Finite Group Databasebeta

← #13150694

Subgroups — #13150694

Order 1728.

1116 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88671C212yesmaximal, complemented
2#11721830C211yesmaximal, characteristic, complemented
3C4×#638⋊C3C211yesmaximal, characteristic, complemented
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5#11721301C212yesmaximal, complemented
6#11721156C211yesmaximal, characteristic, complemented
7#11721150C212yesmaximal, complemented
8#11721149C211yesmaximal, characteristic, complemented
9#11721130C212yesmaximal, complemented
10#11721129C211yesmaximal, characteristic, complemented
11#11721116C211yesmaximal, characteristic, complemented
12#10594928—31nomaximal
13#10594907—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6235C2212yes—
16C2×C4⋊#272C2211yescharacteristic
17C2×#638⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2212yes—
19C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
20C2×#6200C2212yes—
21C2×#6199C2212yes—
22#88671—28no—
23C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
24C2×#6196C2212yes—
25#88542C2212yes—
26C4⋊#1084C2211yescharacteristic
27#3757⋊C3C2211yescharacteristic
28#3752⋊C3C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31C4⋊((C3⋊Dic3)⋊C3)C2212yes—
32#3750⋊C3C2211yescharacteristic
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34#88451C2212yes—
35#88451C2212yes—
36#88448C2212yes—
37#88448C2212yes—
38C2×#3805—32no—
39C2×#3804—32no—
40C4×#638D311yescharacteristic, complemented
41C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
42C3×#930—31no—
43C3×#930—31no—
44C3×#930—61no—
45C3×C4×#67—31no—
46C2×C3×C4⋊Dic3—31no—
47#63962—32no—
48#63961—32no—
49#63919—31no—
50#63918—31no—
51#63916—31no—
52#63908—31no—
53#63900—31no—
54#63899—32no—
55#63898—31no—
56#63893—32no—
57Dic3×D12—31no—
58#63891—32no—
59D3×C4⋊Dic3—31no—
60#63854—32no—
61C23×H(3,𝔽3)C2311yescharacteristic
62C22×#272C2312yes—
63C22×(C3⋊D3)⋊C3C2311yescharacteristic
64C2×C4×H(3,𝔽3)—22no—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67#6235—24no—
68C2×#1084C2312yes—
69C4⋊#272—22no—
70#638⋊C3D414yescomplemented
71C2×(C3⋊Dic3)⋊C3—24no—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
76C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
77C4×(C3⋊D3)⋊C3—22no—
78#6200—24no—
79#6199—24no—
80((C3⋊D3)⋊C3)⋊C4—24no—
81#6196—24no—
82#5611—91no—
83#5555—91no—
84#5507—91no—
85C22×C32×C4—21no—
86C22×C32×C4C2×D311yescharacteristic
87C22×C32×C4C2×D311yescharacteristic
88C2×#638C2×D311yescharacteristic
89C22×C3⋊Dic3C2×D312yes—
90C2×C4×C3⋊D3C2×D311yescharacteristic
91C2×C3×#67—31no—
92C2×C3×#67—32no—
93C2×C3×#67—32no—
94C2×C3×#67—62no—
95C22×C3×Dic3—32no—
96C2×C3×D12—31no—
97C2×C3×D12—31no—
98C2×C3×D12—61no—
99C2×C3×C4×D3—31no—
100C2×#627—32no—
101C2×#627—32no—
102C2×#626—32no—
103C2×#625—32no—
104C2×#624—32no—
105#3805—68no—
106#3804—68no—
107C2×D3×Dic3—32no—
108C2×D3×Dic3—32no—
109C2×D3×Dic3—32no—
110#3757C2×D311yescharacteristic
111#3752C2×D311yescharacteristic
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic
113C4⋊(C3⋊Dic3)C2×D312yes—
114C4⋊(C3⋊Dic3)C2×D311yescharacteristic
115#3750C2×D311yescharacteristic
116C4×C3⋊Dic3C2×D311yescharacteristic
117C3×#217—31no—
118C3×(C2×D3)⋊C4—31no—
119C3×(C2×D3)⋊C4—32no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—62no—
122C3×C4⋊Dic3—32no—
123C3×C4⋊Dic3—31no—
124C3×C4⋊Dic3—31no—
125C3×C4⋊Dic3—31no—
126C3×C4⋊Dic3—31no—
127C3×C4⋊Dic3—61no—
128C3×Dic3⋊C4—31no—
129C3×C4×Dic3—31no—
130#3724—32no—
131#3724—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134(C2×D3)⋊Dic3—32no—
135(C2×D3)⋊Dic3—32no—
136(C2×D3)⋊Dic3—32no—
137(C2×D3)⋊Dic3—32no—
138C22×H(3,𝔽3)—21no—
139C22×H(3,𝔽3)—21no—
140C22×H(3,𝔽3)C2×D412yes—
141C22×H(3,𝔽3)C2411yescharacteristic
142C2×#272—22no—
143C2×#272—22no—
144C2×#272—22no—
145C2×(C3⋊D3)⋊C3—21no—
146C2×(C3⋊D3)⋊C3—21no—
147C2×(C3⋊D3)⋊C3C2×D412yes—
148C4×H(3,𝔽3)—21no—
149C4×H(3,𝔽3)#4112yescomplemented
150#1084—22no—
151(C3⋊Dic3)⋊C3—22no—
152(C3⋊Dic3)⋊C3—22no—
153(C3⋊Dic3)⋊C3—21no—
154(C3⋊Dic3)⋊C3—21no—
155(C3⋊Dic3)⋊C3C2×D412yes—
156C2×#237—92no—
157C2×#237—92no—
158C2×#235—92no—
159C3×#121—91no—
160C3×C4×D4—91no—
161C2×C3×C4⋊C4—91no—
162#938—91no—
163#935—92no—
164D4×Dic3—91no—
165#930—31no—
166#930—31no—
167#930—61no—
168#930—91no—
169#929—92no—
170C4×#67—31no—
171C4×#67—91no—
172C2×C4⋊Dic3—31no—
173C2×C4⋊Dic3—91no—
174C2×C4⋊Dic3—91no—
175#897—92no—
176#896—92no—
177#895—91no—
178#894—92no—
179#893—91no—
180D3×C4⋊C4—91no—
181D3×C4⋊C4—91no—
182#888—91no—
183#888—91no—
184#886—92no—
185#886—92no—
186#883—92no—
187#878—92no—
188C4×D12—91no—
189#869—91no—
190C23×C32—21no—
191C23×C32C22×D311yescharacteristic
192C23×C32C22×D311yescharacteristic
193C22×C3⋊D3C22×D311yescharacteristic
194C22×C3×D3—31no—
195C22×C3×D3—32no—
196C22×C3×D3—32no—
197C22×C3×D3—62no—
198C2×C32×C4—22no—
199C2×C32×C4—22no—
200C2×C32×C4—21no—
201C2×C32×C4—21no—
202C2×C32×C4—42no—
203C2×C32×C4C22×D311yescharacteristic
204C2×C32×C4C22×D311yescharacteristic
205C2×C32×C4C22×D311yescharacteristic
206C2×C32×C4C22×D311yescharacteristic
207#638D1214yescomplemented
208C2×C3⋊Dic3—24no—
209C2×C3⋊Dic3C22×D311yescharacteristic
210C2×C3⋊Dic3C22×D311yescharacteristic
211C2×C3⋊Dic3C22×D312yes—
212C2×C3⋊Dic3C22×D312yes—
213C2×C3⋊Dic3C22×D311yescharacteristic
214C4×C3⋊D3—22no—
215C3×#67—34no—
216C3×#67—64no—
217C3×#67—64no—
218C3×#67—124no—
219C2×C3×Dic3—31no—
220C2×C3×Dic3—31no—
221C2×C3×Dic3—31no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—32no—
225C2×C3×Dic3—32no—
226C2×C3×Dic3—62no—
227C2×C3×Dic3—64no—
228C3×D12—62no—
229C3×D12—62no—
230C3×D12—122no—
231C3×C4×D3—62no—
232#627—64no—
233#627—64no—
234#626—64no—
235#625—64no—
236#624—64no—
237D3×Dic3—64no—
238D3×Dic3—64no—
239D3×Dic3—64no—
240#547—271noSylow
241C2×H(3,𝔽3)—22no—
242C2×H(3,𝔽3)C22×D411yescharacteristic
243C2×H(3,𝔽3)C2×#4111yescharacteristic
244C2×H(3,𝔽3)2−1+411yescharacteristic
245#272—42no—
246(C3⋊D3)⋊C3—22no—
247C2×C3×D4—91no—
248C2×C3×D4—92no—
249C2×C3×D4—91no—
250C22×C3×C4—31no—
251C22×C3×C4—31no—
252C22×C3×C4—61no—
253C22×C3×C4—91no—
254C22×C3×C4—92no—
255C22×C3×C4D3211yescharacteristic
256C2×#67—31no—
257C2×#67—32no—
258C2×#67—32no—
259C2×#67—62no—
260C2×#67—91no—
261C2×#67—92no—
262C2×#67—92no—
263C22×Dic3—32no—
264C22×Dic3—92no—
265C22×Dic3—92no—
266C22×Dic3—92no—
267#237—188no—
268#237—188no—
269#235—188no—
270C2×D12—31no—
271C2×D12—31no—
272C2×D12—61no—
273C2×D12—92no—
274C2×C4×D3—31no—
275C2×C4×D3—91no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—92no—
280C2×Dic6—92no—
281C2×Dic6—92no—
282C2×Dic6—92no—
283C3×C4⋊C4—91no—
284C3×C4⋊C4—91no—
285C3×C4⋊C4—92no—
286C3×C4⋊C4—91no—
287C3×C4⋊C4—91no—
288C3×C22⋊C4—91no—
289C3×C22⋊C4—91no—
290C3×C22⋊C4—92no—
291C3×C42—91no—
292#217—31no—
293#217—91no—
294#217—92no—
295#217—92no—
296(C2×D3)⋊C4—32no—
297(C2×D3)⋊C4—32no—
298(C2×D3)⋊C4—31no—
299(C2×D3)⋊C4—62no—
300(C2×D3)⋊C4—91no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305C4⋊Dic3—31no—
306C4⋊Dic3—31no—
307C4⋊Dic3—32no—
308C4⋊Dic3—31no—
309C4⋊Dic3—31no—
310C4⋊Dic3—61no—
311C4⋊Dic3—91no—
312C4⋊Dic3—91no—
313C4⋊Dic3—92no—
314C4⋊Dic3—92no—
315C4⋊Dic3—92no—
316Dic3⋊C4—31no—
317Dic3⋊C4—91no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322C4×Dic3—31no—
323C4×Dic3—91no—
324C22×C32—21no—
325C22×C32—21no—
326C22×C32—21no—
327C22×C32—21no—
328C22×C32—21no—
329C22×C32—22no—
330C22×C32—41no—
331C22×C32—41no—
332C22×C32C2×D1212yes—
333C22×C32D3×D412yes—
334C22×C32C23×D311yescharacteristic
335C22×C32C23×D311yescharacteristic
336C2×C3⋊D3—21no—
337C2×C3⋊D3—21no—
338C2×C3⋊D3C2×D1212yes—
339C2×C3×D3—32no—
340C2×C3×D3—61no—
341C2×C3×D3—61no—
342C2×C3×D3—62no—
343C2×C3×D3—62no—
344C2×C3×D3—62no—
345C2×C3×D3—62no—
346C2×C3×D3—62no—
347C2×C3×D3—62no—
348C2×C3×D3—122no—
349C2×C3×D3—122no—
350C2×C3×D3—122no—
351C32×C4—21no—
352C32×C4—21no—
353C32×C4—22no—
354C32×C4—41no—
355C32×C4#23512yescomplemented
356C32×C4#23712yescomplemented
357C3⋊Dic3—22no—
358C3⋊Dic3—22no—
359C3⋊Dic3—21no—
360C3⋊Dic3—21no—
361C3⋊Dic3C2×D1212yes—
362C3×Dic3—32no—
363C3×Dic3—62no—
364C3×Dic3—62no—
365C3×Dic3—62no—
366C3×Dic3—61no—
367C3×Dic3—62no—
368C3×Dic3—61no—
369C3×Dic3—122no—
370C2×#41—272no—
371C4⋊Q8—271no—
372#123—272no—
373#123—272no—
374C22⋊Q8—272no—
375#121—91no—
376#121—271no—
377C4×D4—91no—
378C4×D4—271no—
379C2×C4⋊C4—91no—
380C2×C4⋊C4—271no—
381H(3,𝔽3)#54711yescharacteristic, complemented, Sylow
382C23×C3—31no—
383C23×C3—31no—
384C23×C3—61no—
385C23×C3—91no—
386C23×C3—92no—
387C23×C3C2×D3211yescharacteristic
388C22×D3—31no—
389C22×D3—32no—
390C22×D3—32no—
391C22×D3—62no—
392C22×D3—91no—
393C3×D4—94no—
394C3×D4—184no—
395C3×D4—182no—
396C2×C3×C4—22no—
397C2×C3×C4—31no—
398C2×C3×C4—31no—
399C2×C3×C4—31no—
400C2×C3×C4—31no—
401C2×C3×C4—62no—
402C2×C3×C4—62no—
403C2×C3×C4—61no—
404C2×C3×C4—61no—
405C2×C3×C4—91no—
406C2×C3×C4—92no—
407C2×C3×C4—92no—
408C2×C3×C4—91no—
409C2×C3×C4—91no—
410C2×C3×C4—92no—
411C2×C3×C4—122no—
412C2×C3×C4—182no—
413C2×C3×C4—184no—
414C2×C3×C4C2×D3211yescharacteristic
415C2×C3×C4C2×D3211yescharacteristic
416#67—34no—
417#67—64no—
418#67—64no—
419#67—94no—
420#67—124no—
421#67—184no—
422#67—184no—
423C2×Dic3—31no—
424C2×Dic3—32no—
425C2×Dic3—32no—
426C2×Dic3—31no—
427C2×Dic3—31no—
428C2×Dic3—32no—
429C2×Dic3—32no—
430C2×Dic3—62no—
431C2×Dic3—64no—
432C2×Dic3—91no—
433C2×Dic3—91no—
434C2×Dic3—91no—
435C2×Dic3—92no—
436C2×Dic3—92no—
437C2×Dic3—184no—
438C2×Dic3—184no—
439C2×Dic3—184no—
440D12—62no—
441D12—62no—
442D12—122no—
443D12—184no—
444C4×D3—62no—
445C4×D3—182no—
446C4×D3—184no—
447C4×D3—184no—
448C4×D3—184no—
449C4×D3—184no—
450Dic6—184no—
451Dic6—184no—
452Dic6—184no—
453C2×C32—22no—
454C2×C32—22no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—42no—
459C2×C32C22×D1211yescharacteristic
460C2×C32C2×#23511yescharacteristic
461C2×C32C2×D3×D411yescharacteristic
462C2×C32C2×#23711yescharacteristic
463C2×C32#100711yescharacteristic
464C2×C32#101011yescharacteristic
465C3⋊D3—22no—
466C3×D3—62no—
467C3×D3—122no—
468C3×D3—122no—
469C3×D3—242no—
470#41—548no—
471C2×Q8—272no—
472C2×D4—91no—
473C2×D4—92no—
474C2×D4—91no—
475C2×D4—272no—
476C22×C4—92no—
477C22×C4—91no—
478C22×C4—272no—
479C22×C4—272no—
480C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
481C4⋊C4—91no—
482C4⋊C4—91no—
483C4⋊C4—92no—
484C4⋊C4—91no—
485C4⋊C4—91no—
486C4⋊C4—272no—
487C4⋊C4—272no—
488C22⋊C4—91no—
489C22⋊C4—91no—
490C22⋊C4—92no—
491C22⋊C4—272no—
492C22⋊C4—272no—
493C42—91no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—31no—
497C22×C3—31no—
498C22×C3—32no—
499C22×C3—32no—
500C22×C3—61no—
501C22×C3—61no—
502C22×C3—61no—
503C22×C3—61no—
504C22×C3—61no—
505C22×C3—62no—
506C22×C3—92no—
507C22×C3—121no—
508C22×C3—121no—
509C22×C3—181no—
510C22×C3—181no—
511C22×C3—182no—
512C22×C3—182no—
513C22×C3—182no—
514C22×C3D3×D1212yes—
515C22×C3C22×D3211yescharacteristic
516C2×D3—32no—
517C2×D3—61no—
518C2×D3—61no—
519C2×D3—62no—
520C2×D3—62no—
521C2×D3—62no—
522C2×D3—62no—
523C2×D3—62no—
524C2×D3—62no—
525C2×D3—92no—
526C2×D3—122no—
527C2×D3—122no—
528C2×D3—122no—
529C2×D3—181no—
530C2×D3—181no—
531C3×C4—21no—
532C3×C4—32no—
533C3×C4—32no—
534C3×C4—61no—
535C3×C4—61no—
536C3×C4—62no—
537C3×C4—92no—
538C3×C4—121no—
539C3×C4—182no—
540C3×C4—182no—
541C3×C4—181no—
542C3×C4—181no—
543C3×C4—182no—
544C3×C4#380412yes—
545Dic3—32no—
546Dic3—62no—
547Dic3—62no—
548Dic3—61no—
549Dic3—62no—
550Dic3—62no—
551Dic3—61no—
552Dic3—92no—
553Dic3—122no—
554Dic3—182no—
555Dic3—182no—
556Dic3—181no—
557Dic3—181no—
558C32—21no—
559C32#550711yescharacteristic, complemented
560C32#555511yescharacteristic, complemented
561C23—91no—
562C23—92no—
563C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
564Q8—544no—
565D4—94no—
566D4—184no—
567D4—182no—
568D4—544no—
569C2×C4—22no—
570C2×C4—91no—
571C2×C4—91no—
572C2×C4—91no—
573C2×C4—92no—
574C2×C4—92no—
575C2×C4—92no—
576C2×C4—184no—
577C2×C4—182no—
578C2×C4—544no—
579C2×C4—544no—
580C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
581C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
582C2×C3—22no—
583C2×C3—31no—
584C2×C3—31no—
585C2×C3—31no—
586C2×C3—31no—
587C2×C3—31no—
588C2×C3—31no—
589C2×C3—62no—
590C2×C3—62no—
591C2×C3—61no—
592C2×C3—61no—
593C2×C3—61no—
594C2×C3—122no—
595C2×C3—182no—
596C2×C3—362no—
597C2×C3#6429711yescharacteristic
598C2×C3C2×D3×D1211yescharacteristic
599C2×C3C2×#380411yescharacteristic
600D3—62no—
601D3—122no—
602D3—122no—
603D3—182no—
604D3—242no—
605C22—21no—
606C22—21no—
607C22—92no—
608C22—181no—
609C22—181no—
610C22—182no—
611C22—182no—
612C22—182no—
613C22#8865412yes—
614C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
615C4—21no—
616C4—92no—
617C4—181no—
618C4—181no—
619C4—182no—
620C4—182no—
621C4—182no—
622C4#8867112yescomplemented
623C3—31no—
624C3—31no—
625C3—61no—
626C3#1059490611yescharacteristic
627C2—22no—
628C2—182no—
629C2—362no—
630C2#1172248211yescharacteristic
631C2C2×#8865411yescharacteristic
632C2C2×#8867111yescharacteristic
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)#547
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×#88671C2semidirect
2. #11721830C2semidirect
3. C4×#638⋊C3C2semidirect
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #11721301C2semidirect
6. #11721156C2semidirect
7. #11721150C2semidirect
8. #11721149C2semidirect
9. #11721130C2semidirect
10. #11721129C2semidirect
11. #11721116C2semidirect
40. C4×#638D3semidirect
41. C2×C4⋊(C3⋊Dic3)D3semidirect
70. #638⋊C3D4semidirect
149. C4×H(3,𝔽3)#41semidirect
207. #638D12semidirect
355. C32×C4#235semidirect
356. C32×C4#237semidirect
381. H(3,𝔽3)#547semidirect
559. C32#5507semidirect
560. C32#5555semidirect
622. C4#88671semidirect

Maximal subgroups

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

subgroupindex
C2×#886712
#117218302
C4×#638⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
#117213012
#117211562
#117211502
#117211492
#117211302
#117211292
#117211162
#105949283
#105949073