Finite Group Databasebeta

← #74896

Subgroups — #74896

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×#235C211yesmaximal, characteristic, complemented
2C2×#938C211yesmaximal, characteristic, complemented
3C2×#884C211yesmaximal, characteristic, complemented
4C2×#883C211yesmaximal, characteristic, complemented
5C3×#386C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5210C211yesmaximal, characteristic, complemented
8#4961C211yesmaximal, characteristic, complemented
9#4956C211yesmaximal, characteristic, complemented
10#4956C211yesmaximal, characteristic, complemented
11#4947C211yesmaximal, characteristic, complemented
12#4668C211yesmaximal, characteristic, complemented
13#4665C211yesmaximal, characteristic, complemented
14D3×#95C211yesmaximal, characteristic, complemented
15#4656C211yesmaximal, characteristic, complemented
16#2430—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C2×#235—24no—
24C2×#235—24no—
25C2×#235—24no—
26C22×D12C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic
28C22×C4×D3C2211yescharacteristic
29C22×C4×D3C2211yescharacteristic
30C22×C4×D3C2211yescharacteristic
31C22×Dic6C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39#938—24no—
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×C4⋊Dic3C2211yescharacteristic
45C2×C4⋊Dic3C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48#884—24no—
49#883—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×D4—41no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
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×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106#235—232no—
107C2×D12—22no—
108C2×D12—22no—
109C2×D12D414yescomplemented
110C2×C4×D3—22no—
111C2×C4×D3—22no—
112C2×C4×D3—22no—
113C2×C4×D3—22no—
114C2×C4×D3—22no—
115C2×C4×D3—22no—
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3—22no—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C4×D3D414yescomplemented
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#217—22no—
130#217—22no—
131#217—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—
138Dic3⋊C4—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×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
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×D3C2×D412yes—
207C22×D3C2×D412yes—
208C22×D3C2×D412yes—
209C22×D3C2×D412yes—
210C3×D4—44no—
211C2×C3×C4—28no—
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×C4C2×D412yes—
225C2×C3×C4C2×D412yes—
226C2×C3×C4#4112yescomplemented
227#67—28no—
228#67—28no—
229#67—44no—
230#67—44no—
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×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3C2×D412yes—
247C2×Dic3C2×D412yes—
248C2×Dic3#4112yescomplemented
249D12—28no—
250C4×D3—28no—
251C4×D3—28no—
252C4×D3—44no—
253C4×D3—44no—
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—34no—
267C2×D4—41no—
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—62no—
279C2×D4—62no—
280C2×D4—121no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—34no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—31no—
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—121no—
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—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
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—22no—
347C2×D3—22no—
348C2×D3—22no—
349C2×D3—22no—
350C2×D3—22no—
351C2×D3—22no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3#12014yes—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—41no—
361C3×C4—41no—
362Dic3—22no—
363Dic3—22no—
364Dic3—22no—
365Dic3—41no—
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—21no—
379C23—21no—
380C23—21no—
381C23—32no—
382C23—32no—
383C23—32no—
384C23—32no—
385C23—61no—
386C23—61no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23—61no—
391C23—61no—
392C23—61no—
393C23—61no—
394C23—61no—
395C23D3×D412yes—
396C23C23×D311yescharacteristic
397Q8—68no—
398D4—44no—
399D4—68no—
400D4—68no—
401D4—68no—
402D4—124no—
403D4—124no—
404C2×C4—28no—
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—68no—
421C2×C4—68no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
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—124no—
438C2×C4—124no—
439C2×C4#23512yescomplemented
440C2×C4D3×D412yes—
441C2×C4D3×D412yes—
442C2×C3—42no—
443C2×C3—81no—
444C2×C3C2×#12011yescharacteristic
445C2×C3#53211yescharacteristic
446C2×C3#53711yescharacteristic
447C2×C3#54611yescharacteristic
448C2×C3#54611yescharacteristic
449C2×C3#54711yescharacteristic
450C2×C3#54711yescharacteristic
451D3—24no—
452D3—42no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—34no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—62no—
469C22—62no—
470C22—62no—
471C22—62no—
472C22—62no—
473C22—62no—
474C22—62no—
475C22—62no—
476C22—62no—
477C22—121no—
478C22—121no—
479C22—121no—
480C22—121no—
481C22C2×#23511yescharacteristic
482C22C2×D3×D411yescharacteristic
483C22C2×D3×D411yescharacteristic
484C22C2×D3×D411yescharacteristic
485C22#100411yescharacteristic
486C22D3×#4111yescharacteristic
487C22#101011yescharacteristic
488C4—22no—
489C4—22no—
490C4—22no—
491C4—41no—
492C4—41no—
493C4—62no—
494C4—62no—
495C4—62no—
496C4—121no—
497C4—121no—
498C4—121no—
499C3#243011yescharacteristic, complemented, Sylow
500C2—42no—
501C2—64no—
502C2—81no—
503C2—122no—
504C2#549011yescharacteristic
505C2#554811yescharacteristic
506C2#555111yescharacteristic
507C2D3×#12011yescharacteristic
508C2#560811yescharacteristic
509C2#561211yescharacteristic
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×#235C2semidirect
2. C2×#938C2semidirect
3. C2×#884C2semidirect
4. C2×#883C2semidirect
5. C3×#386C2semidirect
6. #5223C2semidirect
7. #5210C2semidirect
8. #4961C2semidirect
9. #4956C2semidirect
10. #4956C2semidirect
11. #4947C2semidirect
12. #4668C2semidirect
13. #4665C2semidirect
14. D3×#95C2semidirect
15. #4656C2semidirect
69. #386D3semidirect
99. C2×#67D4semidirect
109. C2×D12D4semidirect
121. C2×C4×D3D4semidirect
226. C2×C3×C4#41semidirect
248. C2×Dic3#41semidirect
439. C2×C4#235semidirect
499. C3#2430semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C2×#9382
C2×#8842
C2×#8832
C3×#3862
#52232
#52102
#49612
#49562
#49472
#46682
#46652
D3×#952
#46562
#24303