Finite Group Databasebeta

← #13147776

Subgroups — #13147776

Order 1728.

1105 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88649C211yesmaximal, characteristic
2#11721718C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4C4×#634⋊C3C211yesmaximal, characteristic, complemented
5#11721154C212yesmaximal, complemented
6#11721152C211yesmaximal, characteristic, complemented
7#11721136C211yesmaximal, characteristic, complemented
8#11721123C212yesmaximal, complemented
9#11721120C212yesmaximal, complemented
10#11721119C212yesmaximal, complemented
11#11721084C211yesmaximal, characteristic, complemented
12#10593890—31nomaximal
13#10593887—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×#634⋊C3C2211yescharacteristic
18C2×#6200C2212yes—
19C2×#6199C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22#88649C418yescomplemented
23C42×H(3,𝔽3)C2211yescharacteristic
24#88542C2211yescharacteristic
25#88542C2211yescharacteristic
26#88540C2211yescharacteristic
27#88540C2211yescharacteristic
28C4×#1084C2211yescharacteristic
29#3752⋊C3C2212yes—
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
32#3750⋊C3C2212yes—
33C4×(C3⋊Dic3)⋊C3C2212yes—
34#88450C2212yes—
35#88449C2212yes—
36#88448C2212yes—
37#88447C2212yes—
38C2×#3799—31no—
39C2×#3795—31no—
40C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
41C4×#634D311yescharacteristic, complemented
42C3×C4×D12—31no—
43C3×#872—31no—
44C3×#872—31no—
45C3×#872—61no—
46C3×C4×Dic6—31no—
47#63902—32no—
48#63894—31no—
49Dic3×D12—31no—
50D3×C4⋊Dic3—31no—
51D3×C4⋊Dic3—31no—
52#63883—32no—
53#63881—32no—
54#63880—32no—
55#63879—32no—
56#63878—32no—
57D3×Dic3⋊C4—32no—
58#63844—31no—
59Dic3×Dic6—31no—
60#63836—32no—
61C22×#272C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C2×#1084C2311yescharacteristic
68C2×#1084C2311yescharacteristic
69C4×#272C2×C414yescomplemented
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
74C4×(C3⋊D3)⋊C3—42no—
75#634⋊C3C2×C414yescomplemented
76#6200C2×C418yescomplemented
77#6199—42no—
78((C3⋊D3)⋊C3)⋊C4—42no—
79((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
80#5574—91no—
81#5546—91no—
82#5523—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—61no—
91C2×C3×Dic6—31no—
92C2×#626—32no—
93C2×#625—32no—
94C2×#624—32no—
95C2×#624—32no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100#3799—38no—
101#3795—38no—
102C32×C42—21no—
103C32×C42C2×D311yescharacteristic
104C32×C42C2×D311yescharacteristic
105#3752C2×D312yes—
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108#3750C2×D312yes—
109C4×C3⋊Dic3C2×D312yes—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—31no—
112C3×(C2×D3)⋊C4—31no—
113C3×(C2×D3)⋊C4—31no—
114C3×(C2×D3)⋊C4—31no—
115C3×(C2×D3)⋊C4—61no—
116C3×(C2×D3)⋊C4—61no—
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—31no—
119C3×Dic3⋊C4—31no—
120C3×Dic3⋊C4—31no—
121C3×Dic3⋊C4—31no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—31no—
124C3×Dic3⋊C4—61no—
125C3×Dic3⋊C4—61no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—31no—
128C3×C4×Dic3—31no—
129C3×C4×Dic3—61no—
130#3724—32no—
131Dic3⋊Dic3—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134#3722—32no—
135(C2×D3)⋊Dic3—32no—
136Dic32—32no—
137Dic32—32no—
138C22×H(3,𝔽3)C2411yescharacteristic
139C2×#272—21no—
140C2×#272—21no—
141C2×#272C22×C412yes—
142C2×(C3⋊D3)⋊C3—22no—
143C2×(C3⋊D3)⋊C3—22no—
144C2×(C3⋊D3)⋊C3C22×C414yes—
145C4×H(3,𝔽3)—21no—
146C4×H(3,𝔽3)—21no—
147C4×H(3,𝔽3)C22×C412yes—
148#1084—21no—
149#1084—21no—
150#1084C22×C412yes—
151(C3⋊Dic3)⋊C3—22no—
152(C3⋊Dic3)⋊C3—22no—
153(C3⋊Dic3)⋊C3C22×C414yes—
154C2×#239—91no—
155C2×#237—91no—
156C2×#235—91no—
157C3×C4×Q8—91no—
158C3×C4×D4—91no—
159C3×#117—91no—
160Q8×Dic3—91no—
161D4×Dic3—91no—
162C4×#67—92no—
163C2×C4⋊Dic3—91no—
164C2×Dic3⋊C4—92no—
165#893—91no—
166#893—92no—
167#892—92no—
168D3×C4⋊C4—92no—
169D3×C4⋊C4—92no—
170D3×C4⋊C4—91no—
171D3×C4⋊C4—91no—
172#887—91no—
173#67⋊C4—92no—
174#67⋊C4—92no—
175#67⋊C4—92no—
176#877—92no—
177C4×D12—31no—
178C4×D12—91no—
179C4×D12—91no—
180C4×D12—92no—
181#872—31no—
182#872—31no—
183#872—61no—
184#872—92no—
185C4×Dic6—31no—
186C4×Dic6—91no—
187C4×Dic6—91no—
188C22×C3⋊D3C22×D312yes—
189C22×C3×D3—32no—
190C22×C3×D3—31no—
191C22×C3×D3—31no—
192C22×C3×D3—61no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C3⋊Dic3C22×D312yes—
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D312yes—
205C4⋊(C3⋊D3)C4×D314yescomplemented
206C4×C3⋊D3—42no—
207#634C4×D314yescomplemented
208C2×C3×Dic3—32no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—31no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—61no—
220C3×D12—34no—
221C3×C4×D3—34no—
222C3×C4×D3—34no—
223C3×C4×D3—64no—
224C3×C4×D3—122no—
225C3×Dic6—34no—
226#626—38no—
227#625—38no—
228#624—38no—
229#624—122no—
230D3×Dic3—38no—
231D3×Dic3—122no—
232D3×Dic3—122no—
233D3×Dic3—122no—
234#520—271noSylow
235C2×H(3,𝔽3)C23×C411yescharacteristic
236C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
237C2×H(3,𝔽3)2−1+411yescharacteristic
238#272—22no—
239(C3⋊D3)⋊C3—24no—
240C2×C3×Q8—91no—
241C2×C3×D4—91no—
242C22×C3×C4—92no—
243C22×C3×C4—91no—
244C2×#67—92no—
245C2×#67—92no—
246C22×Dic3—92no—
247C22×Dic3—92no—
248C22×Dic3—92no—
249#239—98no—
250#237—98no—
251#235—98no—
252C2×D12—31no—
253C2×D12—91no—
254C2×D12—92no—
255C2×C4×D3—31no—
256C2×C4×D3—31no—
257C2×C4×D3—32no—
258C2×C4×D3—61no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×C4×D3—92no—
262C2×C4×D3—92no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×Dic6—31no—
267C2×Dic6—91no—
268C3×C4⋊C4—91no—
269C3×C4⋊C4—92no—
270C3×C4⋊C4—91no—
271C3×C4⋊C4—91no—
272C3×C4⋊C4—91no—
273C3×C22⋊C4—92no—
274C3×C22⋊C4—91no—
275C3×C22⋊C4—91no—
276C3×C42—31no—
277C3×C42—31no—
278C3×C42—61no—
279C3×C42—92no—
280C3×C42—91no—
281C3×C42D3211yescharacteristic
282#217—92no—
283(C2×D3)⋊C4—31no—
284(C2×D3)⋊C4—31no—
285(C2×D3)⋊C4—31no—
286(C2×D3)⋊C4—31no—
287(C2×D3)⋊C4—32no—
288(C2×D3)⋊C4—61no—
289(C2×D3)⋊C4—61no—
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293C4⋊Dic3—31no—
294C4⋊Dic3—31no—
295C4⋊Dic3—91no—
296C4⋊Dic3—91no—
297C4⋊Dic3—92no—
298C4⋊Dic3—92no—
299Dic3⋊C4—31no—
300Dic3⋊C4—31no—
301Dic3⋊C4—31no—
302Dic3⋊C4—31no—
303Dic3⋊C4—32no—
304Dic3⋊C4—61no—
305Dic3⋊C4—61no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310Dic3⋊C4—92no—
311C4×Dic3—31no—
312C4×Dic3—31no—
313C4×Dic3—32no—
314C4×Dic3—61no—
315C4×Dic3—92no—
316C4×Dic3—92no—
317C4×Dic3—92no—
318C4×Dic3—92no—
319C22×C32—21no—
320C22×C32C23×D311yescharacteristic
321C22×C32C23×D311yescharacteristic
322C2×C3⋊D3—22no—
323C2×C3⋊D3—22no—
324C2×C3⋊D3C2×C4×D314yes—
325C2×C3×D3—34no—
326C2×C3×D3—32no—
327C2×C3×D3—32no—
328C2×C3×D3—62no—
329C2×C3×D3—61no—
330C2×C3×D3—62no—
331C2×C3×D3—61no—
332C2×C3×D3—62no—
333C2×C3×D3—61no—
334C2×C3×D3—61no—
335C2×C3×D3—121no—
336C2×C3×D3—121no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—22no—
342C32×C4—41no—
343C32×C4—41no—
344C32×C4C2×C4×D312yes—
345C32×C4C2×C4×D312yes—
346C3⋊Dic3—22no—
347C3⋊Dic3—22no—
348C3⋊Dic3C2×C4×D314yes—
349C3×Dic3—34no—
350C3×Dic3—32no—
351C3×Dic3—32no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—61no—
355C3×Dic3—61no—
356C3×Dic3—62no—
357C3×Dic3—61no—
358C3×Dic3—61no—
359C3×Dic3—121no—
360C3×Dic3—121no—
361C2×#41—271no—
362C4×Q8—91no—
363C4×Q8—271no—
364C4×D4—91no—
365C4×D4—272no—
366C4×D4—271no—
367C4×D4—272no—
368#117—91no—
369#117—272no—
370C2×C4⋊C4—272no—
371C2×C4⋊C4—271no—
372H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
373C23×C3—92no—
374C23×C3—91no—
375C22×D3—31no—
376C22×D3—31no—
377C22×D3—32no—
378C22×D3—61no—
379C22×D3—92no—
380C3×Q8—94no—
381C3×D4—94no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—31no—
388C2×C3×C4—61no—
389C2×C3×C4—61no—
390C2×C3×C4—61no—
391C2×C3×C4—92no—
392C2×C3×C4—91no—
393C2×C3×C4—94no—
394C2×C3×C4—92no—
395C2×C3×C4—92no—
396C2×C3×C4—91no—
397C2×C3×C4—91no—
398C2×C3×C4—362no—
399C2×C3×C4C2×D3211yescharacteristic
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402#67—98no—
403#67—98no—
404C2×Dic3—31no—
405C2×Dic3—31no—
406C2×Dic3—32no—
407C2×Dic3—31no—
408C2×Dic3—31no—
409C2×Dic3—31no—
410C2×Dic3—31no—
411C2×Dic3—32no—
412C2×Dic3—32no—
413C2×Dic3—61no—
414C2×Dic3—61no—
415C2×Dic3—61no—
416C2×Dic3—92no—
417C2×Dic3—92no—
418C2×Dic3—92no—
419C2×Dic3—98no—
420C2×Dic3—362no—
421C2×Dic3—362no—
422D12—34no—
423D12—94no—
424D12—98no—
425C4×D3—34no—
426C4×D3—34no—
427C4×D3—64no—
428C4×D3—98no—
429C4×D3—98no—
430C4×D3—122no—
431C4×D3—362no—
432C4×D3—362no—
433C4×D3—362no—
434C4×D3—362no—
435C4×D3—362no—
436Dic6—34no—
437Dic6—94no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32C22×C4×D311yescharacteristic
442C2×C32C22×C4×D311yescharacteristic
443C2×C32#100411yescharacteristic
444C2×C32#100711yescharacteristic
445C2×C32#100911yescharacteristic
446C2×C32#101011yescharacteristic
447C3⋊D3—24no—
448C3×D3—62no—
449C3×D3—64no—
450C3×D3—62no—
451C3×D3—122no—
452#41—278no—
453C2×Q8—91no—
454C2×D4—91no—
455C2×D4—272no—
456C22×C4—91no—
457C22×C4—92no—
458C22×C4—272no—
459C22×C4—272no—
460C22×C4—272no—
461C4⋊C4—91no—
462C4⋊C4—91no—
463C4⋊C4—92no—
464C4⋊C4—91no—
465C4⋊C4—91no—
466C4⋊C4—272no—
467C4⋊C4—272no—
468C22⋊C4—91no—
469C22⋊C4—91no—
470C22⋊C4—92no—
471C22⋊C4—272no—
472C42—91no—
473C42—92no—
474C42—272no—
475C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
476C22×C3—31no—
477C22×C3—31no—
478C22×C3—61no—
479C22×C3—94no—
480C22×C3—92no—
481C22×C3—182no—
482C22×C3—182no—
483C22×C3—181no—
484C22×C3—181no—
485C22×C3C22×D3211yescharacteristic
486C2×D3—32no—
487C2×D3—32no—
488C2×D3—34no—
489C2×D3—62no—
490C2×D3—61no—
491C2×D3—61no—
492C2×D3—61no—
493C2×D3—61no—
494C2×D3—62no—
495C2×D3—62no—
496C2×D3—94no—
497C2×D3—121no—
498C2×D3—121no—
499C2×D3—182no—
500C2×D3—182no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—32no—
504C3×C4—32no—
505C3×C4—61no—
506C3×C4—61no—
507C3×C4—61no—
508C3×C4—61no—
509C3×C4—62no—
510C3×C4—94no—
511C3×C4—92no—
512C3×C4—121no—
513C3×C4—121no—
514C3×C4—182no—
515C3×C4—182no—
516C3×C4—181no—
517C3×C4—181no—
518C3×C4C4×D3212yes—
519Dic3—32no—
520Dic3—32no—
521Dic3—34no—
522Dic3—62no—
523Dic3—61no—
524Dic3—61no—
525Dic3—61no—
526Dic3—61no—
527Dic3—62no—
528Dic3—62no—
529Dic3—94no—
530Dic3—121no—
531Dic3—121no—
532Dic3—182no—
533Dic3—182no—
534C32—21no—
535C32#554611yescharacteristic, complemented
536C32#557411yescharacteristic, complemented
537C23—91no—
538C23—92no—
539Q8—94no—
540D4—94no—
541D4—278no—
542C2×C4—92no—
543C2×C4—91no—
544C2×C4—94no—
545C2×C4—91no—
546C2×C4—91no—
547C2×C4—92no—
548C2×C4—92no—
549C2×C4—278no—
550C2×C4—362no—
551C2×C4—1082no—
552C2×C4—1082no—
553C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
554C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
555C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—31no—
559C2×C3—31no—
560C2×C3—31no—
561C2×C3—31no—
562C2×C3—61no—
563C2×C3—61no—
564C2×C3—61no—
565C2×C3—184no—
566C2×C3—182no—
567C2×C3#6429711yescharacteristic
568C2×C3C2×C4×D3211yescharacteristic
569C2×C3#6430711yescharacteristic
570D3—62no—
571D3—62no—
572D3—64no—
573D3—122no—
574D3—184no—
575C22—92no—
576C22—94no—
577C22—181no—
578C22—182no—
579C22—182no—
580C22—181no—
581C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
582C4—21no—
583C4—21no—
584C4—92no—
585C4—94no—
586C4—181no—
587C4—182no—
588C4—181no—
589C4—182no—
590C4C4×((C3⋊D3)⋊C3)⋊C212yes—
591C3—31no—
592C3—31no—
593C3—61no—
594C3#1059388911yescharacteristic
595C2—182no—
596C2—184no—
597C2#1172248211yescharacteristic
598C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
599C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×#88649C2
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#520
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11721718C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C4×#634⋊C3C2semidirect
5. #11721154C2semidirect
6. #11721152C2semidirect
7. #11721136C2semidirect
8. #11721123C2semidirect
9. #11721120C2semidirect
10. #11721119C2semidirect
11. #11721084C2semidirect
22. #88649C4semidirect
40. C4×C4⋊(C3⋊D3)D3semidirect
41. C4×#634D3semidirect
69. C4×#272C2×C4semidirect
73. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
75. #634⋊C3C2×C4semidirect
76. #6200C2×C4semidirect
79. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
205. C4⋊(C3⋊D3)C4×D3semidirect
207. #634C4×D3semidirect
372. H(3,𝔽3)#520semidirect
535. C32#5546semidirect
536. C32#5574semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
#117217182
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#117211542
#117211522
#117211362
#117211232
#117211202
#117211192
#117210842
#105938903
#105938873