Finite Group Databasebeta

← #76672

Subgroups — #76672

Order 384.

952 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#237C211yesmaximal, characteristic, complemented
2C2×#940C211yesmaximal, characteristic, complemented
3C2×#939C211yesmaximal, characteristic, complemented
4C2×#885C211yesmaximal, characteristic, complemented
5C3×#392C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5223C211yesmaximal, characteristic, complemented
8#5220C211yesmaximal, characteristic, complemented
9#4960C211yesmaximal, characteristic, complemented
10#4958C211yesmaximal, characteristic, complemented
11#4957C211yesmaximal, characteristic, complemented
12#4956C211yesmaximal, characteristic, complemented
13Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2431—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic
25C22×#67C2211yescharacteristic
26C23×Dic3C2211yescharacteristic
27C23×Dic3C2211yescharacteristic
28C2×#237—24no—
29C2×#237—24no—
30C2×#237—24no—
31C22×D12C2211yescharacteristic
32C22×C4×D3C2211yescharacteristic
33C22×Dic6C2211yescharacteristic
34C2×C3×C22⋊C4C2211yescharacteristic
35C2×C3×C22⋊C4C2211yescharacteristic
36C2×C3×C22⋊C4C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39C2×#217C2211yescharacteristic
40C2×#217C2211yescharacteristic
41#940—24no—
42#939—24no—
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×(C2×D3)⋊C4C2211yescharacteristic
46C2×(C2×D3)⋊C4C2211yescharacteristic
47C2×(C2×D3)⋊C4C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49C2×C4×Dic3C2211yescharacteristic
50C2×C4×Dic3C2211yescharacteristic
51#885—24no—
52C3×#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C3⋊#95C2211yescharacteristic
57C3⋊#95C2211yescharacteristic
58C22×#41—31no—
59C2×#127—31no—
60C2×#124—31no—
61C2×#121—31no—
62#392—31no—
63#392—31no—
64#392—31no—
65#392D311yescharacteristic, complemented
66#391—31no—
67#390—31no—
68#386—31no—
69#386—31no—
70#386—31no—
71#386—31no—
72C4×C22⋊C4—31no—
73C24×C3C2311yescharacteristic
74C24×C3C2311yescharacteristic
75C24×C3C2311yescharacteristic
76C23×D3C2311yescharacteristic
77C23×D3C2311yescharacteristic
78C2×C3×D4—22no—
79C2×C3×D4—22no—
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4—41no—
85C2×C3×D4—41no—
86C2×C3×D4—41no—
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—22no—
98C2×#67—22no—
99C2×#67—22no—
100C2×#67—22no—
101C2×#67—22no—
102C2×#67—22no—
103C2×#67—22no—
104C2×#67—41no—
105C2×#67—41no—
106C2×#67—41no—
107C2×#67—41no—
108C2×#67D414yescomplemented
109C22×Dic3—22no—
110C22×Dic3—22no—
111C22×Dic3—22no—
112C22×Dic3—22no—
113C22×Dic3—22no—
114C22×Dic3D414yescomplemented
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C22×Dic3C2311yescharacteristic
118C22×Dic3C2311yescharacteristic
119C22×Dic3C2311yescharacteristic
120C22×Dic3C2311yescharacteristic
121#237—232no—
122C2×D12—22no—
123C2×D12—22no—
124C2×D12—41no—
125C2×C4×D3—22no—
126C2×C4×D3—22no—
127C2×C4×D3—22no—
128C2×Dic6—22no—
129C2×Dic6—22no—
130C2×Dic6D414yescomplemented
131C3×C22⋊C4—22no—
132C3×C22⋊C4—22no—
133C3×C22⋊C4—41no—
134#217—22no—
135#217—22no—
136#217—22no—
137#217—41no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140(C2×D3)⋊C4—22no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143Dic3⋊C4—22no—
144C4×Dic3—22no—
145C4×Dic3—22no—
146C2×#41—64no—
147C2×#41—64no—
148C2×#41—64no—
149C22×Q8—31no—
150C22×D4—31no—
151C22×D4—31no—
152C22×D4—31no—
153C22×D4—31no—
154C22×D4—31no—
155C22×D4—31no—
156C22×D4—31no—
157C22×D4C2×D311yescharacteristic
158C22×D4C2×D311yescharacteristic
159C22×D4C2×D311yescharacteristic
160C23×C4—31no—
161C23×C4—31no—
162C23×C4—31no—
163#127—64no—
164#124—64no—
165#121—64no—
166C2×C4⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4—31no—
170C2×C22⋊C4—31no—
171C2×C22⋊C4—31no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—31no—
176C2×C22⋊C4C2×D311yescharacteristic
177C2×C22⋊C4C2×D311yescharacteristic
178C2×C22⋊C4C2×D311yescharacteristic
179C2×C42—31no—
180C2×C42—31no—
181#95—31no—
182#95—31no—
183#95—31no—
184#95—31no—
185#95—31no—
186#95C2×D311yescharacteristic
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3C2×D412yes—
207C23×C3C2×D412yes—
208C23×C3C2411yescharacteristic
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—21no—
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—21no—
219C22×D3—21no—
220C22×D3—21no—
221C22×D3—21no—
222C22×D3C2×D412yes—
223C3×D4—28no—
224C3×D4—44no—
225C3×D4—44no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4C2×D412yes—
237C2×C3×C4#4112yescomplemented
238#67—28no—
239#67—28no—
240#67—44no—
241#67—44no—
242#67—44no—
243#67—44no—
244C2×Dic3—28no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—28no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3C2×D412yes—
259C2×Dic3C2×D412yes—
260C2×Dic3C2×D412yes—
261C2×Dic3C2×D412yes—
262C2×Dic3C2×D412yes—
263C2×Dic3#4112yescomplemented
264D12—44no—
265C4×D3—28no—
266Dic6—28no—
267C24—31no—
268C24—31no—
269C24C22×D311yescharacteristic
270C24C22×D311yescharacteristic
271C24C22×D311yescharacteristic
272#41—632no—
273C2×Q8—34no—
274C2×Q8—62no—
275C2×Q8—62no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—34no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—62no—
287C2×D4—62no—
288C2×D4—62no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—62no—
296C2×D4—62no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4—62no—
300C2×D4—62no—
301C2×D4—121no—
302C2×D4—121no—
303C2×D4—121no—
304C2×D4—121no—
305C2×D4—121no—
306C22×C4—31no—
307C22×C4—34no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—62no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4C22×D311yescharacteristic
322C22×C4C22×D311yescharacteristic
323C22×C4C22×D311yescharacteristic
324C22×C4C22×D311yescharacteristic
325C4⋊C4—62no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—62no—
330C22⋊C4—62no—
331C22⋊C4—62no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C22⋊C4—62no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C42—62no—
339C42—62no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3C22×D411yescharacteristic
362C22×C3C22×D411yescharacteristic
363C22×C3C22×D411yescharacteristic
364C22×C3C2×#4111yescharacteristic
365C22×C3C2×#4111yescharacteristic
366C22×C3H(5,𝔽2)11yescharacteristic
367C22×C3H(5,𝔽2)11yescharacteristic
368C2×D3—22no—
369C2×D3—22no—
370C2×D3—22no—
371C2×D3—41no—
372C2×D3—41no—
373C2×D3—41no—
374C2×D3—41no—
375C2×D3—41no—
376C2×D3—41no—
377C2×D3—41no—
378C2×D3—41no—
379C2×D3—41no—
380C2×D3—41no—
381C2×D3—41no—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—41no—
385C3×C4—41no—
386Dic3—22no—
387Dic3—22no—
388Dic3—22no—
389Dic3—41no—
390Dic3—41no—
391Dic3#12014yes—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—32no—
412C23—61no—
413C23—61no—
414C23—61no—
415C23—61no—
416C23—61no—
417C23—61no—
418C23—61no—
419C23—61no—
420C23—61no—
421C23—61no—
422C23—61no—
423C23—61no—
424C23—61no—
425C23D3×D412yes—
426C23D3×D412yes—
427C23C23×D311yescharacteristic
428Q8—68no—
429D4—28no—
430D4—44no—
431D4—44no—
432D4—68no—
433D4—68no—
434D4—124no—
435D4—124no—
436D4—124no—
437D4—124no—
438D4—124no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—32no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—32no—
454C2×C4—32no—
455C2×C4—68no—
456C2×C4—68no—
457C2×C4—68no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4D3×D412yes—
471C2×C4#23712yescomplemented
472C2×C3—42no—
473C2×C3—42no—
474C2×C3—81no—
475C2×C3C2×#12011yescharacteristic
476C2×C3#53211yescharacteristic
477C2×C3#53511yescharacteristic
478C2×C3#54611yescharacteristic
479C2×C3#54611yescharacteristic
480C2×C3#55011yescharacteristic
481C2×C3#55011yescharacteristic
482D3—42no—
483D3—81no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—62no—
506C22—62no—
507C22—62no—
508C22—121no—
509C22—121no—
510C22—121no—
511C22—121no—
512C22—121no—
513C22—121no—
514C22—121no—
515C22—121no—
516C22—121no—
517C22—121no—
518C22—121no—
519C22C2×D3×D411yescharacteristic
520C22C2×D3×D411yescharacteristic
521C22C2×D3×D411yescharacteristic
522C22C2×#23711yescharacteristic
523C22#100411yescharacteristic
524C22#100411yescharacteristic
525C22D3×#4111yescharacteristic
526C4—22no—
527C4—22no—
528C4—34no—
529C4—41no—
530C4—41no—
531C4—62no—
532C4—62no—
533C4—62no—
534C4—121no—
535C4—121no—
536C3#243111yescharacteristic, complemented, Sylow
537C2—42no—
538C2—42no—
539C2—81no—
540C2—122no—
541C2—241no—
542C2D3×#12011yescharacteristic
543C2#559111yescharacteristic
544C2#559211yescharacteristic
545C2#559611yescharacteristic
546C2#559911yescharacteristic
547C2#561011yescharacteristic
548C2#571811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2431
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C2×#940C2semidirect
3. C2×#939C2semidirect
4. C2×#885C2semidirect
5. C3×#392C2semidirect
6. #5223C2semidirect
7. #5223C2semidirect
8. #5220C2semidirect
9. #4960C2semidirect
10. #4958C2semidirect
11. #4957C2semidirect
12. #4956C2semidirect
13. Dic3×C22⋊C4C2semidirect
14. #4665C2semidirect
15. #4646C2semidirect
65. #392D3semidirect
108. C2×#67D4semidirect
114. C22×Dic3D4semidirect
130. C2×Dic6D4semidirect
237. C2×C3×C4#41semidirect
263. C2×Dic3#41semidirect
471. C2×C4#237semidirect
536. C3#2431semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×#9402
C2×#9392
C2×#8852
C3×#3922
#52232
#52202
#49602
#49582
#49572
#49562
Dic3×C22⋊C42
#46652
#46462
#24313