Finite Group Databasebeta

← #13150861

Subgroups — #13150861

Order 1728.

1007 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88542C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic, complemented
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721323C212yesmaximal, complemented
5#11721315C212yesmaximal, complemented
6#11721155C212yesmaximal, complemented
7#11721130C212yesmaximal, complemented
8#11721127C212yesmaximal, complemented
9#11721088C212yesmaximal, complemented
10#10595038—31nomaximal
11#10594977—31nomaximal
12C23×#272C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
17C2×#634⋊C3C2211yescharacteristic, complemented
18C2×#6200C2212yes—
19C2×#6200C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
21#88542C2212yes—
22#88542C2212yes—
23#3757⋊C3C2212yes—
24#3752⋊C3C2212yescomplemented
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
27#3750⋊C3C2212yescomplemented
28#88451C2212yes—
29#88451C2212yes—
30#88450C2212yescomplemented
31#88448C2212yescomplemented
32#88447C2212yescomplemented
33#64139D311yescharacteristic, complemented
34#64133D311yescharacteristic, complemented
35C3×#930—31no—
36C2×C3×(C2×D3)⋊C4—31no—
37C2×C3×(C2×D3)⋊C4—31no—
38C2×C3×(C2×D3)⋊C4—61no—
39C3×#924—31no—
40#63977—32no—
41#63976—32no—
42Dic3×#67—32no—
43#63952—32no—
44#63913—32no—
45#63910—32no—
46#63907—32no—
47#63891—32no—
48#63885—32no—
49#63861—32no—
50#63858—32no—
51#63854—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C22×#272—22no—
54C22×#272—41no—
55C22×#272—41no—
56C22×#272C2312yes—
57C22×(C3⋊D3)⋊C3C2312yes—
58C2×C4×H(3,𝔽3)—41no—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61C2×#1084—41no—
62C2×#1084C2312yes—
63#638⋊C3—24no—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C4⋊((C3⋊D3)⋊C3)—41no—
68#634⋊C3—41no—
69#6200—42no—
70#6200—42no—
71((C3⋊D3)⋊C3)⋊C4—42no—
72#5644—91no—
73#5617—91no—
74#5494—91no—
75C23×C3×D3—31no—
76C23×C3×D3—31no—
77C23×C3×D3—61no—
78C22×C32×C4—21no—
79C22×C32×C4C2×D311yescharacteristic
80C22×C32×C4C2×D311yescharacteristic
81C2×#638C2×D312yes—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
83C2×#634C2×D311yescharacteristic, complemented
84C2×C3×#67—32no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—31no—
87C22×C3×Dic3—61no—
88C2×C3×D12—31no—
89C2×C3×Dic6—31no—
90C2×#626—32no—
91C2×#626—32no—
92C2×#625—32no—
93C2×#625—32no—
94C2×#624—32no—
95C2×D3×Dic3—32no—
96#3757C2×D312yes—
97#3752C2×D312yescomplemented
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
99C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
100#3750C2×D312yescomplemented
101C3×#217—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4—32no—
107C3×(C2×D3)⋊C4—62no—
108C3×(C2×D3)⋊C4—62no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—31no—
111C3×Dic3⋊C4—32no—
112#3724—32no—
113Dic3⋊Dic3—32no—
114#3722—32no—
115(C2×D3)⋊Dic3—32no—
116(C2×D3)⋊Dic3—32no—
117(C2×D3)⋊Dic3—32no—
118(C2×D3)⋊Dic3—32no—
119(C2×D3)⋊Dic3—32no—
120Dic32—32no—
121Dic32—32no—
122C22×H(3,𝔽3)—21no—
123C22×H(3,𝔽3)—21no—
124C22×H(3,𝔽3)#4112yes—
125C22×H(3,𝔽3)C2411yescharacteristic
126C2×#272—22no—
127C2×#272—22no—
128C2×#272—22no—
129C2×#272—44no—
130C2×(C3⋊D3)⋊C3—22no—
131C2×(C3⋊D3)⋊C3—22no—
132C2×(C3⋊D3)⋊C3—22no—
133C4×H(3,𝔽3)—21no—
134C4×H(3,𝔽3)—21no—
135#1084—22no—
136(C3⋊Dic3)⋊C3—22no—
137(C3⋊Dic3)⋊C3—22no—
138(C3⋊Dic3)⋊C3—22no—
139C3×C22⋊Q8—91no—
140C3×#121—91no—
141C2×C3×C22⋊C4—91no—
142#953—92no—
143#947—92no—
144#939—92no—
145#937—92no—
146D4×Dic3—92no—
147#930—31no—
148#930—91no—
149C4×#67—92no—
150C2×(C2×D3)⋊C4—31no—
151C2×(C2×D3)⋊C4—31no—
152C2×(C2×D3)⋊C4—61no—
153#924—31no—
154#924—91no—
155#898—92no—
156#898—92no—
157#895—92no—
158#894—92no—
159#886—92no—
160#885—92no—
161#885—92no—
162#884—92no—
163#883—92no—
164#882—92no—
165#67⋊C4—92no—
166#879—92no—
167C23×C32—21no—
168C23×C32C22×D311yescharacteristic
169C23×C32C22×D311yescharacteristic
170C22×C3⋊D3C22×D312yes—
171C22×C3×D3—32no—
172C22×C3×D3—32no—
173C22×C3×D3—32no—
174C22×C3×D3—62no—
175C22×C3×D3—64no—
176C22×C3×D3—62no—
177C22×C3×D3—62no—
178C22×C3×D3—121no—
179C22×C3×D3—121no—
180C22×C3×D3—121no—
181C22×C3×D3—121no—
182C22×C3×D3—122no—
183C22×C3×D3—122no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—41no—
187C2×C32×C4—41no—
188C2×C32×C4—42no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193#638—24no—
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D312yes—
197C4⋊(C3⋊D3)—41no—
198#634—41no—
199C3×#67—64no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—121no—
207C2×C3×Dic3—121no—
208C2×C3×Dic3—122no—
209C3×D12—121no—
210C3×Dic6—121no—
211#626—122no—
212#626—122no—
213#625—122no—
214#625—122no—
215#624—122no—
216D3×Dic3—122no—
217#538—271noSylow
218C2×H(3,𝔽3)—22no—
219C2×H(3,𝔽3)C2×#4111yescharacteristic
220C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
221C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
222#272—42no—
223(C3⋊D3)⋊C3—42no—
224C24×C3—91no—
225C23×D3—31no—
226C23×D3—31no—
227C23×D3—61no—
228C2×C3×Q8—91no—
229C2×C3×D4—91no—
230C2×C3×D4—92no—
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—61no—
234C22×C3×C4—91no—
235C22×C3×C4D3211yescharacteristic
236C2×#67—32no—
237C2×#67—92no—
238C2×#67—92no—
239C2×#67—92no—
240C2×#67—92no—
241C2×#67—92no—
242C22×Dic3—31no—
243C22×Dic3—31no—
244C22×Dic3—61no—
245C22×Dic3—92no—
246C2×D12—31no—
247C2×D12—91no—
248C2×D12—92no—
249C2×D12—92no—
250C2×C4×D3—92no—
251C2×C4×D3—92no—
252C2×Dic6—31no—
253C2×Dic6—91no—
254C3×C4⋊C4—91no—
255C3×C4⋊C4—92no—
256C3×C4⋊C4—91no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—92no—
259C3×C22⋊C4—92no—
260C3×C22⋊C4—92no—
261#217—32no—
262#217—92no—
263#217—92no—
264#217—92no—
265#217—92no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—32no—
269(C2×D3)⋊C4—32no—
270(C2×D3)⋊C4—32no—
271(C2×D3)⋊C4—62no—
272(C2×D3)⋊C4—62no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—91no—
283C4⋊Dic3—91no—
284C4⋊Dic3—92no—
285Dic3⋊C4—32no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289C4×Dic3—92no—
290C4×Dic3—92no—
291C4×Dic3—92no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—22no—
300C22×C32#23712yes—
301C22×C32#23912yes—
302C22×C32C23×D311yescharacteristic
303C22×C32C23×D311yescharacteristic
304C2×C3⋊D3—22no—
305C2×C3⋊D3—22no—
306C2×C3⋊D3—22no—
307C2×C3×D3—64no—
308C2×C3×D3—62no—
309C2×C3×D3—62no—
310C2×C3×D3—62no—
311C2×C3×D3—62no—
312C2×C3×D3—62no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—122no—
318C2×C3×D3—122no—
319C2×C3×D3—128no—
320C2×C3×D3—124no—
321C2×C3×D3—124no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—41no—
327C32×C4—41no—
328C3⋊Dic3—22no—
329C3⋊Dic3—22no—
330C3⋊Dic3—22no—
331C3×Dic3—62no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C3×Dic3—122no—
337#126—272no—
338#124—272no—
339#123—272no—
340C22⋊Q8—91no—
341#121—91no—
342#121—272no—
343#120—272no—
344C4×D4—272no—
345C2×C22⋊C4—91no—
346H(3,𝔽3)#53811yescharacteristic, complemented, Sylow
347C23×C3—31no—
348C23×C3—31no—
349C23×C3—61no—
350C23×C3—92no—
351C23×C3—92no—
352C23×C3—182no—
353C23×C3—361no—
354C23×C3—361no—
355C23×C3C2×D3211yescharacteristic
356C22×D3—32no—
357C22×D3—32no—
358C22×D3—32no—
359C22×D3—62no—
360C22×D3—62no—
361C22×D3—62no—
362C22×D3—64no—
363C22×D3—92no—
364C22×D3—121no—
365C22×D3—121no—
366C22×D3—121no—
367C22×D3—121no—
368C22×D3—122no—
369C22×D3—122no—
370C3×Q8—361no—
371C3×D4—184no—
372C3×D4—361no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—41no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—92no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—92no—
384C2×C3×C4—121no—
385C2×C3×C4—121no—
386C2×C3×C4—122no—
387C2×C3×C4—361no—
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390#67—64no—
391#67—184no—
392#67—362no—
393#67—362no—
394#67—362no—
395#67—362no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—32no—
401C2×Dic3—62no—
402C2×Dic3—92no—
403C2×Dic3—92no—
404C2×Dic3—92no—
405C2×Dic3—121no—
406C2×Dic3—121no—
407C2×Dic3—122no—
408C2×Dic3—362no—
409D12—121no—
410D12—361no—
411D12—362no—
412D12—362no—
413C4×D3—362no—
414C4×D3—362no—
415Dic6—121no—
416Dic6—361no—
417C2×C32—24no—
418C2×C32—22no—
419C2×C32—22no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32C2×#23711yescharacteristic
424C2×C32#100411yescharacteristic
425C2×C32#100411yescharacteristic
426C2×C32C2×#23911yescharacteristic
427C2×C32#100911yescharacteristic
428C2×C32#100911yescharacteristic
429C3⋊D3—42no—
430C3×D3—124no—
431C3×D3—122no—
432C3×D3—122no—
433C3×D3—122no—
434C24—91no—
435C2×Q8—91no—
436C2×D4—92no—
437C2×D4—91no—
438C2×D4—272no—
439C2×D4—272no—
440C22×C4—91no—
441C22×C4—272no—
442C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
443C4⋊C4—91no—
444C4⋊C4—92no—
445C4⋊C4—91no—
446C4⋊C4—272no—
447C22⋊C4—92no—
448C22⋊C4—92no—
449C22⋊C4—92no—
450C22⋊C4—92no—
451C22⋊C4—272no—
452C22⋊C4—272no—
453C22⋊C4—272no—
454C42—272no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—31no—
458C22×C3—31no—
459C22×C3—32no—
460C22×C3—32no—
461C22×C3—61no—
462C22×C3—61no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—62no—
466C22×C3—62no—
467C22×C3—61no—
468C22×C3—62no—
469C22×C3—182no—
470C22×C3—182no—
471C22×C3—182no—
472C22×C3—182no—
473C22×C3—182no—
474C22×C3—182no—
475C22×C3—364no—
476C22×C3#379612yes—
477C22×C3C22×D3211yescharacteristic
478C2×D3—62no—
479C2×D3—62no—
480C2×D3—62no—
481C2×D3—62no—
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—64no—
485C2×D3—62no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—122no—
489C2×D3—122no—
490C2×D3—124no—
491C2×D3—124no—
492C2×D3—128no—
493C2×D3—182no—
494C2×D3—182no—
495C2×D3—182no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—61no—
499C3×C4—61no—
500C3×C4—61no—
501C3×C4—61no—
502C3×C4—121no—
503C3×C4—121no—
504C3×C4—182no—
505C3×C4—182no—
506C3×C4—182no—
507C3×C4—182no—
508Dic3—62no—
509Dic3—62no—
510Dic3—62no—
511Dic3—62no—
512Dic3—62no—
513Dic3—122no—
514Dic3—182no—
515Dic3—182no—
516Dic3—182no—
517C32—21no—
518C32#561711yescharacteristic, complemented
519C32#564411yescharacteristic, complemented
520C23—92no—
521C23—92no—
522C23—182no—
523C23—361no—
524C23—361no—
525C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
526Q8—361no—
527D4—184no—
528D4—361no—
529D4—1082no—
530D4—1082no—
531C2×C4—41no—
532C2×C4—92no—
533C2×C4—92no—
534C2×C4—92no—
535C2×C4—92no—
536C2×C4—361no—
537C2×C4—1082no—
538C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
539C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
540C2×C3—22no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—31no—
547C2×C3—62no—
548C2×C3—62no—
549C2×C3—64no—
550C2×C3—61no—
551C2×C3—61no—
552C2×C3—61no—
553C2×C3—362no—
554C2×C3—362no—
555C2×C3C2×#379611yescharacteristic
556C2×C3#6430811yescharacteristic
557C2×C3#6433011yescharacteristic
558D3—122no—
559D3—122no—
560D3—124no—
561D3—122no—
562D3—362no—
563C22—21no—
564C22—21no—
565C22—182no—
566C22—182no—
567C22—182no—
568C22—182no—
569C22—182no—
570C22—182no—
571C22—364no—
572C22#8864912yes—
573C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
574C4—21no—
575C4—21no—
576C4—182no—
577C4—182no—
578C4—182no—
579C4—182no—
580C3—31no—
581C3—31no—
582C3—61no—
583C3#1059497911yescharacteristic
584C2—22no—
585C2—362no—
586C2—362no—
587C2C2×#8864911yescharacteristic
588C2#1172249311yescharacteristic
589C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#538
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88542C2semidirect
2. #64139⋊C3C2semidirect
3. #64133⋊C3C2semidirect
4. #11721323C2semidirect
5. #11721315C2semidirect
6. #11721155C2semidirect
7. #11721130C2semidirect
8. #11721127C2semidirect
9. #11721088C2semidirect
16. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
17. C2×#634⋊C3C22semidirect
20. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
24. #3752⋊C3C22semidirect
25. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
26. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
27. #3750⋊C3C22semidirect
30. #88450C22semidirect
31. #88448C22semidirect
32. #88447C22semidirect
33. #64139D3semidirect
34. #64133D3semidirect
82. C2×C4⋊(C3⋊D3)C2×D3semidirect
83. C2×#634C2×D3semidirect
97. #3752C2×D3semidirect
98. C4⋊(C3⋊Dic3)C2×D3semidirect
99. C4⋊(C3⋊Dic3)C2×D3semidirect
100. #3750C2×D3semidirect
346. H(3,𝔽3)#538semidirect
518. C32#5617semidirect
519. C32#5644semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
#64139⋊C32
#64133⋊C32
#117213232
#117213152
#117211552
#117211302
#117211272
#117210882
#105950383
#105949773