Finite Group Databasebeta

← #13149476

Subgroups — #13149476

Order 1728.

1099 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88648C211yesmaximal, characteristic
2#11721755C211yesmaximal, characteristic, complemented
3#64102⋊C3C211yesmaximal, characteristic, complemented
4C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5C4×#6198C212yesmaximal, complemented
6#11721136C211yesmaximal, characteristic, complemented
7#11721122C212yesmaximal, complemented
8#11721121C212yesmaximal, complemented
9#11721119C212yesmaximal, complemented
10#11721105C211yesmaximal, characteristic, complemented
11#11721083C211yesmaximal, characteristic, complemented
12#10594361—31nomaximal
13#10594327—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×#6198C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22#88648C418yescomplemented
23C4⋊C4×H(3,𝔽3)C2211yescharacteristic
24#88540C2212yes—
25C4×#1084C2211yescharacteristic
26C4×#1084C2212yes—
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2212yes—
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31#88450C2211yescharacteristic
32#88450C2211yescharacteristic
33#88450C2212yes—
34#88449C2212yes—
35#88448C2211yescharacteristic
36#88448C2211yescharacteristic
37#88447C2212yes—
38C2×#3799—31no—
39C2×#3796—31no—
40#64102D311yescharacteristic, complemented
41C4⋊C4×C3⋊D3D311yescharacteristic, complemented
42C3×#893—31no—
43C3×D3×C4⋊C4—31no—
44C3×#887—31no—
45C3×#887—31no—
46C3×#887—61no—
47#63904—32no—
48C4×#626—32no—
49C4×#626—32no—
50#63894—31no—
51Dic3×D12—31no—
52#63883—32no—
53#63881—32no—
54#63877—32no—
55D3×Dic3⋊C4—32no—
56#63874—31no—
57#63855—31no—
58#63843—31no—
59Dic3×Dic6—31no—
60#63836—32no—
61C22×(C3⋊D3)⋊C3C2312yes—
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C2×#1084C2312yes—
68#6229C2×C414yescomplemented
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
74C4×(C3⋊D3)⋊C3—42no—
75C4×(C3⋊D3)⋊C3—42no—
76C4×(C3⋊D3)⋊C3C2×C414yescomplemented
77#6198C2×C418yescomplemented
78((C3⋊D3)⋊C3)⋊C4—42no—
79((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
80#5574—91no—
81#5546—91no—
82#5504—91no—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D312yes—
85C2×C4×C3⋊D3C2×D312yes—
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C2×C3×D12—31no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—31no—
93C2×C3×Dic6—61no—
94C2×#626—32no—
95C2×#626—32no—
96C2×#624—32no—
97C2×#624—32no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100#3799—38no—
101#3796—38no—
102C32×C4⋊C4—21no—
103C32×C4⋊C4C2×D311yescharacteristic
104C32×C4⋊C4C2×D311yescharacteristic
105#3752C2×D312yes—
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107#3750C2×D312yes—
108C4×C3⋊Dic3C2×D311yescharacteristic
109C3×(C2×D3)⋊C4—32no—
110C3×C4⋊Dic3—31no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—62no—
115C3×C4×Dic3—32no—
116C3×C4×Dic3—31no—
117C3×C4×Dic3—31no—
118C3×C4×Dic3—32no—
119C3×C4×Dic3—31no—
120C3×C4×Dic3—62no—
121C3×C4×Dic3—61no—
122#3724—31no—
123#3724—31no—
124Dic3⋊Dic3—31no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—31no—
128#3722—32no—
129#3722—31no—
130#3722—31no—
131(C2×D3)⋊Dic3—31no—
132(C2×D3)⋊Dic3—31no—
133(C2×D3)⋊Dic3—32no—
134Dic32—32no—
135Dic32—32no—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×(C3⋊D3)⋊C3—22no—
138C2×(C3⋊D3)⋊C3—21no—
139C2×(C3⋊D3)⋊C3—22no—
140C2×(C3⋊D3)⋊C3—21no—
141C2×(C3⋊D3)⋊C3C22×C414yes—
142C2×(C3⋊D3)⋊C3C22×C412yes—
143C4×H(3,𝔽3)—22no—
144C4×H(3,𝔽3)C22×C412yes—
145#1084—21no—
146#1084—21no—
147#1084C22×C414yes—
148(C3⋊Dic3)⋊C3—22no—
149(C3⋊Dic3)⋊C3—21no—
150(C3⋊Dic3)⋊C3—21no—
151(C3⋊Dic3)⋊C3C22×C412yes—
152C2×#239—91no—
153C2×#237—91no—
154C2×#235—91no—
155C3×C4×Q8—91no—
156C3×C4×D4—91no—
157C2×C3×C4⋊C4—91no—
158Q8×Dic3—91no—
159D4×Dic3—91no—
160C4×#67—92no—
161C4×#67—92no—
162C4×#67—92no—
163#926—91no—
164C2×Dic3⋊C4—92no—
165#893—31no—
166#893—91no—
167#893—91no—
168#893—92no—
169#892—92no—
170D3×C4⋊C4—31no—
171D3×C4⋊C4—91no—
172D3×C4⋊C4—92no—
173D3×C4⋊C4—92no—
174#887—31no—
175#887—31no—
176#887—61no—
177#887—91no—
178#67⋊C4—92no—
179#877—92no—
180C4×D12—91no—
181C4×D12—92no—
182#872—91no—
183#872—91no—
184#872—92no—
185C4×Dic6—91no—
186C22×C3⋊D3C22×D311yescharacteristic
187C22×C3⋊D3C22×D312yes—
188C22×C3×D3—31no—
189C22×C3×D3—32no—
190C2×C32×C4—21no—
191C2×C32×C4—22no—
192C2×C32×C4C22×D312yes—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C4⋊(C3⋊D3)C4×D314yescomplemented
201C4×C3⋊D3—42no—
202C4×C3⋊D3—42no—
203C4×C3⋊D3C4×D314yescomplemented
204C2×C3×Dic3—31no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—61no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—61no—
217C3×D12—34no—
218C3×C4×D3—34no—
219C3×C4×D3—122no—
220C3×C4×D3—122no—
221C3×Dic6—34no—
222C3×Dic6—34no—
223C3×Dic6—64no—
224#626—38no—
225#626—38no—
226#624—38no—
227#624—122no—
228D3×Dic3—38no—
229D3×Dic3—122no—
230#520—271noSylow
231C2×H(3,𝔽3)C23×C411yescharacteristic
232C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
233C2×H(3,𝔽3)2−1+411yescharacteristic
234(C3⋊D3)⋊C3—24no—
235(C3⋊D3)⋊C3—22no—
236C2×C3×Q8—91no—
237C2×C3×D4—91no—
238C22×C3×C4—91no—
239C22×C3×C4—92no—
240C22×C3×C4—92no—
241C2×#67—92no—
242C2×#67—92no—
243C22×Dic3—92no—
244C22×Dic3—92no—
245#239—98no—
246#237—98no—
247#235—98no—
248C2×D12—31no—
249C2×D12—91no—
250C2×D12—92no—
251C2×C4×D3—31no—
252C2×C4×D3—32no—
253C2×C4×D3—32no—
254C2×C4×D3—91no—
255C2×C4×D3—92no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—92no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×Dic6—31no—
262C2×Dic6—31no—
263C2×Dic6—61no—
264C3×C4⋊C4—31no—
265C3×C4⋊C4—31no—
266C3×C4⋊C4—61no—
267C3×C4⋊C4—91no—
268C3×C4⋊C4—92no—
269C3×C4⋊C4—92no—
270C3×C4⋊C4D3211yescharacteristic
271C3×C22⋊C4—92no—
272C3×C42—91no—
273C3×C42—92no—
274C3×C42—91no—
275#217—92no—
276#217—91no—
277#217—91no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—91no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—91no—
283(C2×D3)⋊C4—91no—
284(C2×D3)⋊C4—91no—
285(C2×D3)⋊C4—92no—
286C4⋊Dic3—31no—
287C4⋊Dic3—91no—
288C4⋊Dic3—92no—
289C4⋊Dic3—91no—
290C4⋊Dic3—91no—
291Dic3⋊C4—32no—
292Dic3⋊C4—32no—
293Dic3⋊C4—32no—
294Dic3⋊C4—62no—
295Dic3⋊C4—92no—
296Dic3⋊C4—91no—
297Dic3⋊C4—91no—
298Dic3⋊C4—92no—
299Dic3⋊C4—91no—
300Dic3⋊C4—91no—
301Dic3⋊C4—92no—
302C4×Dic3—31no—
303C4×Dic3—32no—
304C4×Dic3—32no—
305C4×Dic3—31no—
306C4×Dic3—31no—
307C4×Dic3—62no—
308C4×Dic3—61no—
309C4×Dic3—91no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C4×Dic3—92no—
313C22×C32—21no—
314C22×C32C23×D311yescharacteristic
315C22×C32C23×D311yescharacteristic
316C2×C3⋊D3—22no—
317C2×C3⋊D3—21no—
318C2×C3⋊D3—21no—
319C2×C3⋊D3—22no—
320C2×C3⋊D3C2×C4×D314yes—
321C2×C3⋊D3C2×C4×D312yes—
322C2×C3×D3—34no—
323C2×C3×D3—32no—
324C2×C3×D3—62no—
325C2×C3×D3—61no—
326C2×C3×D3—62no—
327C2×C3×D3—61no—
328C32×C4—22no—
329C32×C4—22no—
330C32×C4—22no—
331C32×C4—42no—
332C32×C4C2×C4×D312yes—
333C32×C4C2×C4×D312yes—
334C3⋊Dic3—22no—
335C3⋊Dic3—21no—
336C3⋊Dic3—21no—
337C3⋊Dic3C2×C4×D312yes—
338C3×Dic3—32no—
339C3×Dic3—34no—
340C3×Dic3—34no—
341C3×Dic3—64no—
342C3×Dic3—62no—
343C3×Dic3—62no—
344C3×Dic3—61no—
345C3×Dic3—61no—
346C3×Dic3—61no—
347C3×Dic3—61no—
348C3×Dic3—61no—
349C3×Dic3—61no—
350C3×Dic3—121no—
351C2×#41—271no—
352C4×Q8—91no—
353C4×Q8—271no—
354C4×D4—91no—
355C4×D4—271no—
356C4×D4—272no—
357C4×D4—272no—
358#117—271no—
359#117—272no—
360C2×C4⋊C4—91no—
361C2×C4⋊C4—272no—
362H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
363C23×C3—91no—
364C23×C3—92no—
365C22×D3—32no—
366C22×D3—31no—
367C22×D3—92no—
368C22×D3—91no—
369C3×Q8—94no—
370C3×D4—94no—
371C2×C3×C4—31no—
372C2×C3×C4—31no—
373C2×C3×C4—32no—
374C2×C3×C4—32no—
375C2×C3×C4—61no—
376C2×C3×C4—62no—
377C2×C3×C4—94no—
378C2×C3×C4—91no—
379C2×C3×C4—92no—
380C2×C3×C4—91no—
381C2×C3×C4—92no—
382C2×C3×C4—91no—
383C2×C3×C4—91no—
384C2×C3×C4—91no—
385C2×C3×C4—362no—
386C2×C3×C4—362no—
387C2×C3×C4C2×D3212yes—
388C2×C3×C4C2×D3211yescharacteristic
389#67—98no—
390#67—98no—
391C2×Dic3—32no—
392C2×Dic3—32no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—31no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—31no—
399C2×Dic3—31no—
400C2×Dic3—32no—
401C2×Dic3—61no—
402C2×Dic3—62no—
403C2×Dic3—61no—
404C2×Dic3—91no—
405C2×Dic3—91no—
406C2×Dic3—91no—
407C2×Dic3—92no—
408C2×Dic3—98no—
409C2×Dic3—362no—
410D12—34no—
411D12—94no—
412D12—98no—
413C4×D3—34no—
414C4×D3—94no—
415C4×D3—98no—
416C4×D3—98no—
417C4×D3—122no—
418C4×D3—122no—
419C4×D3—362no—
420C4×D3—362no—
421C4×D3—362no—
422C4×D3—362no—
423Dic6—34no—
424Dic6—34no—
425Dic6—64no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32C22×C4×D311yescharacteristic
430C2×C32C22×C4×D311yescharacteristic
431C2×C32#100411yescharacteristic
432C2×C32#100411yescharacteristic
433C2×C32#100711yescharacteristic
434C2×C32#101011yescharacteristic
435C3⋊D3—22no—
436C3⋊D3—24no—
437C3×D3—64no—
438C3×D3—62no—
439#41—278no—
440C2×Q8—91no—
441C2×D4—91no—
442C2×D4—272no—
443C22×C4—91no—
444C22×C4—92no—
445C22×C4—92no—
446C22×C4—272no—
447C22×C4—272no—
448C4⋊C4—92no—
449C4⋊C4—91no—
450C4⋊C4—92no—
451C4⋊C4—271no—
452C4⋊C4—272no—
453C4⋊C4—271no—
454C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
455C22⋊C4—92no—
456C22⋊C4—271no—
457C22⋊C4—272no—
458C22⋊C4—271no—
459C42—91no—
460C42—92no—
461C42—91no—
462C42—272no—
463C22×C3—31no—
464C22×C3—31no—
465C22×C3—61no—
466C22×C3—94no—
467C22×C3—92no—
468C22×C3—182no—
469C22×C3—181no—
470C22×C3—181no—
471C22×C3—182no—
472C22×C3C22×D3211yescharacteristic
473C2×D3—32no—
474C2×D3—34no—
475C2×D3—61no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—61no—
479C2×D3—94no—
480C2×D3—92no—
481C2×D3—181no—
482C2×D3—182no—
483C2×D3—182no—
484C2×D3—181no—
485C3×C4—22no—
486C3×C4—32no—
487C3×C4—32no—
488C3×C4—62no—
489C3×C4—62no—
490C3×C4—62no—
491C3×C4—92no—
492C3×C4—94no—
493C3×C4—122no—
494C3×C4—182no—
495C3×C4—181no—
496C3×C4—181no—
497C3×C4—181no—
498C3×C4—181no—
499C3×C4C4×D3212yes—
500Dic3—34no—
501Dic3—34no—
502Dic3—32no—
503Dic3—64no—
504Dic3—61no—
505Dic3—61no—
506Dic3—61no—
507Dic3—61no—
508Dic3—62no—
509Dic3—61no—
510Dic3—62no—
511Dic3—61no—
512Dic3—92no—
513Dic3—121no—
514Dic3—181no—
515Dic3—182no—
516Dic3—181no—
517C32—21no—
518C32#550411yescharacteristic, complemented
519C32#554611yescharacteristic, complemented
520C23—92no—
521C23—91no—
522Q8—94no—
523D4—94no—
524D4—278no—
525C2×C4—92no—
526C2×C4—91no—
527C2×C4—94no—
528C2×C4—91no—
529C2×C4—91no—
530C2×C4—91no—
531C2×C4—91no—
532C2×C4—92no—
533C2×C4—278no—
534C2×C4—362no—
535C2×C4—362no—
536C2×C4—1082no—
537C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
538C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—61no—
546C2×C3—61no—
547C2×C3—61no—
548C2×C3—182no—
549C2×C3—184no—
550C2×C3C2×C4×D3211yescharacteristic
551C2×C3#6431311yescharacteristic
552C2×C3#6431511yescharacteristic
553D3—62no—
554D3—64no—
555D3—184no—
556D3—182no—
557C22—92no—
558C22—94no—
559C22—181no—
560C22—182no—
561C22—182no—
562C22—181no—
563C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
564C4—22no—
565C4—92no—
566C4—94no—
567C4—181no—
568C4—182no—
569C4—181no—
570C4—181no—
571C4—181no—
572C4C4×((C3⋊D3)⋊C3)⋊C212yes—
573C3—31no—
574C3—31no—
575C3—61no—
576C3#1059435911yescharacteristic
577C2—182no—
578C2—184no—
579C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
580C2#1172250611yescharacteristic
581C2#1172251611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC2×#88648C2
cosocleC22×H(3,𝔽3)C24
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#520
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721755C2semidirect
3. #64102⋊C3C2semidirect
4. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
5. C4×#6198C2semidirect
6. #11721136C2semidirect
7. #11721122C2semidirect
8. #11721121C2semidirect
9. #11721119C2semidirect
10. #11721105C2semidirect
11. #11721083C2semidirect
22. #88648C4semidirect
40. #64102D3semidirect
41. C4⋊C4×C3⋊D3D3semidirect
68. #6229C2×C4semidirect
73. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
76. C4×(C3⋊D3)⋊C3C2×C4semidirect
77. #6198C2×C4semidirect
79. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
200. C4⋊(C3⋊D3)C4×D3semidirect
203. C4×C3⋊D3C4×D3semidirect
362. H(3,𝔽3)#520semidirect
518. C32#5504semidirect
519. C32#5546semidirect

Maximal subgroups

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

subgroupindex
C2×#886482
#117217552
#64102⋊C32
C4⋊C4×(C3⋊D3)⋊C32
C4×#61982
#117211362
#117211222
#117211212
#117211192
#117211052
#117210832
#105943613
#105943273