Finite Group Databasebeta

← #77685

Subgroups — #77685

Order 384.

702 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#882C211yesmaximal, characteristic
2C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
3C3×#397C211yesmaximal, characteristic, complemented
4#4991C211yesmaximal, characteristic, complemented
5#4989C211yesmaximal, characteristic, complemented
6#4961C211yesmaximal, characteristic, complemented
7#4959C211yesmaximal, characteristic, complemented
8#4955C211yesmaximal, characteristic, complemented
9#4952C211yesmaximal, characteristic, complemented
10#4675C211yesmaximal, characteristic, complemented
11#4674C211yesmaximal, characteristic, complemented
12#4672C211yesmaximal, characteristic, complemented
13#4668C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2622—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic, complemented
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
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, complemented
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
44#882—24no—
45D3×C22⋊C4—24no—
46C3×#95C2211yescharacteristic, complemented
47C3×#95C2211yescharacteristic, complemented
48C3×#95C2211yescharacteristic, complemented
49C3⋊#95C2211yescharacteristic, complemented
50C3⋊#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C2×#120—31no—
55C22×C22⋊C4—31no—
56#397—31no—
57#397—31no—
58#397—31no—
59#397D311yescharacteristic, complemented
60#394—31no—
61#394—31no—
62#394—31no—
63#394—31no—
64#386—31no—
65#386—31no—
66#385—31no—
67#385—31no—
68#379—31no—
69C24×C3C2311yescharacteristic
70C23×D3—22no—
71C23×D3—22no—
72C23×D3—41no—
73C23×D3—41no—
74C23×D3—41no—
75C23×D3—41no—
76C23×D3D414yes—
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—22no—
87C2×#67—41no—
88C2×#67—41no—
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—22no—
102C2×C4×D3—41no—
103C2×C4×D3—41no—
104C2×C4×D3—41no—
105C2×C4×D3—41no—
106C3×C4⋊C4—41no—
107C3×C22⋊C4—22no—
108C3×C22⋊C4—22no—
109C3×C22⋊C4—41no—
110#217—22no—
111#217—41no—
112(C2×D3)⋊C4—22no—
113(C2×D3)⋊C4—22no—
114(C2×D3)⋊C4—22no—
115(C2×D3)⋊C4—22no—
116(C2×D3)⋊C4—22no—
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—41no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—41no—
124C4⋊Dic3—41no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127C25—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—31no—
131C23×C4—31no—
132C23×C4—31no—
133#120—64no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4C2×D311yescharacteristic, complemented
138C2×C22⋊C4—31no—
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—64no—
153C2×C22⋊C4C2×D311yescharacteristic
154C2×C22⋊C4C2×D311yescharacteristic
155C2×C22⋊C4C2×D311yescharacteristic
156#95—31no—
157#95—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—28no—
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×D3C2×D412yes—
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×Dic3—21no—
234D12—44no—
235D12—44no—
236C4×D3—44no—
237C4×D3—44no—
238C24—31no—
239C24—31no—
240C24—34no—
241C24—31no—
242C24—62no—
243C24—62no—
244C24—121no—
245C24—121no—
246C24—121no—
247C24—121no—
248C24C22×D311yescharacteristic
249C2×D4—62no—
250C2×D4—62no—
251C2×D4—62no—
252C2×D4—62no—
253C2×D4—62no—
254C2×D4—121no—
255C2×D4—121no—
256C2×D4—121no—
257C2×D4—121no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4C22×D311yescharacteristic
270C22×C4C22×D311yescharacteristic
271C22×C4C22×D311yescharacteristic
272C22×C4C22×D311yescharacteristic
273C22×C4C22×D311yescharacteristic
274C22×C4C22×D311yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—41no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C22⋊C4—62no—
285C22⋊C4—62no—
286C22⋊C4—62no—
287C22⋊C4—62no—
288C22⋊C4—62no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3C22×D411yescharacteristic
304C22×C3C2×#4111yescharacteristic
305C22×C3C2×#4111yescharacteristic
306C22×C3H(5,𝔽2)11yescharacteristic
307C22×C3H(5,𝔽2)11yescharacteristic
308C22×C3H(5,𝔽2)11yescharacteristic
309C22×C3H(5,𝔽2)11yescharacteristic
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—22no—
319C2×D3—416no—
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—
331C2×D3#12314yes—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4—41no—
338Dic3—41no—
339Dic3—41no—
340Dic3—41no—
341Dic3—41no—
342Dic3—41no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—32no—
350C23—32no—
351C23—32no—
352C23—32no—
353C23—68no—
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—61no—
370C23—61no—
371C23—124no—
372C23—124no—
373C23—124no—
374C23—124no—
375C23—124no—
376C23—124no—
377C23D3×D412yes—
378C23C23×D311yescharacteristic
379D4—124no—
380D4—124no—
381D4—124no—
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—21no—
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—61no—
415C2×C4—124no—
416C2×C4—124no—
417C2×C3—42no—
418C2×C3C2×#12311yescharacteristic
419C2×C3#53411yescharacteristic
420C2×C3#53411yescharacteristic
421C2×C3#53811yescharacteristic
422C2×C3#53811yescharacteristic
423C2×C3#53811yescharacteristic
424C2×C3#53811yescharacteristic
425D3—24no—
426D3—42no—
427D3—81no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—34no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—62no—
437C22—62no—
438C22—62no—
439C22—62no—
440C22—62no—
441C22—62no—
442C22—62no—
443C22—62no—
444C22—62no—
445C22—1216no—
446C22—121no—
447C22—121no—
448C22—121no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22—121no—
454C22—121no—
455C22—121no—
456C22—121no—
457C22C2×D3×D411yescharacteristic
458C22#100411yescharacteristic
459C22D3×#4111yescharacteristic
460C22D3×#4111yescharacteristic
461C22#100911yescharacteristic
462C22#100911yescharacteristic
463C22#100911yescharacteristic
464C4—41no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—121no—
471C4—121no—
472C4—121no—
473C4—121no—
474C4—121no—
475C3#262211yescharacteristic, complemented, Sylow
476C2—42no—
477C2—64no—
478C2—122no—
479C2—241no—
480C2#560511yescharacteristic
481C2D3×#12311yescharacteristic
482C2#565311yescharacteristic
483C2#565711yescharacteristic
484C2#567811yescharacteristic
485C2#570411yescharacteristic
486C2#570411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radicalC2×#882C2
hypercenter#397D3
hypocenterC3#2622
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×D3×C22⋊C4C2semidirect
3. C3×#397C2semidirect
4. #4991C2semidirect
5. #4989C2semidirect
6. #4961C2semidirect
7. #4959C2semidirect
8. #4955C2semidirect
9. #4952C2semidirect
10. #4675C2semidirect
11. #4674C2semidirect
12. #4672C2semidirect
13. #4668C2semidirect
14. #4665C2semidirect
15. #4665C2semidirect
23. C2×C3×C4⋊C4C22semidirect
37. C2×(C2×D3)⋊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
46. C3×#95C22semidirect
47. C3×#95C22semidirect
48. C3×#95C22semidirect
49. C3⋊#95C22semidirect
50. C3⋊#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
59. #397D3semidirect
137. C2×C4⋊C4C2×D3semidirect
161. #95C2×D3semidirect
162. #95C2×D3semidirect
163. #95C2×D3semidirect
475. C3#2622semidirect

Maximal subgroups

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

subgroupindex
C2×#8822
C2×D3×C22⋊C42
C3×#3972
#49912
#49892
#49612
#49592
#49552
#49522
#46752
#46742
#46722
#46682
#46652
#26223