Finite Group Databasebeta

← #75914

Subgroups — #75914

Order 384.

734 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#894C211yesmaximal, characteristic, complemented
2C2×#884C211yesmaximal, characteristic, complemented
3C2×#884C211yesmaximal, characteristic, complemented
4C2×#874C211yesmaximal, characteristic, complemented
5C3×#388C211yesmaximal, characteristic, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4960C211yesmaximal, characteristic, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4960C211yesmaximal, characteristic, complemented
10#4960C211yesmaximal, characteristic, complemented
11#4933C211yesmaximal, characteristic, complemented
12#4672C211yesmaximal, characteristic, complemented
13#4665C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2673—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C22×D12C2211yescharacteristic
23C22×D12C2211yescharacteristic
24C22×D12C2211yescharacteristic
25C22×D12C2211yescharacteristic
26C22×D12C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic
28C22×C4×D3C2211yescharacteristic
29C2×C3×C4⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C42C2211yescharacteristic
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×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46#894—24no—
47#884—24no—
48#884—24no—
49#874—24no—
50C3×#95C2211yescharacteristic
51C3×#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#127—31no—
57C2×#123—31no—
58C2×#121—31no—
59C2×#121—31no—
60#394—31no—
61#394—31no—
62#392—31no—
63#392—31no—
64#392—31no—
65#392—31no—
66#388D311yescharacteristic, complemented
67#387—31no—
68#386—31no—
69#386—31no—
70#386—31no—
71C24×C3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C23×D3C2311yescharacteristic
74C23×D3C2311yescharacteristic
75C23×D3C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C2×#67—22no—
83C2×#67—22no—
84C2×#67—22no—
85C2×#67—22no—
86C2×#67—41no—
87C2×#67—41no—
88C2×#67—41no—
89C2×#67—41no—
90C2×#67—41no—
91C2×#67—41no—
92C2×#67—41no—
93C2×#67—41no—
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C2×D12—22no—
99C2×D12—22no—
100C2×D12—22no—
101C2×D12—22no—
102C2×D12—22no—
103C2×D12—22no—
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—22no—
107C2×D12—41no—
108C2×D12—41no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12D414yescomplemented
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3—22no—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C3×C4⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—22no—
126C3×C42—22no—
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—41no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—41no—
137Dic3⋊C4—22no—
138Dic3⋊C4D414yescomplemented
139C22×D4—31no—
140C22×D4—31no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—31no—
145C22×D4—31no—
146C22×D4—31no—
147C22×D4—31no—
148C22×D4—31no—
149C23×C4—31no—
150C23×C4—31no—
151#127—64no—
152#123—64no—
153#121—64no—
154#121—64no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4C2×D311yescharacteristic
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊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⋊C4C2×D311yescharacteristic
169C2×C22⋊C4C2×D311yescharacteristic
170C2×C22⋊C4C2×D311yescharacteristic
171C2×C42C2×D311yescharacteristic
172#95—31no—
173#95—31no—
174#95—31no—
175#95—31no—
176#95C2×D311yescharacteristic
177#95C2×D311yescharacteristic
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3C2411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
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×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3C2×D412yes—
213C22×D3C2×D412yes—
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—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4C2×D412yes—
230C2×C3×C4C2×D412yes—
231C2×C3×C4#4112yes—
232#67—44no—
233#67—44no—
234#67—44no—
235#67—44no—
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—
248D12—44no—
249D12—44no—
250D12—44no—
251D12—44no—
252D12—44no—
253D12—44no—
254C4×D3—44no—
255C4×D3—44no—
256C24—31no—
257C24—31no—
258C24—31no—
259C24—31no—
260C24C22×D311yescharacteristic
261C2×D4—34no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—62no—
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—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4—121no—
301C22×C4C22×D311yescharacteristic
302C22×C4C22×D311yescharacteristic
303C22×C4C22×D311yescharacteristic
304C22×C4C22×D311yescharacteristic
305C22×C4C22×D311yescharacteristic
306C22×C4C22×D311yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—34no—
309C4⋊C4—62no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C42—22no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3C22×D411yescharacteristic
332C22×C3C22×D411yescharacteristic
333C22×C3C2×#4111yescharacteristic
334C22×C3H(5,𝔽2)11yescharacteristic
335C22×C3H(5,𝔽2)11yescharacteristic
336C22×C3H(5,𝔽2)11yescharacteristic
337C22×C3H(5,𝔽2)11yescharacteristic
338C2×D3—22no—
339C2×D3—22no—
340C2×D3—22no—
341C2×D3—22no—
342C2×D3—22no—
343C2×D3—22no—
344C2×D3—41no—
345C2×D3—41no—
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—
357C2×D3—41no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—41no—
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—41no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—41no—
372Dic3—22no—
373Dic3—22no—
374Dic3—41no—
375Dic3—41no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—32no—
384C23—32no—
385C23—61no—
386C23—61no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23—61no—
391C23—61no—
392C23—61no—
393C23—61no—
394C23—61no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—61no—
399C23—61no—
400C23—61no—
401C23—61no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23—61no—
407C23—61no—
408C23—61no—
409C23—61no—
410C23—61no—
411C23C23×D311yescharacteristic
412D4—124no—
413D4—124no—
414D4—124no—
415D4—124no—
416D4—124no—
417D4—124no—
418D4—124no—
419D4—124no—
420D4—124no—
421D4—124no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—32no—
438C2×C4—32no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—124no—
450C2×C4—124no—
451C2×C4#23512yes—
452C2×C4D3×D412yes—
453C2×C4D3×D412yes—
454C2×C3—81no—
455C2×C3#53411yescharacteristic
456C2×C3#53511yescharacteristic
457C2×C3#54011yescharacteristic
458C2×C3D4211yescharacteristic
459C2×C3#54611yescharacteristic
460C2×C3#55011yescharacteristic
461C2×C3#56011yescharacteristic
462D3—42no—
463D3—42no—
464D3—81no—
465D3—81no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—62no—
474C22—62no—
475C22—62no—
476C22—62no—
477C22—62no—
478C22—62no—
479C22—121no—
480C22—121no—
481C22—121no—
482C22—121no—
483C22—121no—
484C22—121no—
485C22—121no—
486C22—121no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22—121no—
491C22—121no—
492C22—121no—
493C22—121no—
494C22—121no—
495C22—121no—
496C22—121no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22—121no—
501C22C2×#23511yescharacteristic
502C22C2×D3×D411yescharacteristic
503C22C2×D3×D411yescharacteristic
504C22#100911yescharacteristic
505C22#100911yescharacteristic
506C22#100911yescharacteristic
507C22#100911yescharacteristic
508C4—22no—
509C4—22no—
510C4—22no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—62no—
515C4—62no—
516C4—121no—
517C4—121no—
518C3#267311yescharacteristic, complemented, Sylow
519C2—81no—
520C2—122no—
521C2—122no—
522C2—241no—
523C2—241no—
524C2#553511yescharacteristic
525C2#555011yescharacteristic
526C2#555311yescharacteristic
527C2#560911yescharacteristic
528C2#565311yescharacteristic
529C2#567411yescharacteristic
530C2#571111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#388C2
hypercenter#388D3
hypocenterC3#2673
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#894C2semidirect
2. C2×#884C2semidirect
3. C2×#884C2semidirect
4. C2×#874C2semidirect
5. C3×#388C2semidirect
6. #4991C2semidirect
7. #4960C2semidirect
8. #4960C2semidirect
9. #4960C2semidirect
10. #4960C2semidirect
11. #4933C2semidirect
12. #4672C2semidirect
13. #4665C2semidirect
14. #4665C2semidirect
15. #4665C2semidirect
66. #388D3semidirect
115. C2×D12D4semidirect
138. Dic3⋊C4D4semidirect
518. C3#2673semidirect

Maximal subgroups

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

subgroupindex
C2×#8942
C2×#8842
C2×#8742
C3×#3882
#49912
#49602
#49332
#46722
#46652
#26733