Finite Group Databasebeta

← #77143

Subgroups — #77143

Order 384.

666 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#937C211yesmaximal, characteristic
2C2×#898C211yesmaximal, characteristic, complemented
3C3×#394C211yesmaximal, characteristic, complemented
4#5221C211yesmaximal, characteristic, complemented
5#4994C211yesmaximal, characteristic, complemented
6#4961C211yesmaximal, characteristic, complemented
7#4961C211yesmaximal, characteristic, complemented
8#4959C211yesmaximal, characteristic, complemented
9#4957C211yesmaximal, characteristic, complemented
10#4955C211yesmaximal, characteristic, complemented
11#4955C211yesmaximal, characteristic, complemented
12#4951C211yesmaximal, characteristic, complemented
13#4949C211yesmaximal, characteristic, complemented
14#4670C211yesmaximal, characteristic, complemented
15#4670C211yesmaximal, characteristic, complemented
16#2685—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic, complemented
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic, complemented
25C2×C3×C22⋊C4C2211yescharacteristic, complemented
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic, complemented
30C2×#217C2211yescharacteristic, complemented
31#937—24no—
32C2×(C2×D3)⋊C4C2211yescharacteristic
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×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#898—24no—
50C3×#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C2×#126—31no—
55C2×#120—31no—
56#399—31no—
57#396—31no—
58#394—31no—
59#394—31no—
60#394—31no—
61#394D311yescharacteristic, complemented
62#390—31no—
63#390—31no—
64#388—31no—
65#388—31no—
66#388—31no—
67#379—31no—
68#379—31no—
69C24×C3C2311yescharacteristic
70C24×C3C2311yescharacteristic
71C23×D3—22no—
72C23×D3—22no—
73C23×D3—22no—
74C23×D3—41no—
75C23×D3—41no—
76C23×D3—41no—
77C23×D3—41no—
78C23×D3C2311yescharacteristic
79C23×D3C2311yescharacteristic
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—41no—
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C2×#67—22no—
89C2×#67—22no—
90C2×#67—22no—
91C2×#67—41no—
92C2×#67—41no—
93C2×#67—41no—
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C3×C4⋊C4—22no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4—41no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—41no—
105#217—22no—
106#217—41no—
107#217—41no—
108#217—41no—
109#217—41no—
110(C2×D3)⋊C4—22no—
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—22no—
113(C2×D3)⋊C4—22no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—41no—
116(C2×D3)⋊C4—41no—
117(C2×D3)⋊C4—41no—
118(C2×D3)⋊C4—41no—
119(C2×D3)⋊C4D414yes—
120C4⋊Dic3—22no—
121C4⋊Dic3—41no—
122C4⋊Dic3—41no—
123Dic3⋊C4—22no—
124Dic3⋊C4—41no—
125C4×Dic3—22no—
126C4×Dic3—41no—
127C25—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4C2×D311yescharacteristic
131#126—64no—
132#120—64no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4C2×D311yescharacteristic, complemented
139C2×C22⋊C4—31no—
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊C4—31no—
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⋊C4C2×D311yescharacteristic
155C2×C22⋊C4C2×D311yescharacteristic
156C2×C22⋊C4C2×D311yescharacteristic, complemented
157C2×C22⋊C4C2×D311yescharacteristic, complemented
158C2×C42—31no—
159C2×C42—31no—
160#95—31no—
161#95—31no—
162#95—31no—
163#95C2×D311yescharacteristic, complemented
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3#4112yes—
178C23×C3C2411yescharacteristic
179C22×D3—21no—
180C22×D3—21no—
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—44no—
192C22×D3—44no—
193C22×D3—44no—
194C22×D3—44no—
195C22×D3—44no—
196C22×D3—44no—
197C22×D3—44no—
198C22×D3C2×D412yes—
199C22×D3#4112yes—
200C3×D4—44no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
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×C4C2×D412yes—
216#67—44no—
217#67—44no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—21no—
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×Dic3C2×D412yes—
236C24—31no—
237C24—31no—
238C24—62no—
239C24—62no—
240C24—62no—
241C24—121no—
242C24—121no—
243C24—121no—
244C24—121no—
245C24C22×D311yescharacteristic
246C24C22×D311yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—62no—
251C2×D4—62no—
252C2×D4—62no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4C22×D311yescharacteristic
263C22×C4C22×D311yescharacteristic
264C22×C4C22×D311yescharacteristic
265C22×C4C22×D311yescharacteristic
266C22×C4C22×D311yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—62no—
269C4⋊C4—62no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C22⋊C4—22no—
274C22⋊C4—34no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—62no—
279C22⋊C4—62no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C42—62no—
293C42—121no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3C22×D411yescharacteristic
309C22×C3C2×#4111yescharacteristic
310C22×C3C2×#4111yescharacteristic
311C22×C3H(5,𝔽2)11yescharacteristic
312C22×C3H(5,𝔽2)11yescharacteristic
313C22×C3H(5,𝔽2)11yescharacteristic
314C22×C3H(5,𝔽2)11yescharacteristic
315C2×D3—22no—
316C2×D3—22no—
317C2×D3—22no—
318C2×D3—22no—
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—416no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—41no—
329C2×D3—41no—
330C3×C4—22no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—41no—
335Dic3—22no—
336Dic3—41no—
337Dic3—41no—
338Dic3—41no—
339Dic3—41no—
340Dic3—41no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—32no—
355C23—32no—
356C23—61no—
357C23—61no—
358C23—61no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23—61no—
365C23—61no—
366C23—61no—
367C23—61no—
368C23—124no—
369C23—124no—
370C23—124no—
371C23—124no—
372C23—124no—
373C23—124no—
374C23—124no—
375C23#23712yes—
376C23C23×D311yescharacteristic
377D4—44no—
378D4—124no—
379D4—124no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—32no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4D3×D412yes—
413C2×C3—42no—
414C2×C3—81no—
415C2×C3#53511yescharacteristic
416C2×C3#53811yescharacteristic
417C2×C3#53811yescharacteristic
418C2×C3#54611yescharacteristic
419C2×C3#54611yescharacteristic
420C2×C3#55111yescharacteristic
421C2×C3#56011yescharacteristic
422D3—42no—
423D3—42no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—62no—
439C22—62no—
440C22—62no—
441C22—62no—
442C22—62no—
443C22—62no—
444C22—1216no—
445C22—121no—
446C22—121no—
447C22—121no—
448C22—121no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22C2×D3×D411yescharacteristic
454C22C2×#23711yescharacteristic
455C22#100411yescharacteristic
456C22#100411yescharacteristic
457C22D3×#4111yescharacteristic
458C22#100911yescharacteristic
459C22#100911yescharacteristic
460C4—22no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—62no—
466C4—121no—
467C4—121no—
468C4—121no—
469C4—121no—
470C4—121no—
471C3#268511yescharacteristic, complemented, Sylow
472C2—42no—
473C2—81no—
474C2—122no—
475C2—122no—
476C2#559111yescharacteristic
477C2#560711yescharacteristic
478C2#561311yescharacteristic
479C2#561711yescharacteristic
480C2#565811yescharacteristic
481C2#566611yescharacteristic
482C2#567611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC2×#937C2
hypercenter#394D3
hypocenterC3#2685
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#898C2semidirect
3. C3×#394C2semidirect
4. #5221C2semidirect
5. #4994C2semidirect
6. #4961C2semidirect
7. #4961C2semidirect
8. #4959C2semidirect
9. #4957C2semidirect
10. #4955C2semidirect
11. #4955C2semidirect
12. #4951C2semidirect
13. #4949C2semidirect
14. #4670C2semidirect
15. #4670C2semidirect
21. C2×C3×C4⋊C4C22semidirect
24. C2×C3×C22⋊C4C22semidirect
25. C2×C3×C22⋊C4C22semidirect
29. C2×#217C22semidirect
30. 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
50. C3×#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
61. #394D3semidirect
138. C2×C4⋊C4C2×D3semidirect
156. C2×C22⋊C4C2×D3semidirect
157. C2×C22⋊C4C2×D3semidirect
163. #95C2×D3semidirect
471. C3#2685semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#8982
C3×#3942
#52212
#49942
#49612
#49592
#49572
#49552
#49512
#49492
#46702
#26853