Finite Group Databasebeta

← #13149473

Subgroups — #13149473

Order 1728.

1113 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×((C3⋊D3)⋊C3)⋊Q8C211yesmaximal, characteristic
2#11721755C211yesmaximal, characteristic, complemented
3C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
4#64096⋊C3C211yesmaximal, characteristic, complemented
5#11721119C212yesmaximal
6C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
7C4×((C3⋊D3)⋊C3)⋊C4C212yesmaximal
8#11721084C211yesmaximal, characteristic, complemented
9#11721083C211yesmaximal, characteristic, complemented
10#11721078C212yesmaximal, complemented
11#11721077C212yesmaximal, complemented
12#10594363—31nomaximal
13D3×#887—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
19C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
21C2×#6196C2212yes—
22((C3⋊D3)⋊C3)⋊Q8C418yescomplemented
23C4⋊C4×H(3,𝔽3)C2211yescharacteristic
24#88540C2212yes—
25C4×#1084C2212yes—
26C4×#1084C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30C4×(C3⋊Dic3)⋊C3C2212yes—
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32#88449C2212yes—
33#88448C2212yes—
34#88447C2211yescharacteristic
35#88447C2212yes—
36#88447C2211yescharacteristic
37#88447C2212yes—
38C2×#3797—31no—
39C2×D3×Dic6—31no—
40C4⋊C4×C3⋊D3D311yescharacteristic
41#64096D311yescharacteristic, complemented
42C3×D3×C4⋊C4—31no—
43C3×#887—31no—
44C3×#887—31no—
45C3×#887—31no—
46C3×#887—61no—
47#63883—32no—
48C4×#624—31no—
49C4×#624—32no—
50D3×Dic3⋊C4—32no—
51C4×D3×Dic3—31no—
52C4×D3×Dic3—32no—
53#63844—31no—
54#63843—31no—
55Dic3×Dic6—31no—
56Dic3×Dic6—31no—
57#63838—32no—
58#63837—32no—
59#63837—32no—
60#63837—32no—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2311yescharacteristic
65C2×#1084C2312yes—
66C2×#1084C2311yescharacteristic
67#6229C2×C414yescomplemented
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4×(C3⋊D3)⋊C3—24no—
74C4×(C3⋊D3)⋊C3C2×C414yes—
75#634⋊C3C2×C414yescomplemented
76((C3⋊D3)⋊C3)⋊C4—22no—
77((C3⋊D3)⋊C3)⋊C4C2×C418yes—
78((C3⋊D3)⋊C3)⋊C4Q814yescomplemented
79#6196C2×C418yescomplemented
80C4×D3×Q8—91no—
81C4×D3×Q8—91no—
82C2×#887—91no—
83C2×C4×C3⋊D3C2×D312yes—
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×#634C2×D311yescharacteristic
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—32no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—61no—
92C2×#627—32no—
93C2×#627—32no—
94C2×#624—32no—
95C2×#624—31no—
96C2×#624—31no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—31no—
99C2×D3×Dic3—31no—
100#3797—38no—
101D3×Dic6—38no—
102C32×C4⋊C4—21no—
103C32×C4⋊C4C2×D311yescharacteristic
104C32×C4⋊C4C2×D311yescharacteristic
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106#3750C2×D312yes—
107#3750C2×D312yes—
108C4×C3⋊Dic3C2×D312yes—
109C4×C3⋊Dic3C2×D311yescharacteristic
110C3×C4⋊Dic3—31no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—62no—
116C3×C4×Dic3—32no—
117C3×C4×Dic3—32no—
118C3×C4×Dic3—32no—
119C3×C4×Dic3—31no—
120C3×C4×Dic3—31no—
121C3×C4×Dic3—31no—
122C3×C4×Dic3—62no—
123C3×C4×Dic3—61no—
124#3724—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—32no—
128Dic32—31no—
129Dic32—31no—
130Dic32—31no—
131Dic32—32no—
132Dic32—32no—
133Dic32—32no—
134Dic32—31no—
135Dic32—32no—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×(C3⋊D3)⋊C3C22×C412yes—
138C2×(C3⋊D3)⋊C3C2×Q812yes—
139C2×(C3⋊D3)⋊C3#4112yes—
140C4×H(3,𝔽3)—22no—
141C4×H(3,𝔽3)C22×C412yes—
142#1084—21no—
143#1084C22×C414yes—
144#1084C2×Q812yescomplemented
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3C22×C412yes—
148(C3⋊Dic3)⋊C3C22×C414yes—
149(C3⋊Dic3)⋊C3C2×Q812yescomplemented
150C2×D3×Q8—91no—
151C2×D3×Q8—91no—
152C22×Dic6—91no—
153C3×C4×Q8—91no—
154C3×C4×Q8—91no—
155C2×C3×C4⋊C4—91no—
156Q8×Dic3—91no—
157Q8×Dic3—91no—
158C2×Dic3⋊C4—92no—
159C2×C4×Dic3—92no—
160C2×C4×Dic3—91no—
161D3×C4⋊C4—31no—
162D3×C4⋊C4—91no—
163D3×C4⋊C4—92no—
164D3×C4⋊C4—92no—
165#887—31no—
166#887—31no—
167#887—31no—
168#887—61no—
169#887—91no—
170#887—92no—
171#887—92no—
172#887—92no—
173#887—91no—
174#887—92no—
175#887—91no—
176C42×D3—92no—
177C42×D3—91no—
178C42×D3—92no—
179C42×D3—91no—
180C4×Dic6—92no—
181C4×Dic6—92no—
182C4×Dic6—91no—
183C4×Dic6—91no—
184C22×C3⋊D3C22×D311yescharacteristic
185C22×C3×D3—31no—
186C2×C32×C4—21no—
187C2×C32×C4—22no—
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D311yescharacteristic
197C4×C3⋊D3—24no—
198C4×C3⋊D3C4×D314yes—
199#634C4×D314yescomplemented
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—61no—
214C3×C4×D3—34no—
215C3×C4×D3—64no—
216C3×Dic6—34no—
217C3×Dic6—34no—
218C3×Dic6—34no—
219C3×Dic6—64no—
220#627—38no—
221#627—38no—
222#624—38no—
223#624—34no—
224#624—62no—
225D3×Dic3—38no—
226D3×Dic3—34no—
227D3×Dic3—62no—
228C2×C4×Q8—271noSylow
229C2×H(3,𝔽3)C23×C411yescharacteristic
230C2×H(3,𝔽3)C22×Q811yescharacteristic
231C2×H(3,𝔽3)C2×#4111yescharacteristic
232(C3⋊D3)⋊C3C4×Q814yescomplemented
233C2×C3×Q8—91no—
234C2×C3×Q8—91no—
235C22×C3×C4—91no—
236C22×C3×C4—92no—
237C22×Dic3—91no—
238C22×Dic3—92no—
239C22×Dic3—91no—
240D3×Q8—98no—
241D3×Q8—98no—
242C2×C4×D3—31no—
243C2×C4×D3—32no—
244C2×C4×D3—91no—
245C2×C4×D3—92no—
246C2×C4×D3—91no—
247C2×C4×D3—92no—
248C2×C4×D3—91no—
249C2×C4×D3—92no—
250C2×C4×D3—91no—
251C2×C4×D3—91no—
252C2×Dic6—31no—
253C2×Dic6—31no—
254C2×Dic6—31no—
255C2×Dic6—61no—
256C2×Dic6—91no—
257C2×Dic6—98no—
258C2×Dic6—92no—
259C2×Dic6—92no—
260C2×Dic6—92no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—31no—
263C3×C4⋊C4—61no—
264C3×C4⋊C4—92no—
265C3×C4⋊C4—91no—
266C3×C4⋊C4—92no—
267C3×C4⋊C4—92no—
268C3×C4⋊C4D3211yescharacteristic
269C3×C42—91no—
270C3×C42—92no—
271C3×C42—92no—
272C3×C42—91no—
273C4⋊Dic3—31no—
274C4⋊Dic3—91no—
275C4⋊Dic3—92no—
276C4⋊Dic3—92no—
277Dic3⋊C4—32no—
278Dic3⋊C4—32no—
279Dic3⋊C4—32no—
280Dic3⋊C4—32no—
281Dic3⋊C4—62no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288C4×Dic3—32no—
289C4×Dic3—31no—
290C4×Dic3—32no—
291C4×Dic3—31no—
292C4×Dic3—32no—
293C4×Dic3—31no—
294C4×Dic3—62no—
295C4×Dic3—61no—
296C4×Dic3—92no—
297C4×Dic3—91no—
298C4×Dic3—92no—
299C4×Dic3—91no—
300C4×Dic3—91no—
301C4×Dic3—92no—
302C4×Dic3—92no—
303C4×Dic3—91no—
304C4×Dic3—91no—
305C4×Dic3—92no—
306C4×Dic3—92no—
307C4×Dic3—91no—
308C4×Dic3—92no—
309C4×Dic3—91no—
310C22×C32—21no—
311C22×C32C23×D311yescharacteristic
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3C2×C4×D312yes—
314C2×C3⋊D3#23712yes—
315C2×C3⋊D3D3×Q812yes—
316C2×C3×D3—32no—
317C2×C3×D3—32no—
318C2×C3×D3—32no—
319C32×C4—22no—
320C32×C4—22no—
321C32×C4—22no—
322C32×C4—42no—
323C32×C4C2×C4×D312yes—
324C32×C4C2×C4×D312yes—
325C3⋊Dic3—22no—
326C3⋊Dic3—21no—
327C3⋊Dic3C2×C4×D314yes—
328C3⋊Dic3C2×C4×D312yes—
329C3⋊Dic3D3×Q812yescomplemented
330C3×Dic3—34no—
331C3×Dic3—32no—
332C3×Dic3—34no—
333C3×Dic3—34no—
334C3×Dic3—32no—
335C3×Dic3—32no—
336C3×Dic3—32no—
337C3×Dic3—64no—
338C3×Dic3—62no—
339C3×Dic3—61no—
340C3×Dic3—61no—
341C3×Dic3—62no—
342C3×Dic3—61no—
343C3×Dic3—121no—
344C22×Q8—271no—
345C4×Q8—91no—
346C4×Q8—91no—
347C4×Q8—272no—
348C4×Q8—272no—
349C4×Q8—271no—
350C4×Q8—271no—
351C2×C4⋊C4—91no—
352C2×C4⋊C4—272no—
353C2×C42—271no—
354C2×C42—272no—
355H(3,𝔽3)C2×C4×Q811yescharacteristic, complemented, Sylow
356C23×C3—91no—
357C22×D3—31no—
358C22×D3—91no—
359C3×Q8—94no—
360C3×Q8—94no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—32no—
364C2×C3×C4—32no—
365C2×C3×C4—61no—
366C2×C3×C4—62no—
367C2×C3×C4—94no—
368C2×C3×C4—91no—
369C2×C3×C4—91no—
370C2×C3×C4—92no—
371C2×C3×C4—91no—
372C2×C3×C4—92no—
373C2×C3×C4—92no—
374C2×C3×C4—91no—
375C2×C3×C4—91no—
376C2×C3×C4—184no—
377C2×C3×C4C2×D3212yes—
378C2×C3×C4C2×D3211yescharacteristic
379C2×Dic3—31no—
380C2×Dic3—31no—
381C2×Dic3—32no—
382C2×Dic3—32no—
383C2×Dic3—31no—
384C2×Dic3—32no—
385C2×Dic3—31no—
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—31no—
389C2×Dic3—32no—
390C2×Dic3—61no—
391C2×Dic3—62no—
392C2×Dic3—61no—
393C2×Dic3—91no—
394C2×Dic3—91no—
395C2×Dic3—92no—
396C2×Dic3—91no—
397C2×Dic3—92no—
398C2×Dic3—98no—
399C2×Dic3—94no—
400C2×Dic3—182no—
401C4×D3—34no—
402C4×D3—64no—
403C4×D3—94no—
404C4×D3—94no—
405C4×D3—98no—
406C4×D3—94no—
407C4×D3—98no—
408C4×D3—184no—
409C4×D3—182no—
410C4×D3—182no—
411Dic6—34no—
412Dic6—34no—
413Dic6—34no—
414Dic6—64no—
415Dic6—94no—
416Dic6—98no—
417Dic6—98no—
418Dic6—98no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32C22×C4×D311yescharacteristic
423C2×C32C22×C4×D311yescharacteristic
424C2×C32C2×#23711yescharacteristic
425C2×C32C2×D3×Q811yescharacteristic
426C2×C32C2×D3×Q811yescharacteristic
427C2×C32D3×#4111yescharacteristic
428C3⋊D3#88714yescomplemented
429C3×D3—34no—
430C2×Q8—91no—
431C2×Q8—91no—
432C2×Q8—272no—
433C2×Q8—278no—
434C22×C4—91no—
435C22×C4—92no—
436C22×C4—271no—
437C22×C4—272no—
438C22×C4—271no—
439C4⋊C4—92no—
440C4⋊C4—91no—
441C4⋊C4—92no—
442C4⋊C4—92no—
443C4⋊C4—272no—
444C4⋊C4—272no—
445C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
446C42—92no—
447C42—91no—
448C42—92no—
449C42—91no—
450C42—272no—
451C42—271no—
452C42—272no—
453C42—271no—
454C22×C3—31no—
455C22×C3—31no—
456C22×C3—61no—
457C22×C3—92no—
458C22×C3—92no—
459C22×C3—92no—
460C22×C3C22×D3211yescharacteristic
461C2×D3—32no—
462C2×D3—32no—
463C2×D3—32no—
464C2×D3—92no—
465C2×D3—92no—
466C2×D3—92no—
467C3×C4—22no—
468C3×C4—32no—
469C3×C4—32no—
470C3×C4—62no—
471C3×C4—62no—
472C3×C4—62no—
473C3×C4—92no—
474C3×C4—94no—
475C3×C4—92no—
476C3×C4—94no—
477C3×C4—92no—
478C3×C4—122no—
479C3×C4—181no—
480C3×C4—182no—
481C3×C4—181no—
482C3×C4C4×D3212yes—
483Dic3—34no—
484Dic3—34no—
485Dic3—32no—
486Dic3—32no—
487Dic3—34no—
488Dic3—32no—
489Dic3—32no—
490Dic3—64no—
491Dic3—62no—
492Dic3—61no—
493Dic3—61no—
494Dic3—62no—
495Dic3—61no—
496Dic3—92no—
497Dic3—94no—
498Dic3—92no—
499Dic3—121no—
500Dic3—182no—
501Dic3—181no—
502C32—21no—
503C32C2×#88711yescharacteristic, complemented
504C32C4×D3×Q811yescharacteristic, complemented
505C23—91no—
506Q8—94no—
507Q8—94no—
508Q8—278no—
509C2×C4—91no—
510C2×C4—91no—
511C2×C4—92no—
512C2×C4—92no—
513C2×C4—91no—
514C2×C4—94no—
515C2×C4—91no—
516C2×C4—91no—
517C2×C4—92no—
518C2×C4—184no—
519C2×C4—278no—
520C2×C4—274no—
521C2×C4—542no—
522C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
523C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
524C2×C3—31no—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—61no—
531C2×C3—61no—
532C2×C3—61no—
533C2×C3—94no—
534C2×C3C2×C4×D3211yescharacteristic
535C2×C3D3×#23711yescharacteristic
536C2×C3D32×Q811yescharacteristic
537D3—34no—
538D3—94no—
539C22—92no—
540C22—92no—
541C22—92no—
542C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
543C4—22no—
544C4—92no—
545C4—92no—
546C4—92no—
547C4—94no—
548C4—94no—
549C4—182no—
550C4—181no—
551C4—181no—
552C4C4×((C3⋊D3)⋊C3)⋊C212yes—
553C3—31no—
554C3—31no—
555C3—61no—
556C3D3×#88711yescharacteristic
557C2—94no—
558C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
559C2#1172250411yescharacteristic
560C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
cosocleC22×H(3,𝔽3)C24
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×Q8
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
4. #64096⋊C3C2semidirect
8. #11721084C2semidirect
9. #11721083C2semidirect
10. #11721078C2semidirect
11. #11721077C2semidirect
22. ((C3⋊D3)⋊C3)⋊Q8C4semidirect
41. #64096D3semidirect
67. #6229C2×C4semidirect
75. #634⋊C3C2×C4semidirect
78. ((C3⋊D3)⋊C3)⋊C4Q8semidirect
79. #6196C2×C4semidirect
144. #1084C2×Q8semidirect
149. (C3⋊Dic3)⋊C3C2×Q8semidirect
199. #634C4×D3semidirect
232. (C3⋊D3)⋊C3C4×Q8semidirect
329. C3⋊Dic3D3×Q8semidirect
355. H(3,𝔽3)C2×C4×Q8semidirect
428. C3⋊D3#887semidirect
503. C32C2×#887semidirect
504. C32C4×D3×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊Q82
#117217552
C4⋊C4×(C3⋊D3)⋊C32
#64096⋊C32
#117211192
C4×((C3⋊D3)⋊C3)⋊C42
#117210842
#117210832
#117210782
#117210772
#105943633
D3×#8873