Finite Group Databasebeta

← #13162008

Subgroups — #13162008

Order 1728.

730 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#64153C211yesmaximal, characteristic, complemented
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C3×#3755C211yesmaximal, characteristic
4#11723677C211yesmaximal, characteristic
5C2×#88812C211yesmaximal, characteristic, complemented
6C2×#88811C211yesmaximal, characteristic
7#11722071C211yesmaximal, characteristic, complemented
8#10591611—31nomaximal
9#10591608—31nomaximal
10#10591605—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C3×#3756—22no—
16C3×#3755—22no—
17C3×#3750—22no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#88814—22no—
22#88812—22no—
23#88811—22no—
24#64153D311yescharacteristic, complemented
25C2×#3756D311yescharacteristic, complemented
26C2×#3755D311yescharacteristic
27C3×#944—31no—
28C3×#944—31no—
29C3×#944—61no—
30C2×C3×#216—31no—
31C2×C3×#216—31no—
32C2×C3×#216—61no—
33C2×C3×#215—31no—
34C2×C3×#215—31no—
35C2×C3×#215—61no—
36#63864—31no—
37#63863—31no—
38#63860—31no—
39C2×#3719—31no—
40C2×#3719—31no—
41C2×#3718—31no—
42C2×#3717—31no—
43C2×#3716—31no—
44C2×#3715—31no—
45#63555—31no—
46#63554—31no—
47#63554—31no—
48C22×C3×C3⋊D3—21no—
49C33×Q8—21no—
50C33×Q8—21no—
51C2×C33×C4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D412yes—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×#634—21no—
59C3×#634—21no—
60C3×#634—21no—
61C3×#634—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C3×C3⋊(C3⋊C8)D412yes—
64#5190—91no—
65#5167—91no—
66#5086—91no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—41no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C2×D311yescharacteristic
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×#634C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—61no—
85C2×C3×Dic6—61no—
86#3756—22no—
87#3755—22no—
88#3750—22no—
89C4×C3⋊Dic3C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#216—62no—
93C3×#216—62no—
94C3×#216—64no—
95C3×#215—62no—
96C3×#215—62no—
97C3×#215—64no—
98C3×Dic3⋊C4—62no—
99C3×Dic3⋊C4—62no—
100C3×Dic3⋊C4—64no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—61no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—61no—
109C2×C3×C3⋊C8—61no—
110#3722—62no—
111(C2×D3)⋊Dic3—62no—
112(C2×D3)⋊Dic3—62no—
113#3719—62no—
114#3719—62no—
115#3718—62no—
116#3717—62no—
117#3716—62no—
118#3715—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—42no—
122C33×C4—41no—
123C33×C4C2×D411yescharacteristic
124C33×C4C2×D411yescharacteristic
125C3×C3⋊Dic3—22no—
126C3×C3⋊Dic3—41no—
127C3×C3⋊Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×SD16—91no—
130C3×C4⋊Q8—91no—
131#947—91no—
132#944—31no—
133#944—31no—
134#944—61no—
135C2×#216—31no—
136C2×#216—31no—
137C2×#216—61no—
138C2×#216—91no—
139C2×#216—91no—
140C2×#215—31no—
141C2×#215—31no—
142C2×#215—61no—
143C2×#215—91no—
144#936—91no—
145C2×#214—91no—
146C2×Dic12—91no—
147C2×#204—91no—
148#875—91no—
149#815—91no—
150#815—91no—
151#803—91no—
152C22×C3⋊D3—21no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—121no—
156C32×Q8—21no—
157C32×Q8—21no—
158C32×Q8—22no—
159C32×Q8—22no—
160C32×Q8—21no—
161C32×Q8—21no—
162C32×Q8—21no—
163C32×Q8—21no—
164C32×Q8—41no—
165C32×Q8—41no—
166C32×Q8—41no—
167C32×Q8—41no—
168C32×Q8—42no—
169C32×Q8—42no—
170C2×C32×C4—21no—
171C2×C32×C4—22no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—41no—
178C2×C32×C4—41no—
179C2×C32×C4—41no—
180C2×C32×C4—42no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3#6712yes—
187C4⋊(C3⋊D3)—21no—
188C4⋊(C3⋊D3)—21no—
189#634—21no—
190#634—21no—
191#634—21no—
192#634—21no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—121no—
202C3×D12—62no—
203C3×D12—61no—
204C3×D12—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—121no—
208C3×Dic6—62no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—121no—
218C3×Dic6—121no—
219C3×Dic6—121no—
220C3⋊(C3⋊C8)#6712yes—
221C3⋊(C3⋊C8)#6712yes—
222C3×C3⋊C8—32no—
223C3×C3⋊C8—32no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8—32no—
226C3×C3⋊C8—62no—
227C3×C3⋊C8—62no—
228#497—271noSylow
229C2×C33#12711yescharacteristic
230C2×C33#13712yes—
231C3×C3⋊D3—81no—
232C2×C3×Q8—21no—
233C2×C3×Q8—21no—
234C2×C3×Q8—21no—
235C2×C3×Q8—41no—
236C2×C3×Q8—91no—
237C2×C3×Q8—91no—
238C2×C3×Q8D3211yescharacteristic
239C2×C3×Q8D3211yescharacteristic
240C2×C3×Q8D3211yescharacteristic
241C2×C3×D4—91no—
242C2×D12—31no—
243C2×D12—31no—
244C2×D12—61no—
245C2×Dic6—31no—
246C2×Dic6—31no—
247C2×Dic6—31no—
248C2×Dic6—31no—
249C2×Dic6—61no—
250C2×Dic6—61no—
251C3×Q16—182no—
252C3×SD16—182no—
253C2×C3×C8—91no—
254C2×C3×C8—91no—
255C3×C4⋊C4—182no—
256C3×C42—91no—
257#217—182no—
258#216—62no—
259#216—62no—
260#216—64no—
261#216—182no—
262#216—182no—
263#215—62no—
264#215—62no—
265#215—64no—
266#215—182no—
267#214—182no—
268(C2×D3)⋊C4—182no—
269(C2×D3)⋊C4—182no—
270Dic3⋊C4—62no—
271Dic3⋊C4—62no—
272Dic3⋊C4—64no—
273C4×Dic3—31no—
274C4×Dic3—31no—
275C4×Dic3—61no—
276C2×C3⋊C8—31no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—61no—
281C2×C3⋊C8—61no—
282Dic12—182no—
283#204—182no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—41no—
288C22×C32D3×D411yescharacteristic
289C22×C32D3×D411yescharacteristic
290C22×C32C2×#6711yescharacteristic
291C2×C3⋊D3—41no—
292C2×C3⋊D3—42no—
293C2×C3×D3—122no—
294C2×C3×D3—122no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—122no—
298C2×C3×D3—242no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—42no—
308C32×C4—42no—
309C32×C4—42no—
310C32×C4—44no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—41no—
314C32×C4D3×D411yescharacteristic
315C32×C4D3×D411yescharacteristic
316C32×C4D3×D411yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4C2×#6711yescharacteristic
319C32×C4C2×#6711yescharacteristic
320C3⋊Dic3—22no—
321C3⋊Dic3—41no—
322C3⋊Dic3—41no—
323C3×Dic3—64no—
324C3×Dic3—62no—
325C3×Dic3—62no—
326C3×Dic3—122no—
327C3×Dic3—122no—
328C3×Dic3—121no—
329C3×Dic3—121no—
330C3×Dic3—121no—
331C3×Dic3—121no—
332C2×Q16—91no—
333C2×Q16—271no—
334C2×SD16—91no—
335C2×SD16—271no—
336C4⋊Q8—91no—
337#124—271no—
338#97—271no—
339C33#49711yescharacteristic, complemented, Sylow
340C23×C3—181no—
341C22×D3—61no—
342C22×D3—61no—
343C22×D3—121no—
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—21no—
347C3×Q8—21no—
348C3×Q8—21no—
349C3×Q8—21no—
350C3×Q8—21no—
351C3×Q8—21no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—42no—
357C3×Q8—42no—
358C3×Q8—181no—
359C3×Q8—181no—
360C3×Q8—181no—
361C3×Q8—181no—
362C3×D4—181no—
363C3×D4—181no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—41no—
375C2×C3×C4—92no—
376C2×C3×C4—181no—
377C2×C3×C4—181no—
378C2×C3×C4C2×D3211yescharacteristic
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381C2×Dic3—32no—
382C2×Dic3—32no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—61no—
389C2×Dic3—121no—
390D12—62no—
391D12—61no—
392D12—61no—
393D12—61no—
394D12—61no—
395D12—121no—
396Dic6—62no—
397Dic6—61no—
398Dic6—61no—
399Dic6—61no—
400Dic6—61no—
401Dic6—61no—
402Dic6—61no—
403Dic6—61no—
404Dic6—61no—
405Dic6—121no—
406Dic6—121no—
407Dic6—121no—
408C3×C8—92no—
409C3×C8—92no—
410C3⋊C8—32no—
411C3⋊C8—32no—
412C3⋊C8—32no—
413C3⋊C8—32no—
414C3⋊C8—62no—
415C3⋊C8—62no—
416C2×C32—22no—
417C2×C32—22no—
418C2×C32—22no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—41no—
423C2×C32—42no—
424C2×C32#91512yes—
425C2×C32#91812yes—
426C2×C32#94011yescharacteristic
427C2×C32#94011yescharacteristic
428C2×C32#94011yescharacteristic
429C2×C32#94212yes—
430C3⋊D3—81no—
431C3×D3—242no—
432C3×D3—241no—
433C3×D3—241no—
434C2×Q8—91no—
435C2×Q8—91no—
436C2×Q8#111311yescharacteristic
437C2×D4—91no—
438Q16—182no—
439Q16—542no—
440SD16—182no—
441SD16—542no—
442C2×C8—91no—
443C2×C8—91no—
444C4⋊C4—182no—
445C22⋊C4—542no—
446C42—91no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—362no—
452C22×C3—361no—
453C22×C3D3×#6711yescharacteristic
454C22×C3D3×#6711yescharacteristic
455C22×C3#381111yescharacteristic
456C2×D3—122no—
457C2×D3—122no—
458C2×D3—122no—
459C2×D3—121no—
460C2×D3—121no—
461C2×D3—242no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—44no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—182no—
478C3×C4—361no—
479C3×C4—361no—
480C3×C4D3×#6711yescharacteristic
481C3×C4D3×#6711yescharacteristic
482C3×C4D3×#6711yescharacteristic
483C3×C4D3×#6711yescharacteristic
484C3×C4#381111yescharacteristic
485C3×C4#381111yescharacteristic
486Dic3—64no—
487Dic3—62no—
488Dic3—62no—
489Dic3—122no—
490Dic3—122no—
491Dic3—121no—
492Dic3—121no—
493Dic3—121no—
494Dic3—121no—
495C32—21no—
496C32—21no—
497C32—21no—
498C32—41no—
499C32#508611yescharacteristic, complemented
500C32#516711yescharacteristic, complemented
501C32#519011yescharacteristic, complemented
502C23—181no—
503Q8—21no—
504Q8—21no—
505Q8—181no—
506Q8—181no—
507Q8—181no—
508Q8—181no—
509D4—181no—
510D4—181no—
511C2×C4—21no—
512C2×C4—92no—
513C2×C4—181no—
514C2×C4—181no—
515C2×C4C2×#111311yescharacteristic
516C8—92no—
517C8—92no—
518C2×C3—22no—
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—41no—
526C2×C3—721no—
527C2×C3#6394612yes—
528C2×C3#6394812yes—
529C2×C3#6395112yes—
530C2×C3#6397911yescharacteristic
531C2×C3#6397911yescharacteristic
532C2×C3#6397911yescharacteristic
533D3—242no—
534D3—241no—
535D3—241no—
536C22—362no—
537C22—361no—
538C22#8904911yescharacteristic
539C4—41no—
540C4—182no—
541C4—361no—
542C4—361no—
543C4#8904911yescharacteristic
544C4#8904911yescharacteristic
545C3—21no—
546C3—21no—
547C3—21no—
548C3—41no—
549C3#1059160511yescharacteristic, complemented
550C3#1059160811yescharacteristic, complemented
551C3#1059161111yescharacteristic, complemented
552C2—721no—
553C2#1172371212yes—
554C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×Q8#1113
hypocenterC33#497
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64153C2semidirect
2. C2×C3×#3756C2semidirect
5. C2×#88812C2semidirect
7. #11722071C2semidirect
24. #64153D3semidirect
25. C2×#3756D3semidirect
339. C33#497semidirect
499. C32#5086semidirect
500. C32#5167semidirect
501. C32#5190semidirect
549. C3#10591605semidirect
550. C3#10591608semidirect
551. C3#10591611semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C2×C3×#37562
C2×C3×#37552
#117236772
C2×#888122
C2×#888112
#117220712
#105916113
#105916083
#105916053