Finite Group Databasebeta

← #77696

Subgroups — #77696

Order 384.

680 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#883C211yesmaximal, characteristic, complemented
2C2×#882C211yesmaximal, characteristic
3C3×#397C211yesmaximal, characteristic, complemented
4#4994C211yesmaximal, characteristic, complemented
5#4991C211yesmaximal, characteristic, complemented
6#4957C211yesmaximal, characteristic, complemented
7#4955C211yesmaximal, characteristic, complemented
8#4955C211yesmaximal, characteristic, complemented
9#4952C211yesmaximal, characteristic, complemented
10#4676C211yesmaximal, characteristic, complemented
11#4672C211yesmaximal, characteristic, complemented
12#4670C211yesmaximal, characteristic, complemented
13#4669C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2697—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic, complemented
22C2×C3×C4⋊C4C2211yescharacteristic, complemented
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
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, complemented
39C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
41C2×C4⋊Dic3C2211yescharacteristic, complemented
42C2×Dic3⋊C4C2211yescharacteristic, complemented
43C2×Dic3⋊C4C2211yescharacteristic, complemented
44C2×Dic3⋊C4C2211yescharacteristic, complemented
45C2×C4×Dic3C2211yescharacteristic, complemented
46#883—24no—
47#882—24no—
48C3×#95C2211yescharacteristic, complemented
49C3×#95C2211yescharacteristic, complemented
50C3×#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C2×#123—31no—
55C2×#120—31no—
56#397—31no—
57#397D311yescharacteristic, complemented
58#396—31no—
59#396—31no—
60#394—31no—
61#394—31no—
62#390—31no—
63#388—31no—
64#388—31no—
65#386—31no—
66#386—31no—
67#379—31no—
68#379—31no—
69C24×C3C2311yescharacteristic
70C23×D3—22no—
71C23×D3—22no—
72C23×D3—22no—
73C23×D3—41no—
74C23×D3—41no—
75C23×D3—41no—
76C23×D3—41no—
77C23×D3C2311yescharacteristic
78C23×D3C2311yescharacteristic
79C23×D3C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C2×#67—41no—
87C2×#67—41no—
88C2×#67D414yes—
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C2×D12—22no—
95C2×D12—22no—
96C2×D12—22no—
97C2×D12—22no—
98C2×D12—41no—
99C2×D12—41no—
100C2×C4×D3—22no—
101C2×C4×D3—41no—
102C2×C4×D3—41no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C22⋊C4—41no—
107#217—22no—
108#217—41no—
109(C2×D3)⋊C4—22no—
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)⋊C4—41no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122C4⋊Dic3—22no—
123Dic3⋊C4—22no—
124Dic3⋊C4—41no—
125Dic3⋊C4—41no—
126C4×Dic3—41no—
127C25—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—31no—
131C23×C4—31no—
132#123—64no—
133#120—64no—
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
157C2×C42—31no—
158#95—31no—
159#95—31no—
160#95—31no—
161#95C2×D311yescharacteristic, complemented
162#95C2×D311yescharacteristic, complemented
163#95C2×D311yescharacteristic, complemented
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3C2×D412yes—
171C23×C3C2411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
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—44no—
191C22×D3—44no—
192C22×D3—44no—
193C22×D3—44no—
194C22×D3—44no—
195C22×D3—44no—
196C22×D3—44no—
197C22×D3C2×D412yes—
198C22×D3#4112yes—
199C22×D3#4112yes—
200C2×C3×C4—21no—
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×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218#67—44no—
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×Dic3C2×D412yes—
234D12—44no—
235D12—44no—
236C4×D3—44no—
237C24—31no—
238C24—31no—
239C24—31no—
240C24—62no—
241C24—62no—
242C24—62no—
243C24—121no—
244C24—121no—
245C24—121no—
246C24—121no—
247C24C22×D311yescharacteristic
248C2×D4—34no—
249C2×D4—62no—
250C2×D4—62no—
251C2×D4—62no—
252C2×D4—62no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C2×D4—121no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—62no—
263C22×C4—121no—
264C22×C4—121no—
265C22×C4C22×D311yescharacteristic
266C22×C4C22×D311yescharacteristic
267C22×C4C22×D311yescharacteristic
268C22×C4C22×D311yescharacteristic
269C22×C4C22×D311yescharacteristic
270C22×C4C22×D311yescharacteristic
271C4⋊C4—41no—
272C4⋊C4—62no—
273C4⋊C4—62no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—41no—
279C22⋊C4—62no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C22⋊C4—62no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C42—121no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3C22×D411yescharacteristic
303C22×C3C2×#4111yescharacteristic
304C22×C3C2×#4111yescharacteristic
305C22×C3H(5,𝔽2)11yescharacteristic
306C22×C3H(5,𝔽2)11yescharacteristic
307C22×C3H(5,𝔽2)11yescharacteristic
308C22×C3H(5,𝔽2)11yescharacteristic
309C2×D3—22no—
310C2×D3—22no—
311C2×D3—22no—
312C2×D3—22no—
313C2×D3—22no—
314C2×D3—22no—
315C2×D3—22no—
316C2×D3—22no—
317C2×D3—22no—
318C2×D3—416no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—41no—
329C2×D3—41no—
330C2×D3—41no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337Dic3—22no—
338Dic3—41no—
339Dic3—41no—
340Dic3—41no—
341Dic3—41no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—32no—
349C23—32no—
350C23—32no—
351C23—61no—
352C23—61no—
353C23—61no—
354C23—61no—
355C23—61no—
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—61no—
369C23—124no—
370C23—124no—
371C23—124no—
372C23—124no—
373C23—124no—
374C23—124no—
375C23—124no—
376C23D3×D412yes—
377C23C23×D311yescharacteristic
378D4—124no—
379D4—124no—
380D4—124no—
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—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—32no—
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×C4—61no—
413C2×C4—61no—
414C2×C4—124no—
415C2×C3—42no—
416C2×C3#53411yescharacteristic
417C2×C3#53811yescharacteristic
418C2×C3#53811yescharacteristic
419C2×C3#54611yescharacteristic
420C2×C3#54611yescharacteristic
421C2×C3#55111yescharacteristic
422C2×C3#56011yescharacteristic
423D3—42no—
424D3—42no—
425D3—42no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—62no—
438C22—62no—
439C22—62no—
440C22—62no—
441C22—62no—
442C22—1216no—
443C22—121no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22—121no—
448C22—121no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22—121no—
454C22—121no—
455C22C2×D3×D411yescharacteristic
456C22#100411yescharacteristic
457C22D3×#4111yescharacteristic
458C22D3×#4111yescharacteristic
459C22#100911yescharacteristic
460C22#100911yescharacteristic
461C22#100911yescharacteristic
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—62no—
469C4—121no—
470C4—121no—
471C4—121no—
472C4—121no—
473C3#269711yescharacteristic, complemented, Sylow
474C2—42no—
475C2—122no—
476C2—122no—
477C2—122no—
478C2#561011yescharacteristic
479C2#565311yescharacteristic
480C2#565411yescharacteristic
481C2#566611yescharacteristic
482C2#567811yescharacteristic
483C2#570411yescharacteristic
484C2#570511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radicalC2×#882C2
hypercenter#397D3
hypocenterC3#2697
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#883C2semidirect
3. C3×#397C2semidirect
4. #4994C2semidirect
5. #4991C2semidirect
6. #4957C2semidirect
7. #4955C2semidirect
8. #4955C2semidirect
9. #4952C2semidirect
10. #4676C2semidirect
11. #4672C2semidirect
12. #4670C2semidirect
13. #4669C2semidirect
14. #4665C2semidirect
15. #4665C2semidirect
21. C22×C4×D3C22semidirect
22. C2×C3×C4⋊C4C22semidirect
38. C2×(C2×D3)⋊C4C22semidirect
39. C2×(C2×D3)⋊C4C22semidirect
40. C2×(C2×D3)⋊C4C22semidirect
41. C2×C4⋊Dic3C22semidirect
42. C2×Dic3⋊C4C22semidirect
43. C2×Dic3⋊C4C22semidirect
44. C2×Dic3⋊C4C22semidirect
45. C2×C4×Dic3C22semidirect
48. C3×#95C22semidirect
49. C3×#95C22semidirect
50. C3×#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
57. #397D3semidirect
138. C2×C4⋊C4C2×D3semidirect
161. #95C2×D3semidirect
162. #95C2×D3semidirect
163. #95C2×D3semidirect
473. C3#2697semidirect

Maximal subgroups

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

subgroupindex
C2×#8832
C2×#8822
C3×#3972
#49942
#49912
#49572
#49552
#49522
#46762
#46722
#46702
#46692
#46652
#26973