Finite Group Databasebeta

← #13156622

Subgroups — #13156622

Order 1728.

858 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic, complemented
2#11723441C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5C3×#63911C211yesmaximal, characteristic, complemented
6C2×C3×#3716C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10591472—31nomaximal
9#10590881—31nomaximal
10#10590371—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×C3×#626—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88811—22no—
20#88804—22no—
21#88791—22no—
22#88790—22no—
23C3×#3716—22no—
24C2×#3753—31no—
25C2×#3745—31no—
26#64084—31no—
27C3×#940—31no—
28C2×C3×#214—31no—
29C2×C3×#204—31no—
30C3×#874—31no—
31#63911D311yescharacteristic, complemented
32#63864—31no—
33#63864—31no—
34#63864—61no—
35C2×#3717—31no—
36C2×#3716—31no—
37C2×#3716D311yescharacteristic, complemented
38C2×#3715—31no—
39C2×#3714—31no—
40C2×#3714—31no—
41C2×#3714—31no—
42C2×#3714—31no—
43C2×#3714—61no—
44C2×#3714—61no—
45C3×#815—31no—
46C3×#803—31no—
47#63554D311yescharacteristic, complemented
48C22×C3⋊(C3⋊D3)—21no—
49C22×C3×C3⋊D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C4⋊(C3⋊(C3⋊D3))—21no—
53C4⋊(C3⋊(C3⋊D3))—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×#634—21no—
58C3×#634—21no—
59C2×C32×Dic3D412yes—
60C32×D12—21no—
61C32×D12—21no—
62C3×#626—44no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65#5176—91no—
66#5074—91no—
67#4712—91no—
68C2×C32×D4—31no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C4⋊(C3⋊D3)—121no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×#634C2×D311yescharacteristic
78C2×C3×#67—62no—
79C2×C3×D12—21no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×D12—62no—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×D12C2×D311yescharacteristic
86C2×C3×Dic6—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—61no—
89C2×#626—22no—
90C2×C32×C8—31no—
91C32×C42—31no—
92#3753—62no—
93#3752—62no—
94C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
95#3745—62no—
96C3×#214—62no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C2×D311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C3×#204—62no—
107#3722—64no—
108#3722—62no—
109#3722—62no—
110#3717—62no—
111#3716—22no—
112#3716—62no—
113#3715—62no—
114#3714—64no—
115#3714—62no—
116#3714—62no—
117#3714—64no—
118#3714—62no—
119#3714—62no—
120C22×C33C2×D411yescharacteristic
121C2×C3⋊(C3⋊D3)—41no—
122C2×C3⋊(C3⋊D3)—42no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—42no—
125C2×C32×D3—41no—
126C2×C32×D3—42no—
127C33×C4C2×D411yescharacteristic
128C33×C4C2×D411yescharacteristic
129C3×C3⋊Dic3—41no—
130C32×Dic3—22no—
131C2×C3×SD16—91no—
132C3×#127—91no—
133#947—91no—
134C2×#215—91no—
135#940—31no—
136C2×#214—31no—
137C2×#214—91no—
138C2×#213—91no—
139C2×#213—91no—
140C2×#213—91no—
141C2×#213—181no—
142C2×D24—91no—
143C2×D24—91no—
144C2×D24—91no—
145C2×D24—181no—
146C2×#204—31no—
147C2×#204—91no—
148#875—91no—
149#875—91no—
150#875—181no—
151#874—31no—
152C3×#97—91no—
153#815—31no—
154#803—31no—
155C23×C32—61no—
156C22×C3⋊D3—21no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—62no—
159C22×C3⋊D3—61no—
160C22×C3⋊D3—61no—
161C22×C3⋊D3—121no—
162C22×C3⋊D3—121no—
163C22×C3⋊D3—122no—
164C22×C3×D3—21no—
165C22×C3×D3—21no—
166C22×C3×D3—41no—
167C22×C3×D3—61no—
168C22×C3×D3—61no—
169C22×C3×D3—121no—
170C32×D4—61no—
171C32×D4—61no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—32no—
176C2×C32×C4—41no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3—21no—
181C4⋊(C3⋊D3)—21no—
182C4⋊(C3⋊D3)—21no—
183C4⋊(C3⋊D3)—61no—
184C4⋊(C3⋊D3)—61no—
185C4⋊(C3⋊D3)—62no—
186C4⋊(C3⋊D3)—62no—
187C4⋊(C3⋊D3)—61no—
188C4⋊(C3⋊D3)—61no—
189C4⋊(C3⋊D3)—61no—
190C4⋊(C3⋊D3)—61no—
191C4⋊(C3⋊D3)—121no—
192C4⋊(C3⋊D3)—121no—
193C4⋊(C3⋊D3)—121no—
194C4⋊(C3⋊D3)—121no—
195C4⋊(C3⋊D3)—122no—
196C4⋊(C3⋊D3)—122no—
197#634—21no—
198#634—21no—
199C3×#67—124no—
200C2×C3×Dic3—22no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3D1212yes—
205C2×C3×Dic3#6712yes—
206C3×D12—22no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—41no—
212C3×D12—62no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—121no—
218C3×D12—124no—
219C3×Dic6—61no—
220C3×Dic6—61no—
221C3×Dic6—61no—
222C3×Dic6—61no—
223C3×Dic6—121no—
224C3×Dic6—121no—
225#626—44no—
226C32×C8—32no—
227C3⋊(C3⋊C8)D1212yescomplemented
228C3×C3⋊C8—22no—
229C3×C3⋊C8—32no—
230C3×C3⋊C8—32no—
231C3×C3⋊C8—62no—
232C3×C3⋊C8D1212yescomplemented
233C3×C3⋊C8#6712yescomplemented
234#496—271noSylow
235C2×C33#12711yescharacteristic
236C2×C33AGL(1,ℤ/8)12yes—
237C3⋊(C3⋊D3)—81no—
238C3×C3⋊D3—81no—
239C32×D3—81no—
240C2×C3×Q8—91no—
241C2×C3×D4—31no—
242C2×C3×D4—31no—
243C2×C3×D4—61no—
244C2×C3×D4—91no—
245C2×C3×D4—182no—
246C2×#67—62no—
247C2×D12—31no—
248C2×D12—31no—
249C2×D12—61no—
250C2×D12—62no—
251C2×D12—91no—
252C2×D12—91no—
253C2×D12—91no—
254C2×D12—181no—
255C2×D12—181no—
256C2×D12—181no—
257C2×D12—361no—
258C2×D12D3211yescharacteristic
259C2×Dic6—31no—
260C2×Dic6—31no—
261C2×Dic6—61no—
262C3×SD16—182no—
263C2×C3×C8—31no—
264C2×C3×C8—31no—
265C2×C3×C8—61no—
266C2×C3×C8—91no—
267C3×C42—31no—
268C3×C42—31no—
269C3×C42—61no—
270#215—182no—
271#214—62no—
272#214—182no—
273#213—182no—
274#213—182no—
275#213—184no—
276#213—182no—
277(C2×D3)⋊C4—182no—
278(C2×D3)⋊C4—182no—
279(C2×D3)⋊C4—184no—
280(C2×D3)⋊C4—182no—
281C4×Dic3D3211yescharacteristic
282C2×C3⋊C8—31no—
283C2×C3⋊C8—31no—
284C2×C3⋊C8—61no—
285C2×C3⋊C8D3211yescharacteristic
286D24—182no—
287D24—182no—
288D24—184no—
289D24—182no—
290#204—62no—
291#204—182no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—41no—
296C22×C32—121no—
297C22×C32—122no—
298C22×C32C2×D1211yescharacteristic
299C22×C32D3×D411yescharacteristic
300C22×C32C2×#6711yescharacteristic
301C2×C3⋊D3—41no—
302C2×C3⋊D3—42no—
303C2×C3⋊D3—122no—
304C2×C3⋊D3—122no—
305C2×C3⋊D3—122no—
306C2×C3⋊D3—124no—
307C2×C3⋊D3—121no—
308C2×C3⋊D3—122no—
309C2×C3⋊D3—121no—
310C2×C3⋊D3—121no—
311C2×C3⋊D3—124no—
312C2×C3⋊D3—122no—
313C2×C3⋊D3—122no—
314C2×C3⋊D3—242no—
315C2×C3⋊D3—242no—
316C2×C3⋊D3—244no—
317C2×C3×D3—42no—
318C2×C3×D3—42no—
319C2×C3×D3—42no—
320C2×C3×D3—41no—
321C2×C3×D3—41no—
322C2×C3×D3—82no—
323C2×C3×D3—122no—
324C2×C3×D3—121no—
325C2×C3×D3—122no—
326C2×C3×D3—122no—
327C2×C3×D3—121no—
328C2×C3×D3—242no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—41no—
336C32×C4—41no—
337C32×C4—62no—
338C32×C4C2×D1211yescharacteristic
339C32×C4C2×D1211yescharacteristic
340C32×C4D3×D411yescharacteristic
341C32×C4D3×D411yescharacteristic
342C32×C4D3×D411yescharacteristic
343C32×C4C2×#6711yescharacteristic
344C3⋊Dic3—41no—
345C3×Dic3—22no—
346C3×Dic3—22no—
347C3×Dic3—42no—
348C3×Dic3—122no—
349C3×Dic3—121no—
350C3×Dic3—121no—
351C2×SD16—91no—
352C2×SD16—271no—
353C2×D8—271no—
354C2×D8—271no—
355#127—91no—
356#124—271no—
357#97—91no—
358C33#49611yescharacteristic, complemented, Sylow
359C23×C3—61no—
360C23×C3—61no—
361C23×C3—121no—
362C23×C3—181no—
363C22×D3—21no—
364C22×D3—61no—
365C22×D3—61no—
366C22×D3—121no—
367C22×D3—182no—
368C22×D3—181no—
369C22×D3—181no—
370C22×D3—181no—
371C22×D3—361no—
372C22×D3—361no—
373C22×D3—362no—
374C3×Q8—181no—
375C3×Q8—181no—
376C3×D4—62no—
377C3×D4—61no—
378C3×D4—61no—
379C3×D4—61no—
380C3×D4—61no—
381C3×D4—121no—
382C3×D4—181no—
383C3×D4—181no—
384C3×D4—364no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—32no—
389C2×C3×C4—32no—
390C2×C3×C4—41no—
391C2×C3×C4—62no—
392C2×C3×C4—181no—
393C2×C3×C4C2×D3211yescharacteristic
394C2×C3×C4C2×D3211yescharacteristic
395C2×C3×C4C2×D3211yescharacteristic
396#67—124no—
397C2×Dic3—62no—
398C2×Dic3—61no—
399C2×Dic3—61no—
400C2×Dic3#62612yes—
401D12—21no—
402D12—21no—
403D12—62no—
404D12—61no—
405D12—61no—
406D12—61no—
407D12—61no—
408D12—121no—
409D12—124no—
410D12—182no—
411D12—182no—
412D12—181no—
413D12—181no—
414D12—181no—
415D12—181no—
416D12—181no—
417D12—181no—
418D12—361no—
419D12—361no—
420D12—361no—
421D12—361no—
422D12—362no—
423D12—362no—
424Dic6—61no—
425Dic6—61no—
426Dic6—61no—
427Dic6—61no—
428Dic6—121no—
429Dic6—121no—
430C3×C8—32no—
431C3×C8—32no—
432C3×C8—62no—
433C3×C8—92no—
434C3⋊C8—32no—
435C3⋊C8—32no—
436C3⋊C8—62no—
437C3⋊C8#62612yescomplemented
438C2×C32—22no—
439C2×C32—22no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—22no—
443C2×C32—21no—
444C2×C32—41no—
445C2×C32—42no—
446C2×C32—241no—
447C2×C32#87411yescharacteristic
448C2×C32#90812yes—
449C2×C32#91412yes—
450C2×C32#93212yes—
451C2×C32#94011yescharacteristic
452C2×C32#94011yescharacteristic
453C3⋊D3—81no—
454C3⋊D3—242no—
455C3⋊D3—242no—
456C3⋊D3—242no—
457C3⋊D3—244no—
458C3⋊D3—241no—
459C3⋊D3—241no—
460C3⋊D3—241no—
461C3×D3—82no—
462C3×D3—81no—
463C3×D3—81no—
464C3×D3—242no—
465C3×D3—241no—
466C3×D3—241no—
467C2×Q8—91no—
468C2×D4—31no—
469C2×D4—91no—
470C2×D4—182no—
471C2×D4—271no—
472SD16—182no—
473SD16—542no—
474D8—542no—
475D8—542no—
476C2×C8—31no—
477C2×C8—91no—
478C22⋊C4—542no—
479C42—31no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—41no—
484C22×C3—122no—
485C22×C3—122no—
486C22×C3—122no—
487C22×C3—121no—
488C22×C3—121no—
489C22×C3—242no—
490C22×C3—362no—
491C22×C3—361no—
492C22×C3D3×D1211yescharacteristic
493C22×C3C2×#62611yescharacteristic
494C22×C3D3×#6711yescharacteristic
495C2×D3—42no—
496C2×D3—41no—
497C2×D3—122no—
498C2×D3—122no—
499C2×D3—121no—
500C2×D3—122no—
501C2×D3—121no—
502C2×D3—242no—
503C2×D3—362no—
504C2×D3—362no—
505C2×D3—362no—
506C2×D3—364no—
507C2×D3—364no—
508C2×D3—362no—
509C2×D3—362no—
510C2×D3—361no—
511C2×D3—361no—
512C2×D3—362no—
513C2×D3—361no—
514C2×D3—722no—
515C2×D3—722no—
516C2×D3—724no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—41no—
524C3×C4—41no—
525C3×C4—62no—
526C3×C4—62no—
527C3×C4—122no—
528C3×C4—361no—
529C3×C4D3×D1211yescharacteristic
530C3×C4D3×D1211yescharacteristic
531C3×C4D3×D1211yescharacteristic
532C3×C4C2×#62611yescharacteristic
533C3×C4D3×#6711yescharacteristic
534C3×C4#381111yescharacteristic
535Dic3—22no—
536Dic3—122no—
537Dic3—121no—
538Dic3—121no—
539C32—21no—
540C32—21no—
541C32—21no—
542C32—41no—
543C32#471211yescharacteristic, complemented
544C32#507411yescharacteristic, complemented
545C32#517611yescharacteristic, complemented
546C23—61no—
547C23—181no—
548C23—541no—
549Q8—181no—
550Q8—181no—
551D4—61no—
552D4—61no—
553D4—181no—
554D4—181no—
555D4—364no—
556D4—541no—
557D4—541no—
558C2×C4—32no—
559C2×C4—181no—
560C2×C4D3311yescharacteristic
561C8—32no—
562C8—92no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—22no—
567C2×C3—22no—
568C2×C3—22no—
569C2×C3—41no—
570C2×C3—42no—
571C2×C3—242no—
572C2×C3—241no—
573C2×C3—241no—
574C2×C3—721no—
575C2×C3#6379412yes—
576C2×C3#6383012yes—
577C2×C3#6391111yescharacteristic
578C2×C3#6391111yescharacteristic
579C2×C3#6393412yes—
580C2×C3#6397911yescharacteristic
581D3—81no—
582D3—242no—
583D3—241no—
584D3—241no—
585D3—722no—
586D3—722no—
587D3—722no—
588D3—724no—
589D3—721no—
590D3—721no—
591D3—721no—
592C22—121no—
593C22—122no—
594C22—361no—
595C22—362no—
596C22—1081no—
597C22—1082no—
598C22D3×#62611yescharacteristic
599C4—62no—
600C4—361no—
601C4D3×#62611yescharacteristic
602C4#8895511yescharacteristic
603C3—21no—
604C3—21no—
605C3—21no—
606C3—41no—
607C3#1059037111yescharacteristic, complemented
608C3#1059088111yescharacteristic, complemented
609C3#1059147211yescharacteristic, complemented
610C2—241no—
611C2—721no—
612C2—2161no—
613C2#1172286812yes—
614C2#1172303011yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#496
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88811C2semidirect
2. #11723441C2semidirect
3. C2×#88791C2semidirect
4. C2×#88790C2semidirect
5. C3×#63911C2semidirect
6. C2×C3×#3716C2semidirect
7. C3×#63554C2semidirect
31. #63911D3semidirect
37. C2×#3716D3semidirect
47. #63554D3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
64. C32×C3⋊C8D4semidirect
227. C3⋊(C3⋊C8)D12semidirect
232. C3×C3⋊C8D12semidirect
233. C3×C3⋊C8#67semidirect
358. C33#496semidirect
437. C3⋊C8#626semidirect
543. C32#4712semidirect
544. C32#5074semidirect
545. C32#5176semidirect
607. C3#10590371semidirect
608. C3#10590881semidirect
609. C3#10591472semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117234412
C2×#887912
C2×#887902
C3×#639112
C2×C3×#37162
C3×#635542
#105914723
#105908813
#105903713