Finite Group Databasebeta

← #11803870

Subgroups — #11803870

Order 1152.

1693 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×C3×C4×D12C211yesmaximal, characteristic, complemented
3C2×#63911C211yesmaximal, characteristic
4C3×#4932C211yesmaximal, characteristic, complemented
5#10591387C214yesmaximal, complemented
6#10591382C214yesmaximal, complemented
7#10591320C211yesmaximal, characteristic, complemented
8#10591318C211yesmaximal, characteristic, complemented
9#10591224C211yesmaximal, characteristic, complemented
10#82329—31nomaximal
11#73389—31nomaximal
12C22×C4⋊(C3⋊D3)C2211yescharacteristic
13C22×C4×C3⋊D3C2212yes—
14C22×C3×D12C2211yescharacteristic
15C22×C3×C4×D3C2212yes—
16C22×#626C2214yes—
17C2×C32×C42C2211yescharacteristic
18C2×#3752C2212yes—
19C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C4×C4⋊(C3⋊D3)—24no—
21C2×C3×(C2×D3)⋊C4C2212yes—
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2211yescharacteristic
25C3×C4×D12—24no—
26C2×#3722C2214yes—
27C2×(C2×D3)⋊Dic3C2214yes—
28#63911C418yescomplemented
29C3×C3⋊#95C2212yes—
30C3×C3⋊#95C2212yes—
31#63579C2214yes—
32C2×C3×C4×D4—31no—
33C2×#940—31no—
34C2×#874—31no—
35C2×C4×D12—31no—
36C2×C4×D12—31no—
37C2×C4×D12—61no—
38C2×C4×D12D311yescharacteristic, complemented
39C3×#376—31no—
40#5221—31no—
41#5213—34no—
42#4988—31no—
43#4977—31no—
44#4956—34no—
45#4939—31no—
46#4939—31no—
47#4936—31no—
48#4932D311yescharacteristic, complemented
49#4665—34no—
50#4665—34no—
51C23×C3⋊D3C2312yes—
52C23×C3×D3C2312yes—
53C22×C32×C4C2311yescharacteristic
54C22×C32×C4C2311yescharacteristic
55C22×C32×C4C2311yescharacteristic
56C22×C3⋊Dic3C2312yes—
57C2×C4⋊(C3⋊D3)—22no—
58C2×C4⋊(C3⋊D3)—22no—
59C2×C4⋊(C3⋊D3)C2×C414yescomplemented
60C2×C4×C3⋊D3—24no—
61C2×C4×C3⋊D3—44no—
62C22×C3×Dic3C2312yes—
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2312yes—
65C2×C3×D12—22no—
66C2×C3×D12—22no—
67C2×C3×D12C2×C414yescomplemented
68C2×C3×C4×D3—24no—
69C2×C3×C4×D3—44no—
70C2×#626—44no—
71C2×#626—44no—
72C2×#626C2×C4116yescomplemented
73C32×C42D414yescomplemented
74#3752—24no—
75C4⋊(C3⋊Dic3)—22no—
76C3×(C2×D3)⋊C4—24no—
77C3×C4⋊Dic3—22no—
78C3×C4×Dic3—22no—
79C3×C4×Dic3C2×C414yescomplemented
80#3722—44no—
81(C2×D3)⋊Dic3—44no—
82#2328—91noSylow
83C22×C3×D4—31no—
84C23×C3×C4—32no—
85C22×#67—34no—
86C22×D12—31no—
87C22×D12—31no—
88C22×D12—34no—
89C22×D12—61no—
90C22×D12C2×D311yescharacteristic
91C22×C4×D3—32no—
92C22×C4×D3—32no—
93C22×C4×D3—62no—
94C22×C4×D3C2×D312yes—
95C3×C4×D4—64no—
96C2×C3×C4⋊C4—31no—
97C2×C3×C22⋊C4—32no—
98C2×C3×C42—21no—
99C2×C3×C42—31no—
100C2×C3×C42—31no—
101C2×C3×C42C2×D311yescharacteristic
102C2×C3×C42C2×D311yescharacteristic
103C2×#217—34no—
104#940—38no—
105C2×(C2×D3)⋊C4—32no—
106C2×(C2×D3)⋊C4—32no—
107C2×(C2×D3)⋊C4—34no—
108C2×(C2×D3)⋊C4—34no—
109C2×(C2×D3)⋊C4—34no—
110C2×(C2×D3)⋊C4—62no—
111C2×(C2×D3)⋊C4C2×D312yes—
112C2×C4⋊Dic3—31no—
113C2×C4⋊Dic3—31no—
114C2×C4⋊Dic3—61no—
115C2×C4⋊Dic3C2×D311yescharacteristic
116C2×C4×Dic3C2×D311yescharacteristic
117C2×C4×Dic3C2×D311yescharacteristic
118#874—38no—
119C4×D12—24no—
120C4×D12—68no—
121C4×D12—64no—
122C4×D12—64no—
123C3×#95—32no—
124C3×#95—32no—
125C3⋊#95—34no—
126C3⋊#95—34no—
127C3⋊#95C2×D312yes—
128C3⋊#95C2×D312yes—
129C23×C32C2411yescharacteristic
130C22×C3⋊D3—22no—
131C22×C3⋊D3—22no—
132C22×C3⋊D3—24no—
133C22×C3⋊D3—24no—
134C22×C3⋊D3C22×C414yes—
135C22×C3×D3—22no—
136C22×C3×D3—22no—
137C22×C3×D3—24no—
138C22×C3×D3—24no—
139C22×C3×D3C22×C414yes—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4C22×C412yes—
143C2×C32×C4C2×D412yes—
144C2×C32×C4C2×D412yes—
145C2×C32×C4C2×D412yes—
146C2×C32×C4#4112yes—
147C2×C3⋊Dic3—22no—
148C2×C3⋊Dic3—24no—
149C4⋊(C3⋊D3)—28no—
150C4×C3⋊D3—48no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3C22×C414yes—
159C3×D12—28no—
160C3×C4×D3—48no—
161#626—416no—
162C2×#127—91no—
163C2×C4×D4—31no—
164C2×C4×D4—91no—
165#390—91no—
166#390—91no—
167#386—94no—
168#386—94no—
169#382—91no—
170#376—31no—
171C24×C3—32no—
172C23×D3—32no—
173C23×D3—32no—
174C23×D3—62no—
175C23×D3C22×D312yes—
176C2×C3×D4—34no—
177C2×C3×D4—62no—
178C2×C3×D4—62no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—32no—
183C22×C3×C4—32no—
184C22×C3×C4—32no—
185C22×C3×C4—64no—
186C22×C3×C4—124no—
187C22×C3×C4C22×D311yescharacteristic
188C22×C3×C4C22×D311yescharacteristic
189C22×C3×C4C22×D311yescharacteristic
190C22×C3×C4C22×D311yescharacteristic
191C22×C3×C4C22×D311yescharacteristic
192C22×C3×C4C22×D311yescharacteristic
193C2×#67—316no—
194C2×#67—124no—
195C2×#67—124no—
196C22×Dic3—32no—
197C22×Dic3—32no—
198C22×Dic3—62no—
199C22×Dic3C22×D312yes—
200C22×Dic3C22×D312yes—
201C22×Dic3C22×D312yes—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—34no—
205C2×D12—34no—
206C2×D12—316no—
207C2×D12—64no—
208C2×D12—64no—
209C2×D12—64no—
210C2×D12—62no—
211C2×D12—62no—
212C2×D12—62no—
213C2×D12—62no—
214C2×D12—124no—
215C2×D12—124no—
216C2×D12C4×D314yescomplemented
217C2×C4×D3—24no—
218C2×C4×D3—44no—
219C2×C4×D3—68no—
220C2×C4×D3—64no—
221C2×C4×D3—64no—
222C2×C4×D3—124no—
223C2×C4×D3—124no—
224C2×C4×D3—128no—
225C3×C4⋊C4—62no—
226C3×C22⋊C4—64no—
227C3×C42—24no—
228C3×C42—34no—
229C3×C42—62no—
230C3×C42D1214yescomplemented
231C3×C42#6714yescomplemented
232#217—124no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—68no—
235(C2×D3)⋊C4—64no—
236(C2×D3)⋊C4—64no—
237(C2×D3)⋊C4—124no—
238(C2×D3)⋊C4—124no—
239(C2×D3)⋊C4—124no—
240C4⋊Dic3—22no—
241C4⋊Dic3—64no—
242C4⋊Dic3—62no—
243C4⋊Dic3—62no—
244C4×Dic3—22no—
245C4×Dic3C4×D314yescomplemented
246C22×C32C23×C411yescharacteristic
247C22×C32C22×D411yescharacteristic
248C22×C32C2×#4111yescharacteristic
249C22×C32H(5,𝔽2)12yes—
250C22×C32H(5,𝔽2)12yes—
251C2×C3⋊D3—24no—
252C2×C3⋊D3—28no—
253C2×C3⋊D3—42no—
254C2×C3⋊D3—42no—
255C2×C3⋊D3—44no—
256C2×C3×D3—24no—
257C2×C3×D3—28no—
258C2×C3×D3—42no—
259C2×C3×D3—42no—
260C2×C3×D3—44no—
261C32×C4—22no—
262C32×C4—22no—
263C32×C4C4×D414yes—
264C3⋊Dic3—42no—
265C3×Dic3—24no—
266C3×Dic3—42no—
267C3×Dic3—42no—
268C22×D4—31no—
269C22×D4—94no—
270C22×D4—91no—
271C23×C4—32no—
272C23×C4—92no—
273#127—98no—
274C4×D4—64no—
275C4×D4—184no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—91no—
278C2×C22⋊C4—32no—
279C2×C22⋊C4—92no—
280C2×C22⋊C4—94no—
281C2×C22⋊C4—94no—
282C2×C42—31no—
283C2×C42—31no—
284C2×C42D3211yescharacteristic
285#95—32no—
286#95—32no—
287#95—94no—
288C23×C3—21no—
289C23×C3—34no—
290C23×C3—62no—
291C23×C3—62no—
292C23×C3—64no—
293C23×C3—64no—
294C23×C3C23×D311yescharacteristic
295C23×C3C23×D311yescharacteristic
296C22×D3—22no—
297C22×D3—24no—
298C22×D3—24no—
299C22×D3—22no—
300C22×D3—34no—
301C22×D3—34no—
302C22×D3—64no—
303C22×D3—64no—
304C22×D3—64no—
305C22×D3—62no—
306C22×D3—62no—
307C22×D3—62no—
308C22×D3—64no—
309C22×D3—64no—
310C22×D3—64no—
311C22×D3—64no—
312C22×D3—62no—
313C22×D3—124no—
314C22×D3—124no—
315C22×D3C2×C4×D314yes—
316C3×D4—68no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—34no—
328C2×C3×C4—42no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—64no—
336C2×C3×C4—128no—
337C2×C3×C4C2×C4×D312yes—
338C2×C3×C4C2×C4×D312yes—
339C2×C3×C4C2×D1212yes—
340C2×C3×C4C2×D1212yes—
341C2×C3×C4C2×D1212yes—
342C2×C3×C4#23512yes—
343C2×C3×C4#23512yes—
344C2×C3×C4C2×#6712yes—
345C2×C3×C4C2×#6712yes—
346C2×C3×C4C2×#6712yes—
347#67—1216no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—24no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—64no—
356C2×Dic3—62no—
357C2×Dic3—64no—
358C2×Dic3—64no—
359C2×Dic3—62no—
360C2×Dic3—124no—
361C2×Dic3C2×C4×D314yes—
362D12—28no—
363D12—616no—
364D12—68no—
365D12—68no—
366D12—1216no—
367C4×D3—48no—
368C4×D3—1216no—
369C4×D3—128no—
370C4×D3—128no—
371C2×C32C2×C4×D411yescharacteristic
372C2×C32#51812yes—
373C2×C32#53511yescharacteristic
374C2×C32#53511yescharacteristic
375C2×C32#54012yes—
376C3⋊D3—44no—
377C3×D3—44no—
378C24—32no—
379C24—92no—
380C2×D4—34no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—916no—
384C2×D4—94no—
385C2×D4—182no—
386C2×D4—182no—
387C2×D4—364no—
388C2×D4—364no—
389C22×C4—32no—
390C22×C4—32no—
391C22×C4—32no—
392C22×C4—64no—
393C22×C4—92no—
394C22×C4—124no—
395C22×C4—184no—
396C22×C4—364no—
397C22×C4C2×D3211yescharacteristic
398C22×C4C2×D3211yescharacteristic
399C22×C4C2×D3211yescharacteristic
400C4⋊C4—62no—
401C4⋊C4—182no—
402C22⋊C4—64no—
403C22⋊C4—184no—
404C22⋊C4—364no—
405C22⋊C4—364no—
406C42—34no—
407C42—62no—
408C42#62614yescomplemented
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—21no—
414C22×C3—64no—
415C22×C3—68no—
416C22×C3—122no—
417C22×C3—122no—
418C22×C3—124no—
419C22×C3C22×C4×D311yescharacteristic
420C22×C3C22×C4×D311yescharacteristic
421C22×C3C22×D1211yescharacteristic
422C22×C3C2×#23511yescharacteristic
423C22×C3C2×#23511yescharacteristic
424C22×C3#100412yes—
425C22×C3#100912yes—
426C22×C3#100912yes—
427C22×C3#100912yes—
428C22×C3C22×#6711yescharacteristic
429C2×D3—24no—
430C2×D3—28no—
431C2×D3—42no—
432C2×D3—44no—
433C2×D3—42no—
434C2×D3—68no—
435C2×D3—64no—
436C2×D3—68no—
437C2×D3—64no—
438C2×D3—68no—
439C2×D3—128no—
440C2×D3—124no—
441C2×D3—128no—
442C2×D3—124no—
443C2×D3—122no—
444C2×D3—122no—
445C2×D3—124no—
446C2×D3—122no—
447C2×D3—124no—
448C2×D3—122no—
449C3×C4—24no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—24no—
455C3×C4—42no—
456C3×C4—64no—
457C3×C4—122no—
458C3×C4—122no—
459C3×C4C4×D1214yes—
460C3×C4C4×#6714yes—
461Dic3—24no—
462Dic3—42no—
463Dic3—42no—
464Dic3—124no—
465Dic3—122no—
466Dic3—122no—
467C32#232811yescharacteristic, complemented, Sylow
468C23—34no—
469C23—62no—
470C23—64no—
471C23—64no—
472C23—62no—
473C23—94no—
474C23—182no—
475C23—184no—
476C23—184no—
477C23—182no—
478C23C22×D3211yescharacteristic
479D4—68no—
480D4—188no—
481D4—3616no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—34no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—64no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—128no—
493C2×C4—184no—
494C2×C4—182no—
495C2×C4—368no—
496C2×C4#379812yes—
497C2×C4C4×D3212yes—
498C2×C4C2×#62612yes—
499C2×C4C2×#62612yes—
500C2×C4C2×#62612yes—
501C2×C3—21no—
502C2×C3—22no—
503C2×C3—22no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—124no—
507C2×C3C2×C4×D1211yescharacteristic
508C2×C3#552112yes—
509C2×C3#552511yescharacteristic
510C2×C3#552511yescharacteristic
511C2×C3#553512yes—
512C2×C3#554512yes—
513C2×C3#556212yes—
514C2×C3C2×C4×#6711yescharacteristic
515C2×C3#580511yescharacteristic
516C2×C3#583011yescharacteristic
517D3—44no—
518D3—128no—
519D3—124no—
520D3—124no—
521C22—64no—
522C22—68no—
523C22—124no—
524C22—122no—
525C22—122no—
526C22—184no—
527C22—188no—
528C22—364no—
529C22—362no—
530C22—362no—
531C22C2×#379811yescharacteristic
532C22C2×C4×D3211yescharacteristic
533C22#6430712yes—
534C22#6430812yes—
535C22C22×#62611yescharacteristic
536C4—22no—
537C4—22no—
538C4—64no—
539C4—122no—
540C4—122no—
541C4—362no—
542C4C4×#62614yes—
543C3—21no—
544C3#7338911yescharacteristic, complemented
545C3#8232911yescharacteristic, complemented
546C2—124no—
547C2—364no—
548C2#1059398612yes—
549C2#1059399412yes—
550C2C2×C4×#62611yescharacteristic
551C2#1059504211yescharacteristic
552C2#1059504511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C42C22
socle, cosocleC23×C32C24
prime radicalC2×#63911C2
hypercenterC2×C42D32
hypocenterC32#2328
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×C4⋊(C3⋊D3)C2semidirect
2. C2×C3×C4×D12C2semidirect
4. C3×#4932C2semidirect
5. #10591387C2semidirect
6. #10591382C2semidirect
7. #10591320C2semidirect
8. #10591318C2semidirect
9. #10591224C2semidirect
28. #63911C4semidirect
38. C2×C4×D12D3semidirect
48. #4932D3semidirect
59. C2×C4⋊(C3⋊D3)C2×C4semidirect
67. C2×C3×D12C2×C4semidirect
72. C2×#626C2×C4semidirect
73. C32×C42D4semidirect
79. C3×C4×Dic3C2×C4semidirect
216. C2×D12C4×D3semidirect
230. C3×C42D12semidirect
231. C3×C42#67semidirect
245. C4×Dic3C4×D3semidirect
408. C42#626semidirect
467. C32#2328semidirect
544. C3#73389semidirect
545. C3#82329semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊(C3⋊D3)2
C2×C3×C4×D122
C2×#639112
C3×#49322
#105913872
#105913822
#105913202
#105913182
#105912242
#823293
#733893