Finite Group Databasebeta

← #13149422

Subgroups — #13149422

Order 1728.

1083 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×((C3⋊D3)⋊C3)⋊Q8C212yesmaximal
2#11721756C211yesmaximal, characteristic, complemented
3C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
4#64097⋊C3C211yesmaximal, characteristic, complemented
5#11721129C211yesmaximal, characteristic
6#11721119C212yesmaximal
7C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
8#11721117C211yesmaximal, characteristic, complemented
9#11721116C211yesmaximal, characteristic, complemented
10#11721080C212yesmaximal, complemented
11#11721079C212yesmaximal, complemented
12#10594374—31nomaximal
13D3×#888—31nomaximal
14C2×#6229C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2212yes—
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
20C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
21C2×#6196C2212yes—
22C2×#6196C2212yes—
23((C3⋊D3)⋊C3)⋊Q8—28no—
24C4⋊C4×H(3,𝔽3)C2211yescharacteristic
25C4⋊#1084C2211yescharacteristic
26#88540C2212yes—
27C4×#1084C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33#88449C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2211yescharacteristic
37#88447C2211yescharacteristic
38C2×#3797—32no—
39C2×D3×Dic6—32no—
40C4⋊C4×C3⋊D3D311yescharacteristic
41#64097D311yescharacteristic, complemented
42C3×D3×C4⋊C4—31no—
43C3×#888—31no—
44C3×#888—31no—
45C3×#888—31no—
46C3×#888—61no—
47#63919—31no—
48#63918—31no—
49#63918—31no—
50#63918—31no—
51#63900—31no—
52D3×C4⋊Dic3—31no—
53#63883—32no—
54C4×#624—31no—
55D3×Dic3⋊C4—32no—
56C4×D3×Dic3—31no—
57#63840—32no—
58#63839—32no—
59#63839—32no—
60#63839—32no—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2312yes—
65C2×#1084C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67#6229—24no—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4×(C3⋊D3)⋊C3—24no—
74C4×(C3⋊D3)⋊C3Q814yescomplemented
75#634⋊C3—24no—
76((C3⋊D3)⋊C3)⋊C4—24no—
77((C3⋊D3)⋊C3)⋊C4D414yes—
78((C3⋊D3)⋊C3)⋊C4Q814yescomplemented
79#6196—24no—
80#6196—24no—
81D3×C4⋊Q8—91no—
82D3×C4⋊Q8—91no—
83C2×#888—91no—
84C2×C4×C3⋊D3C2×D312yes—
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×#634C2×D312yes—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—32no—
89C2×C3×Dic6—32no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—32no—
92C2×C3×Dic6—62no—
93C2×#627—32no—
94C2×#627—32no—
95C2×#627—32no—
96C2×#627—32no—
97C2×#624—31no—
98C2×#624—31no—
99C2×#624—32no—
100C2×D3×Dic3—31no—
101C2×D3×Dic3—31no—
102C2×D3×Dic3—32no—
103#3797—68no—
104D3×Dic6—68no—
105C32×C4⋊C4—21no—
106C32×C4⋊C4C2×D311yescharacteristic
107C32×C4⋊C4C2×D311yescharacteristic
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110#3750C2×D312yes—
111#3750C2×D312yes—
112C4×C3⋊Dic3C2×D311yescharacteristic
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—61no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—62no—
123C3×C4×Dic3—31no—
124C3×C4×Dic3—31no—
125C3×C4×Dic3—31no—
126C3×C4×Dic3—61no—
127#3724—32no—
128#3724—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132Dic3⋊Dic3—32no—
133Dic32—31no—
134Dic32—31no—
135Dic32—31no—
136Dic32—31no—
137C22×H(3,𝔽3)C2411yescharacteristic
138C2×(C3⋊D3)⋊C3C2×D412yes—
139C2×(C3⋊D3)⋊C3C2×Q812yes—
140C2×(C3⋊D3)⋊C3C2×Q812yes—
141C4×H(3,𝔽3)—22no—
142C4×H(3,𝔽3)C2×Q812yescomplemented
143#1084—22no—
144#1084C2×D412yes—
145#1084C2×Q812yescomplemented
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3—22no—
148(C3⋊Dic3)⋊C3C2×D412yes—
149(C3⋊Dic3)⋊C3C2×Q812yescomplemented
150(C3⋊Dic3)⋊C3C2×Q812yescomplemented
151C2×D3×Q8—92no—
152C2×D3×Q8—92no—
153C22×Dic6—92no—
154C3×C4⋊Q8—91no—
155C3×C4⋊Q8—91no—
156C2×C3×C4⋊C4—91no—
157#944—92no—
158#944—92no—
159C2×C4⋊Dic3—91no—
160C2×Dic3⋊C4—92no—
161C2×C4×Dic3—91no—
162D3×C4⋊C4—31no—
163D3×C4⋊C4—91no—
164D3×C4⋊C4—92no—
165D3×C4⋊C4—92no—
166D3×C4⋊C4—91no—
167D3×C4⋊C4—91no—
168#888—31no—
169#888—31no—
170#888—31no—
171#888—61no—
172#888—91no—
173#888—91no—
174#888—91no—
175#888—91no—
176#888—92no—
177#888—92no—
178#888—91no—
179#888—92no—
180#888—92no—
181C42×D3—91no—
182C42×D3—91no—
183#869—91no—
184#869—91no—
185C22×C3⋊D3C22×D311yescharacteristic
186C22×C3×D3—31no—
187C2×C32×C4—21no—
188C2×C32×C4—22no—
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D312yes—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D311yescharacteristic
198C4×C3⋊D3—24no—
199C4×C3⋊D3Dic614yescomplemented
200#634—24no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—61no—
213C2×C3×Dic3—61no—
214C2×C3×Dic3—62no—
215C3×C4×D3—34no—
216C3×C4×D3—64no—
217C3×Dic6—64no—
218C3×Dic6—64no—
219C3×Dic6—64no—
220C3×Dic6—124no—
221#627—64no—
222#627—64no—
223#627—64no—
224#627—64no—
225#624—34no—
226#624—34no—
227#624—64no—
228D3×Dic3—34no—
229D3×Dic3—34no—
230D3×Dic3—64no—
231C2×C4⋊Q8—271noSylow
232C2×H(3,𝔽3)C22×D411yescharacteristic
233C2×H(3,𝔽3)C22×Q811yescharacteristic
234C2×H(3,𝔽3)C22×Q811yescharacteristic
235(C3⋊D3)⋊C3C4⋊Q814yescomplemented
236C2×C3×Q8—92no—
237C2×C3×Q8—92no—
238C22×C3×C4—91no—
239C22×C3×C4—92no—
240C22×Dic3—91no—
241C22×Dic3—91no—
242C22×Dic3—92no—
243D3×Q8—188no—
244D3×Q8—188no—
245C2×C4×D3—31no—
246C2×C4×D3—32no—
247C2×C4×D3—91no—
248C2×C4×D3—92no—
249C2×C4×D3—91no—
250C2×C4×D3—91no—
251C2×C4×D3—91no—
252C2×C4×D3—91no—
253C2×C4×D3—92no—
254C2×C4×D3—92no—
255C2×Dic6—32no—
256C2×Dic6—32no—
257C2×Dic6—32no—
258C2×Dic6—62no—
259C2×Dic6—92no—
260C2×Dic6—92no—
261C2×Dic6—92no—
262C2×Dic6—92no—
263C2×Dic6—92no—
264C2×Dic6—92no—
265C2×Dic6—92no—
266C2×Dic6—188no—
267C3×C4⋊C4—31no—
268C3×C4⋊C4—31no—
269C3×C4⋊C4—61no—
270C3×C4⋊C4—92no—
271C3×C4⋊C4—91no—
272C3×C4⋊C4—92no—
273C3×C4⋊C4—91no—
274C3×C4⋊C4—92no—
275C3×C4⋊C4—91no—
276C3×C4⋊C4D3211yescharacteristic
277C3×C42—91no—
278C3×C42—91no—
279C4⋊Dic3—31no—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—31no—
283C4⋊Dic3—61no—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286C4⋊Dic3—92no—
287C4⋊Dic3—92no—
288C4⋊Dic3—92no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—32no—
292Dic3⋊C4—32no—
293Dic3⋊C4—62no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302C4×Dic3—31no—
303C4×Dic3—31no—
304C4×Dic3—31no—
305C4×Dic3—61no—
306C4×Dic3—91no—
307C4×Dic3—91no—
308C4×Dic3—91no—
309C4×Dic3—91no—
310C4×Dic3—91no—
311C4×Dic3—91no—
312C4×Dic3—91no—
313C22×C32—21no—
314C22×C32C23×D311yescharacteristic
315C22×C32C23×D311yescharacteristic
316C2×C3⋊D3C2×Dic612yes—
317C2×C3⋊D3D3×D412yes—
318C2×C3⋊D3D3×Q812yes—
319C2×C3×D3—32no—
320C2×C3×D3—32no—
321C2×C3×D3—32no—
322C32×C4—22no—
323C32×C4—22no—
324C32×C4—22no—
325C32×C4—42no—
326C32×C4C2×Dic612yescomplemented
327C32×C4D3×Q812yescomplemented
328C3⋊Dic3—22no—
329C3⋊Dic3—22no—
330C3⋊Dic3C2×Dic612yescomplemented
331C3⋊Dic3D3×D412yes—
332C3⋊Dic3D3×Q812yescomplemented
333C3×Dic3—32no—
334C3×Dic3—32no—
335C3×Dic3—32no—
336C3×Dic3—32no—
337C3×Dic3—32no—
338C3×Dic3—32no—
339C3×Dic3—32no—
340C3×Dic3—62no—
341C3×Dic3—62no—
342C3×Dic3—62no—
343C3×Dic3—62no—
344C3×Dic3—62no—
345C3×Dic3—62no—
346C3×Dic3—122no—
347C22×Q8—272no—
348C4⋊Q8—91no—
349C4⋊Q8—91no—
350C4⋊Q8—271no—
351C4⋊Q8—271no—
352C4⋊Q8—272no—
353C4⋊Q8—272no—
354C2×C4⋊C4—91no—
355C2×C4⋊C4—272no—
356C2×C4⋊C4—271no—
357C2×C42—271no—
358H(3,𝔽3)C2×C4⋊Q811yescharacteristic, complemented, Sylow
359C23×C3—91no—
360C22×D3—31no—
361C22×D3—91no—
362C3×Q8—184no—
363C3×Q8—184no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—32no—
367C2×C3×C4—32no—
368C2×C3×C4—61no—
369C2×C3×C4—62no—
370C2×C3×C4—94no—
371C2×C3×C4—91no—
372C2×C3×C4—91no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—91no—
376C2×C3×C4—92no—
377C2×C3×C4—92no—
378C2×C3×C4—92no—
379C2×C3×C4—184no—
380C2×C3×C4C2×D3212yes—
381C2×C3×C4C2×D3211yescharacteristic
382C2×Dic3—31no—
383C2×Dic3—31no—
384C2×Dic3—31no—
385C2×Dic3—31no—
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—31no—
389C2×Dic3—32no—
390C2×Dic3—32no—
391C2×Dic3—32no—
392C2×Dic3—32no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—62no—
396C2×Dic3—91no—
397C2×Dic3—91no—
398C2×Dic3—91no—
399C2×Dic3—92no—
400C2×Dic3—92no—
401C2×Dic3—94no—
402C2×Dic3—94no—
403C2×Dic3—184no—
404C4×D3—34no—
405C4×D3—64no—
406C4×D3—94no—
407C4×D3—94no—
408C4×D3—94no—
409C4×D3—94no—
410C4×D3—94no—
411C4×D3—184no—
412C4×D3—184no—
413C4×D3—184no—
414Dic6—64no—
415Dic6—64no—
416Dic6—64no—
417Dic6—124no—
418Dic6—184no—
419Dic6—184no—
420Dic6—184no—
421Dic6—184no—
422Dic6—184no—
423Dic6—184no—
424Dic6—184no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32C22×Dic611yescharacteristic
429C2×C32C2×D3×D411yescharacteristic
430C2×C32C2×D3×D411yescharacteristic
431C2×C32C2×D3×Q811yescharacteristic
432C2×C32C2×D3×Q811yescharacteristic
433C2×C32C2×D3×Q811yescharacteristic
434C3⋊D3#88814yescomplemented
435C3×D3—34no—
436C2×Q8—92no—
437C2×Q8—92no—
438C2×Q8—272no—
439C2×Q8—272no—
440C2×Q8—548no—
441C22×C4—91no—
442C22×C4—92no—
443C22×C4—271no—
444C22×C4—271no—
445C22×C4—272no—
446C4⋊C4—92no—
447C4⋊C4—91no—
448C4⋊C4—92no—
449C4⋊C4—92no—
450C4⋊C4—91no—
451C4⋊C4—91no—
452C4⋊C4—272no—
453C4⋊C4—272no—
454C4⋊C4—272no—
455C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
456C42—91no—
457C42—91no—
458C42—271no—
459C42—271no—
460C22×C3—31no—
461C22×C3—31no—
462C22×C3—61no—
463C22×C3—92no—
464C22×C3—92no—
465C22×C3—92no—
466C22×C3C22×D3211yescharacteristic
467C2×D3—32no—
468C2×D3—32no—
469C2×D3—32no—
470C2×D3—92no—
471C2×D3—92no—
472C2×D3—92no—
473C3×C4—22no—
474C3×C4—32no—
475C3×C4—32no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—62no—
479C3×C4—92no—
480C3×C4—92no—
481C3×C4—92no—
482C3×C4—92no—
483C3×C4—92no—
484C3×C4—122no—
485C3×C4—182no—
486C3×C4—182no—
487C3×C4—182no—
488C3×C4D3×Dic612yes—
489Dic3—32no—
490Dic3—32no—
491Dic3—32no—
492Dic3—32no—
493Dic3—32no—
494Dic3—32no—
495Dic3—32no—
496Dic3—62no—
497Dic3—62no—
498Dic3—62no—
499Dic3—62no—
500Dic3—62no—
501Dic3—62no—
502Dic3—92no—
503Dic3—92no—
504Dic3—92no—
505Dic3—122no—
506Dic3—182no—
507Dic3—182no—
508C32—21no—
509C32C2×#88811yescharacteristic, complemented
510C32D3×C4⋊Q811yescharacteristic, complemented
511C23—91no—
512Q8—184no—
513Q8—184no—
514Q8—544no—
515Q8—544no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—91no—
519C2×C4—91no—
520C2×C4—91no—
521C2×C4—94no—
522C2×C4—92no—
523C2×C4—92no—
524C2×C4—92no—
525C2×C4—184no—
526C2×C4—274no—
527C2×C4—274no—
528C2×C4—544no—
529C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
530C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
531C2×C3—31no—
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—61no—
538C2×C3—61no—
539C2×C3—61no—
540C2×C3—94no—
541C2×C3C2×D3×Dic611yescharacteristic
542C2×C3D32×D411yescharacteristic
543C2×C3D32×Q811yescharacteristic
544D3—34no—
545D3—94no—
546C22—92no—
547C22—92no—
548C22—92no—
549C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
550C4—22no—
551C4—92no—
552C4—92no—
553C4—92no—
554C4—92no—
555C4—92no—
556C4—182no—
557C4—182no—
558C4—182no—
559C4((C3⋊D3)⋊C3)⋊Q812yescomplemented
560C3—31no—
561C3—31no—
562C3—61no—
563C3D3×#88811yescharacteristic
564C2—94no—
565C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
566C2D4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
567C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4⋊Q8
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
2. #11721756C2semidirect
4. #64097⋊C3C2semidirect
8. #11721117C2semidirect
9. #11721116C2semidirect
10. #11721080C2semidirect
11. #11721079C2semidirect
41. #64097D3semidirect
74. C4×(C3⋊D3)⋊C3Q8semidirect
78. ((C3⋊D3)⋊C3)⋊C4Q8semidirect
142. C4×H(3,𝔽3)C2×Q8semidirect
145. #1084C2×Q8semidirect
149. (C3⋊Dic3)⋊C3C2×Q8semidirect
150. (C3⋊Dic3)⋊C3C2×Q8semidirect
199. C4×C3⋊D3Dic6semidirect
235. (C3⋊D3)⋊C3C4⋊Q8semidirect
326. C32×C4C2×Dic6semidirect
327. C32×C4D3×Q8semidirect
330. C3⋊Dic3C2×Dic6semidirect
332. C3⋊Dic3D3×Q8semidirect
358. H(3,𝔽3)C2×C4⋊Q8semidirect
434. C3⋊D3#888semidirect
509. C32C2×#888semidirect
510. C32D3×C4⋊Q8semidirect
559. C4((C3⋊D3)⋊C3)⋊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
#117217562
C4⋊C4×(C3⋊D3)⋊C32
#64097⋊C32
#117211292
#117211192
C4×((C3⋊D3)⋊C3)⋊C42
#117211172
#117211162
#117210802
#117210792
#105943743
D3×#8883