Finite Group Databasebeta

← #13149560

Subgroups — #13149560

Order 1728.

909 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721760C211yesmaximal, characteristic, complemented
2#64102⋊C3C211yesmaximal, characteristic, complemented
3C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#11721159C211yesmaximal, characteristic, complemented
5C4×#6200C211yesmaximal, characteristic, complemented
6#11721141C211yesmaximal, characteristic, complemented
7#11721140C212yesmaximal, complemented
8C4×#6198C212yesmaximal, complemented
9#11721112C212yesmaximal, complemented
10#11721100C212yesmaximal, complemented
11#11721093C211yesmaximal, characteristic, complemented
12#10594583—31nomaximal
13#10594559—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#6200C2211yescharacteristic
20C2×#6200C2212yes—
21C2×#6198C2212yes—
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23#88542C2212yes—
24C4⋊#1084C2211yescharacteristic
25C4×#1084C2212yes—
26#3752⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88451C2211yescharacteristic
31#88450C2212yes—
32#88450C2212yes—
33#88450C2211yescharacteristic
34#88449C2212yes—
35#88448C2212yes—
36#88448C2211yescharacteristic
37#64102D311yescharacteristic, complemented
38C4⋊C4×C3⋊D3D311yescharacteristic, complemented
39C3×#893—31no—
40C3×#892—31no—
41C3×#892—31no—
42C3×#892—61no—
43C3×D3×C4⋊C4—31no—
44#63916—31no—
45#63912—32no—
46#63909—31no—
47#63909—31no—
48#63908—31no—
49#63906—32no—
50#63905—32no—
51C4×#626—32no—
52C4×#626—31no—
53C4×#626—32no—
54C4×#625—31no—
55#63872—31no—
56#63869—31no—
57#63868—32no—
58#63859—32no—
59#63853—32no—
60C22×#272C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2312yes—
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2312yes—
66C2×#1084C2311yescharacteristic
67C4×#272—22no—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4⋊((C3⋊D3)⋊C3)—22no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3—42no—
75C4×(C3⋊D3)⋊C3—42no—
76#6200—22no—
77#6200—42no—
78#6198—24no—
79#5558—91no—
80#5530—91no—
81#5514—91no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D312yes—
85C2×C4×C3⋊D3C2×D312yes—
86C2×C3×D12—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—61no—
93C2×#626—32no—
94C2×#626—32no—
95C2×#626—31no—
96C2×#626—32no—
97C2×#625—31no—
98C2×#625—32no—
99C32×C4⋊C4—21no—
100C32×C4⋊C4C2×D311yescharacteristic
101C32×C4⋊C4C2×D311yescharacteristic
102#3752C2×D312yes—
103C4⋊(C3⋊Dic3)C2×D311yescharacteristic
104#3750C2×D312yes—
105C4×C3⋊Dic3C2×D311yescharacteristic
106C3×(C2×D3)⋊C4—32no—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—62no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—61no—
114C3×Dic3⋊C4—32no—
115C3×C4×Dic3—32no—
116C3×C4×Dic3—32no—
117C3×C4×Dic3—31no—
118C3×C4×Dic3—62no—
119#3724—32no—
120#3724—31no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—31no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125#3722—32no—
126#3722—32no—
127#3722—31no—
128(C2×D3)⋊Dic3—31no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131(C2×D3)⋊Dic3—31no—
132(C2×D3)⋊Dic3—31no—
133C22×H(3,𝔽3)C2411yescharacteristic
134C2×#272—21no—
135C2×#272—21no—
136C2×#272—21no—
137C2×(C3⋊D3)⋊C3—21no—
138C2×(C3⋊D3)⋊C3—21no—
139C2×(C3⋊D3)⋊C3—22no—
140C2×(C3⋊D3)⋊C3—22no—
141C2×(C3⋊D3)⋊C3—22no—
142C2×(C3⋊D3)⋊C3#4112yes—
143C4×H(3,𝔽3)—22no—
144C4×H(3,𝔽3)#4112yes—
145#1084—21no—
146#1084—22no—
147(C3⋊Dic3)⋊C3—21no—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—22no—
150(C3⋊Dic3)⋊C3—21no—
151C3×C4×D4—91no—
152C3×#117—91no—
153C2×C3×C4⋊C4—91no—
154#938—91no—
155#938—91no—
156#930—91no—
157#930—91no—
158#930—92no—
159#929—92no—
160C4×#67—91no—
161C4×#67—92no—
162C4×#67—92no—
163C4×#67—91no—
164#898—92no—
165#895—91no—
166#895—91no—
167#894—92no—
168#893—31no—
169#893—91no—
170#892—31no—
171#892—31no—
172#892—61no—
173D3×C4⋊C4—31no—
174D3×C4⋊C4—91no—
175#890—91no—
176#890—91no—
177#884—92no—
178#884—92no—
179#883—92no—
180#879—92no—
181#876—92no—
182C4×D12—92no—
183C4×D12—91no—
184#870—91no—
185C22×C3⋊D3C22×D312yes—
186C22×C3⋊D3C22×D311yescharacteristic
187C22×C3×D3—31no—
188C22×C3×D3—32no—
189C22×C3×D3—31no—
190C22×C3×D3—31no—
191C22×C3×D3—61no—
192C2×C32×C4—21no—
193C2×C32×C4—22no—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D312yes—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D312yes—
201C2×C3⋊Dic3C22×D311yescharacteristic
202C4⋊(C3⋊D3)—22no—
203C4×C3⋊D3—22no—
204C4×C3⋊D3—42no—
205C4×C3⋊D3—42no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—61no—
215C2×C3×Dic3—62no—
216C3×D12—62no—
217C3×C4×D3—64no—
218C3×C4×D3—62no—
219C3×C4×D3—62no—
220C3×C4×D3—62no—
221C3×C4×D3—122no—
222C3×C4×D3—122no—
223#626—64no—
224#626—64no—
225#626—62no—
226#626—122no—
227#625—62no—
228#625—122no—
229#553—271noSylow
230C2×H(3,𝔽3)C2×#4111yescharacteristic
231C2×H(3,𝔽3)C2×#4111yescharacteristic
232C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
233#272—41no—
234(C3⋊D3)⋊C3—22no—
235(C3⋊D3)⋊C3—42no—
236C2×C3×D4—91no—
237C22×C3×C4—91no—
238C22×C3×C4—92no—
239C22×C3×C4—92no—
240C22×C3×C4—91no—
241C2×#67—91no—
242C2×#67—92no—
243C2×#67—92no—
244C2×#67—92no—
245C2×#67—91no—
246C2×#67—92no—
247C2×D12—31no—
248C2×D12—91no—
249C2×D12—92no—
250C2×D12—91no—
251C2×D12—92no—
252C2×C4×D3—31no—
253C2×C4×D3—31no—
254C2×C4×D3—31no—
255C2×C4×D3—32no—
256C2×C4×D3—32no—
257C2×C4×D3—61no—
258C2×C4×D3—91no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—31no—
263C3×C4⋊C4—61no—
264C3×C4⋊C4—91no—
265C3×C4⋊C4—92no—
266C3×C4⋊C4—91no—
267C3×C4⋊C4D3211yescharacteristic
268C3×C22⋊C4—92no—
269C3×C22⋊C4—92no—
270C3×C42—91no—
271C3×C42—92no—
272#217—91no—
273#217—92no—
274#217—92no—
275#217—91no—
276(C2×D3)⋊C4—32no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—62no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—91no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—91no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—91no—
288(C2×D3)⋊C4—91no—
289C4⋊Dic3—31no—
290C4⋊Dic3—31no—
291C4⋊Dic3—31no—
292C4⋊Dic3—61no—
293C4⋊Dic3—91no—
294C4⋊Dic3—92no—
295C4⋊Dic3—91no—
296C4⋊Dic3—92no—
297Dic3⋊C4—32no—
298Dic3⋊C4—92no—
299Dic3⋊C4—91no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—91no—
305C4×Dic3—32no—
306C4×Dic3—32no—
307C4×Dic3—31no—
308C4×Dic3—62no—
309C4×Dic3—91no—
310C22×C32—21no—
311C22×C32C23×D311yescharacteristic
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3—21no—
314C2×C3⋊D3—21no—
315C2×C3⋊D3—22no—
316C2×C3⋊D3—22no—
317C2×C3⋊D3—22no—
318C2×C3⋊D3#23912yes—
319C2×C3×D3—32no—
320C2×C3×D3—61no—
321C2×C3×D3—61no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—61no—
325C2×C3×D3—61no—
326C2×C3×D3—62no—
327C2×C3×D3—61no—
328C2×C3×D3—61no—
329C2×C3×D3—62no—
330C2×C3×D3—61no—
331C2×C3×D3—61no—
332C2×C3×D3—121no—
333C2×C3×D3—121no—
334C32×C4—22no—
335C32×C4—22no—
336C32×C4—22no—
337C32×C4—42no—
338C32×C4#23512yes—
339C32×C4#23512yes—
340C3⋊Dic3—21no—
341C3⋊Dic3—22no—
342C3⋊Dic3—21no—
343C3⋊Dic3—21no—
344C3×Dic3—62no—
345C3×Dic3—61no—
346C3×Dic3—61no—
347C3×Dic3—61no—
348C3×Dic3—62no—
349C3×Dic3—61no—
350C3×Dic3—61no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—122no—
354#126—272no—
355#125—271no—
356#123—272no—
357#121—272no—
358#121—271no—
359#121—271no—
360C4×D4—91no—
361C4×D4—272no—
362C4×D4—271no—
363#117—91no—
364C2×C4⋊C4—91no—
365H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
366C23×C3—91no—
367C23×C3—92no—
368C23×C3—91no—
369C22×D3—31no—
370C22×D3—31no—
371C22×D3—31no—
372C22×D3—32no—
373C22×D3—61no—
374C22×D3—91no—
375C22×D3—92no—
376C3×D4—182no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—32no—
380C2×C3×C4—32no—
381C2×C3×C4—61no—
382C2×C3×C4—62no—
383C2×C3×C4—91no—
384C2×C3×C4—91no—
385C2×C3×C4—92no—
386C2×C3×C4—91no—
387C2×C3×C4—92no—
388C2×C3×C4—91no—
389C2×C3×C4—182no—
390C2×C3×C4—182no—
391C2×C3×C4—362no—
392C2×C3×C4—362no—
393C2×C3×C4C2×D3212yes—
394C2×C3×C4C2×D3211yescharacteristic
395#67—182no—
396#67—184no—
397#67—184no—
398#67—182no—
399#67—362no—
400#67—362no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—32no—
404C2×Dic3—32no—
405C2×Dic3—31no—
406C2×Dic3—31no—
407C2×Dic3—31no—
408C2×Dic3—32no—
409C2×Dic3—61no—
410C2×Dic3—62no—
411C2×Dic3—91no—
412C2×Dic3—91no—
413C2×Dic3—92no—
414C2×Dic3—91no—
415D12—62no—
416D12—182no—
417D12—184no—
418D12—182no—
419D12—362no—
420C4×D3—62no—
421C4×D3—62no—
422C4×D3—64no—
423C4×D3—62no—
424C4×D3—122no—
425C4×D3—122no—
426C4×D3—182no—
427C4×D3—362no—
428C4×D3—362no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32C2×#23511yescharacteristic
433C2×C32C2×#23511yescharacteristic
434C2×C32#100411yescharacteristic
435C2×C32#100411yescharacteristic
436C2×C32C2×#23911yescharacteristic
437C2×C32D3×#4111yescharacteristic
438C3⋊D3—22no—
439C3⋊D3—42no—
440C3×D3—62no—
441C3×D3—122no—
442C3×D3—122no—
443C3×D3—121no—
444C3×D3—121no—
445C2×D4—91no—
446C2×D4—271no—
447C2×D4—272no—
448C2×D4—272no—
449C22×C4—91no—
450C22×C4—91no—
451C22×C4—92no—
452C22×C4—92no—
453C4⋊C4—91no—
454C4⋊C4—92no—
455C4⋊C4—91no—
456C4⋊C4—272no—
457C4⋊C4—271no—
458C4⋊C4—272no—
459C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
460C22⋊C4—92no—
461C22⋊C4—92no—
462C22⋊C4—271no—
463C22⋊C4—272no—
464C22⋊C4—271no—
465C22⋊C4—272no—
466C42—92no—
467C42—91no—
468C22×C3—31no—
469C22×C3—31no—
470C22×C3—61no—
471C22×C3—92no—
472C22×C3—181no—
473C22×C3—181no—
474C22×C3—182no—
475C22×C3—182no—
476C22×C3—182no—
477C22×C3—181no—
478C22×C3—181no—
479C22×C3—181no—
480C22×C3C22×D3211yescharacteristic
481C2×D3—32no—
482C2×D3—61no—
483C2×D3—61no—
484C2×D3—61no—
485C2×D3—61no—
486C2×D3—61no—
487C2×D3—61no—
488C2×D3—61no—
489C2×D3—61no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—62no—
494C2×D3—92no—
495C2×D3—121no—
496C2×D3—121no—
497C2×D3—181no—
498C2×D3—181no—
499C2×D3—182no—
500C2×D3—182no—
501C2×D3—182no—
502C3×C4—22no—
503C3×C4—32no—
504C3×C4—32no—
505C3×C4—62no—
506C3×C4—62no—
507C3×C4—62no—
508C3×C4—122no—
509C3×C4—181no—
510C3×C4—182no—
511C3×C4—181no—
512C3×C4—181no—
513C3×C4—181no—
514C3×C4—182no—
515C3×C4#379812yes—
516Dic3—61no—
517Dic3—61no—
518Dic3—62no—
519Dic3—62no—
520Dic3—62no—
521Dic3—61no—
522Dic3—61no—
523Dic3—62no—
524Dic3—61no—
525Dic3—122no—
526Dic3—181no—
527Dic3—181no—
528Dic3—182no—
529Dic3—181no—
530C32—21no—
531C32#551411yescharacteristic, complemented
532C32#555811yescharacteristic, complemented
533C23—91no—
534C23—91no—
535C23—92no—
536D4—182no—
537D4—544no—
538D4—542no—
539D4—1082no—
540C2×C4—91no—
541C2×C4—91no—
542C2×C4—92no—
543C2×C4—91no—
544C2×C4—91no—
545C2×C4—92no—
546C2×C4—182no—
547C2×C4—182no—
548C2×C4—362no—
549C2×C4—362no—
550C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
551C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
552C2×C3—31no—
553C2×C3—31no—
554C2×C3—31no—
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—61no—
559C2×C3—61no—
560C2×C3—61no—
561C2×C3—182no—
562C2×C3—362no—
563C2×C3—361no—
564C2×C3C2×#379811yescharacteristic
565C2×C3#6431311yescharacteristic
566C2×C3D3×#23911yescharacteristic
567D3—62no—
568D3—121no—
569D3—121no—
570D3—122no—
571D3—122no—
572D3—182no—
573D3—362no—
574C22—92no—
575C22—181no—
576C22—181no—
577C22—182no—
578C22—181no—
579C22—181no—
580C22—182no—
581C22—181no—
582C22—182no—
583C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
584C4—22no—
585C4—181no—
586C4—182no—
587C4—182no—
588C4—181no—
589C4—181no—
590C4—181no—
591C4#8865012yes—
592C3—31no—
593C3—31no—
594C3—61no—
595C3#1059456111yescharacteristic
596C2—182no—
597C2—361no—
598C2—362no—
599C2C2×#8865011yescharacteristic
600C2#1172250611yescharacteristic
601C2#1172251911yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#553
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721760C2semidirect
2. #64102⋊C3C2semidirect
3. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
4. #11721159C2semidirect
5. C4×#6200C2semidirect
6. #11721141C2semidirect
7. #11721140C2semidirect
8. C4×#6198C2semidirect
9. #11721112C2semidirect
10. #11721100C2semidirect
11. #11721093C2semidirect
37. #64102D3semidirect
38. C4⋊C4×C3⋊D3D3semidirect
365. H(3,𝔽3)#553semidirect
531. C32#5514semidirect
532. C32#5558semidirect

Maximal subgroups

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

subgroupindex
#117217602
#64102⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#117211592
C4×#62002
#117211412
#117211402
C4×#61982
#117211122
#117211002
#117210932
#105945833
#105945593