Finite Group Databasebeta

← #13156616

Subgroups — #13156616

Order 1728.

782 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 108 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
2C2×#88792C211yesmaximal, characteristic, complemented
3C2×#88789C211yesmaximal, characteristic, complemented
4C3×#63911C211yesmaximal, characteristic, complemented
5C3×#63860C211yesmaximal, characteristic, complemented
6C2×C3×#3713C211yesmaximal, characteristic, complemented
7#11722019C211yesmaximal, characteristic, complemented
8#10591483—31nomaximal
9#10591469—31nomaximal
10#10590354—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×C3×#626—22no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19#88811—22no—
20#88792—22no—
21#88789—22no—
22C3×(C2×D3)⋊Dic3—22no—
23C3×#3713—22no—
24C2×#3754—31no—
25C2×#3753—31no—
26C3×#940—31no—
27C3×#936—31no—
28C2×C3×#213—31no—
29C2×C3×#213—31no—
30C3×#875—31no—
31C3×#874—31no—
32#63911D311yescharacteristic, complemented
33#63860D311yescharacteristic, complemented
34C2×#3717—31no—
35C2×#3716—31no—
36C2×#3715—31no—
37C2×#3715—31no—
38C2×#3715—31no—
39C2×#3715—61no—
40C2×#3713—31no—
41C2×#3713—31no—
42C2×#3713—61no—
43C2×#3713D311yescharacteristic, complemented
44#63630—31no—
45#63554—31no—
46#63554—31no—
47#63554—61no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×#634—21no—
56C3×#634—21no—
57C2×C32×Dic3D412yes—
58C32×D12—21no—
59C32×D12—21no—
60C32×D12—21no—
61C32×D12—21no—
62C3×#626—44no—
63C3⋊(C3⋊(C3⋊C8))D412yescomplemented
64C3×C3⋊(C3⋊C8)D412yescomplemented
65#5154—91no—
66#5074—91no—
67#5058—91no—
68C2×C32×D4—31no—
69C2×C32×D4—31no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×#634C2×D311yescharacteristic
72C2×C3×#67—62no—
73C2×C3×D12—21no—
74C2×C3×D12—21no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×D12—62no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C2×#626—22no—
87C32×C42—31no—
88#3754—62no—
89#3753—62no—
90C2×C3⋊(C3⋊C8)—31no—
91C2×C3⋊(C3⋊C8)—31no—
92C2×C3⋊(C3⋊C8)—31no—
93C2×C3⋊(C3⋊C8)—61no—
94C2×C3⋊(C3⋊C8)—61no—
95C2×C3⋊(C3⋊C8)—61no—
96C2×C3⋊(C3⋊C8)—121no—
97C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
98C3×#217—62no—
99C3×#213—62no—
100C3×#213—62no—
101C3×(C2×D3)⋊C4—62no—
102C3×C4×Dic3—21no—
103C3×C4×Dic3C2×D311yescharacteristic
104C3×C4×Dic3C2×D311yescharacteristic
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108(C2×D3)⋊Dic3—22no—
109#3717—62no—
110#3716—62no—
111#3715—62no—
112#3715—62no—
113#3715—64no—
114#3715—62no—
115#3713—22no—
116#3713—62no—
117#3713—62no—
118#3713—64no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—42no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3—42no—
125C2×C32×D3—42no—
126C33×C4C2×D411yescharacteristic
127C33×C4C2×D411yescharacteristic
128C3×C3⋊Dic3—41no—
129C32×Dic3—22no—
130C2×C3×D8—91no—
131C3×#127—91no—
132C3×#124—91no—
133C2×#215—91no—
134C2×#215—91no—
135#940—31no—
136#936—31no—
137C2×#214—91no—
138C2×#214—91no—
139C2×#214—91no—
140C2×#214—181no—
141C2×#213—31no—
142C2×#213—31no—
143C2×#213—91no—
144C2×#213—91no—
145C2×#213—91no—
146C2×#213—181no—
147C2×#204—91no—
148#875—31no—
149#874—31no—
150#815—91no—
151#803—91no—
152#803—91no—
153#803—181no—
154C23×C32—61no—
155C23×C32—61no—
156C22×C3⋊D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—21no—
161C22×C3×D3—41no—
162C22×C3×D3—41no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—121no—
166C32×D4—61no—
167C32×D4—61no—
168C32×D4—61no—
169C32×D4—61no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—32no—
174C2×C32×C4—41no—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—21no—
179C4⋊(C3⋊D3)—21no—
180C4⋊(C3⋊D3)—21no—
181#634—21no—
182#634—21no—
183C3×#67—124no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—121no—
188C2×C3×Dic3#6712yes—
189C2×C3×Dic3#6712yes—
190C3×D12—22no—
191C3×D12—22no—
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—21no—
195C3×D12—21no—
196C3×D12—21no—
197C3×D12—21no—
198C3×D12—21no—
199C3×D12—21no—
200C3×D12—41no—
201C3×D12—41no—
202C3×D12—62no—
203C3×D12—61no—
204C3×D12—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—121no—
208C3×D12—124no—
209C3×Dic6—62no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—121no—
215#626—44no—
216C3⋊(C3⋊C8)—32no—
217C3⋊(C3⋊C8)—32no—
218C3⋊(C3⋊C8)—32no—
219C3⋊(C3⋊C8)—62no—
220C3⋊(C3⋊C8)—62no—
221C3⋊(C3⋊C8)—62no—
222C3⋊(C3⋊C8)—122no—
223C3⋊(C3⋊C8)#6712yescomplemented
224C3×C3⋊C8—32no—
225C3×C3⋊C8—32no—
226C3×C3⋊C8—62no—
227#496—271noSylow
228C2×C33#12711yescharacteristic
229C2×C33AGL(1,ℤ/8)12yes—
230C3×C3⋊D3—81no—
231C32×D3—81no—
232C32×D3—81no—
233C2×C3×Q8—91no—
234C2×C3×D4—31no—
235C2×C3×D4—31no—
236C2×C3×D4—31no—
237C2×C3×D4—31no—
238C2×C3×D4—61no—
239C2×C3×D4—61no—
240C2×C3×D4—91no—
241C2×C3×D4—182no—
242C2×#67—62no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—61no—
246C2×D12—62no—
247C2×D12D3211yescharacteristic
248C2×D12D3211yescharacteristic
249C2×Dic6—31no—
250C2×Dic6—31no—
251C2×Dic6—61no—
252C3×D8—182no—
253C2×C3×C8—91no—
254C3×C22⋊C4—182no—
255C3×C42—31no—
256C3×C42—31no—
257C3×C42—61no—
258#217—62no—
259#215—182no—
260#215—182no—
261#214—182no—
262#214—182no—
263#214—182no—
264#214—184no—
265#213—62no—
266#213—62no—
267#213—182no—
268#213—182no—
269#213—182no—
270#213—184no—
271(C2×D3)⋊C4—62no—
272C4×Dic3D3211yescharacteristic
273C2×C3⋊C8—31no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—61no—
276C2×C3⋊C8—91no—
277C2×C3⋊C8—91no—
278C2×C3⋊C8—91no—
279C2×C3⋊C8—181no—
280C2×C3⋊C8—181no—
281C2×C3⋊C8—181no—
282C2×C3⋊C8—361no—
283#204—182no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—41no—
288C22×C32—121no—
289C22×C32—121no—
290C22×C32—122no—
291C22×C32—122no—
292C22×C32D3×D411yescharacteristic
293C22×C32C2×#6711yescharacteristic
294C22×C32C2×#6711yescharacteristic
295C2×C3⋊D3—41no—
296C2×C3⋊D3—42no—
297C2×C3×D3—42no—
298C2×C3×D3—42no—
299C2×C3×D3—42no—
300C2×C3×D3—42no—
301C2×C3×D3—42no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3—42no—
307C2×C3×D3—82no—
308C2×C3×D3—82no—
309C2×C3×D3—122no—
310C2×C3×D3—122no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C2×C3×D3—122no—
314C2×C3×D3—242no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—62no—
324C32×C4D3×D411yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4D3×D411yescharacteristic
327C32×C4D3×D411yescharacteristic
328C32×C4D3×D411yescharacteristic
329C32×C4C2×#6711yescharacteristic
330C3⋊Dic3—41no—
331C3×Dic3—22no—
332C3×Dic3—22no—
333C3×Dic3—42no—
334C3×Dic3—122no—
335C3×Dic3—121no—
336C3×Dic3—121no—
337C2×SD16—271no—
338C2×SD16—271no—
339C2×D8—91no—
340C2×D8—271no—
341#127—91no—
342#124—91no—
343#97—271no—
344C33#49611yescharacteristic, complemented, Sylow
345C23×C3—61no—
346C23×C3—61no—
347C23×C3—61no—
348C23×C3—61no—
349C23×C3—121no—
350C23×C3—121no—
351C23×C3—181no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—61no—
355C22×D3—61no—
356C22×D3—121no—
357C3×Q8—181no—
358C3×Q8—181no—
359C3×D4—62no—
360C3×D4—62no—
361C3×D4—61no—
362C3×D4—61no—
363C3×D4—61no—
364C3×D4—61no—
365C3×D4—61no—
366C3×D4—61no—
367C3×D4—61no—
368C3×D4—61no—
369C3×D4—121no—
370C3×D4—121no—
371C3×D4—181no—
372C3×D4—181no—
373C3×D4—364no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—32no—
378C2×C3×C4—32no—
379C2×C3×C4—41no—
380C2×C3×C4—62no—
381C2×C3×C4—181no—
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385#67—124no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—121no—
389C2×Dic3#62512yes—
390D12—21no—
391D12—21no—
392D12—21no—
393D12—21no—
394D12—62no—
395D12—61no—
396D12—61no—
397D12—61no—
398D12—61no—
399D12—121no—
400D12—124no—
401Dic6—62no—
402Dic6—61no—
403Dic6—61no—
404Dic6—61no—
405Dic6—61no—
406Dic6—121no—
407C3×C8—92no—
408C3⋊C8—32no—
409C3⋊C8—32no—
410C3⋊C8—62no—
411C3⋊C8—92no—
412C3⋊C8—92no—
413C3⋊C8—92no—
414C3⋊C8—182no—
415C3⋊C8—182no—
416C3⋊C8—182no—
417C3⋊C8—362no—
418C2×C32—22no—
419C2×C32—21no—
420C2×C32—22no—
421C2×C32—22no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—41no—
425C2×C32—42no—
426C2×C32—241no—
427C2×C32—241no—
428C2×C32#91112yes—
429C2×C32#93212yes—
430C2×C32#94011yescharacteristic
431C2×C32#94011yescharacteristic
432C2×C32#94011yescharacteristic
433C2×C32#94912yes—
434C3⋊D3—81no—
435C3×D3—82no—
436C3×D3—82no—
437C3×D3—81no—
438C3×D3—81no—
439C3×D3—81no—
440C3×D3—81no—
441C3×D3—242no—
442C3×D3—241no—
443C3×D3—241no—
444C2×Q8—91no—
445C2×D4—31no—
446C2×D4—31no—
447C2×D4—91no—
448C2×D4—182no—
449SD16—542no—
450SD16—542no—
451D8—182no—
452D8—542no—
453C2×C8—91no—
454C2×C8—271no—
455C22⋊C4—182no—
456C42—31no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—41no—
461C22×C3—122no—
462C22×C3—122no—
463C22×C3—122no—
464C22×C3—122no—
465C22×C3—121no—
466C22×C3—121no—
467C22×C3—122no—
468C22×C3—122no—
469C22×C3—121no—
470C22×C3—121no—
471C22×C3—242no—
472C22×C3—242no—
473C22×C3—362no—
474C22×C3—361no—
475C22×C3C2×#62511yescharacteristic
476C22×C3D3×#6711yescharacteristic
477C22×C3D3×#6711yescharacteristic
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—41no—
481C2×D3—41no—
482C2×D3—122no—
483C2×D3—122no—
484C2×D3—121no—
485C2×D3—122no—
486C2×D3—121no—
487C2×D3—242no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—62no—
497C3×C4—62no—
498C3×C4—122no—
499C3×C4—361no—
500C3×C4#380211yescharacteristic
501C3×C4#380211yescharacteristic
502C3×C4D3×#6711yescharacteristic
503C3×C4D3×#6711yescharacteristic
504C3×C4#381111yescharacteristic
505C3×C4#381111yescharacteristic
506Dic3—22no—
507Dic3—122no—
508Dic3—121no—
509Dic3—121no—
510C32—21no—
511C32—21no—
512C32—21no—
513C32—41no—
514C32#505811yescharacteristic, complemented
515C32#507411yescharacteristic, complemented
516C32#515411yescharacteristic, complemented
517C23—61no—
518C23—61no—
519C23—181no—
520Q8—181no—
521Q8—181no—
522D4—61no—
523D4—61no—
524D4—61no—
525D4—61no—
526D4—181no—
527D4—181no—
528D4—364no—
529C2×C4—32no—
530C2×C4—181no—
531C2×C4D3311yescharacteristic
532C8—92no—
533C8—272no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—22no—
538C2×C3—22no—
539C2×C3—22no—
540C2×C3—41no—
541C2×C3—42no—
542C2×C3—242no—
543C2×C3—242no—
544C2×C3—241no—
545C2×C3—241no—
546C2×C3—241no—
547C2×C3—241no—
548C2×C3—721no—
549C2×C3#6382312yes—
550C2×C3#6391411yescharacteristic
551C2×C3#6392512yes—
552C2×C3#6393212yes—
553C2×C3#6397911yescharacteristic
554C2×C3#6397911yescharacteristic
555D3—81no—
556D3—81no—
557D3—242no—
558D3—241no—
559D3—241no—
560C22—121no—
561C22—121no—
562C22—122no—
563C22—122no—
564C22—361no—
565C22—362no—
566C22D3×#62511yescharacteristic
567C4—62no—
568C4—361no—
569C4#8895311yescharacteristic
570C4#8895411yescharacteristic
571C3—21no—
572C3—21no—
573C3—21no—
574C3—41no—
575C3#1059035411yescharacteristic, complemented
576C3#1059146911yescharacteristic, complemented
577C3#1059148311yescharacteristic, complemented
578C2—241no—
579C2—241no—
580C2—721no—
581C2#1172286612yes—
582C2#1172302911yescharacteristic
subgroupG/H
centerC22D3×#625
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. C2×#88792C2semidirect
3. C2×#88789C2semidirect
4. C3×#63911C2semidirect
5. C3×#63860C2semidirect
6. C2×C3×#3713C2semidirect
7. #11722019C2semidirect
32. #63911D3semidirect
33. #63860D3semidirect
43. C2×#3713D3semidirect
63. C3⋊(C3⋊(C3⋊C8))D4semidirect
64. C3×C3⋊(C3⋊C8)D4semidirect
223. C3⋊(C3⋊C8)#67semidirect
344. C33#496semidirect
514. C32#5058semidirect
515. C32#5074semidirect
516. C32#5154semidirect
575. C3#10590354semidirect
576. C3#10591469semidirect
577. C3#10591483semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
C2×#887922
C2×#887892
C3×#639112
C3×#638602
C2×C3×#37132
#117220192
#105914833
#105914693
#105903543