Finite Group Databasebeta

← #13150751

Subgroups — #13150751

Order 1728.

925 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊#1084C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic, complemented
3C4×#638⋊C3C211yesmaximal, characteristic, complemented
4#11721318C212yesmaximal, complemented
5#11721315C212yesmaximal, complemented
6#11721141C211yesmaximal, characteristic, complemented
7#11721132C211yesmaximal, characteristic, complemented
8#11721131C212yesmaximal, complemented
9#11721111C212yesmaximal, complemented
10#11721093C211yesmaximal, characteristic, complemented
11#11721081C211yesmaximal, characteristic, complemented
12#10594989—31nomaximal
13#10594843—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C22×#1084C2212yes—
16C2×#638⋊C3C2212yes—
17C2×#638⋊C3C2211yescharacteristic
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
20C2×#6198C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22C4⋊#1084C2211yescharacteristic
23C4⋊#1084C2211yescharacteristic
24C4⋊#1084C2212yes—
25#3757⋊C3C2211yescharacteristic
26#3752⋊C3C2211yescharacteristic
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31#88450C2212yes—
32#88450C2212yes—
33#88450C2212yes—
34#88449C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64139D311yescharacteristic, complemented
38C4×#638D311yescharacteristic, complemented
39C3×#930—31no—
40C3×C4×#67—31no—
41C2×C3×C4⋊Dic3—31no—
42C2×C3×C4⋊Dic3—31no—
43C2×C3×C4⋊Dic3—61no—
44#63978—32no—
45#63978—32no—
46Dic3×#67—32no—
47#63952—32no—
48#63916—31no—
49#63909—31no—
50#63898—31no—
51#63894—31no—
52#63890—32no—
53#63872—31no—
54#63869—31no—
55#63858—32no—
56#63854—32no—
57#63849—31no—
58#63848—32no—
59#63841—31no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2312yes—
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C2×C4×H(3,𝔽3)—22no—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084—42no—
67C2×#1084C2312yes—
68C2×#1084C2312yes—
69#638⋊C3—24no—
70#638⋊C3—22no—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C4⋊((C3⋊D3)⋊C3)—22no—
76C4×(C3⋊D3)⋊C3—22no—
77#6198—42no—
78((C3⋊D3)⋊C3)⋊C4—42no—
79#5618—91no—
80#5565—91no—
81#5514—91no—
82C22×C32×C4—21no—
83C22×C32×C4C2×D311yescharacteristic
84C22×C32×C4C2×D311yescharacteristic
85C2×#638C2×D311yescharacteristic
86C2×#638C2×D312yes—
87C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
88C2×C4×C3⋊D3C2×D311yescharacteristic
89C2×C3×#67—31no—
90C2×C3×#67—32no—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—32no—
93C22×C3×Dic3—62no—
94C2×C3×D12—31no—
95C2×C3×C4×D3—31no—
96C2×#626—32no—
97C2×#626—32no—
98C2×#624—32no—
99C2×D3×Dic3—32no—
100#3757C2×D311yescharacteristic
101#3752C2×D312yes—
102#3752C2×D311yescharacteristic
103C4⋊(C3⋊Dic3)C2×D311yescharacteristic
104#3750C2×D311yescharacteristic
105C4×C3⋊Dic3C2×D311yescharacteristic
106C3×#217—31no—
107C3×(C2×D3)⋊C4—31no—
108C3×(C2×D3)⋊C4—32no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—32no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—32no—
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—61no—
117C3×C4⋊Dic3—61no—
118C3×C4⋊Dic3—62no—
119C3×Dic3⋊C4—31no—
120C3×C4×Dic3—31no—
121#3724—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125#3722—32no—
126#3722—32no—
127#3722—32no—
128(C2×D3)⋊Dic3—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131Dic32—32no—
132Dic32—32no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)—21no—
135C22×H(3,𝔽3)#4112yes—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×(C3⋊D3)⋊C3—21no—
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3—22no—
140C2×(C3⋊D3)⋊C3—21no—
141C2×(C3⋊D3)⋊C3—22no—
142C2×(C3⋊D3)⋊C3—21no—
143C4×H(3,𝔽3)—21no—
144C4×H(3,𝔽3)#4112yes—
145#1084—22no—
146#1084—22no—
147(C3⋊Dic3)⋊C3—21no—
148(C3⋊Dic3)⋊C3—22no—
149(C3⋊Dic3)⋊C3—21no—
150(C3⋊Dic3)⋊C3—21no—
151C3×#121—91no—
152C3×C4×D4—91no—
153C2×C3×C4⋊C4—91no—
154#939—92no—
155#939—92no—
156#938—91no—
157D4×Dic3—91no—
158D4×Dic3—92no—
159#930—31no—
160#930—91no—
161#930—91no—
162#930—92no—
163C4×#67—31no—
164C4×#67—91no—
165C4×#67—92no—
166#926—91no—
167C2×C4⋊Dic3—31no—
168C2×C4⋊Dic3—31no—
169C2×C4⋊Dic3—61no—
170#898—92no—
171#898—92no—
172#895—91no—
173#894—92no—
174#893—91no—
175#892—91no—
176#892—91no—
177#890—91no—
178#890—91no—
179#886—92no—
180#883—92no—
181#67⋊C4—92no—
182#879—92no—
183C4×D12—91no—
184#870—91no—
185C23×C32—21no—
186C23×C32C22×D311yescharacteristic
187C23×C32C22×D311yescharacteristic
188C22×C3⋊D3C22×D312yes—
189C22×C3⋊D3C22×D311yescharacteristic
190C22×C3×D3—32no—
191C22×C3×D3—31no—
192C2×C32×C4—24no—
193C2×C32×C4—22no—
194C2×C32×C4—22no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201#638—22no—
202#638—24no—
203C2×C3⋊Dic3C22×D311yescharacteristic
204C2×C3⋊Dic3C22×D312yes—
205C2×C3⋊Dic3C22×D311yescharacteristic
206C2×C3⋊Dic3C22×D311yescharacteristic
207C4⋊(C3⋊D3)—22no—
208C4×C3⋊D3—22no—
209C3×#67—64no—
210C3×#67—62no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—62no—
220C2×C3×Dic3—62no—
221C2×C3×Dic3—122no—
222C2×C3×Dic3—122no—
223C2×C3×Dic3—124no—
224C3×D12—62no—
225C3×C4×D3—62no—
226#626—122no—
227#626—122no—
228#624—122no—
229D3×Dic3—122no—
230#553—271noSylow
231C2×H(3,𝔽3)—22no—
232C2×H(3,𝔽3)C2×#4111yescharacteristic
233C2×H(3,𝔽3)C2×#4111yescharacteristic
234C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
235(C3⋊D3)⋊C3—42no—
236(C3⋊D3)⋊C3—41no—
237C2×C3×D4—91no—
238C2×C3×D4—92no—
239C2×C3×D4—91no—
240C22×C3×C4—31no—
241C22×C3×C4—31no—
242C22×C3×C4—61no—
243C22×C3×C4—91no—
244C22×C3×C4—92no—
245C22×C3×C4D3211yescharacteristic
246C2×#67—31no—
247C2×#67—32no—
248C2×#67—92no—
249C2×#67—91no—
250C2×#67—92no—
251C2×#67—92no—
252C22×Dic3—32no—
253C22×Dic3—32no—
254C22×Dic3—62no—
255C22×Dic3—92no—
256C2×D12—31no—
257C2×D12—91no—
258C2×D12—92no—
259C2×C4×D3—31no—
260C2×C4×D3—91no—
261C2×C4×D3—92no—
262C2×C4×D3—92no—
263C3×C4⋊C4—91no—
264C3×C4⋊C4—91no—
265C3×C4⋊C4—91no—
266C3×C4⋊C4—91no—
267C3×C4⋊C4—92no—
268C3×C22⋊C4—92no—
269C3×C22⋊C4—91no—
270C3×C22⋊C4—91no—
271C3×C42—91no—
272#217—31no—
273#217—91no—
274#217—92no—
275#217—92no—
276#217—92no—
277(C2×D3)⋊C4—31no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—91no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287C4⋊Dic3—31no—
288C4⋊Dic3—31no—
289C4⋊Dic3—32no—
290C4⋊Dic3—32no—
291C4⋊Dic3—31no—
292C4⋊Dic3—31no—
293C4⋊Dic3—31no—
294C4⋊Dic3—61no—
295C4⋊Dic3—61no—
296C4⋊Dic3—62no—
297C4⋊Dic3—91no—
298C4⋊Dic3—92no—
299C4⋊Dic3—92no—
300Dic3⋊C4—31no—
301Dic3⋊C4—91no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306C4×Dic3—31no—
307C4×Dic3—91no—
308C4×Dic3—92no—
309C4×Dic3—92no—
310C4×Dic3—92no—
311C22×C32—22no—
312C22×C32—22no—
313C22×C32—21no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32—22no—
319C22×C32#23512yes—
320C22×C32#23712yes—
321C22×C32C23×D311yescharacteristic
322C22×C32C23×D311yescharacteristic
323C2×C3⋊D3—21no—
324C2×C3⋊D3—21no—
325C2×C3⋊D3—22no—
326C2×C3⋊D3—22no—
327C2×C3⋊D3—22no—
328C2×C3⋊D3—21no—
329C2×C3×D3—62no—
330C2×C3×D3—61no—
331C2×C3×D3—62no—
332C2×C3×D3—61no—
333C2×C3×D3—62no—
334C2×C3×D3—61no—
335C32×C4—22no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—22no—
339C32×C4#23712yes—
340C32×C4#23712yes—
341C3⋊Dic3—21no—
342C3⋊Dic3—21no—
343C3⋊Dic3—22no—
344C3⋊Dic3—21no—
345C3×Dic3—62no—
346C3×Dic3—61no—
347C3×Dic3—61no—
348C3×Dic3—61no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—122no—
354C3×Dic3—122no—
355#126—272no—
356#125—271no—
357#123—272no—
358#121—91no—
359#121—271no—
360#121—272no—
361C4×D4—91no—
362C4×D4—271no—
363C4×D4—272no—
364#117—271no—
365C2×C4⋊C4—91no—
366H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
367C23×C3—31no—
368C23×C3—31no—
369C23×C3—61no—
370C23×C3—92no—
371C23×C3—91no—
372C23×C3C2×D3211yescharacteristic
373C22×D3—31no—
374C22×D3—32no—
375C22×D3—92no—
376C22×D3—91no—
377C3×D4—182no—
378C3×D4—182no—
379C3×D4—184no—
380C2×C3×C4—22no—
381C2×C3×C4—31no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—62no—
386C2×C3×C4—62no—
387C2×C3×C4—64no—
388C2×C3×C4—61no—
389C2×C3×C4—61no—
390C2×C3×C4—91no—
391C2×C3×C4—92no—
392C2×C3×C4—91no—
393C2×C3×C4—91no—
394C2×C3×C4—92no—
395C2×C3×C4—92no—
396C2×C3×C4—182no—
397C2×C3×C4—362no—
398C2×C3×C4C2×D3211yescharacteristic
399C2×C3×C4C2×D3211yescharacteristic
400#67—64no—
401#67—62no—
402#67—184no—
403#67—182no—
404#67—362no—
405#67—362no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—31no—
411C2×Dic3—31no—
412C2×Dic3—31no—
413C2×Dic3—32no—
414C2×Dic3—62no—
415C2×Dic3—62no—
416C2×Dic3—91no—
417C2×Dic3—91no—
418C2×Dic3—92no—
419C2×Dic3—91no—
420C2×Dic3—122no—
421C2×Dic3—122no—
422C2×Dic3—124no—
423C2×Dic3—362no—
424D12—62no—
425D12—182no—
426D12—362no—
427C4×D3—62no—
428C4×D3—182no—
429C4×D3—362no—
430C4×D3—362no—
431C2×C32—24no—
432C2×C32—22no—
433C2×C32—22no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32C2×#23511yescharacteristic
438C2×C32C2×#23711yescharacteristic
439C2×C32C2×#23711yescharacteristic
440C2×C32C2×#23711yescharacteristic
441C2×C32#100911yescharacteristic
442C2×C32#100911yescharacteristic
443C3⋊D3—41no—
444C3⋊D3—42no—
445C3×D3—122no—
446C3×D3—121no—
447C2×D4—91no—
448C2×D4—92no—
449C2×D4—91no—
450C2×D4—272no—
451C22×C4—92no—
452C22×C4—91no—
453C22×C4—272no—
454C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
455C4⋊C4—91no—
456C4⋊C4—91no—
457C4⋊C4—92no—
458C4⋊C4—91no—
459C4⋊C4—91no—
460C4⋊C4—272no—
461C4⋊C4—272no—
462C22⋊C4—91no—
463C22⋊C4—91no—
464C22⋊C4—92no—
465C22⋊C4—272no—
466C22⋊C4—272no—
467C22⋊C4—272no—
468C42—91no—
469C42—272no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—31no—
473C22×C3—31no—
474C22×C3—32no—
475C22×C3—32no—
476C22×C3—61no—
477C22×C3—61no—
478C22×C3—61no—
479C22×C3—61no—
480C22×C3—62no—
481C22×C3—62no—
482C22×C3—61no—
483C22×C3—62no—
484C22×C3—182no—
485C22×C3—181no—
486C22×C3—181no—
487C22×C3—182no—
488C22×C3—181no—
489C22×C3—182no—
490C22×C3#379512yes—
491C22×C3C22×D3211yescharacteristic
492C2×D3—61no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—61no—
496C2×D3—61no—
497C2×D3—62no—
498C2×D3—181no—
499C2×D3—182no—
500C2×D3—182no—
501C2×D3—182no—
502C2×D3—181no—
503C2×D3—181no—
504C3×C4—21no—
505C3×C4—32no—
506C3×C4—32no—
507C3×C4—61no—
508C3×C4—61no—
509C3×C4—62no—
510C3×C4—62no—
511C3×C4—181no—
512C3×C4—181no—
513C3×C4—181no—
514C3×C4—182no—
515C3×C4—182no—
516C3×C4—182no—
517C3×C4#380512yes—
518Dic3—62no—
519Dic3—62no—
520Dic3—62no—
521Dic3—62no—
522Dic3—61no—
523Dic3—62no—
524Dic3—61no—
525Dic3—61no—
526Dic3—122no—
527Dic3—122no—
528Dic3—181no—
529Dic3—181no—
530Dic3—182no—
531Dic3—181no—
532C32—21no—
533C32#556511yescharacteristic, complemented
534C32#561811yescharacteristic, complemented
535C23—91no—
536C23—92no—
537C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
538D4—182no—
539D4—184no—
540D4—182no—
541D4—1082no—
542C2×C4—22no—
543C2×C4—91no—
544C2×C4—91no—
545C2×C4—91no—
546C2×C4—92no—
547C2×C4—92no—
548C2×C4—92no—
549C2×C4—182no—
550C2×C4—362no—
551C2×C4—1082no—
552C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
553C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
554C2×C3—22no—
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—31no—
559C2×C3—31no—
560C2×C3—31no—
561C2×C3—62no—
562C2×C3—62no—
563C2×C3—64no—
564C2×C3—61no—
565C2×C3—61no—
566C2×C3—61no—
567C2×C3—361no—
568C2×C3—362no—
569C2×C3C2×#379511yescharacteristic
570C2×C3#6430811yescharacteristic
571C2×C3C2×#380511yescharacteristic
572D3—122no—
573D3—121no—
574D3—362no—
575D3—361no—
576C22—21no—
577C22—21no—
578C22—181no—
579C22—182no—
580C22—181no—
581C22—182no—
582C22—182no—
583C22—181no—
584C22#8864812yes—
585C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
586C4—21no—
587C4—181no—
588C4—181no—
589C4—182no—
590C4—182no—
591C4—182no—
592C4—181no—
593C4#8867212yes—
594C3—31no—
595C3—31no—
596C3—61no—
597C3#1059499511yescharacteristic
598C2—22no—
599C2—361no—
600C2—362no—
601C2C2×#8864811yescharacteristic
602C2#1172249311yescharacteristic
603C2C2×#8867211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#553
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#1084C2semidirect
2. #64139⋊C3C2semidirect
3. C4×#638⋊C3C2semidirect
4. #11721318C2semidirect
5. #11721315C2semidirect
6. #11721141C2semidirect
7. #11721132C2semidirect
8. #11721131C2semidirect
9. #11721111C2semidirect
10. #11721093C2semidirect
11. #11721081C2semidirect
37. #64139D3semidirect
38. C4×#638D3semidirect
366. H(3,𝔽3)#553semidirect
533. C32#5565semidirect
534. C32#5618semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
#64139⋊C32
C4×#638⋊C32
#117213182
#117213152
#117211412
#117211322
#117211312
#117211112
#117210932
#117210812
#105949893
#105948433