Finite Group Databasebeta

← #13149419

Subgroups — #13149419

Order 1728.

1121 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88649C211yesmaximal, characteristic
2C4⋊C4×#272C211yesmaximal, characteristic, complemented
3#64102⋊C3C211yesmaximal, characteristic, complemented
4#64096⋊C3C211yesmaximal, characteristic, complemented
5C4×#6200C212yesmaximal, complemented
6#11721132C211yesmaximal, characteristic, complemented
7#11721124C212yesmaximal, complemented
8#11721120C212yesmaximal, complemented
9#11721119C212yesmaximal, complemented
10#11721097C211yesmaximal, characteristic, complemented
11#11721082C211yesmaximal, characteristic, complemented
12#10594359—31nomaximal
13#10594327—31nomaximal
14C2×C4×#272C2212yes—
15C2×C4×#272C2211yescharacteristic
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#634⋊C3C2211yescharacteristic
19C2×#6200C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22#88649C418yescomplemented
23C4⋊C4×H(3,𝔽3)C2211yescharacteristic
24C4⋊#1084C2211yescharacteristic
25#88540C2212yes—
26#3752⋊C3C2212yes—
27#3750⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2212yes—
31#88451C2211yescharacteristic
32#88451C2211yescharacteristic
33#88450C2212yes—
34#88449C2211yescharacteristic
35#88449C2211yescharacteristic
36#88448C2212yes—
37#88447C2212yes—
38C2×#3799—31no—
39C2×#3795—31no—
40#64102D311yescharacteristic, complemented
41#64096D311yescharacteristic, complemented
42C3×#893—31no—
43C3×D3×C4⋊C4—31no—
44C3×D3×C4⋊C4—31no—
45C3×D3×C4⋊C4—61no—
46C3×#887—31no—
47#63904—32no—
48C4×#626—32no—
49C4×#625—32no—
50#63898—31no—
51#63894—31no—
52#63883—32no—
53#63879—32no—
54#63878—32no—
55D3×Dic3⋊C4—32no—
56#63855—31no—
57#63855—31no—
58#63844—31no—
59#63843—31no—
60#63836—32no—
61C22×#272C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2312yes—
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2312yes—
66C2×#1084C2311yescharacteristic
67C4×#272—42no—
68C4×#272C2×C414yescomplemented
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
75C4×(C3⋊D3)⋊C3—42no—
76#634⋊C3C2×C414yescomplemented
77#6200C2×C418yescomplemented
78((C3⋊D3)⋊C3)⋊C4—42no—
79((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
80#5574—91no—
81#5546—91no—
82#5504—91no—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D312yes—
85C2×#634C2×D311yescharacteristic
86C2×C3×D12—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—61no—
93C2×C3×C4×D3—62no—
94C2×C3×Dic6—31no—
95C2×#626—32no—
96C2×#625—32no—
97C2×#624—32no—
98C2×#624—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101#3799—38no—
102#3795—38no—
103C32×C4⋊C4—21no—
104C32×C4⋊C4C2×D311yescharacteristic
105C32×C4⋊C4C2×D311yescharacteristic
106#3752C2×D312yes—
107#3750C2×D312yes—
108C4×C3⋊Dic3C2×D311yescharacteristic
109C4×C3⋊Dic3C2×D312yes—
110C4×C3⋊Dic3C2×D311yescharacteristic
111C3×(C2×D3)⋊C4—32no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—61no—
115C3×Dic3⋊C4—32no—
116C3×Dic3⋊C4—32no—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4—62no—
119C3×C4×Dic3—32no—
120C3×C4×Dic3—31no—
121C3×C4×Dic3—31no—
122#3724—32no—
123Dic3⋊Dic3—31no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—31no—
126Dic3⋊Dic3—31no—
127Dic3⋊Dic3—31no—
128#3722—32no—
129(C2×D3)⋊Dic3—31no—
130(C2×D3)⋊Dic3—31no—
131(C2×D3)⋊Dic3—31no—
132(C2×D3)⋊Dic3—31no—
133(C2×D3)⋊Dic3—32no—
134Dic32—32no—
135Dic32—32no—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×#272—21no—
138C2×#272—21no—
139C2×#272C22×C412yes—
140C2×(C3⋊D3)⋊C3—22no—
141C2×(C3⋊D3)⋊C3—22no—
142C2×(C3⋊D3)⋊C3C22×C414yes—
143C4×H(3,𝔽3)—22no—
144C4×H(3,𝔽3)C22×C412yes—
145#1084—22no—
146#1084C22×C412yes—
147(C3⋊Dic3)⋊C3—21no—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—21no—
150(C3⋊Dic3)⋊C3—21no—
151(C3⋊Dic3)⋊C3C22×C414yes—
152C2×#239—91no—
153C2×#237—91no—
154C2×#235—91no—
155C3×C4×Q8—91no—
156C3×C4×D4—91no—
157C2×C3×C4⋊C4—91no—
158Q8×Dic3—91no—
159D4×Dic3—91no—
160C4×#67—92no—
161C4×#67—92no—
162C4×#67—92no—
163#926—91no—
164C2×Dic3⋊C4—92no—
165#893—31no—
166#893—91no—
167#893—91no—
168#893—92no—
169#892—92no—
170D3×C4⋊C4—31no—
171D3×C4⋊C4—31no—
172D3×C4⋊C4—61no—
173D3×C4⋊C4—92no—
174D3×C4⋊C4—92no—
175#887—31no—
176#887—91no—
177#887—91no—
178#67⋊C4—92no—
179#877—92no—
180C4×D12—91no—
181C4×D12—92no—
182#872—92no—
183#872—91no—
184#872—91no—
185C4×Dic6—91no—
186C22×C3⋊D3C22×D312yes—
187C22×C3×D3—32no—
188C22×C3×D3—31no—
189C22×C3×D3—31no—
190C22×C3×D3—61no—
191C2×C32×C4—21no—
192C2×C32×C4—22no—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D311yescharacteristic
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D312yes—
202C4⋊(C3⋊D3)C4×D314yescomplemented
203C4×C3⋊D3—42no—
204#634C4×D314yescomplemented
205C2×C3×Dic3—32no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—61no—
215C2×C3×Dic3—62no—
216C3×D12—34no—
217C3×C4×D3—34no—
218C3×C4×D3—34no—
219C3×C4×D3—64no—
220C3×C4×D3—122no—
221C3×C4×D3—122no—
222C3×C4×D3—122no—
223C3×C4×D3—124no—
224C3×Dic6—34no—
225#626—38no—
226#625—38no—
227#624—38no—
228#624—122no—
229D3×Dic3—38no—
230D3×Dic3—122no—
231#520—271noSylow
232C2×H(3,𝔽3)C23×C411yescharacteristic
233C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
234C2×H(3,𝔽3)2−1+411yescharacteristic
235#272—22no—
236(C3⋊D3)⋊C3—24no—
237C2×C3×Q8—91no—
238C2×C3×D4—91no—
239C22×C3×C4—92no—
240C22×C3×C4—91no—
241C22×C3×C4—92no—
242C2×#67—92no—
243C2×#67—92no—
244C22×Dic3—92no—
245C22×Dic3—92no—
246#239—98no—
247#237—98no—
248#235—98no—
249C2×D12—31no—
250C2×D12—91no—
251C2×D12—92no—
252C2×C4×D3—31no—
253C2×C4×D3—31no—
254C2×C4×D3—32no—
255C2×C4×D3—32no—
256C2×C4×D3—32no—
257C2×C4×D3—61no—
258C2×C4×D3—62no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×C4×D3—92no—
262C2×C4×D3—92no—
263C2×C4×D3—92no—
264C2×Dic6—31no—
265C2×Dic6—91no—
266C3×C4⋊C4—31no—
267C3×C4⋊C4—31no—
268C3×C4⋊C4—61no—
269C3×C4⋊C4—92no—
270C3×C4⋊C4—91no—
271C3×C4⋊C4—92no—
272C3×C4⋊C4D3211yescharacteristic
273C3×C22⋊C4—92no—
274C3×C42—91no—
275C3×C42—91no—
276C3×C42—92no—
277#217—91no—
278#217—91no—
279#217—92no—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—91no—
285(C2×D3)⋊C4—91no—
286(C2×D3)⋊C4—91no—
287(C2×D3)⋊C4—91no—
288C4⋊Dic3—31no—
289C4⋊Dic3—31no—
290C4⋊Dic3—61no—
291C4⋊Dic3—91no—
292C4⋊Dic3—91no—
293C4⋊Dic3—92no—
294Dic3⋊C4—32no—
295Dic3⋊C4—32no—
296Dic3⋊C4—32no—
297Dic3⋊C4—62no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—91no—
302Dic3⋊C4—91no—
303Dic3⋊C4—91no—
304Dic3⋊C4—91no—
305C4×Dic3—32no—
306C4×Dic3—31no—
307C4×Dic3—31no—
308C4×Dic3—91no—
309C4×Dic3—91no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C4×Dic3—92no—
313C4×Dic3—92no—
314C22×C32—21no—
315C22×C32C23×D311yescharacteristic
316C22×C32C23×D311yescharacteristic
317C2×C3⋊D3—22no—
318C2×C3⋊D3—22no—
319C2×C3⋊D3C2×C4×D314yes—
320C2×C3×D3—34no—
321C2×C3×D3—32no—
322C2×C3×D3—32no—
323C2×C3×D3—62no—
324C2×C3×D3—62no—
325C2×C3×D3—62no—
326C2×C3×D3—61no—
327C2×C3×D3—62no—
328C2×C3×D3—61no—
329C2×C3×D3—62no—
330C2×C3×D3—61no—
331C2×C3×D3—61no—
332C32×C4—22no—
333C32×C4—22no—
334C32×C4—22no—
335C32×C4—42no—
336C32×C4C2×C4×D312yes—
337C32×C4C2×C4×D312yes—
338C3⋊Dic3—21no—
339C3⋊Dic3—21no—
340C3⋊Dic3—21no—
341C3⋊Dic3—21no—
342C3⋊Dic3C2×C4×D314yes—
343C3×Dic3—34no—
344C3×Dic3—32no—
345C3×Dic3—32no—
346C3×Dic3—62no—
347C3×Dic3—61no—
348C3×Dic3—62no—
349C3×Dic3—61no—
350C3×Dic3—61no—
351C3×Dic3—62no—
352C3×Dic3—61no—
353C3×Dic3—122no—
354C2×#41—271no—
355C4×Q8—91no—
356C4×Q8—271no—
357C4×D4—91no—
358C4×D4—271no—
359C4×D4—272no—
360C4×D4—272no—
361#117—271no—
362#117—272no—
363C2×C4⋊C4—91no—
364C2×C4⋊C4—272no—
365H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
366C23×C3—92no—
367C23×C3—91no—
368C22×D3—31no—
369C22×D3—31no—
370C22×D3—32no—
371C22×D3—61no—
372C22×D3—92no—
373C3×Q8—94no—
374C3×D4—94no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—32no—
378C2×C3×C4—32no—
379C2×C3×C4—61no—
380C2×C3×C4—62no—
381C2×C3×C4—92no—
382C2×C3×C4—91no—
383C2×C3×C4—94no—
384C2×C3×C4—91no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—92no—
389C2×C3×C4—362no—
390C2×C3×C4—362no—
391C2×C3×C4C2×D3212yes—
392C2×C3×C4C2×D3211yescharacteristic
393#67—98no—
394#67—98no—
395C2×Dic3—31no—
396C2×Dic3—31no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—31no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—61no—
405C2×Dic3—62no—
406C2×Dic3—92no—
407C2×Dic3—91no—
408C2×Dic3—91no—
409C2×Dic3—91no—
410C2×Dic3—91no—
411C2×Dic3—98no—
412C2×Dic3—362no—
413D12—34no—
414D12—94no—
415D12—98no—
416C4×D3—34no—
417C4×D3—34no—
418C4×D3—64no—
419C4×D3—98no—
420C4×D3—98no—
421C4×D3—122no—
422C4×D3—122no—
423C4×D3—122no—
424C4×D3—124no—
425C4×D3—362no—
426C4×D3—362no—
427C4×D3—362no—
428Dic6—34no—
429Dic6—94no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32C22×C4×D311yescharacteristic
434C2×C32C22×C4×D311yescharacteristic
435C2×C32#100411yescharacteristic
436C2×C32#100711yescharacteristic
437C2×C32#100911yescharacteristic
438C2×C32#101011yescharacteristic
439C3⋊D3—24no—
440C3×D3—64no—
441C3×D3—62no—
442C3×D3—64no—
443C3×D3—62no—
444#41—278no—
445C2×Q8—91no—
446C2×D4—91no—
447C2×D4—272no—
448C22×C4—91no—
449C22×C4—92no—
450C22×C4—92no—
451C22×C4—272no—
452C22×C4—272no—
453C4⋊C4—92no—
454C4⋊C4—91no—
455C4⋊C4—92no—
456C4⋊C4—271no—
457C4⋊C4—272no—
458C4⋊C4—271no—
459C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
460C22⋊C4—92no—
461C22⋊C4—271no—
462C22⋊C4—272no—
463C22⋊C4—271no—
464C42—91no—
465C42—92no—
466C42—91no—
467C42—272no—
468C22×C3—31no—
469C22×C3—31no—
470C22×C3—61no—
471C22×C3—94no—
472C22×C3—92no—
473C22×C3—182no—
474C22×C3—182no—
475C22×C3—181no—
476C22×C3—181no—
477C22×C3C22×D3211yescharacteristic
478C2×D3—32no—
479C2×D3—32no—
480C2×D3—34no—
481C2×D3—62no—
482C2×D3—61no—
483C2×D3—61no—
484C2×D3—61no—
485C2×D3—61no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—62no—
490C2×D3—94no—
491C2×D3—182no—
492C2×D3—182no—
493C3×C4—22no—
494C3×C4—32no—
495C3×C4—32no—
496C3×C4—62no—
497C3×C4—62no—
498C3×C4—62no—
499C3×C4—94no—
500C3×C4—92no—
501C3×C4—122no—
502C3×C4—181no—
503C3×C4—181no—
504C3×C4—181no—
505C3×C4—181no—
506C3×C4—182no—
507C3×C4C4×D3212yes—
508Dic3—32no—
509Dic3—32no—
510Dic3—34no—
511Dic3—62no—
512Dic3—62no—
513Dic3—62no—
514Dic3—61no—
515Dic3—61no—
516Dic3—61no—
517Dic3—61no—
518Dic3—94no—
519Dic3—122no—
520Dic3—181no—
521Dic3—181no—
522Dic3—181no—
523Dic3—181no—
524C32—21no—
525C32#554611yescharacteristic, complemented
526C32#557411yescharacteristic, complemented
527C23—92no—
528C23—91no—
529Q8—94no—
530D4—94no—
531D4—278no—
532C2×C4—92no—
533C2×C4—91no—
534C2×C4—94no—
535C2×C4—91no—
536C2×C4—91no—
537C2×C4—91no—
538C2×C4—91no—
539C2×C4—92no—
540C2×C4—278no—
541C2×C4—362no—
542C2×C4—362no—
543C2×C4—1082no—
544C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
545C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
546C2×C3—31no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—31no—
550C2×C3—31no—
551C2×C3—31no—
552C2×C3—61no—
553C2×C3—61no—
554C2×C3—61no—
555C2×C3—184no—
556C2×C3—182no—
557C2×C3C2×C4×D3211yescharacteristic
558C2×C3#6431411yescharacteristic
559C2×C3#6431511yescharacteristic
560D3—62no—
561D3—62no—
562D3—64no—
563D3—64no—
564D3—184no—
565C22—92no—
566C22—94no—
567C22—181no—
568C22—182no—
569C22—181no—
570C22—182no—
571C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
572C4—22no—
573C4—92no—
574C4—94no—
575C4—181no—
576C4—182no—
577C4—181no—
578C4—181no—
579C4—181no—
580C4C4×((C3⋊D3)⋊C3)⋊C212yes—
581C3—31no—
582C3—31no—
583C3—61no—
584C3#1059436111yescharacteristic
585C2—182no—
586C2—184no—
587C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
588C2#1172250711yescharacteristic
589C2#1172251611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC2×#88649C2
cosocleC22×H(3,𝔽3)C24
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#520
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×#272C2semidirect
3. #64102⋊C3C2semidirect
4. #64096⋊C3C2semidirect
5. C4×#6200C2semidirect
6. #11721132C2semidirect
7. #11721124C2semidirect
8. #11721120C2semidirect
9. #11721119C2semidirect
10. #11721097C2semidirect
11. #11721082C2semidirect
22. #88649C4semidirect
40. #64102D3semidirect
41. #64096D3semidirect
68. C4×#272C2×C4semidirect
74. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
76. #634⋊C3C2×C4semidirect
77. #6200C2×C4semidirect
79. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
202. C4⋊(C3⋊D3)C4×D3semidirect
204. #634C4×D3semidirect
365. H(3,𝔽3)#520semidirect
525. C32#5546semidirect
526. C32#5574semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
C4⋊C4×#2722
#64102⋊C32
#64096⋊C32
C4×#62002
#117211322
#117211242
#117211202
#117211192
#117210972
#117210822
#105943593
#105943273