Finite Group Databasebeta

← #76926

Subgroups — #76926

Order 384.

786 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#937C211yesmaximal, characteristic
2C2×#936C211yesmaximal, characteristic, complemented
3C2×#896C211yesmaximal, characteristic, complemented
4C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#394C211yesmaximal, characteristic, complemented
6#4990C211yesmaximal, characteristic, complemented
7#4959C211yesmaximal, characteristic, complemented
8#4957C211yesmaximal, characteristic, complemented
9#4956C211yesmaximal, characteristic, complemented
10#4956C211yesmaximal, characteristic, complemented
11#4955C211yesmaximal, characteristic, complemented
12#4952C211yesmaximal, characteristic, complemented
13#4944C211yesmaximal, characteristic, complemented
14#4673C211yesmaximal, characteristic, complemented
15#4666C211yesmaximal, characteristic, complemented
16#2472—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×Dic6C2211yescharacteristic, complemented
24C2×C3×C4⋊C4C2211yescharacteristic, complemented
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic, complemented
28C2×C3×C22⋊C4C2211yescharacteristic, complemented
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic, complemented
33C2×#217C2211yescharacteristic, complemented
34#937—24no—
35#936—24no—
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×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic, complemented
45C2×Dic3⋊C4C2211yescharacteristic, complemented
46C2×C4×Dic3C2211yescharacteristic, complemented
47C2×C4×Dic3C2211yescharacteristic, complemented
48#896—24no—
49D3×C22⋊C4—24no—
50C3×#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#124—31no—
57C2×C22⋊Q8—31no—
58C2×#120—31no—
59C22×C22⋊C4—31no—
60#397—31no—
61#394—31no—
62#394D311yescharacteristic, complemented
63#393—31no—
64#390—31no—
65#388—31no—
66#388—31no—
67#386—31no—
68#386—31no—
69#379—31no—
70C4×C22⋊C4—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—22no—
106C2×C4×D3—41no—
107C2×C4×D3—41no—
108C2×Dic6—22no—
109C2×Dic6—22no—
110C2×Dic6—41no—
111C3×C4⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116#217—22no—
117#217—22no—
118#217—22no—
119#217—22no—
120#217D414yescomplemented
121(C2×D3)⋊C4—22no—
122(C2×D3)⋊C4—22no—
123(C2×D3)⋊C4—22no—
124(C2×D3)⋊C4—22no—
125(C2×D3)⋊C4—22no—
126(C2×D3)⋊C4—22no—
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4D414yes—
129Dic3⋊C4—22no—
130Dic3⋊C4—22no—
131C4×Dic3—22no—
132C4×Dic3—41no—
133C25—31no—
134C22×Q8—31no—
135C22×D4—31no—
136C22×D4—31no—
137C22×D4C2×D311yescharacteristic
138C23×C4—31no—
139C23×C4—31no—
140#124—64no—
141C22⋊Q8—64no—
142#120—64no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4C2×D311yescharacteristic, complemented
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⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—31no—
157C2×C22⋊C4—31no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—64no—
160C2×C22⋊C4C2×D311yescharacteristic
161C2×C22⋊C4C2×D311yescharacteristic
162C2×C22⋊C4C2×D311yescharacteristic, complemented
163C2×C22⋊C4C2×D311yescharacteristic, complemented
164C2×C42—31no—
165C2×C42—31no—
166#95—31no—
167#95—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×C3C2×D412yes—
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×D3#4112yes—
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×C4—21no—
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×Dic3C2×D412yes—
242C2×Dic3C2×D412yes—
243C2×Dic3C2×D412yes—
244C4×D3—44no—
245C4×D3—44no—
246Dic6—44no—
247C24—31no—
248C24—31no—
249C24—62no—
250C24—62no—
251C24—62no—
252C24—62no—
253C24—62no—
254C24—121no—
255C24—121no—
256C24C22×D311yescharacteristic
257C24C22×D311yescharacteristic
258C2×Q8—62no—
259C2×Q8—62no—
260C2×Q8—121no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—121no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—62no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—121no—
281C22×C4—121no—
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D311yescharacteristic
285C22×C4C22×D311yescharacteristic
286C22×C4C22×D311yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—62no—
289C4⋊C4—62no—
290C22⋊C4—22no—
291C22⋊C4—34no—
292C22⋊C4—34no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C22⋊C4—62no—
303C22⋊C4—62no—
304C22⋊C4—62no—
305C22⋊C4—62no—
306C22⋊C4—121no—
307C42—62no—
308C42—121no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
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×C3—41no—
323C22×C3C22×D411yescharacteristic
324C22×C3C22×D411yescharacteristic
325C22×C3C2×#4111yescharacteristic
326C22×C3C2×#4111yescharacteristic
327C22×C3C2×#4111yescharacteristic
328C22×C3H(5,𝔽2)11yescharacteristic
329C22×C3H(5,𝔽2)11yescharacteristic
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—22no—
339C2×D3—416no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—41no—
344C2×D3#12414yescomplemented
345C3×C4—22no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—41no—
350Dic3—22no—
351Dic3—22no—
352Dic3—22no—
353Dic3—41no—
354Dic3—41no—
355Dic3—41no—
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—21no—
369C23—32no—
370C23—32no—
371C23—32no—
372C23—32no—
373C23—68no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23—61no—
384C23—124no—
385C23—124no—
386C23—124no—
387C23—124no—
388C23—124no—
389C23—124no—
390C23D3×D412yes—
391C23C23×D311yescharacteristic
392Q8—124no—
393D4—44no—
394D4—124no—
395D4—124no—
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—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—32no—
411C2×C4—32no—
412C2×C4—32no—
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—61no—
427C2×C4—61no—
428C2×C4—124no—
429C2×C4—124no—
430C2×C4D3×D412yes—
431C2×C3—42no—
432C2×C3—81no—
433C2×C3#52511yescharacteristic
434C2×C3C2×#12411yescharacteristic
435C2×C3#53811yescharacteristic
436C2×C3D4211yescharacteristic
437C2×C3#54611yescharacteristic
438C2×C3#54611yescharacteristic
439C2×C3#55111yescharacteristic
440D3—24no—
441D3—42no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—34no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—62no—
458C22—62no—
459C22—62no—
460C22—62no—
461C22—62no—
462C22—62no—
463C22—62no—
464C22—62no—
465C22—62no—
466C22—1216no—
467C22—121no—
468C22—121no—
469C22—121no—
470C22—121no—
471C22C2×D3×D411yescharacteristic
472C22C2×D3×D411yescharacteristic
473C22#100411yescharacteristic
474C22#100411yescharacteristic
475C22D3×#4111yescharacteristic
476C22D3×#4111yescharacteristic
477C22D3×#4111yescharacteristic
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—62no—
484C4—62no—
485C4—62no—
486C4—121no—
487C4—121no—
488C4—121no—
489C3#247211yescharacteristic, complemented, Sylow
490C2—42no—
491C2—64no—
492C2—81no—
493C2—122no—
494C2#559011yescharacteristic
495C2#561011yescharacteristic
496C2#561211yescharacteristic
497C2#561311yescharacteristic
498C2#565611yescharacteristic
499C2#565911yescharacteristic
500C2D3×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC2×#937C2
hypercenter#394D3
hypocenterC3#2472
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#936C2semidirect
3. C2×#896C2semidirect
4. C2×D3×C22⋊C4C2semidirect
5. C3×#394C2semidirect
6. #4990C2semidirect
7. #4959C2semidirect
8. #4957C2semidirect
9. #4956C2semidirect
10. #4956C2semidirect
11. #4955C2semidirect
12. #4952C2semidirect
13. #4944C2semidirect
14. #4673C2semidirect
15. #4666C2semidirect
23. C22×Dic6C22semidirect
24. C2×C3×C4⋊C4C22semidirect
27. C2×C3×C22⋊C4C22semidirect
28. C2×C3×C22⋊C4C22semidirect
32. C2×#217C22semidirect
33. C2×#217C22semidirect
44. C2×Dic3⋊C4C22semidirect
45. C2×Dic3⋊C4C22semidirect
46. C2×C4×Dic3C22semidirect
47. C2×C4×Dic3C22semidirect
50. C3×#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
62. #394D3semidirect
120. #217D4semidirect
145. C2×C4⋊C4C2×D3semidirect
162. C2×C22⋊C4C2×D3semidirect
163. C2×C22⋊C4C2×D3semidirect
171. #95C2×D3semidirect
344. C2×D3#124semidirect
489. C3#2472semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#9362
C2×#8962
C2×D3×C22⋊C42
C3×#3942
#49902
#49592
#49572
#49562
#49552
#49522
#49442
#46732
#46662
#24723