Finite Group Databasebeta

← #13156623

Subgroups — #13156623

Order 1728.

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

normal subgroups only

This group has 826 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×#88810C211yesmaximal, characteristic, complemented
2#11723439C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5C3×#63911C211yesmaximal, characteristic, complemented
6C2×C3×#3714C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10591469—31nomaximal
9#10590883—31nomaximal
10#10590371—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×C3×#626—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88810—22no—
20#88803—22no—
21#88796—22no—
22#88794—22no—
23C3×#3714—22no—
24C2×#3754—31no—
25C2×#3744—31no—
26#64084—31no—
27C3×#940—31no—
28C2×C3×#213—31no—
29C2×C3×D24—31no—
30C3×#874—31no—
31#63911D311yescharacteristic, complemented
32#63860—31no—
33#63860—31no—
34#63860—61no—
35C2×#3716—31no—
36C2×#3716—31no—
37C2×#3716—31no—
38C2×#3716—31no—
39C2×#3716—61no—
40C2×#3716—61no—
41C2×#3714—31no—
42C2×#3714—31no—
43C2×#3714D311yescharacteristic, complemented
44C2×#3713—31no—
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
52C2×C3⋊(C3⋊Dic3)—21no—
53#6308—21no—
54#6308—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C2×C32×Dic3D412yes—
60C32×D12—21no—
61C32×D12—21no—
62C3×#626—44no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65#5154—91no—
66#5074—91no—
67#4712—91no—
68C2×C32×D4—31no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×#634—31no—
72C2×#634—31no—
73C2×#634—31no—
74C2×#634—61no—
75C2×#634—61no—
76C2×#634—61no—
77C2×#634—121no—
78C2×C3×#67—62no—
79C2×C3×D12—21no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—61no—
85C2×C3×D12—61no—
86C2×C3×D12—62no—
87C2×C3×D12C2×D311yescharacteristic
88C2×C3×D12C2×D311yescharacteristic
89C2×#626—22no—
90C2×C32×C8—31no—
91C32×C42—31no—
92#3754—62no—
93#3752—62no—
94C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
95#3744—62no—
96C3×#213—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×D24—62no—
107(C2×D3)⋊Dic3—64no—
108(C2×D3)⋊Dic3—62no—
109(C2×D3)⋊Dic3—62no—
110#3716—64no—
111#3716—62no—
112#3716—62no—
113#3716—64no—
114#3716—62no—
115#3716—62no—
116#3714—22no—
117#3714—62no—
118#3714—62no—
119#3713—62no—
120C22×C33C2×D411yescharacteristic
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—42no—
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×D8—91no—
132C3×#127—91no—
133#940—31no—
134#936—91no—
135C2×#214—91no—
136C2×#214—91no—
137C2×#214—91no—
138C2×#214—181no—
139C2×#213—31no—
140C2×#213—91no—
141C2×#213—91no—
142C2×D24—31no—
143C2×D24—91no—
144C2×#204—91no—
145C2×#204—91no—
146C2×#204—91no—
147C2×#204—181no—
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—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—41no—
161C22×C3×D3—62no—
162C22×C3×D3—61no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C22×C3×D3—121no—
167C32×D4—61no—
168C32×D4—61no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—32no—
173C2×C32×C4—41no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3—61no—
178C2×C3⋊Dic3—62no—
179C2×C3⋊Dic3—61no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—121no—
182C2×C3⋊Dic3—121no—
183C2×C3⋊Dic3—122no—
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)—21no—
186C4⋊(C3⋊D3)—21no—
187C4⋊(C3⋊D3)—21no—
188#634—61no—
189#634—61no—
190#634—62no—
191#634—62no—
192#634—61no—
193#634—61no—
194#634—61no—
195#634—61no—
196#634—121no—
197#634—121no—
198#634—121no—
199#634—121no—
200#634—122no—
201#634—122no—
202C3×#67—124no—
203C2×C3×Dic3—22no—
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—61no—
218C3×D12—61no—
219C3×D12—61no—
220C3×D12—61no—
221C3×D12—121no—
222C3×D12—121no—
223C3×D12—121no—
224C3×D12—124no—
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×D4—31no—
241C2×C3×D4—31no—
242C2×C3×D4—61no—
243C2×C3×D4—91no—
244C2×C3×D4—91no—
245C2×C3×D4—182no—
246C2×#67—62no—
247C2×D12—31no—
248C2×D12—31no—
249C2×D12—31no—
250C2×D12—31no—
251C2×D12—61no—
252C2×D12—61no—
253C2×D12—62no—
254C2×D12D3211yescharacteristic
255C2×Dic6—91no—
256C2×Dic6—91no—
257C2×Dic6—91no—
258C2×Dic6—181no—
259C2×Dic6—181no—
260C2×Dic6—181no—
261C2×Dic6—361no—
262C3×D8—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#217—182no—
271#214—182no—
272#214—182no—
273#214—184no—
274#214—182no—
275#213—62no—
276#213—182no—
277#213—182no—
278(C2×D3)⋊C4—182no—
279(C2×D3)⋊C4—182no—
280(C2×D3)⋊C4—184no—
281C4×Dic3D3211yescharacteristic
282C2×C3⋊C8—31no—
283C2×C3⋊C8—31no—
284C2×C3⋊C8—61no—
285C2×C3⋊C8D3211yescharacteristic
286D24—62no—
287D24—182no—
288#204—182no—
289#204—182no—
290#204—184no—
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—42no—
302C2×C3⋊D3—41no—
303C2×C3⋊D3—41no—
304C2×C3⋊D3—42no—
305C2×C3×D3—42no—
306C2×C3×D3—42no—
307C2×C3×D3—42no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—82no—
311C2×C3×D3—122no—
312C2×C3×D3—122no—
313C2×C3×D3—124no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C2×C3×D3—122no—
317C2×C3×D3—122no—
318C2×C3×D3—122no—
319C2×C3×D3—122no—
320C2×C3×D3—121no—
321C2×C3×D3—121no—
322C2×C3×D3—242no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—41no—
330C32×C4—41no—
331C32×C4—62no—
332C32×C4C2×D1211yescharacteristic
333C32×C4C2×D1211yescharacteristic
334C32×C4D3×D411yescharacteristic
335C32×C4D3×D411yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4C2×#6711yescharacteristic
338C3⋊Dic3—122no—
339C3⋊Dic3—122no—
340C3⋊Dic3—122no—
341C3⋊Dic3—124no—
342C3⋊Dic3—121no—
343C3⋊Dic3—121no—
344C3⋊Dic3—121no—
345C3×Dic3—22no—
346C3×Dic3—22no—
347C3×Dic3—42no—
348C2×SD16—271no—
349C2×SD16—271no—
350C2×D8—91no—
351C2×D8—271no—
352#127—91no—
353#124—271no—
354#97—91no—
355C33#49611yescharacteristic, complemented, Sylow
356C23×C3—61no—
357C23×C3—61no—
358C23×C3—121no—
359C23×C3—181no—
360C23×C3—181no—
361C22×D3—21no—
362C22×D3—62no—
363C22×D3—61no—
364C22×D3—61no—
365C22×D3—61no—
366C22×D3—61no—
367C22×D3—121no—
368C3×D4—62no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—61no—
372C3×D4—61no—
373C3×D4—121no—
374C3×D4—181no—
375C3×D4—181no—
376C3×D4—181no—
377C3×D4—181no—
378C3×D4—364no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—32no—
383C2×C3×C4—32no—
384C2×C3×C4—41no—
385C2×C3×C4—62no—
386C2×C3×C4C2×D3211yescharacteristic
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3211yescharacteristic
389#67—124no—
390C2×Dic3—182no—
391C2×Dic3—181no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—361no—
395C2×Dic3—361no—
396C2×Dic3—362no—
397C2×Dic3#62612yes—
398D12—21no—
399D12—21no—
400D12—62no—
401D12—61no—
402D12—61no—
403D12—61no—
404D12—61no—
405D12—61no—
406D12—61no—
407D12—61no—
408D12—61no—
409D12—121no—
410D12—121no—
411D12—121no—
412D12—124no—
413Dic6—182no—
414Dic6—182no—
415Dic6—181no—
416Dic6—181no—
417Dic6—181no—
418Dic6—181no—
419Dic6—181no—
420Dic6—181no—
421Dic6—361no—
422Dic6—361no—
423Dic6—361no—
424Dic6—361no—
425Dic6—362no—
426Dic6—362no—
427C3×C8—32no—
428C3×C8—32no—
429C3×C8—62no—
430C3×C8—92no—
431C3⋊C8—32no—
432C3⋊C8—32no—
433C3⋊C8—62no—
434C3⋊C8#62612yescomplemented
435C2×C32—22no—
436C2×C32—22no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—22no—
440C2×C32—21no—
441C2×C32—41no—
442C2×C32—42no—
443C2×C32—241no—
444C2×C32#87411yescharacteristic
445C2×C32#90812yes—
446C2×C32#91112yes—
447C2×C32#93212yes—
448C2×C32#94011yescharacteristic
449C2×C32#94011yescharacteristic
450C3⋊D3—81no—
451C3⋊D3—81no—
452C3×D3—82no—
453C3×D3—81no—
454C3×D3—81no—
455C3×D3—242no—
456C3×D3—242no—
457C3×D3—241no—
458C3×D3—241no—
459C3×D3—241no—
460C3×D3—241no—
461C2×Q8—271no—
462C2×D4—31no—
463C2×D4—91no—
464C2×D4—91no—
465C2×D4—182no—
466SD16—542no—
467SD16—542no—
468D8—182no—
469D8—542no—
470C2×C8—31no—
471C2×C8—91no—
472C22⋊C4—542no—
473C42—31no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—41no—
478C22×C3—122no—
479C22×C3—122no—
480C22×C3—122no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3—242no—
484C22×C3—362no—
485C22×C3—362no—
486C22×C3—361no—
487C22×C3—361no—
488C22×C3D3×D1211yescharacteristic
489C22×C3C2×#62611yescharacteristic
490C22×C3D3×#6711yescharacteristic
491C2×D3—42no—
492C2×D3—41no—
493C2×D3—122no—
494C2×D3—122no—
495C2×D3—124no—
496C2×D3—122no—
497C2×D3—122no—
498C2×D3—121no—
499C2×D3—121no—
500C2×D3—122no—
501C2×D3—122no—
502C2×D3—121no—
503C2×D3—121no—
504C2×D3—242no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—62no—
514C3×C4—62no—
515C3×C4—122no—
516C3×C4D3×D1211yescharacteristic
517C3×C4D3×D1211yescharacteristic
518C3×C4D3×D1211yescharacteristic
519C3×C4C2×#62611yescharacteristic
520C3×C4D3×#6711yescharacteristic
521C3×C4#381111yescharacteristic
522Dic3—22no—
523Dic3—362no—
524Dic3—362no—
525Dic3—362no—
526Dic3—364no—
527Dic3—361no—
528Dic3—361no—
529Dic3—361no—
530C32—21no—
531C32—21no—
532C32—21no—
533C32—41no—
534C32#471211yescharacteristic, complemented
535C32#507411yescharacteristic, complemented
536C32#515411yescharacteristic, complemented
537C23—61no—
538C23—181no—
539C23—181no—
540Q8—541no—
541Q8—541no—
542D4—61no—
543D4—61no—
544D4—181no—
545D4—181no—
546D4—181no—
547D4—181no—
548D4—364no—
549C2×C4—32no—
550C2×C4—541no—
551C2×C4D3311yescharacteristic
552C8—32no—
553C8—92no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—22no—
558C2×C3—22no—
559C2×C3—22no—
560C2×C3—41no—
561C2×C3—42no—
562C2×C3—242no—
563C2×C3—241no—
564C2×C3—241no—
565C2×C3—721no—
566C2×C3—721no—
567C2×C3#6379512yes—
568C2×C3#6383012yes—
569C2×C3#6391111yescharacteristic
570C2×C3#6391111yescharacteristic
571C2×C3#6393212yes—
572C2×C3#6397911yescharacteristic
573D3—81no—
574D3—242no—
575D3—242no—
576D3—241no—
577D3—241no—
578D3—241no—
579D3—241no—
580C22—121no—
581C22—122no—
582C22—361no—
583C22—361no—
584C22—362no—
585C22—362no—
586C22D3×#62611yescharacteristic
587C4—62no—
588C4—1081no—
589C4D3×#62611yescharacteristic
590C4#8895511yescharacteristic
591C3—21no—
592C3—21no—
593C3—21no—
594C3—41no—
595C3#1059037111yescharacteristic, complemented
596C3#1059088311yescharacteristic, complemented
597C3#1059146911yescharacteristic, complemented
598C2—241no—
599C2—721no—
600C2—721no—
601C2#1172286912yes—
602C2#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×#88810C2semidirect
2. #11723439C2semidirect
3. C2×#88796C2semidirect
4. C2×#88794C2semidirect
5. C3×#63911C2semidirect
6. C2×C3×#3714C2semidirect
7. C3×#63554C2semidirect
31. #63911D3semidirect
43. C2×#3714D3semidirect
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
355. C33#496semidirect
434. C3⋊C8#626semidirect
534. C32#4712semidirect
535. C32#5074semidirect
536. C32#5154semidirect
595. C3#10590371semidirect
596. C3#10590883semidirect
597. C3#10591469semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
#117234392
C2×#887962
C2×#887942
C3×#639112
C2×C3×#37142
C3×#635542
#105914693
#105908833
#105903713