Finite Group Databasebeta

← #13150695

Subgroups — #13150695

Order 1728.

1114 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88672C212yesmaximal, complemented
2C2×C4⋊#1084C211yesmaximal, characteristic, complemented
3#64139⋊C3C211yesmaximal, characteristic, complemented
4C4×#638⋊C3C211yesmaximal, characteristic, complemented
5#11721307C212yesmaximal, complemented
6#11721141C211yesmaximal, characteristic, complemented
7#11721133C212yesmaximal, complemented
8#11721132C211yesmaximal, characteristic, complemented
9#11721131C212yesmaximal, complemented
10#11721129C211yesmaximal, characteristic, complemented
11#11721116C211yesmaximal, characteristic, complemented
12#10594928—31nomaximal
13#10594906—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—
21#88672—28no—
22C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
23C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
24C2×#6196C2212yes—
25C4⋊#1084C2211yescharacteristic
26C4⋊#1084C2211yescharacteristic
27C4⋊#1084C2212yes—
28#3757⋊C3C2211yescharacteristic
29#3752⋊C3C2211yescharacteristic
30#3752⋊C3C2212yes—
31C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
32#3750⋊C3C2211yescharacteristic
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34#88450C2212yes—
35#88450C2212yes—
36#88448C2212yes—
37#88448C2212yes—
38C2×#3804—32no—
39C2×#3804—32no—
40#64139D311yescharacteristic, complemented
41C4×#638D311yescharacteristic, complemented
42C3×#930—31no—
43C3×C4×#67—31no—
44C2×C3×C4⋊Dic3—31no—
45C2×C3×C4⋊Dic3—31no—
46C2×C3×C4⋊Dic3—61no—
47#63966—32no—
48#63961—32no—
49#63919—31no—
50#63918—31no—
51#63916—31no—
52#63909—31no—
53#63900—31no—
54#63899—32no—
55#63898—31no—
56#63895—32no—
57#63894—31no—
58#63890—32no—
59D3×C4⋊Dic3—31no—
60#63854—32no—
61C23×H(3,𝔽3)C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2312yes—
63C22×(C3⋊D3)⋊C3C2311yescharacteristic
64C2×C4×H(3,𝔽3)—22no—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×#1084—24no—
68C2×#1084C2312yes—
69C2×#1084C2312yes—
70#638⋊C3—24no—
71#638⋊C3D414yescomplemented
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
76C4⋊((C3⋊D3)⋊C3)—22no—
77C4×(C3⋊D3)⋊C3—22no—
78#6198—24no—
79((C3⋊D3)⋊C3)⋊C4—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×D312yes—
89C2×#638C2×D311yescharacteristic
90C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
91C2×C4×C3⋊D3C2×D311yescharacteristic
92C2×C3×#67—31no—
93C2×C3×#67—32no—
94C22×C3×Dic3—32no—
95C22×C3×Dic3—32no—
96C22×C3×Dic3—62no—
97C2×C3×D12—31no—
98C2×C3×C4×D3—31no—
99C2×#627—32no—
100C2×#627—32no—
101C2×#626—32no—
102C2×#626—32no—
103C2×#624—32no—
104C2×#624—32no—
105#3804—68no—
106#3804—68no—
107C2×D3×Dic3—32no—
108C2×D3×Dic3—32no—
109#3757C2×D311yescharacteristic
110#3752C2×D311yescharacteristic
111#3752C2×D312yes—
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic
113#3750C2×D311yescharacteristic
114C4×C3⋊Dic3C2×D311yescharacteristic
115C3×#217—31no—
116C3×(C2×D3)⋊C4—31no—
117C3×(C2×D3)⋊C4—32no—
118C3×C4⋊Dic3—32no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—32no—
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3—31no—
124C3×C4⋊Dic3—31no—
125C3×C4⋊Dic3—61no—
126C3×C4⋊Dic3—61no—
127C3×C4⋊Dic3—62no—
128C3×Dic3⋊C4—31no—
129C3×C4×Dic3—31no—
130#3724—32no—
131#3724—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134#3722—32no—
135#3722—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×(C3⋊D3)⋊C3—21no—
143C2×(C3⋊D3)⋊C3—21no—
144C2×(C3⋊D3)⋊C3—22no—
145C2×(C3⋊D3)⋊C3—22no—
146C2×(C3⋊D3)⋊C3—22no—
147C2×(C3⋊D3)⋊C3C2×D412yes—
148C4×H(3,𝔽3)—21no—
149C4×H(3,𝔽3)#4112yescomplemented
150#1084—22no—
151#1084—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—91no—
167#930—91no—
168#929—92no—
169C4×#67—31no—
170C4×#67—91no—
171C2×C4⋊Dic3—31no—
172C2×C4⋊Dic3—31no—
173C2×C4⋊Dic3—61no—
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×D312yes—
194C22×C3⋊D3C22×D311yescharacteristic
195C22×C3×D3—31no—
196C22×C3×D3—32no—
197C2×C32×C4—24no—
198C2×C32×C4—22no—
199C2×C32×C4—22no—
200C2×C32×C4—21no—
201C2×C32×C4—21no—
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C22×D311yescharacteristic
204C2×C32×C4C22×D311yescharacteristic
205C2×C32×C4C22×D311yescharacteristic
206#638—24no—
207#638D1214yescomplemented
208C2×C3⋊Dic3C22×D311yescharacteristic
209C2×C3⋊Dic3C22×D311yescharacteristic
210C2×C3⋊Dic3C22×D312yes—
211C2×C3⋊Dic3C22×D311yescharacteristic
212C4⋊(C3⋊D3)—22no—
213C4×C3⋊D3—22no—
214C3×#67—34no—
215C3×#67—64no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—31no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—62no—
225C2×C3×Dic3—62no—
226C2×C3×Dic3—64no—
227C2×C3×Dic3—64no—
228C2×C3×Dic3—124no—
229C3×D12—62no—
230C3×C4×D3—62no—
231#627—64no—
232#627—64no—
233#626—64no—
234#626—64no—
235#624—64no—
236#624—64no—
237D3×Dic3—64no—
238D3×Dic3—64no—
239#547—271noSylow
240C2×H(3,𝔽3)—22no—
241C2×H(3,𝔽3)C22×D411yescharacteristic
242C2×H(3,𝔽3)C2×#4111yescharacteristic
243C2×H(3,𝔽3)2−1+411yescharacteristic
244(C3⋊D3)⋊C3—22no—
245(C3⋊D3)⋊C3—42no—
246C2×C3×D4—91no—
247C2×C3×D4—91no—
248C2×C3×D4—92no—
249C22×C3×C4—31no—
250C22×C3×C4—31no—
251C22×C3×C4—61no—
252C22×C3×C4—91no—
253C22×C3×C4—92no—
254C22×C3×C4D3211yescharacteristic
255C2×#67—31no—
256C2×#67—32no—
257C2×#67—91no—
258C2×#67—92no—
259C2×#67—92no—
260C2×#67—92no—
261C22×Dic3—32no—
262C22×Dic3—32no—
263C22×Dic3—62no—
264C22×Dic3—92no—
265C22×Dic3—92no—
266#237—188no—
267#237—188no—
268#235—188no—
269C2×D12—31no—
270C2×D12—91no—
271C2×D12—92no—
272C2×C4×D3—31no—
273C2×C4×D3—91no—
274C2×C4×D3—92no—
275C2×C4×D3—92no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×Dic6—92no—
279C2×Dic6—92no—
280C2×Dic6—92no—
281C3×C4⋊C4—91no—
282C3×C4⋊C4—91no—
283C3×C4⋊C4—91no—
284C3×C4⋊C4—91no—
285C3×C4⋊C4—92no—
286C3×C22⋊C4—92no—
287C3×C22⋊C4—91no—
288C3×C22⋊C4—91no—
289C3×C42—91no—
290#217—31no—
291#217—91no—
292#217—92no—
293#217—92no—
294(C2×D3)⋊C4—31no—
295(C2×D3)⋊C4—32no—
296(C2×D3)⋊C4—91no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302C4⋊Dic3—31no—
303C4⋊Dic3—31no—
304C4⋊Dic3—32no—
305C4⋊Dic3—32no—
306C4⋊Dic3—31no—
307C4⋊Dic3—31no—
308C4⋊Dic3—31no—
309C4⋊Dic3—61no—
310C4⋊Dic3—61no—
311C4⋊Dic3—62no—
312C4⋊Dic3—91no—
313C4⋊Dic3—92no—
314C4⋊Dic3—92no—
315Dic3⋊C4—31no—
316Dic3⋊C4—91no—
317Dic3⋊C4—92no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321C4×Dic3—31no—
322C4×Dic3—91no—
323C22×C32—22no—
324C22×C32—22no—
325C22×C32—21no—
326C22×C32—21no—
327C22×C32—21no—
328C22×C32—21no—
329C22×C32—21no—
330C22×C32—22no—
331C22×C32C2×D1212yes—
332C22×C32D3×D412yes—
333C22×C32C23×D311yescharacteristic
334C22×C32C23×D311yescharacteristic
335C2×C3⋊D3—21no—
336C2×C3⋊D3—21no—
337C2×C3⋊D3—22no—
338C2×C3⋊D3—22no—
339C2×C3⋊D3—22no—
340C2×C3⋊D3C2×D1212yes—
341C2×C3×D3—32no—
342C2×C3×D3—61no—
343C2×C3×D3—61no—
344C2×C3×D3—62no—
345C2×C3×D3—62no—
346C2×C3×D3—62no—
347C32×C4—22no—
348C32×C4—21no—
349C32×C4—21no—
350C32×C4—22no—
351C32×C4#23712yescomplemented
352C32×C4#23712yescomplemented
353C3⋊Dic3—22no—
354C3⋊Dic3—21no—
355C3⋊Dic3—21no—
356C3⋊Dic3C2×D1212yes—
357C3×Dic3—32no—
358C3×Dic3—62no—
359C3×Dic3—62no—
360C3×Dic3—62no—
361C3×Dic3—62no—
362C3×Dic3—62no—
363C3×Dic3—61no—
364C3×Dic3—61no—
365C3×Dic3—122no—
366C3×Dic3—122no—
367C2×#41—272no—
368C4⋊Q8—271no—
369#123—272no—
370#123—272no—
371C22⋊Q8—272no—
372#121—91no—
373#121—271no—
374C4×D4—91no—
375C4×D4—271no—
376C2×C4⋊C4—91no—
377C2×C4⋊C4—271no—
378H(3,𝔽3)#54711yescharacteristic, complemented, Sylow
379C23×C3—31no—
380C23×C3—31no—
381C23×C3—61no—
382C23×C3—91no—
383C23×C3—92no—
384C23×C3C2×D3211yescharacteristic
385C22×D3—31no—
386C22×D3—32no—
387C22×D3—91no—
388C22×D3—92no—
389C3×D4—94no—
390C3×D4—182no—
391C3×D4—184no—
392C2×C3×C4—22no—
393C2×C3×C4—31no—
394C2×C3×C4—31no—
395C2×C3×C4—31no—
396C2×C3×C4—31no—
397C2×C3×C4—62no—
398C2×C3×C4—62no—
399C2×C3×C4—64no—
400C2×C3×C4—61no—
401C2×C3×C4—61no—
402C2×C3×C4—91no—
403C2×C3×C4—92no—
404C2×C3×C4—91no—
405C2×C3×C4—91no—
406C2×C3×C4—92no—
407C2×C3×C4—92no—
408C2×C3×C4—182no—
409C2×C3×C4—184no—
410C2×C3×C4C2×D3211yescharacteristic
411C2×C3×C4C2×D3211yescharacteristic
412#67—34no—
413#67—64no—
414#67—94no—
415#67—184no—
416#67—184no—
417#67—184no—
418C2×Dic3—31no—
419C2×Dic3—32no—
420C2×Dic3—32no—
421C2×Dic3—32no—
422C2×Dic3—32no—
423C2×Dic3—31no—
424C2×Dic3—31no—
425C2×Dic3—32no—
426C2×Dic3—62no—
427C2×Dic3—62no—
428C2×Dic3—64no—
429C2×Dic3—64no—
430C2×Dic3—91no—
431C2×Dic3—91no—
432C2×Dic3—91no—
433C2×Dic3—92no—
434C2×Dic3—124no—
435C2×Dic3—184no—
436C2×Dic3—184no—
437D12—62no—
438D12—182no—
439D12—184no—
440C4×D3—62no—
441C4×D3—182no—
442C4×D3—184no—
443C4×D3—184no—
444C4×D3—184no—
445C4×D3—184no—
446Dic6—184no—
447Dic6—184no—
448Dic6—184no—
449C2×C32—24no—
450C2×C32—22no—
451C2×C32—22no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32C22×D1211yescharacteristic
456C2×C32C2×D3×D411yescharacteristic
457C2×C32C2×#23711yescharacteristic
458C2×C32C2×#23711yescharacteristic
459C2×C32#101011yescharacteristic
460C2×C32#101011yescharacteristic
461C3⋊D3—22no—
462C3⋊D3—42no—
463C3×D3—62no—
464C3×D3—122no—
465#41—548no—
466C2×Q8—272no—
467C2×D4—91no—
468C2×D4—92no—
469C2×D4—91no—
470C2×D4—272no—
471C22×C4—92no—
472C22×C4—91no—
473C22×C4—272no—
474C22×C4—272no—
475C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
476C4⋊C4—91no—
477C4⋊C4—91no—
478C4⋊C4—92no—
479C4⋊C4—91no—
480C4⋊C4—91no—
481C4⋊C4—272no—
482C4⋊C4—272no—
483C22⋊C4—91no—
484C22⋊C4—91no—
485C22⋊C4—92no—
486C22⋊C4—272no—
487C22⋊C4—272no—
488C42—91no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—31no—
492C22×C3—31no—
493C22×C3—32no—
494C22×C3—32no—
495C22×C3—61no—
496C22×C3—61no—
497C22×C3—61no—
498C22×C3—61no—
499C22×C3—62no—
500C22×C3—62no—
501C22×C3—61no—
502C22×C3—62no—
503C22×C3—92no—
504C22×C3—181no—
505C22×C3—181no—
506C22×C3—182no—
507C22×C3—182no—
508C22×C3—182no—
509C22×C3D3×D1212yes—
510C22×C3C22×D3211yescharacteristic
511C2×D3—32no—
512C2×D3—61no—
513C2×D3—61no—
514C2×D3—62no—
515C2×D3—62no—
516C2×D3—62no—
517C2×D3—92no—
518C2×D3—181no—
519C2×D3—181no—
520C2×D3—182no—
521C2×D3—182no—
522C2×D3—182no—
523C3×C4—21no—
524C3×C4—32no—
525C3×C4—32no—
526C3×C4—61no—
527C3×C4—61no—
528C3×C4—62no—
529C3×C4—62no—
530C3×C4—92no—
531C3×C4—182no—
532C3×C4—181no—
533C3×C4—181no—
534C3×C4—182no—
535C3×C4—182no—
536C3×C4#380512yes—
537Dic3—32no—
538Dic3—62no—
539Dic3—62no—
540Dic3—62no—
541Dic3—62no—
542Dic3—61no—
543Dic3—62no—
544Dic3—61no—
545Dic3—92no—
546Dic3—122no—
547Dic3—122no—
548Dic3—182no—
549Dic3—181no—
550Dic3—181no—
551C32—21no—
552C32#555511yescharacteristic, complemented
553C32#561111yescharacteristic, complemented
554C23—91no—
555C23—92no—
556C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
557Q8—544no—
558D4—94no—
559D4—184no—
560D4—182no—
561D4—544no—
562C2×C4—22no—
563C2×C4—91no—
564C2×C4—91no—
565C2×C4—91no—
566C2×C4—92no—
567C2×C4—92no—
568C2×C4—92no—
569C2×C4—184no—
570C2×C4—182no—
571C2×C4—544no—
572C2×C4—544no—
573C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
574C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
575C2×C3—22no—
576C2×C3—31no—
577C2×C3—31no—
578C2×C3—31no—
579C2×C3—31no—
580C2×C3—31no—
581C2×C3—31no—
582C2×C3—62no—
583C2×C3—62no—
584C2×C3—64no—
585C2×C3—61no—
586C2×C3—61no—
587C2×C3—61no—
588C2×C3—182no—
589C2×C3—362no—
590C2×C3#6429811yescharacteristic
591C2×C3C2×D3×D1211yescharacteristic
592C2×C3C2×#380511yescharacteristic
593D3—62no—
594D3—122no—
595D3—182no—
596D3—362no—
597C22—21no—
598C22—21no—
599C22—92no—
600C22—181no—
601C22—181no—
602C22—182no—
603C22—182no—
604C22—182no—
605C22#8865412yes—
606C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
607C4—21no—
608C4—92no—
609C4—181no—
610C4—181no—
611C4—182no—
612C4—182no—
613C4—182no—
614C4#8867212yescomplemented
615C3—31no—
616C3—31no—
617C3—61no—
618C3#1059490711yescharacteristic
619C2—22no—
620C2—182no—
621C2—362no—
622C2#1172248311yescharacteristic
623C2C2×#8865411yescharacteristic
624C2C2×#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)#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×#88672C2semidirect
2. C2×C4⋊#1084C2semidirect
3. #64139⋊C3C2semidirect
4. C4×#638⋊C3C2semidirect
5. #11721307C2semidirect
6. #11721141C2semidirect
7. #11721133C2semidirect
8. #11721132C2semidirect
9. #11721131C2semidirect
10. #11721129C2semidirect
11. #11721116C2semidirect
40. #64139D3semidirect
41. C4×#638D3semidirect
71. #638⋊C3D4semidirect
149. C4×H(3,𝔽3)#41semidirect
207. #638D12semidirect
351. C32×C4#237semidirect
352. C32×C4#237semidirect
378. H(3,𝔽3)#547semidirect
552. C32#5555semidirect
553. C32#5611semidirect
614. C4#88672semidirect

Maximal subgroups

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

subgroupindex
C2×#886722
C2×C4⋊#10842
#64139⋊C32
C4×#638⋊C32
#117213072
#117211412
#117211332
#117211322
#117211312
#117211292
#117211162
#105949283
#105949063