Finite Group Databasebeta

← #13156645

Subgroups — #13156645

Order 1728.

730 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 106 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
1#11723455C211yesmaximal, characteristic, complemented
2C2×#88799C211yesmaximal, characteristic, complemented
3C2×#88794C211yesmaximal, characteristic
4C3×#63863C211yesmaximal, characteristic
5C2×C3×#3718C211yesmaximal, characteristic, complemented
6C2×C3×#3715C211yesmaximal, characteristic
7#11722071C211yesmaximal, characteristic, complemented
8#10591627—31nomaximal
9#10591611—31nomaximal
10#10591491—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18#88807—22no—
19#88799—22no—
20#88794—22no—
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3718—22no—
23C3×#3715—22no—
24#64153—31no—
25C2×#3756—31no—
26C2×#3754—31no—
27C3×#947—31no—
28C2×C3×#216—31no—
29C2×C3×#216—31no—
30C2×C3×#215—31no—
31C3×#936—31no—
32C2×C3×#214—31no—
33#63866—31no—
34#63866—31no—
35#63866—61no—
36#63863D311yescharacteristic
37C2×#3719—31no—
38C2×#3719—31no—
39C2×#3719—61no—
40C2×#3718D311yescharacteristic, complemented
41C2×#3716—31no—
42C2×#3716—31no—
43C2×#3716—61no—
44C2×#3715D311yescharacteristic
45#63555—31no—
46#63554—31no—
47#63554—31no—
48C22×C32×D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)—21no—
51#6308—21no—
52#6308—21no—
53C2×C3×C3⋊Dic3D412yes—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C3×C3⋊(C3⋊C8)D412yes—
64#5190—91no—
65#5167—91no—
66#5060—91no—
67C2×C32×Q8—31no—
68C2×C32×Q8—31no—
69C2×C32×D4—31no—
70C2×#634—31no—
71C2×#634—31no—
72C2×#634—31no—
73C2×#634—61no—
74C2×#634—61no—
75C2×#634—61no—
76C2×#634—121no—
77C2×C3×D12—21no—
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×Dic6—21no—
81C2×C3×Dic6—21no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C3×Dic6C2×D311yescharacteristic
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C2×D311yescharacteristic
86#3756—62no—
87#3754—62no—
88#3750—62no—
89C4×C3⋊Dic3C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#217—62no—
93C3×#216—62no—
94C3×#216—62no—
95C3×#215—62no—
96C3×#214—62no—
97C3×(C2×D3)⋊C4—62no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—61no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8—61no—
107Dic3⋊Dic3—62no—
108Dic3⋊Dic3—62no—
109Dic3⋊Dic3—64no—
110(C2×D3)⋊Dic3—22no—
111#3719—62no—
112#3719—62no—
113#3719—64no—
114#3718—22no—
115#3716—62no—
116#3716—62no—
117#3716—64no—
118#3715—22no—
119C22×C33C2×D411yescharacteristic
120C2×C32×D3—41no—
121C2×C32×D3—42no—
122C33×C4C2×D411yescharacteristic
123C33×C4C2×D411yescharacteristic
124C3⋊(C3⋊Dic3)—41no—
125C3×C3⋊Dic3—22no—
126C32×Dic3—41no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×SD16—91no—
130C3×#124—91no—
131#947—31no—
132#944—91no—
133#944—91no—
134#944—181no—
135C2×#216—31no—
136C2×#216—31no—
137C2×#216—91no—
138C2×#216—91no—
139C2×#216—181no—
140C2×#215—31no—
141#936—31no—
142C2×#214—31no—
143C2×#214—91no—
144C2×#214—91no—
145C2×#214—181no—
146C2×Dic12—91no—
147C2×#204—91no—
148#869—91no—
149#815—91no—
150#815—91no—
151#803—91no—
152C23×C32—61no—
153C22×C3×D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—41no—
156C32×Q8—61no—
157C32×Q8—61no—
158C32×Q8—61no—
159C32×Q8—61no—
160C32×D4—61no—
161C32×D4—61no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—41no—
166C2×C32×C4—61no—
167C2×C32×C4—61no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—62no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—121no—
176C2×C3⋊Dic3—121no—
177C2×C3⋊Dic3—122no—
178C2×C3⋊Dic3#6712yes—
179#634—61no—
180#634—61no—
181#634—62no—
182#634—62no—
183#634—61no—
184#634—61no—
185#634—61no—
186#634—61no—
187#634—121no—
188#634—121no—
189#634—121no—
190#634—121no—
191#634—122no—
192#634—122no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—41no—
201C2×C3×Dic3—62no—
202C3×D12—22no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—41no—
208C3×Dic6—22no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—21no—
215C3×Dic6—21no—
216C3×Dic6—21no—
217C3×Dic6—41no—
218C3×Dic6—41no—
219C3×Dic6—41no—
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—
231C32×D3—81no—
232C2×C3×Q8—31no—
233C2×C3×Q8—31no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×Q8—61no—
238C2×C3×D4—31no—
239C2×C3×D4—31no—
240C2×C3×D4—61no—
241C2×D12D3211yescharacteristic
242C2×Dic6—91no—
243C2×Dic6—91no—
244C2×Dic6—91no—
245C2×Dic6—181no—
246C2×Dic6—181no—
247C2×Dic6—181no—
248C2×Dic6—361no—
249C2×Dic6D3211yescharacteristic
250C2×Dic6D3211yescharacteristic
251C3×Q16—182no—
252C3×SD16—182no—
253C2×C3×C8—91no—
254C2×C3×C8—91no—
255C3×C22⋊C4—182no—
256C3×C42—91no—
257#217—62no—
258#216—62no—
259#216—62no—
260#216—182no—
261#216—182no—
262#216—184no—
263#215—62no—
264#214—62no—
265#214—182no—
266#214—182no—
267#214—184no—
268(C2×D3)⋊C4—62no—
269C4⋊Dic3—182no—
270Dic3⋊C4—182no—
271Dic3⋊C4—182no—
272Dic3⋊C4—184no—
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×C32—121no—
289C22×C32—122no—
290C22×C32D3×D411yescharacteristic
291C22×C32D3×D411yescharacteristic
292C22×C32C2×#6711yescharacteristic
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—42no—
298C2×C3×D3—82no—
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—121no—
308C32×C4—121no—
309C32×C4D3×D411yescharacteristic
310C32×C4D3×D411yescharacteristic
311C32×C4D3×D411yescharacteristic
312C32×C4D3×D411yescharacteristic
313C32×C4C2×#6711yescharacteristic
314C32×C4C2×#6711yescharacteristic
315C3⋊Dic3—22no—
316C3⋊Dic3—122no—
317C3⋊Dic3—122no—
318C3⋊Dic3—122no—
319C3⋊Dic3—124no—
320C3⋊Dic3—121no—
321C3⋊Dic3—121no—
322C3⋊Dic3—121no—
323C3×Dic3—42no—
324C3×Dic3—42no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3—41no—
328C3×Dic3—41no—
329C3×Dic3—64no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C2×Q16—91no—
333C2×Q16—271no—
334C2×SD16—91no—
335C2×SD16—271no—
336C4⋊Q8—271no—
337#124—91no—
338#97—271no—
339C33#49711yescharacteristic, complemented, Sylow
340C23×C3—61no—
341C23×C3—61no—
342C23×C3—121no—
343C22×D3—21no—
344C3×Q8—62no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—61no—
349C3×Q8—61no—
350C3×Q8—61no—
351C3×Q8—61no—
352C3×Q8—61no—
353C3×Q8—121no—
354C3×Q8—121no—
355C3×Q8—121no—
356C3×D4—62no—
357C3×D4—61no—
358C3×D4—61no—
359C3×D4—61no—
360C3×D4—61no—
361C3×D4—121no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—62no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—92no—
372C2×C3×C4—121no—
373C2×C3×C4C2×D3211yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—32no—
379C2×Dic3—32no—
380C2×Dic3—62no—
381C2×Dic3—182no—
382C2×Dic3—181no—
383C2×Dic3—181no—
384C2×Dic3—181no—
385C2×Dic3—361no—
386C2×Dic3—361no—
387C2×Dic3—362no—
388D12—21no—
389D12—21no—
390Dic6—21no—
391Dic6—21no—
392Dic6—21no—
393Dic6—21no—
394Dic6—182no—
395Dic6—182no—
396Dic6—181no—
397Dic6—181no—
398Dic6—181no—
399Dic6—181no—
400Dic6—181no—
401Dic6—181no—
402Dic6—361no—
403Dic6—361no—
404Dic6—361no—
405Dic6—361no—
406Dic6—362no—
407Dic6—362no—
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—241no—
425C2×C32#91512yes—
426C2×C32#91812yes—
427C2×C32#94011yescharacteristic
428C2×C32#94011yescharacteristic
429C2×C32#94011yescharacteristic
430C2×C32#95112yes—
431C3×D3—82no—
432C3×D3—81no—
433C3×D3—81no—
434C2×Q8—31no—
435C2×Q8—31no—
436C2×Q8—271no—
437C2×D4—31no—
438Q16—182no—
439Q16—542no—
440SD16—182no—
441SD16—542no—
442C2×C8—91no—
443C2×C8—91no—
444C4⋊C4—542no—
445C22⋊C4—182no—
446C42—91no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—122no—
452C22×C3—122no—
453C22×C3—122no—
454C22×C3—121no—
455C22×C3—121no—
456C22×C3—242no—
457C22×C3D3×#6711yescharacteristic
458C22×C3D3×#6711yescharacteristic
459C22×C3#381111yescharacteristic
460C2×D3—42no—
461C2×D3—41no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—122no—
471C3×C4—122no—
472C3×C4—121no—
473C3×C4—121no—
474C3×C4—121no—
475C3×C4—121no—
476C3×C4—182no—
477C3×C4D3×#6711yescharacteristic
478C3×C4D3×#6711yescharacteristic
479C3×C4D3×#6711yescharacteristic
480C3×C4D3×#6711yescharacteristic
481C3×C4#381111yescharacteristic
482C3×C4#381111yescharacteristic
483Dic3—41no—
484Dic3—41no—
485Dic3—64no—
486Dic3—62no—
487Dic3—62no—
488Dic3—362no—
489Dic3—362no—
490Dic3—362no—
491Dic3—364no—
492Dic3—361no—
493Dic3—361no—
494Dic3—361no—
495C32—21no—
496C32—21no—
497C32—21no—
498C32—41no—
499C32#506011yescharacteristic, complemented
500C32#516711yescharacteristic, complemented
501C32#519011yescharacteristic, complemented
502C23—61no—
503Q8—61no—
504Q8—61no—
505Q8—61no—
506Q8—61no—
507Q8—541no—
508Q8—541no—
509D4—61no—
510D4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—92no—
514C2×C4—541no—
515C2×C4D3311yescharacteristic
516C8—92no—
517C8—92no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—41no—
525C2×C3—42no—
526C2×C3—242no—
527C2×C3—241no—
528C2×C3—241no—
529C2×C3#6392912yes—
530C2×C3#6394312yes—
531C2×C3#6395112yes—
532C2×C3#6397911yescharacteristic
533C2×C3#6397911yescharacteristic
534C2×C3#6397911yescharacteristic
535D3—81no—
536C22—121no—
537C22—122no—
538C22#8895211yescharacteristic
539C4—121no—
540C4—121no—
541C4—182no—
542C4—1081no—
543C4#8895211yescharacteristic
544C4#8895211yescharacteristic
545C3—21no—
546C3—21no—
547C3—21no—
548C3—41no—
549C3#1059149111yescharacteristic, complemented
550C3#1059161111yescharacteristic, complemented
551C3#1059162711yescharacteristic, complemented
552C2—241no—
553C2#1172288812yes—
554C2#1172303111yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#497
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. #11723455C2semidirect
2. C2×#88799C2semidirect
5. C2×C3×#3718C2semidirect
7. #11722071C2semidirect
40. C2×#3718D3semidirect
339. C33#497semidirect
499. C32#5060semidirect
500. C32#5167semidirect
501. C32#5190semidirect
549. C3#10591491semidirect
550. C3#10591611semidirect
551. C3#10591627semidirect

Maximal subgroups

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

subgroupindex
#117234552
C2×#887992
C2×#887942
C3×#638632
C2×C3×#37182
C2×C3×#37152
#117220712
#105916273
#105916113
#105914913