Finite Group Databasebeta

← #13150858

Subgroups — #13150858

Order 1728.

969 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721830C211yesmaximal, characteristic, complemented
2C2×#3752⋊C3C211yesmaximal, characteristic, complemented
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721321C212yesmaximal, complemented
5#11721293C212yesmaximal, complemented
6#11721157C212yesmaximal, complemented
7#11721155C212yesmaximal, complemented
8#11721111C212yesmaximal, complemented
9#11721102C212yesmaximal, complemented
10#10594979—31nomaximal
11#10594977—31nomaximal
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6235C2212yes—
15C2×C4⋊#272C2211yescharacteristic, complemented
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic, complemented
18C2×#6200C2212yes—
19C2×#6200C2212yes—
20C2×#6199C2212yescomplemented
21#88542C2212yescomplemented
22C4⋊#1084C2211yescharacteristic, complemented
23#3757⋊C3C2212yes—
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
27#3750⋊C3C2212yescomplemented
28#88451C2212yescomplemented
29#88450C2212yes—
30#88450C2212yes—
31#88449C2212yescomplemented
32#88447C2212yescomplemented
33C2×#3752D311yescharacteristic, complemented
34#64133D311yescharacteristic, complemented
35C3×#930—31no—
36C3×#930—31no—
37C3×#930—61no—
38C2×C3×(C2×D3)⋊C4—31no—
39C3×#924—31no—
40#63981—32no—
41#63976—32no—
42#67⋊Dic3—32no—
43Dic3×#67—32no—
44#63913—32no—
45#63907—32no—
46#63891—32no—
47#63890—32no—
48#63886—32no—
49#63885—32no—
50#63858—32no—
51#63848—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C22×#272C2312yes—
54C22×(C3⋊D3)⋊C3—22no—
55C22×(C3⋊D3)⋊C3—41no—
56C22×(C3⋊D3)⋊C3—41no—
57C22×(C3⋊D3)⋊C3C2312yes—
58C2×C4×H(3,𝔽3)—41no—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61#6235—24no—
62C2×#1084C2312yes—
63C4⋊#272—41no—
64C2×(C3⋊Dic3)⋊C3—41no—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68#634⋊C3—41no—
69#6200—42no—
70#6200—42no—
71#6199—42no—
72#5644—91no—
73#5617—91no—
74#5494—91no—
75C23×C3⋊D3C2×D311yescharacteristic
76C23×C3×D3—31no—
77C22×C32×C4—21no—
78C22×C32×C4C2×D311yescharacteristic
79C22×C32×C4C2×D311yescharacteristic
80C22×C3⋊Dic3C2×D311yescharacteristic
81C2×#634C2×D311yescharacteristic, complemented
82C2×C3×#67—32no—
83C2×C3×#67—32no—
84C2×C3×#67—62no—
85C22×C3×Dic3—31no—
86C2×C3×D12—31no—
87C2×C3×D12—31no—
88C2×C3×D12—61no—
89C2×C3×Dic6—31no—
90C2×#626—32no—
91C2×#626—32no—
92C2×#625—32no—
93C2×#625—32no—
94C2×D3×Dic3—32no—
95C2×D3×Dic3—32no—
96#3757C2×D312yes—
97#3752C2×D312yes—
98#3752C2×D312yes—
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—62no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—61no—
111C3×Dic3⋊C4—32no—
112Dic3⋊Dic3—32no—
113Dic3⋊Dic3—32no—
114#3722—32no—
115#3722—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×(C3⋊D3)⋊C3—22no—
130C2×(C3⋊D3)⋊C3—22no—
131C2×(C3⋊D3)⋊C3—22no—
132C2×(C3⋊D3)⋊C3—44no—
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—31no—
149#930—61no—
150C4×#67—92no—
151C2×(C2×D3)⋊C4—31no—
152C2×(C2×D3)⋊C4—91no—
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⋊D3—22no—
171C22×C3⋊D3—41no—
172C22×C3⋊D3—41no—
173C22×C3⋊D3C22×D312yes—
174C22×C3×D3—32no—
175C22×C3×D3—32no—
176C22×C3×D3—32no—
177C22×C3×D3—62no—
178C22×C3×D3—62no—
179C22×C3×D3—121no—
180C22×C3×D3—121no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—41no—
184C2×C32×C4—41no—
185C2×C32×C4—42no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3—41no—
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D312yes—
194#634—41no—
195C3×#67—64no—
196C3×#67—64no—
197C3×#67—124no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—121no—
205C3×D12—121no—
206C3×D12—121no—
207C3×D12—122no—
208C3×Dic6—121no—
209#626—122no—
210#626—122no—
211#625—122no—
212#625—122no—
213D3×Dic3—122no—
214D3×Dic3—122no—
215#538—271noSylow
216C2×H(3,𝔽3)—22no—
217C2×H(3,𝔽3)C2×#4111yescharacteristic
218C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
219C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
220#272—42no—
221(C3⋊D3)⋊C3—42no—
222C24×C3—91no—
223C23×D3—31no—
224C23×D3—91no—
225C2×C3×Q8—91no—
226C2×C3×D4—92no—
227C2×C3×D4—91no—
228C22×C3×C4—31no—
229C22×C3×C4—31no—
230C22×C3×C4—61no—
231C22×C3×C4—91no—
232C22×C3×C4D3211yescharacteristic
233C2×#67—32no—
234C2×#67—32no—
235C2×#67—62no—
236C2×#67—92no—
237C2×#67—92no—
238C2×#67—92no—
239C2×#67—92no—
240C22×Dic3—31no—
241C22×Dic3—91no—
242C22×Dic3—92no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—61no—
246C2×D12—92no—
247C2×D12—92no—
248C2×C4×D3—92no—
249C2×C4×D3—92no—
250C2×Dic6—31no—
251C2×Dic6—91no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—92no—
254C3×C4⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—92no—
259#217—32no—
260#217—92no—
261#217—92no—
262#217—92no—
263#217—92no—
264(C2×D3)⋊C4—32no—
265(C2×D3)⋊C4—32no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—62no—
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277C4⋊Dic3—31no—
278C4⋊Dic3—31no—
279C4⋊Dic3—31no—
280C4⋊Dic3—61no—
281C4⋊Dic3—91no—
282C4⋊Dic3—92no—
283Dic3⋊C4—32no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287C4×Dic3—92no—
288C4×Dic3—92no—
289C4×Dic3—92no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—22no—
296C22×C32—41no—
297C22×C32—41no—
298C22×C32#23512yes—
299C22×C32#23912yes—
300C22×C32C23×D311yescharacteristic
301C22×C32C23×D311yescharacteristic
302C2×C3⋊D3—22no—
303C2×C3⋊D3—22no—
304C2×C3⋊D3—22no—
305C2×C3⋊D3—44no—
306C2×C3×D3—62no—
307C2×C3×D3—62no—
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—122no—
316C2×C3×D3—122no—
317C2×C3×D3—122no—
318C2×C3×D3—124no—
319C32×C4—22no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—41no—
325C3⋊Dic3—22no—
326C3⋊Dic3—22no—
327C3⋊Dic3—22no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3—62no—
333C3×Dic3—122no—
334#126—272no—
335#124—272no—
336#123—272no—
337C22⋊Q8—91no—
338#121—91no—
339#121—272no—
340#120—272no—
341C4×D4—272no—
342C2×C22⋊C4—91no—
343H(3,𝔽3)#53811yescharacteristic, complemented, Sylow
344C23×C3—31no—
345C23×C3—31no—
346C23×C3—61no—
347C23×C3—92no—
348C23×C3—92no—
349C23×C3—182no—
350C23×C3—361no—
351C23×C3—361no—
352C23×C3C2×D3211yescharacteristic
353C22×D3—32no—
354C22×D3—32no—
355C22×D3—32no—
356C22×D3—62no—
357C22×D3—62no—
358C22×D3—92no—
359C22×D3—121no—
360C22×D3—121no—
361C22×D3—182no—
362C22×D3—361no—
363C22×D3—361no—
364C3×Q8—361no—
365C3×D4—184no—
366C3×D4—361no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—41no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—92no—
375C2×C3×C4—92no—
376C2×C3×C4—92no—
377C2×C3×C4—92no—
378C2×C3×C4—121no—
379C2×C3×C4—121no—
380C2×C3×C4—122no—
381C2×C3×C4—361no—
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384#67—64no—
385#67—64no—
386#67—124no—
387#67—362no—
388#67—362no—
389#67—362no—
390#67—362no—
391C2×Dic3—32no—
392C2×Dic3—32no—
393C2×Dic3—32no—
394C2×Dic3—32no—
395C2×Dic3—32no—
396C2×Dic3—62no—
397C2×Dic3—92no—
398C2×Dic3—92no—
399C2×Dic3—92no—
400C2×Dic3—121no—
401C2×Dic3—361no—
402C2×Dic3—362no—
403D12—121no—
404D12—121no—
405D12—122no—
406D12—362no—
407D12—362no—
408C4×D3—362no—
409C4×D3—362no—
410Dic6—121no—
411Dic6—361no—
412C2×C32—22no—
413C2×C32—22no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—42no—
418C2×C32C2×#23511yescharacteristic
419C2×C32#100411yescharacteristic
420C2×C32#100411yescharacteristic
421C2×C32#100411yescharacteristic
422C2×C32C2×#23911yescharacteristic
423C2×C32#100911yescharacteristic
424C3⋊D3—42no—
425C3×D3—122no—
426C3×D3—122no—
427C3×D3—122no—
428C3×D3—242no—
429C24—91no—
430C2×Q8—91no—
431C2×D4—92no—
432C2×D4—91no—
433C2×D4—272no—
434C2×D4—272no—
435C22×C4—91no—
436C22×C4—272no—
437C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
438C4⋊C4—91no—
439C4⋊C4—92no—
440C4⋊C4—91no—
441C4⋊C4—272no—
442C22⋊C4—92no—
443C22⋊C4—92no—
444C22⋊C4—92no—
445C22⋊C4—92no—
446C22⋊C4—272no—
447C22⋊C4—272no—
448C22⋊C4—272no—
449C42—272no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—31no—
453C22×C3—31no—
454C22×C3—32no—
455C22×C3—32no—
456C22×C3—61no—
457C22×C3—61no—
458C22×C3—61no—
459C22×C3—61no—
460C22×C3—61no—
461C22×C3—62no—
462C22×C3—121no—
463C22×C3—121no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3—182no—
470C22×C3—364no—
471C22×C3#379912yes—
472C22×C3C22×D3211yescharacteristic
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—62no—
479C2×D3—62no—
480C2×D3—62no—
481C2×D3—62no—
482C2×D3—122no—
483C2×D3—122no—
484C2×D3—122no—
485C2×D3—124no—
486C2×D3—182no—
487C2×D3—182no—
488C2×D3—182no—
489C2×D3—364no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—61no—
493C3×C4—61no—
494C3×C4—61no—
495C3×C4—61no—
496C3×C4—62no—
497C3×C4—121no—
498C3×C4—182no—
499C3×C4—182no—
500C3×C4—182no—
501C3×C4—182no—
502Dic3—62no—
503Dic3—62no—
504Dic3—62no—
505Dic3—62no—
506Dic3—62no—
507Dic3—122no—
508Dic3—182no—
509Dic3—182no—
510Dic3—182no—
511C32—21no—
512C32#549411yescharacteristic, complemented
513C32#564411yescharacteristic, complemented
514C23—92no—
515C23—92no—
516C23—182no—
517C23—361no—
518C23—361no—
519C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
520Q8—361no—
521D4—184no—
522D4—361no—
523D4—1082no—
524D4—1082no—
525C2×C4—41no—
526C2×C4—92no—
527C2×C4—92no—
528C2×C4—92no—
529C2×C4—92no—
530C2×C4—361no—
531C2×C4—1082no—
532C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
533C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
534C2×C3—22no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—62no—
542C2×C3—62no—
543C2×C3—61no—
544C2×C3—61no—
545C2×C3—61no—
546C2×C3—122no—
547C2×C3—362no—
548C2×C3—362no—
549C2×C3C2×#379911yescharacteristic
550C2×C3#6430711yescharacteristic
551C2×C3#6433011yescharacteristic
552D3—122no—
553D3—122no—
554D3—122no—
555D3—242no—
556D3—362no—
557C22—21no—
558C22—21no—
559C22—182no—
560C22—182no—
561C22—182no—
562C22—182no—
563C22—182no—
564C22—182no—
565C22—364no—
566C22#8865212yes—
567C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
568C4—21no—
569C4—21no—
570C4—182no—
571C4—182no—
572C4—182no—
573C4—182no—
574C3—31no—
575C3—31no—
576C3—61no—
577C3#1059503811yescharacteristic
578C2—22no—
579C2—362no—
580C2—362no—
581C2C2×#8865211yescharacteristic
582C2#1172249211yescharacteristic
583C2#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. #11721830C2semidirect
2. C2×#3752⋊C3C2semidirect
3. #64133⋊C3C2semidirect
4. #11721321C2semidirect
5. #11721293C2semidirect
6. #11721157C2semidirect
7. #11721155C2semidirect
8. #11721111C2semidirect
9. #11721102C2semidirect
15. C2×C4⋊#272C22semidirect
17. C2×#634⋊C3C22semidirect
20. C2×#6199C22semidirect
21. #88542C22semidirect
22. C4⋊#1084C22semidirect
26. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
27. #3750⋊C3C22semidirect
28. #88451C22semidirect
31. #88449C22semidirect
32. #88447C22semidirect
33. C2×#3752D3semidirect
34. #64133D3semidirect
81. C2×#634C2×D3semidirect
99. C4⋊(C3⋊Dic3)C2×D3semidirect
100. #3750C2×D3semidirect
343. H(3,𝔽3)#538semidirect
512. C32#5494semidirect
513. C32#5644semidirect

Maximal subgroups

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

subgroupindex
#117218302
C2×#3752⋊C32
#64133⋊C32
#117213212
#117212932
#117211572
#117211552
#117211112
#117211022
#105949793
#105949773