Finite Group Databasebeta

← #74919

Subgroups — #74919

Order 384.

688 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×#937C211yesmaximal, characteristic
2C2×#885C211yesmaximal, characteristic, complemented
3C3×#386C211yesmaximal, characteristic, complemented
4#5222C211yesmaximal, characteristic, complemented
5#5213C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4959C211yesmaximal, characteristic, complemented
8#4955C211yesmaximal, characteristic, complemented
9#4955C211yesmaximal, characteristic, complemented
10#4947C211yesmaximal, characteristic, complemented
11#4945C211yesmaximal, characteristic, complemented
12#4675C211yesmaximal, characteristic, complemented
13#4675C211yesmaximal, characteristic, complemented
14#4673C211yesmaximal, characteristic, complemented
15#4669C211yesmaximal, characteristic, complemented
16#2684—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic, complemented
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×Dic6C2211yescharacteristic, complemented
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
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×Dic3⋊C4C2211yescharacteristic, complemented
43C2×Dic3⋊C4C2211yescharacteristic, complemented
44C2×C4×Dic3C2211yescharacteristic, complemented
45#885—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×#124—31no—
55C2×#120—31no—
56#397—31no—
57#397—31no—
58#394—31no—
59#394—31no—
60#393—31no—
61#393—31no—
62#388—31no—
63#388—31no—
64#386—31no—
65#386—31no—
66#386D311yescharacteristic, complemented
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×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4—41no—
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—41no—
94C2×#67—41no—
95C2×#67—41no—
96C2×#67D414yes—
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C2×Dic6—22no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C3×C22⋊C4—22no—
107C3×C22⋊C4—41no—
108C3×C22⋊C4—41no—
109#217—22no—
110#217—22no—
111#217—22no—
112#217—41no—
113#217—41no—
114(C2×D3)⋊C4—22no—
115(C2×D3)⋊C4—22no—
116(C2×D3)⋊C4—22no—
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—22no—
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—
124Dic3⋊C4—41no—
125Dic3⋊C4—41no—
126C4×Dic3—22no—
127C25—31no—
128C22×Q8—31no—
129C22×D4—31no—
130C22×D4—31no—
131C22×D4C2×D311yescharacteristic
132C23×C4C2×D311yescharacteristic, complemented
133#124—64no—
134#120—64no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C22⋊C4—31no—
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⋊C4C2×D311yescharacteristic
153C2×C22⋊C4C2×D311yescharacteristic
154C2×C22⋊C4C2×D311yescharacteristic, complemented
155C2×C42—31no—
156#95—31no—
157#95—31no—
158#95—31no—
159#95—31no—
160#95—31no—
161#95—31no—
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×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3C2×D412yes—
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×C4—21no—
216C2×C3×C4—44no—
217#67—44no—
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—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3C2×D412yes—
237Dic6—44no—
238C24—31no—
239C24—31no—
240C24—62no—
241C24—62no—
242C24—62no—
243C24—121no—
244C24—121no—
245C24—121no—
246C24—121no—
247C24C22×D311yescharacteristic
248C24C22×D311yescharacteristic
249C2×Q8—62no—
250C2×Q8—121no—
251C2×Q8—121no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—34no—
255C2×D4—41no—
256C2×D4—62no—
257C2×D4—62no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—121no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4C22×D311yescharacteristic
271C22×C4C22×D311yescharacteristic
272C22×C4C22×D311yescharacteristic
273C22×C4C22×D311yescharacteristic
274C22×C4C22×D311yescharacteristic
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—22no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C22⋊C4—62no—
285C22⋊C4—62no—
286C22⋊C4—62no—
287C22⋊C4—62no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C22⋊C4—121no—
295C42—62no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3C22×D411yescharacteristic
311C22×C3C2×#4111yescharacteristic
312C22×C3C2×#4111yescharacteristic
313C22×C3H(5,𝔽2)11yescharacteristic
314C22×C3H(5,𝔽2)11yescharacteristic
315C22×C3H(5,𝔽2)11yescharacteristic
316C22×C3H(5,𝔽2)11yescharacteristic
317C2×D3—22no—
318C2×D3—22no—
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—22no—
322C2×D3—22no—
323C2×D3—416no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—41no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—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—
342Dic3—41no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—32no—
356C23—32no—
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—
376C23#23512yes—
377C23D3×D412yes—
378C23C23×D311yescharacteristic
379Q8—124no—
380D4—44no—
381D4—124no—
382D4—124no—
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—32no—
399C2×C4—44no—
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—61no—
416C2×C4—61no—
417C2×C3—42no—
418C2×C3—42no—
419C2×C3#53411yescharacteristic
420C2×C3#53811yescharacteristic
421C2×C3#53811yescharacteristic
422C2×C3#54611yescharacteristic
423C2×C3#54611yescharacteristic
424C2×C3#55111yescharacteristic
425C2×C3H(3,𝔽4)11yescharacteristic
426D3—42no—
427D3—42no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—62no—
443C22—62no—
444C22—62no—
445C22—62no—
446C22—62no—
447C22—62no—
448C22—1216no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22—121no—
454C22—121no—
455C22—121no—
456C22—121no—
457C22C2×#23511yescharacteristic
458C22C2×D3×D411yescharacteristic
459C22#100411yescharacteristic
460C22#100411yescharacteristic
461C22#100411yescharacteristic
462C22D3×#4111yescharacteristic
463C22#100911yescharacteristic
464C4—41no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—62no—
470C4—121no—
471C4—121no—
472C4—121no—
473C4—121no—
474C4—121no—
475C3#268411yescharacteristic, complemented, Sylow
476C2—42no—
477C2—42no—
478C2—122no—
479C2—122no—
480C2#549011yescharacteristic
481C2#555611yescharacteristic
482C2#556011yescharacteristic
483C2#559411yescharacteristic
484C2#560511yescharacteristic
485C2#561011yescharacteristic
486C2#565711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC2×#937C2
hypercenter#386D3
hypocenterC3#2684
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#885C2semidirect
3. C3×#386C2semidirect
4. #5222C2semidirect
5. #5213C2semidirect
6. #5213C2semidirect
7. #4959C2semidirect
8. #4955C2semidirect
9. #4955C2semidirect
10. #4947C2semidirect
11. #4945C2semidirect
12. #4675C2semidirect
13. #4675C2semidirect
14. #4673C2semidirect
15. #4669C2semidirect
19. C23×C3×C4C22semidirect
22. C22×Dic6C22semidirect
25. C2×C3×C22⋊C4C22semidirect
29. C2×#217C22semidirect
30. C2×#217C22semidirect
42. C2×Dic3⋊C4C22semidirect
43. C2×Dic3⋊C4C22semidirect
44. C2×C4×Dic3C22semidirect
46. C3×#95C22semidirect
47. C3×#95C22semidirect
48. C3⋊#95C22semidirect
49. C3⋊#95C22semidirect
50. C3⋊#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
66. #386D3semidirect
132. C23×C4C2×D3semidirect
154. C2×C22⋊C4C2×D3semidirect
162. #95C2×D3semidirect
163. #95C2×D3semidirect
475. C3#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#8852
C3×#3862
#52222
#52132
#49592
#49552
#49472
#49452
#46752
#46732
#46692
#26843