Finite Group Databasebeta

← #74699

Subgroups — #74699

Order 384.

914 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#237C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic, complemented
3C2×#930C211yesmaximal, characteristic, complemented
4C2×#886C211yesmaximal, characteristic, complemented
5C3×#386C211yesmaximal, characteristic, complemented
6C2×C3⋊#95C211yesmaximal, characteristic, complemented
7#4961C211yesmaximal, characteristic, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4958C211yesmaximal, characteristic, complemented
10#4958C211yesmaximal, characteristic, complemented
11#4950C211yesmaximal, characteristic, complemented
12#4673C211yesmaximal, characteristic, complemented
13#4668C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4656C211yesmaximal, characteristic, complemented
16#2430—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C23×Dic3C2211yescharacteristic
25C23×Dic3C2211yescharacteristic
26C2×#237—24no—
27C2×#237—24no—
28C2×#237—24no—
29C22×D12C2211yescharacteristic
30C22×C4×D3C2211yescharacteristic
31C22×Dic6C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2211yescharacteristic
35C2×#217C2211yescharacteristic
36#939—24no—
37#930—24no—
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×(C2×D3)⋊C4C2211yescharacteristic
45C2×C4⋊Dic3C2211yescharacteristic
46C2×C4⋊Dic3C2211yescharacteristic
47C2×C4⋊Dic3C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49#886—24no—
50C3×#95C2211yescharacteristic
51C3×#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C3⋊#95C2211yescharacteristic
57C3⋊#95C2211yescharacteristic
58C22×#41—31no—
59C2×#123—31no—
60C2×#121—31no—
61C2×#121—31no—
62#395—31no—
63#394—31no—
64#393—31no—
65#392—31no—
66#386—31no—
67#386—31no—
68#386—31no—
69#386D311yescharacteristic, complemented
70#385—31no—
71#385—31no—
72C2×#95—31no—
73C24×C3C2311yescharacteristic
74C24×C3C2311yescharacteristic
75C23×D3C2311yescharacteristic
76C23×D3C2311yescharacteristic
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4D414yescomplemented
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—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—22no—
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—22no—
98C2×#67—41no—
99C2×#67D414yescomplemented
100C22×Dic3—22no—
101C22×Dic3—22no—
102C22×Dic3—22no—
103C22×Dic3—22no—
104C22×Dic3—22no—
105C22×Dic3—22no—
106C22×Dic3—22no—
107C22×Dic3—41no—
108C22×Dic3D414yescomplemented
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115#237—232no—
116C2×D12—22no—
117C2×D12—22no—
118C2×D12—41no—
119C2×C4×D3—22no—
120C2×C4×D3—22no—
121C2×C4×D3—22no—
122C2×Dic6—22no—
123C2×Dic6—22no—
124C2×Dic6—22no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—41no—
128#217—22no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—22no—
137C4⋊Dic3—22no—
138C4⋊Dic3—22no—
139Dic3⋊C4—22no—
140C2×#41—64no—
141C2×#41—64no—
142C2×#41—64no—
143C22×Q8—31no—
144C22×D4—31no—
145C22×D4—31no—
146C22×D4—31no—
147C22×D4—31no—
148C22×D4—31no—
149C22×D4C2×D311yescharacteristic
150C23×C4—31no—
151C23×C4—31no—
152C23×C4—31no—
153C23×C4—31no—
154C23×C4C2×D311yescharacteristic
155#123—64no—
156#121—64no—
157#121—64no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C22⋊C4—31no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4—31no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4—31no—
170C2×C22⋊C4C2×D311yescharacteristic
171C2×C22⋊C4C2×D311yescharacteristic
172C2×C22⋊C4C2×D311yescharacteristic
173#95—31no—
174#95—31no—
175#95—31no—
176#95—31no—
177#95—31no—
178#95—31no—
179#95C2×D311yescharacteristic
180#95C2×D311yescharacteristic
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3C2×D412yes—
192C23×C3C2×D412yes—
193C23×C3C2×D412yes—
194C23×C3C2×D412yes—
195C23×C3C2411yescharacteristic
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
199C22×D3—21no—
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—21no—
207C22×D3—21no—
208C22×D3—21no—
209C22×D3C2×D412yes—
210C3×D4—28no—
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×C4—21no—
224C2×C3×C4—44no—
225C2×C3×C4C2×D412yes—
226C2×C3×C4#4112yescomplemented
227#67—28no—
228#67—28no—
229#67—44no—
230#67—44no—
231C2×Dic3—28no—
232C2×Dic3—28no—
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×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—44no—
248C2×Dic3C2×D412yes—
249C2×Dic3C2×D412yes—
250C2×Dic3C2×D412yes—
251C2×Dic3#4112yescomplemented
252D12—44no—
253C4×D3—28no—
254Dic6—28no—
255C24—31no—
256C24—31no—
257C24C22×D311yescharacteristic
258C24C22×D311yescharacteristic
259#41—632no—
260C2×Q8—62no—
261C2×Q8—62no—
262C2×Q8—62no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—34no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—62no—
271C2×D4—62no—
272C2×D4—62no—
273C2×D4—62no—
274C2×D4—62no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4D1214yescomplemented
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—34no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—41no—
291C22×C4—62no—
292C22×C4—62no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4—121no—
302C22×C4C22×D311yescharacteristic
303C22×C4C22×D311yescharacteristic
304C22×C4C22×D311yescharacteristic
305C22×C4C22×D311yescharacteristic
306C22×C4C22×D311yescharacteristic
307C4⋊C4—62no—
308C4⋊C4—62no—
309C4⋊C4—62no—
310C4⋊C4—62no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—41no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—62no—
319C22⋊C4—62no—
320C22⋊C4—62no—
321C22⋊C4—121no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3#12014yes—
336C22×C3C22×D411yescharacteristic
337C22×C3C22×D411yescharacteristic
338C22×C3C22×D411yescharacteristic
339C22×C3C2×#4111yescharacteristic
340C22×C3C2×#4111yescharacteristic
341C22×C3H(5,𝔽2)11yescharacteristic
342C22×C32−1+411yescharacteristic
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—41no—
347C2×D3—41no—
348C2×D3—41no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—41no—
362Dic3—22no—
363Dic3—22no—
364Dic3—22no—
365Dic3—22no—
366Dic3—41no—
367Dic3—41no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—32no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23—61no—
384C23—61no—
385C23—61no—
386C23—61no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23—61no—
391C23—61no—
392C23C2×D1212yes—
393C23C2×D1212yes—
394C23D3×D412yes—
395C23D3×D412yes—
396C23C23×D311yescharacteristic
397Q8—68no—
398D4—28no—
399D4—68no—
400D4—68no—
401D4—124no—
402D4—124no—
403D4—124no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—32no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—32no—
421C2×C4—44no—
422C2×C4—68no—
423C2×C4—68no—
424C2×C4—68no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—124no—
440C2×C4C2×D1212yes—
441C2×C4#23712yescomplemented
442C2×C3—24no—
443C2×C3—42no—
444C2×C3C2×#12011yescharacteristic
445C2×C3#53211yescharacteristic
446C2×C3#53711yescharacteristic
447C2×C3#54611yescharacteristic
448C2×C3#54611yescharacteristic
449C2×C3#54711yescharacteristic
450C2×C3#54711yescharacteristic
451D3—42no—
452D3—81no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—62no—
467C22—62no—
468C22—62no—
469C22—121no—
470C22—121no—
471C22—121no—
472C22—121no—
473C22—121no—
474C22—121no—
475C22—121no—
476C22—121no—
477C22—121no—
478C22—121no—
479C22—121no—
480C22#88214yes—
481C22C22×D1211yescharacteristic
482C22C2×D3×D411yescharacteristic
483C22C2×D3×D411yescharacteristic
484C22C2×#23711yescharacteristic
485C22#100411yescharacteristic
486C22D3×#4111yescharacteristic
487C22#101011yescharacteristic
488C4—22no—
489C4—22no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—62no—
494C4—62no—
495C4—62no—
496C4—62no—
497C4—121no—
498C4—121no—
499C3#243011yescharacteristic, complemented, Sylow
500C2—24no—
501C2—42no—
502C2—122no—
503C2—241no—
504C2C2×#88211yescharacteristic
505C2#555411yescharacteristic
506C2#555511yescharacteristic
507C2#559111yescharacteristic
508C2#559611yescharacteristic
509C2#559711yescharacteristic
510C2#565511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
hypercenter#386D3
hypocenterC3#2430
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C2×#939C2semidirect
3. C2×#930C2semidirect
4. C2×#886C2semidirect
5. C3×#386C2semidirect
6. C2×C3⋊#95C2semidirect
7. #4961C2semidirect
8. #4960C2semidirect
9. #4958C2semidirect
10. #4958C2semidirect
11. #4950C2semidirect
12. #4673C2semidirect
13. #4668C2semidirect
14. #4665C2semidirect
15. #4656C2semidirect
69. #386D3semidirect
79. C2×C3×D4D4semidirect
99. C2×#67D4semidirect
108. C22×Dic3D4semidirect
226. C2×C3×C4#41semidirect
251. C2×Dic3#41semidirect
280. C2×D4D12semidirect
441. C2×C4#237semidirect
499. C3#2430semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×#9392
C2×#9302
C2×#8862
C3×#3862
C2×C3⋊#952
#49612
#49602
#49582
#49502
#46732
#46682
#46652
#46562
#24303