Finite Group Databasebeta

← #77134

Subgroups — #77134

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×#940C211yesmaximal, characteristic, complemented
2C2×#895C211yesmaximal, characteristic, complemented
3C2×#884C211yesmaximal, characteristic, complemented
4C2×#883C211yesmaximal, characteristic, complemented
5C3×#394C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#4978C211yesmaximal, characteristic, complemented
8#4961C211yesmaximal, characteristic, complemented
9#4960C211yesmaximal, characteristic, complemented
10#4960C211yesmaximal, characteristic, complemented
11#4960C211yesmaximal, characteristic, complemented
12#4956C211yesmaximal, characteristic, complemented
13#4956C211yesmaximal, characteristic, complemented
14#4670C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2673—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
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×C22⋊C4C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36#940—24no—
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×C4⋊Dic3C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×C4×Dic3C2211yescharacteristic
47#895—24no—
48#884—24no—
49#883—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#394D311yescharacteristic, complemented
62#392—31no—
63#392—31no—
64#392—31no—
65#392—31no—
66#388—31no—
67#387—31no—
68#386—31no—
69#386—31no—
70#386—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C23×D3C2311yescharacteristic
74C23×D3C2311yescharacteristic
75C23×D3C2311yescharacteristic
76C2×C3×D4—22no—
77C2×C3×D4—41no—
78C2×C3×D4—41no—
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C2×#67—22no—
85C2×#67—22no—
86C2×#67—22no—
87C2×#67—22no—
88C2×#67—22no—
89C2×#67—22no—
90C2×#67—22no—
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—41no—
94C2×#67—41no—
95C2×#67—41no—
96C2×#67—41no—
97C2×#67—41no—
98C2×#67—41no—
99C2×#67—41no—
100C2×#67—41no—
101C2×#67—41no—
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C2×D12—22no—
108C2×D12—22no—
109C2×D12—22no—
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—41no—
126C3×C22⋊C4—41no—
127#217—22no—
128#217—41no—
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—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—22no—
137Dic3⋊C4D414yescomplemented
138C4×Dic3—22no—
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×D4C2×D311yescharacteristic
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⋊C4C2×D311yescharacteristic
168C2×C22⋊C4C2×D311yescharacteristic
169C2×C22⋊C4C2×D311yescharacteristic
170C2×C22⋊C4C2×D311yescharacteristic
171C2×C42—31no—
172#95—31no—
173#95—31no—
174#95—31no—
175#95—31no—
176#95—31no—
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×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3C2411yescharacteristic
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—
214C3×D4—44no—
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×C4C2×D412yes—
229C2×C3×C4C2×D412yes—
230#67—44no—
231#67—44no—
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×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3C2×D412yes—
249C2×Dic3C2×D412yes—
250C2×Dic3#4112yes—
251D12—44no—
252D12—44no—
253D12—44no—
254C4×D3—44no—
255C4×D3—44no—
256C24—31no—
257C24—31no—
258C24—31no—
259C24C22×D311yescharacteristic
260C24C22×D311yescharacteristic
261C2×D4—22no—
262C2×D4—34no—
263C2×D4—41no—
264C2×D4—41no—
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—62no—
276C2×D4—62no—
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—31no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4—121no—
300C22×C4—121no—
301C22×C4—121no—
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—41no—
313C22⋊C4—41no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—62no—
319C22⋊C4—62no—
320C22⋊C4—121no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C42—62no—
324C22×C3—41no—
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×C3—41no—
337C22×C3—41no—
338C22×C3C22×D411yescharacteristic
339C22×C3C22×D411yescharacteristic
340C22×C3C2×#4111yescharacteristic
341C22×C3H(5,𝔽2)11yescharacteristic
342C22×C3H(5,𝔽2)11yescharacteristic
343C22×C3H(5,𝔽2)11yescharacteristic
344C22×C3H(5,𝔽2)11yescharacteristic
345C2×D3—22no—
346C2×D3—22no—
347C2×D3—22no—
348C2×D3—22no—
349C2×D3—22no—
350C2×D3—22no—
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—41no—
369C3×C4—41no—
370C3×C4—41no—
371Dic3—22no—
372Dic3—22no—
373Dic3—22no—
374Dic3—41no—
375Dic3—41no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—32no—
391C23—32no—
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—44no—
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—32no—
436C2×C4—32no—
437C2×C4—32no—
438C2×C4—61no—
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—61no—
450C2×C4—124no—
451C2×C4—124no—
452C2×C4D3×D412yes—
453C2×C4D3×D412yes—
454C2×C3—81no—
455C2×C3—81no—
456C2×C3#53411yescharacteristic
457C2×C3#53511yescharacteristic
458C2×C3#54011yescharacteristic
459C2×C3D4211yescharacteristic
460C2×C3#54611yescharacteristic
461C2×C3#55011yescharacteristic
462C2×C3#56011yescharacteristic
463D3—42no—
464D3—42no—
465D3—81no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—62no—
481C22—62no—
482C22—62no—
483C22—62no—
484C22—62no—
485C22—62no—
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×D3×D411yescharacteristic
502C22C2×D3×D411yescharacteristic
503C22#100411yescharacteristic
504C22#100411yescharacteristic
505C22D3×#4111yescharacteristic
506C22#100911yescharacteristic
507C22#100911yescharacteristic
508C4—22no—
509C4—22no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—62no—
514C4—62no—
515C4—62no—
516C4—121no—
517C4—121no—
518C3#267311yescharacteristic, complemented, Sylow
519C2—81no—
520C2—81no—
521C2—122no—
522C2—122no—
523C2—241no—
524C2#558911yescharacteristic
525C2#559911yescharacteristic
526C2#560711yescharacteristic
527C2#560911yescharacteristic
528C2#566311yescharacteristic
529C2#566611yescharacteristic
530C2#567611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2673
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#940C2semidirect
2. C2×#895C2semidirect
3. C2×#884C2semidirect
4. C2×#883C2semidirect
5. C3×#394C2semidirect
6. #5223C2semidirect
7. #4978C2semidirect
8. #4961C2semidirect
9. #4960C2semidirect
10. #4960C2semidirect
11. #4960C2semidirect
12. #4956C2semidirect
13. #4956C2semidirect
14. #4670C2semidirect
15. #4665C2semidirect
61. #394D3semidirect
115. C2×D12D4semidirect
137. 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×#9402
C2×#8952
C2×#8842
C2×#8832
C3×#3942
#52232
#49782
#49612
#49602
#49562
#46702
#46652
#26733