#947C2semidirect
3. C22×(C2×D3)⋊C4C2semidirect
5. #5231C2semidirect
6. #5213C2semidirect
7. #4992C2semidirect
8. #4991C2semidirect
9. #4991C2semidirect
10. #4990C2semidirect
11. #4990C2semidirect
12. #4988C2semidirect
13. #4958C2semidirect
14. #4955C2semidirect
15. Dic3×C22⋊C4C2semidirect
18. C22×C3×Q8C22semidirect
25. C2×C3×C4⋊C4C22semidirect
26. C2×C3×C4⋊C4C22semidirect
27. C2×C3×C4⋊C4C22semidirect
43. C2×(C2×D3)⋊C4C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×(C2×D3)⋊C4C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×C4×Dic3C22semidirect
48. C2×C4×Dic3C22semidirect
50. C3⋊#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
57. C2×C22⋊Q8D3semidirect
130. (C2×D3)⋊C4D4semidirect
134. C22×Q8C2×D3semidirect
143. C2×C4⋊C4C2×D3semidirect
144. C2×C4⋊C4C2×D3semidirect
145. C2×C4⋊C4C2×D3semidirect
315. C22×C3#124semidirect
470. C22#947semidirect
489. C3#2472semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C2×#9472
C22×(C2×D3)⋊C42
C2×#8822
#52312
#52132
#49922
#49912
#49902
#49882
#49582
#49552
Dic3×C22⋊C42
#24723