Finite Group Databasebeta

← #77109

Subgroups — #77109

Order 384.

786 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#937C211yesmaximal, characteristic
2C2×#892C211yesmaximal, characteristic, complemented
3C2×#883C211yesmaximal, characteristic, complemented
4C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#394C211yesmaximal, characteristic, complemented
6#5221C211yesmaximal, characteristic, complemented
7#4994C211yesmaximal, characteristic, complemented
8#4961C211yesmaximal, characteristic, complemented
9#4957C211yesmaximal, characteristic, complemented
10#4956C211yesmaximal, characteristic, complemented
11#4955C211yesmaximal, characteristic, complemented
12#4951C211yesmaximal, characteristic, complemented
13#4948C211yesmaximal, characteristic, complemented
14#4670C211yesmaximal, characteristic, complemented
15#4668C211yesmaximal, characteristic, complemented
16#2411—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic, complemented
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic, complemented
27C2×C3×C22⋊C4C2211yescharacteristic, complemented
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic, complemented
32C2×#217C2211yescharacteristic, complemented
33#937—24no—
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×C4⋊Dic3C2211yescharacteristic, complemented
43C2×C4⋊Dic3C2211yescharacteristic, complemented
44C2×C4⋊Dic3C2211yescharacteristic, complemented
45C2×Dic3⋊C4C2211yescharacteristic, complemented
46C2×Dic3⋊C4C2211yescharacteristic, complemented
47C2×C4×Dic3C2211yescharacteristic, complemented
48C2×C4×Dic3C2211yescharacteristic, complemented
49#892—24no—
50#883—24no—
51D3×C22⋊C4—24no—
52C3×#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#123—31no—
57C2×#120—31no—
58C2×#117—31no—
59C22×C22⋊C4—31no—
60#396—31no—
61#396—31no—
62#394—31no—
63#394D311yescharacteristic, complemented
64#390—31no—
65#390—31no—
66#388—31no—
67#388—31no—
68#386—31no—
69#385—31no—
70#379—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C23×D3—22no—
74C23×D3—22no—
75C23×D3—22no—
76C23×D3—22no—
77C23×D3—22no—
78C23×D3—41no—
79C23×D3—41no—
80C23×D3C2311yescharacteristic
81C23×D3C2311yescharacteristic
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4—41no—
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C2×#67—22no—
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—41no—
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C2×C4×D3—22no—
103C2×C4×D3—22no—
104C2×C4×D3—22no—
105C2×C4×D3—41no—
106C2×C4×D3—41no—
107C2×C4×D3D414yescomplemented
108C3×C4⋊C4—22no—
109C3×C22⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—41no—
113#217—22no—
114#217—22no—
115#217—22no—
116#217—41no—
117#217—41no—
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—41no—
125(C2×D3)⋊C4D414yes—
126C4⋊Dic3—22no—
127C4⋊Dic3—22no—
128C4⋊Dic3—41no—
129Dic3⋊C4—22no—
130Dic3⋊C4—41no—
131C4×Dic3—22no—
132C4×Dic3—22no—
133C25—31no—
134C22×D4—31no—
135C22×D4—31no—
136C22×D4C2×D311yescharacteristic
137C23×C4—31no—
138C23×C4—31no—
139#123—64no—
140#120—64no—
141#117—64no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4C2×D311yescharacteristic, complemented
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×C22⋊C4C2×D311yescharacteristic, complemented
165C2×C22⋊C4C2×D311yescharacteristic, complemented
166C2×C42—31no—
167C2×C42—31no—
168#95—31no—
169#95—31no—
170#95—31no—
171#95C2×D311yescharacteristic, complemented
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
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#4112yes—
186C23×C3C2411yescharacteristic
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—28no—
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—44no—
199C22×D3—44no—
200C22×D3—44no—
201C22×D3—44no—
202C22×D3—44no—
203C22×D3—44no—
204C22×D3C2×D412yes—
205C22×D3C2×D412yes—
206C22×D3#4112yes—
207C22×D3#4112yes—
208C3×D4—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×C4C2×D412yes—
223C2×C3×C4C2×D412yes—
224#67—44no—
225#67—44no—
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×Dic3C2×D412yes—
243C2×Dic3C2×D412yes—
244C4×D3—28no—
245C4×D3—44no—
246C24—31no—
247C24—31no—
248C24—62no—
249C24—62no—
250C24—62no—
251C24—62no—
252C24—62no—
253C24—121no—
254C24—121no—
255C24C22×D311yescharacteristic
256C24C22×D311yescharacteristic
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—121no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—34no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4C22×D311yescharacteristic
279C22×C4C22×D311yescharacteristic
280C22×C4C22×D311yescharacteristic
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—62no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—34no—
293C22⋊C4—41no—
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—62no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C42—62no—
307C42—62no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3C22×D411yescharacteristic
323C22×C3C22×D411yescharacteristic
324C22×C3C2×#4111yescharacteristic
325C22×C3C2×#4111yescharacteristic
326C22×C3C2×#4111yescharacteristic
327C22×C3H(5,𝔽2)11yescharacteristic
328C22×C3H(5,𝔽2)11yescharacteristic
329C2×D3—22no—
330C2×D3—22no—
331C2×D3—22no—
332C2×D3—22no—
333C2×D3—22no—
334C2×D3—22no—
335C2×D3—22no—
336C2×D3—22no—
337C2×D3—22no—
338C2×D3—416no—
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3#12114yes—
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—41no—
349Dic3—22no—
350Dic3—22no—
351Dic3—41no—
352Dic3—41no—
353Dic3—41no—
354Dic3—41no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—32no—
369C23—32no—
370C23—32no—
371C23—32no—
372C23—68no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23—124no—
384C23—124no—
385C23—124no—
386C23—124no—
387C23—124no—
388C23—124no—
389C23#23712yes—
390C23C23×D311yescharacteristic
391D4—44no—
392D4—124no—
393D4—124no—
394C2×C4—21no—
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—21no—
406C2×C4—21no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—68no—
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—124no—
427C2×C4D3×D412yes—
428C2×C4D3×D412yes—
429C2×C3—42no—
430C2×C3—81no—
431C2×C3C2×#12111yescharacteristic
432C2×C3#52511yescharacteristic
433C2×C3#53511yescharacteristic
434C2×C3#54611yescharacteristic
435C2×C3#54611yescharacteristic
436C2×C3#55111yescharacteristic
437C2×C3#55111yescharacteristic
438D3—24no—
439D3—42no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—34no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—62no—
456C22—62no—
457C22—62no—
458C22—62no—
459C22—62no—
460C22—62no—
461C22—62no—
462C22—62no—
463C22—62no—
464C22—1216no—
465C22—121no—
466C22—121no—
467C22—121no—
468C22—121no—
469C22C2×D3×D411yescharacteristic
470C22C2×D3×D411yescharacteristic
471C22C2×#23711yescharacteristic
472C22#100411yescharacteristic
473C22D3×#4111yescharacteristic
474C22D3×#4111yescharacteristic
475C22#100911yescharacteristic
476C4—22no—
477C4—22no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—62no—
482C4—62no—
483C4—121no—
484C4—121no—
485C4—121no—
486C4—121no—
487C3#241111yescharacteristic, complemented, Sylow
488C2—42no—
489C2—64no—
490C2—81no—
491C2—122no—
492C2#559111yescharacteristic
493C2D3×#12111yescharacteristic
494C2#560611yescharacteristic
495C2#560711yescharacteristic
496C2#565811yescharacteristic
497C2#565911yescharacteristic
498C2#567611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC2×#937C2
hypercenter#394D3
hypocenterC3#2411
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#892C2semidirect
3. C2×#883C2semidirect
4. C2×D3×C22⋊C4C2semidirect
5. C3×#394C2semidirect
6. #5221C2semidirect
7. #4994C2semidirect
8. #4961C2semidirect
9. #4957C2semidirect
10. #4956C2semidirect
11. #4955C2semidirect
12. #4951C2semidirect
13. #4948C2semidirect
14. #4670C2semidirect
15. #4668C2semidirect
23. C2×C3×C4⋊C4C22semidirect
26. C2×C3×C22⋊C4C22semidirect
27. C2×C3×C22⋊C4C22semidirect
31. C2×#217C22semidirect
32. C2×#217C22semidirect
42. C2×C4⋊Dic3C22semidirect
43. C2×C4⋊Dic3C22semidirect
44. C2×C4⋊Dic3C22semidirect
45. C2×Dic3⋊C4C22semidirect
46. C2×Dic3⋊C4C22semidirect
47. C2×C4×Dic3C22semidirect
48. C2×C4×Dic3C22semidirect
52. C3×#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
63. #394D3semidirect
107. C2×C4×D3D4semidirect
147. C2×C4⋊C4C2×D3semidirect
164. C2×C22⋊C4C2×D3semidirect
165. C2×C22⋊C4C2×D3semidirect
171. #95C2×D3semidirect
487. C3#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#8922
C2×#8832
C2×D3×C22⋊C42
C3×#3942
#52212
#49942
#49612
#49572
#49562
#49552
#49512
#49482
#46702
#46682
#24113