Finite Group Databasebeta

← #13147840

Subgroups — #13147840

Order 1728.

889 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721716C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4#11721141C211yesmaximal, characteristic, complemented
5#11721141C211yesmaximal, characteristic, complemented
6#11721139C212yesmaximal, complemented
7#11721129C211yesmaximal, characteristic, complemented
8#11721126C212yesmaximal, complemented
9#11721122C212yesmaximal, complemented
10#11721108C212yesmaximal, complemented
11#11721081C211yesmaximal, characteristic, complemented
12#10593982—31nomaximal
13#10593925—31nomaximal
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#6198C2212yes—
20C2×#6198C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22C42×H(3,𝔽3)C2211yescharacteristic
23C4⋊#1084C2211yescharacteristic
24C4⋊#1084C2211yescharacteristic
25#88540C2212yes—
26#88540C2212yes—
27#3752⋊C3C2212yes—
28#3752⋊C3C2211yescharacteristic
29#3752⋊C3C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
32#88450C2212yes—
33#88450C2212yes—
34#88450C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
38C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×C4×D12—31no—
41C3×#870—31no—
42C3×#870—31no—
43C3×#870—61no—
44#63916—31no—
45#63916—31no—
46#63910—32no—
47#63909—31no—
48#63909—31no—
49#63904—32no—
50#63900—31no—
51D3×C4⋊Dic3—31no—
52#63886—32no—
53#63881—32no—
54#63861—32no—
55#63853—32no—
56#63850—32no—
57#63849—31no—
58#63848—32no—
59#63841—31no—
60C22×(C3⋊D3)⋊C3C2312yes—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084C2312yes—
67C2×#1084C2312yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C4⋊((C3⋊D3)⋊C3)—22no—
72C4⋊((C3⋊D3)⋊C3)—22no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3—22no—
75C4×(C3⋊D3)⋊C3—42no—
76#6198—24no—
77#6198—42no—
78((C3⋊D3)⋊C3)⋊C4—42no—
79#5695—91no—
80#5565—91no—
81#5558—91no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×C4×C3⋊D3C2×D312yes—
87C2×C3×D12—31no—
88C2×C3×D12—31no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3—32no—
92C2×#626—32no—
93C2×#626—32no—
94C2×#626—32no—
95C2×#626—32no—
96C2×#624—32no—
97C2×D3×Dic3—32no—
98C32×C42—21no—
99C32×C42C2×D311yescharacteristic
100C32×C42C2×D311yescharacteristic
101#3752C2×D312yes—
102#3752C2×D311yescharacteristic
103#3752C2×D311yescharacteristic
104C4⋊(C3⋊Dic3)C2×D311yescharacteristic
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106C3×(C2×D3)⋊C4—31no—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—31no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—61no—
116C3×C4⋊Dic3—61no—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—62no—
122C3×Dic3⋊C4—62no—
123#3724—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)C2411yescharacteristic
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3—22no—
137C2×(C3⋊D3)⋊C3—21no—
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3—21no—
140C2×(C3⋊D3)⋊C3—21no—
141C2×(C3⋊D3)⋊C3—22no—
142C2×(C3⋊D3)⋊C3#4112yes—
143C4×H(3,𝔽3)—21no—
144C4×H(3,𝔽3)—21no—
145C4×H(3,𝔽3)#4112yes—
146#1084—22no—
147#1084—22no—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—21no—
150(C3⋊Dic3)⋊C3—22no—
151C3×#125—91no—
152C3×C4×D4—91no—
153C3×C4×D4—91no—
154#939—92no—
155#938—91no—
156#938—91no—
157#930—91no—
158#930—91no—
159#929—92no—
160C4×#67—92no—
161#926—91no—
162C2×C4⋊Dic3—91no—
163#898—92no—
164#898—92no—
165#895—92no—
166#895—91no—
167#895—91no—
168#894—92no—
169#893—92no—
170#892—91no—
171#892—91no—
172D3×C4⋊C4—91no—
173D3×C4⋊C4—91no—
174#884—92no—
175#883—92no—
176#67⋊C4—92no—
177#879—92no—
178C4×D12—31no—
179C4×D12—31no—
180C4×D12—91no—
181C4×D12—91no—
182#870—31no—
183#870—31no—
184#870—61no—
185C22×C3⋊D3C22×D311yescharacteristic
186C22×C3⋊D3C22×D312yes—
187C22×C3⋊D3C22×D311yescharacteristic
188C22×C3×D3—31no—
189C22×C3×D3—32no—
190C22×C3×D3—31no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D312yes—
202C2×C3⋊Dic3C22×D311yescharacteristic
203C4⋊(C3⋊D3)—22no—
204C4⋊(C3⋊D3)—22no—
205C4×C3⋊D3—22no—
206C4×C3⋊D3—22no—
207C4×C3⋊D3—42no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—62no—
217C3×D12—62no—
218C3×D12—62no—
219C3×C4×D3—62no—
220C3×C4×D3—62no—
221C3×C4×D3—122no—
222#626—64no—
223#626—64no—
224#626—122no—
225#626—122no—
226#624—122no—
227D3×Dic3—122no—
228#553—271noSylow
229C2×H(3,𝔽3)C2×#4111yescharacteristic
230C2×H(3,𝔽3)C2×#4111yescharacteristic
231C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
232(C3⋊D3)⋊C3—22no—
233(C3⋊D3)⋊C3—41no—
234(C3⋊D3)⋊C3—42no—
235C2×C3×D4—91no—
236C2×C3×D4—91no—
237C22×C3×C4—91no—
238C22×C3×C4—91no—
239C22×C3×C4—92no—
240C2×#67—92no—
241C2×#67—92no—
242C2×#67—92no—
243C2×#67—92no—
244C22×Dic3—92no—
245C2×D12—31no—
246C2×D12—31no—
247C2×D12—91no—
248C2×D12—91no—
249C2×D12—92no—
250C2×D12—92no—
251C2×C4×D3—31no—
252C2×C4×D3—31no—
253C2×C4×D3—32no—
254C2×C4×D3—91no—
255C2×C4×D3—91no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—92no—
259C3×C4⋊C4—91no—
260C3×C4⋊C4—91no—
261C3×C4⋊C4—91no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4—92no—
265C3×C22⋊C4—92no—
266C3×C22⋊C4—91no—
267C3×C22⋊C4—91no—
268C3×C42—31no—
269C3×C42—31no—
270C3×C42—61no—
271C3×C42D3211yescharacteristic
272#217—92no—
273#217—92no—
274#217—92no—
275(C2×D3)⋊C4—31no—
276(C2×D3)⋊C4—31no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—91no—
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—31no—
290C4⋊Dic3—31no—
291C4⋊Dic3—31no—
292C4⋊Dic3—31no—
293C4⋊Dic3—61no—
294C4⋊Dic3—61no—
295C4⋊Dic3—91no—
296C4⋊Dic3—91no—
297C4⋊Dic3—92no—
298Dic3⋊C4—32no—
299Dic3⋊C4—32no—
300Dic3⋊C4—32no—
301Dic3⋊C4—32no—
302Dic3⋊C4—62no—
303Dic3⋊C4—62no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306C4×Dic3—92no—
307C4×Dic3—92no—
308C4×Dic3—92no—
309C22×C32—21no—
310C22×C32C23×D311yescharacteristic
311C22×C32C23×D311yescharacteristic
312C2×C3⋊D3—21no—
313C2×C3⋊D3—21no—
314C2×C3⋊D3—22no—
315C2×C3⋊D3—21no—
316C2×C3⋊D3—21no—
317C2×C3⋊D3—22no—
318C2×C3⋊D3—22no—
319C2×C3⋊D3—21no—
320C2×C3⋊D3#23512yes—
321C2×C3×D3—32no—
322C2×C3×D3—61no—
323C2×C3×D3—61no—
324C2×C3×D3—62no—
325C2×C3×D3—61no—
326C2×C3×D3—62no—
327C2×C3×D3—61no—
328C2×C3×D3—61no—
329C2×C3×D3—62no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—22no—
335C32×C4—41no—
336C32×C4—41no—
337C32×C4#23512yes—
338C32×C4#23712yes—
339C3⋊Dic3—21no—
340C3⋊Dic3—21no—
341C3⋊Dic3—22no—
342C3×Dic3—61no—
343C3×Dic3—61no—
344C3×Dic3—62no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—122no—
350C3×Dic3—122no—
351#126—272no—
352#125—91no—
353#123—272no—
354#121—272no—
355#121—271no—
356#121—271no—
357C4×D4—91no—
358C4×D4—91no—
359C4×D4—272no—
360#117—271no—
361C2×C4⋊C4—271no—
362H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
363C23×C3—91no—
364C23×C3—92no—
365C23×C3—91no—
366C22×D3—31no—
367C22×D3—31no—
368C22×D3—32no—
369C22×D3—91no—
370C22×D3—91no—
371C22×D3—92no—
372C3×D4—182no—
373C3×D4—182no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—31no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—91no—
384C2×C3×C4—92no—
385C2×C3×C4—91no—
386C2×C3×C4—92no—
387C2×C3×C4—92no—
388C2×C3×C4—182no—
389C2×C3×C4—182no—
390C2×C3×C4—362no—
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C2×D3211yescharacteristic
393C2×C3×C4C2×D3211yescharacteristic
394#67—184no—
395#67—184no—
396#67—362no—
397#67—362no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—32no—
401C2×Dic3—32no—
402C2×Dic3—31no—
403C2×Dic3—32no—
404C2×Dic3—31no—
405C2×Dic3—62no—
406C2×Dic3—62no—
407C2×Dic3—91no—
408C2×Dic3—92no—
409C2×Dic3—91no—
410C2×Dic3—362no—
411D12—62no—
412D12—62no—
413D12—182no—
414D12—182no—
415D12—184no—
416D12—362no—
417C4×D3—62no—
418C4×D3—62no—
419C4×D3—122no—
420C4×D3—182no—
421C4×D3—182no—
422C4×D3—362no—
423C4×D3—362no—
424C4×D3—362no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32C2×#23511yescharacteristic
429C2×C32C2×#23511yescharacteristic
430C2×C32C2×#23711yescharacteristic
431C2×C32#100411yescharacteristic
432C2×C32D3×#4111yescharacteristic
433C2×C32#100911yescharacteristic
434C3⋊D3—22no—
435C3⋊D3—42no—
436C3⋊D3—41no—
437C3×D3—62no—
438C3×D3—121no—
439C3×D3—122no—
440C2×D4—91no—
441C2×D4—91no—
442C2×D4—272no—
443C2×D4—272no—
444C22×C4—91no—
445C22×C4—91no—
446C22×C4—92no—
447C22×C4—272no—
448C4⋊C4—91no—
449C4⋊C4—91no—
450C4⋊C4—91no—
451C4⋊C4—91no—
452C4⋊C4—92no—
453C4⋊C4—92no—
454C4⋊C4—272no—
455C22⋊C4—91no—
456C22⋊C4—92no—
457C22⋊C4—91no—
458C22⋊C4—272no—
459C22⋊C4—272no—
460C22⋊C4—272no—
461C42—272no—
462C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
463C22×C3—31no—
464C22×C3—31no—
465C22×C3—61no—
466C22×C3—92no—
467C22×C3—181no—
468C22×C3—181no—
469C22×C3—182no—
470C22×C3—181no—
471C22×C3—182no—
472C22×C3—181no—
473C22×C3—182no—
474C22×C3—181no—
475C22×C3C22×D3211yescharacteristic
476C2×D3—32no—
477C2×D3—61no—
478C2×D3—61no—
479C2×D3—61no—
480C2×D3—62no—
481C2×D3—61no—
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—61no—
485C2×D3—92no—
486C2×D3—181no—
487C2×D3—181no—
488C2×D3—181no—
489C2×D3—182no—
490C2×D3—182no—
491C2×D3—181no—
492C2×D3—182no—
493C2×D3—181no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—32no—
497C3×C4—32no—
498C3×C4—61no—
499C3×C4—61no—
500C3×C4—61no—
501C3×C4—61no—
502C3×C4—62no—
503C3×C4—121no—
504C3×C4—121no—
505C3×C4—182no—
506C3×C4—181no—
507C3×C4—181no—
508C3×C4—182no—
509C3×C4—182no—
510C3×C4#379512yes—
511Dic3—62no—
512Dic3—62no—
513Dic3—62no—
514Dic3—62no—
515Dic3—61no—
516Dic3—61no—
517Dic3—62no—
518Dic3—122no—
519Dic3—122no—
520Dic3—181no—
521Dic3—181no—
522Dic3—182no—
523C32—21no—
524C32#555811yescharacteristic, complemented
525C32#556511yescharacteristic, complemented
526C23—91no—
527C23—91no—
528C23—92no—
529D4—182no—
530D4—182no—
531D4—544no—
532D4—1082no—
533C2×C4—91no—
534C2×C4—92no—
535C2×C4—92no—
536C2×C4—91no—
537C2×C4—92no—
538C2×C4—182no—
539C2×C4—182no—
540C2×C4—362no—
541C2×C4—1082no—
542C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
543C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
544C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
545C2×C3—31no—
546C2×C3—31no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—31no—
550C2×C3—31no—
551C2×C3—61no—
552C2×C3—61no—
553C2×C3—61no—
554C2×C3—182no—
555C2×C3—362no—
556C2×C3—361no—
557C2×C3C2×#379511yescharacteristic
558C2×C3D3×#23511yescharacteristic
559C2×C3#6430711yescharacteristic
560D3—62no—
561D3—121no—
562D3—122no—
563D3—182no—
564D3—361no—
565D3—362no—
566C22—92no—
567C22—181no—
568C22—181no—
569C22—181no—
570C22—182no—
571C22—181no—
572C22—182no—
573C22—182no—
574C22—181no—
575C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
576C4—21no—
577C4—21no—
578C4—181no—
579C4—181no—
580C4—182no—
581C4—182no—
582C4—182no—
583C4#8864812yes—
584C3—31no—
585C3—31no—
586C3—61no—
587C3#1059392311yescharacteristic
588C2—182no—
589C2—361no—
590C2—362no—
591C2C2×#8864811yescharacteristic
592C2#1172249011yescharacteristic
593C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#553
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721716C2semidirect
2. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #11721141C2semidirect
5. #11721141C2semidirect
6. #11721139C2semidirect
7. #11721129C2semidirect
8. #11721126C2semidirect
9. #11721122C2semidirect
10. #11721108C2semidirect
11. #11721081C2semidirect
37. C4×C4⋊(C3⋊D3)D3semidirect
38. C4×C4⋊(C3⋊D3)D3semidirect
362. H(3,𝔽3)#553semidirect
524. C32#5558semidirect
525. C32#5565semidirect

Maximal subgroups

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

subgroupindex
#117217162
C4×C4⋊((C3⋊D3)⋊C3)2
#117211412
#117211392
#117211292
#117211262
#117211222
#117211082
#117210812
#105939823
#105939253