Finite Group Databasebeta

← #13156578

Subgroups — #13156578

Order 1728.

770 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 124 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×#88793C211yesmaximal, characteristic, complemented
3C2×#88792C211yesmaximal, characteristic, complemented
4C3×#63911C211yesmaximal, characteristic
5C3×#63866C211yesmaximal, characteristic, complemented
6C2×C3×#3715C211yesmaximal, characteristic, complemented
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591618—31nomaximal
9#10591490—31nomaximal
10#10590375—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#626—22no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19#88811—22no—
20#88793—22no—
21#88792—22no—
22C3×Dic3⋊Dic3—22no—
23C3×#3715—22no—
24C2×#3755—31no—
25C2×#3754—31no—
26C3×#944—31no—
27C2×C3×#215—31no—
28C3×#940—31no—
29C2×C3×#214—31no—
30C3×#874—31no—
31C3×#869—31no—
32#63911D311yescharacteristic
33#63866D311yescharacteristic, complemented
34C2×#3717—31no—
35C2×#3716—31no—
36C2×#3715—31no—
37C2×#3715—31no—
38C2×#3715—31no—
39C2×#3715—31no—
40C2×#3715—31no—
41C2×#3715—61no—
42C2×#3715—61no—
43C2×#3715D311yescharacteristic, complemented
44C4×C3⋊(C3⋊C8)—31no—
45Dic3×C3⋊C8—31no—
46Dic3×C3⋊C8—31no—
47Dic3×C3⋊C8—61no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D412yes—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×#626—44no—
63C3⋊(C3⋊(C3⋊C8))D412yescomplemented
64C3×C3⋊(C3⋊C8)D412yescomplemented
65#5175—91no—
66#5085—91no—
67#5076—91no—
68C2×C32×Q8—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—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×D12—62no—
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×Dic6—21no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C2×D311yescharacteristic
86C2×#626—22no—
87C32×C42—31no—
88#3755—62no—
89#3754—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×#215—62no—
99C3×#214—62no—
100C3×C4⋊Dic3—62no—
101C3×Dic3⋊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—
108Dic3⋊Dic3—22no—
109#3717—62no—
110#3716—62no—
111#3715—22no—
112#3715—62no—
113#3715—62no—
114#3715—62no—
115#3715—62no—
116#3715—64no—
117#3715—64no—
118#3715—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—42no—
122C2×C32×D3—41no—
123C2×C32×D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3×C3⋊Dic3—41no—
127C32×Dic3—41no—
128C32×Dic3SD1614yescomplemented
129C2×C3×SD16—91no—
130C3×C4⋊Q8—91no—
131C3×#127—91no—
132#944—31no—
133C2×#215—31no—
134C2×#215—91no—
135C2×#215—91no—
136C2×#215—91no—
137C2×#215—91no—
138C2×#215—181no—
139#940—31no—
140C2×#214—31no—
141C2×#214—91no—
142C2×#214—91no—
143C2×#214—91no—
144C2×#214—91no—
145C2×#214—181no—
146C2×#204—91no—
147#874—31no—
148#869—31no—
149C8×Dic3—91no—
150C4×C3⋊C8—91no—
151C4×C3⋊C8—91no—
152C4×C3⋊C8—181no—
153C23×C32—61no—
154C22×C3⋊D3—21no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—41no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—121no—
161C32×Q8—61no—
162C32×Q8—61no—
163C32×D4—61no—
164C32×D4—61no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—32no—
169C2×C32×C4—41no—
170C2×C32×C4—61no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—21no—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—21no—
177#634—21no—
178#634—21no—
179C3×#67—124no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—121no—
187C2×C3×Dic3#6712yes—
188C2×C3×Dic3#6712yes—
189C3×D12—22no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—41no—
195C3×D12—62no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—121no—
201C3×D12—124no—
202C3×Dic6—22no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—41no—
208C3×Dic6—62no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—121no—
214#626—44no—
215C3⋊(C3⋊C8)—32no—
216C3⋊(C3⋊C8)—32no—
217C3⋊(C3⋊C8)—32no—
218C3⋊(C3⋊C8)—62no—
219C3⋊(C3⋊C8)—62no—
220C3⋊(C3⋊C8)—62no—
221C3⋊(C3⋊C8)—122no—
222C3⋊(C3⋊C8)#6712yescomplemented
223C3×C3⋊C8—32no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8—62no—
226#492—271noSylow
227C2×C33#12711yescharacteristic
228C2×C33C2×SD1612yes—
229C3×C3⋊D3—81no—
230C32×D3—81no—
231C2×C3×Q8—31no—
232C2×C3×Q8—31no—
233C2×C3×Q8—61no—
234C2×C3×Q8—91no—
235C2×C3×D4—31no—
236C2×C3×D4—31no—
237C2×C3×D4—61no—
238C2×C3×D4—91no—
239C2×C3×D4—182no—
240C2×#67—62no—
241C2×D12—31no—
242C2×D12—31no—
243C2×D12—61no—
244C2×D12—62no—
245C2×D12D3211yescharacteristic
246C2×Dic6—31no—
247C2×Dic6—31no—
248C2×Dic6—61no—
249C2×Dic6D3211yescharacteristic
250C3×SD16—182no—
251C2×C3×C8—91no—
252C3×C4⋊C4—182no—
253C3×C42—31no—
254C3×C42—31no—
255C3×C42—61no—
256#215—62no—
257#215—182no—
258#215—182no—
259#215—182no—
260#215—182no—
261#215—184no—
262#214—62no—
263#214—182no—
264#214—182no—
265#214—182no—
266#214—182no—
267#214—184no—
268C4⋊Dic3—62no—
269Dic3⋊C4—62no—
270C4×Dic3D3211yescharacteristic
271C2×C3⋊C8—31no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—61no—
274C2×C3⋊C8—91no—
275C2×C3⋊C8—91no—
276C2×C3⋊C8—91no—
277C2×C3⋊C8—181no—
278C2×C3⋊C8—181no—
279C2×C3⋊C8—181no—
280C2×C3⋊C8—361no—
281#204—182no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32—121no—
287C22×C32—122no—
288C22×C32D3×D411yescharacteristic
289C22×C32C2×#6711yescharacteristic
290C22×C32C2×#6711yescharacteristic
291C2×C3⋊D3—41no—
292C2×C3⋊D3—42no—
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C2×C3×D3—42no—
296C2×C3×D3—41no—
297C2×C3×D3—41no—
298C2×C3×D3—82no—
299C2×C3×D3—122no—
300C2×C3×D3—122no—
301C2×C3×D3—121no—
302C2×C3×D3—121no—
303C2×C3×D3—122no—
304C2×C3×D3—242no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—34no—
312C32×C4—41no—
313C32×C4—41no—
314C32×C4—121no—
315C32×C4D3×D411yescharacteristic
316C32×C4D3×D411yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4D3×D411yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4C2×#6711yescharacteristic
321C3⋊Dic3—41no—
322C3×Dic3—24no—
323C3×Dic3—42no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—122no—
327C3×Dic3—121no—
328C3×Dic3—121no—
329C3×Dic3#21414yescomplemented
330C3×Dic3#21514yescomplemented
331C2×SD16—91no—
332C2×SD16—271no—
333C2×SD16—271no—
334C2×SD16—271no—
335C4⋊Q8—91no—
336#127—91no—
337C4×C8—271no—
338C33#49211yescharacteristic, complemented, Sylow
339C23×C3—61no—
340C23×C3—61no—
341C23×C3—121no—
342C23×C3—181no—
343C22×D3—21no—
344C22×D3—61no—
345C22×D3—61no—
346C22×D3—121no—
347C3×Q8—62no—
348C3×Q8—61no—
349C3×Q8—61no—
350C3×Q8—61no—
351C3×Q8—61no—
352C3×Q8—121no—
353C3×Q8—181no—
354C3×Q8—181no—
355C3×D4—62no—
356C3×D4—61no—
357C3×D4—61no—
358C3×D4—61no—
359C3×D4—61no—
360C3×D4—121no—
361C3×D4—181no—
362C3×D4—181no—
363C3×D4—364no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—32no—
368C2×C3×C4—32no—
369C2×C3×C4—41no—
370C2×C3×C4—62no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—121no—
374C2×C3×C4—181no—
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377C2×C3×C4C2×D3211yescharacteristic
378#67—124no—
379C2×Dic3—21no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—121no—
383C2×Dic3#62512yes—
384D12—21no—
385D12—21no—
386D12—62no—
387D12—61no—
388D12—61no—
389D12—61no—
390D12—61no—
391D12—121no—
392D12—124no—
393Dic6—21no—
394Dic6—21no—
395Dic6—62no—
396Dic6—61no—
397Dic6—61no—
398Dic6—61no—
399Dic6—61no—
400Dic6—121no—
401C3×C8—92no—
402C3⋊C8—32no—
403C3⋊C8—32no—
404C3⋊C8—62no—
405C3⋊C8—92no—
406C3⋊C8—92no—
407C3⋊C8—92no—
408C3⋊C8—182no—
409C3⋊C8—182no—
410C3⋊C8—182no—
411C3⋊C8—362no—
412C2×C32—22no—
413C2×C32—21no—
414C2×C32—22no—
415C2×C32—22no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—41no—
419C2×C32—42no—
420C2×C32—241no—
421C2×C32D3×SD1612yes—
422C2×C32C2×#21412yes—
423C2×C32#94011yescharacteristic
424C2×C32#94011yescharacteristic
425C2×C32#94011yescharacteristic
426C2×C32C2×#21512yes—
427C3⋊D3—81no—
428C3×D3—82no—
429C3×D3—81no—
430C3×D3—81no—
431C3×D3—242no—
432C3×D3—241no—
433C3×D3—241no—
434C2×Q8—31no—
435C2×Q8—91no—
436C2×D4—31no—
437C2×D4—91no—
438C2×D4—182no—
439SD16—182no—
440SD16—542no—
441SD16—542no—
442SD16—542no—
443C2×C8—91no—
444C2×C8—271no—
445C4⋊C4—182no—
446C42—31no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—122no—
452C22×C3—122no—
453C22×C3—121no—
454C22×C3—122no—
455C22×C3—121no—
456C22×C3—242no—
457C22×C3—362no—
458C22×C3—361no—
459C22×C3C2×#62511yescharacteristic
460C22×C3D3×#6711yescharacteristic
461C22×C3D3×#6711yescharacteristic
462C2×D3—42no—
463C2×D3—41no—
464C2×D3—122no—
465C2×D3—122no—
466C2×D3—121no—
467C2×D3—122no—
468C2×D3—121no—
469C2×D3—242no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—34no—
477C3×C4—34no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—64no—
481C3×C4—122no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—361no—
485C3×C4#380211yescharacteristic
486C3×C4#380211yescharacteristic
487C3×C4D3×#6711yescharacteristic
488C3×C4D3×#6711yescharacteristic
489C3×C4#381111yescharacteristic
490C3×C4#381111yescharacteristic
491Dic3—41no—
492Dic3—122no—
493Dic3—121no—
494Dic3—121no—
495Dic3#371514yescomplemented
496C32—21no—
497C32—21no—
498C32—21no—
499C32—41no—
500C32#507611yescharacteristic, complemented
501C32#508511yescharacteristic, complemented
502C32#517511yescharacteristic, complemented
503C23—61no—
504C23—181no—
505Q8—61no—
506Q8—61no—
507Q8—181no—
508Q8—181no—
509D4—61no—
510D4—61no—
511D4—181no—
512D4—181no—
513D4—364no—
514C2×C4—32no—
515C2×C4—61no—
516C2×C4—181no—
517C2×C4D3311yescharacteristic
518C8—92no—
519C8—272no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—22no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—41no—
527C2×C3—42no—
528C2×C3—242no—
529C2×C3—241no—
530C2×C3—241no—
531C2×C3—721no—
532C2×C3C2×#371512yes—
533C2×C3#6391411yescharacteristic
534C2×C3D3×#21412yes—
535C2×C3D3×#21512yes—
536C2×C3#6397911yescharacteristic
537C2×C3#6397911yescharacteristic
538D3—81no—
539D3—242no—
540D3—241no—
541D3—241no—
542C22—121no—
543C22—122no—
544C22—361no—
545C22—362no—
546C22D3×#62511yescharacteristic
547C4—34no—
548C4—121no—
549C4—361no—
550C4#8895311yescharacteristic
551C4#8895411yescharacteristic
552C3—21no—
553C3—21no—
554C3—21no—
555C3—41no—
556C3#1059037511yescharacteristic, complemented
557C3#1059149011yescharacteristic, complemented
558C3#1059161811yescharacteristic, complemented
559C2—241no—
560C2—721no—
561C2D3×#371512yes—
562C2#1172302911yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypocenterC33#492
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×#88793C2semidirect
3. C2×#88792C2semidirect
5. C3×#63866C2semidirect
6. C2×C3×#3715C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
33. #63866D3semidirect
43. C2×#3715D3semidirect
63. C3⋊(C3⋊(C3⋊C8))D4semidirect
64. C3×C3⋊(C3⋊C8)D4semidirect
128. C32×Dic3SD16semidirect
222. C3⋊(C3⋊C8)#67semidirect
329. C3×Dic3#214semidirect
330. C3×Dic3#215semidirect
338. C33#492semidirect
495. Dic3#3715semidirect
500. C32#5076semidirect
501. C32#5085semidirect
502. C32#5175semidirect
556. C3#10590375semidirect
557. C3#10591490semidirect
558. C3#10591618semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
C2×#887932
C2×#887922
C3×#639112
C3×#638662
C2×C3×#37152
Dic3×C3⋊(C3⋊C8)2
#105916183
#105914903
#105903753