Finite Group Databasebeta

← #13156634

Subgroups — #13156634

Order 1728.

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

normal subgroups only

This group has 686 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
1#11723678C211yesmaximal, characteristic, complemented
2C2×#88798C211yesmaximal, characteristic, complemented
3C2×#88792C211yesmaximal, characteristic
4C3×#63863C211yesmaximal, characteristic
5C2×C3×#3719C211yesmaximal, characteristic, complemented
6C2×C3×#3716C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic, complemented
8#10591627—31nomaximal
9#10591491—31nomaximal
10#10590870—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88815—22no—
19#88798—22no—
20#88792—22no—
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3719—22no—
23C3×#3716—22no—
24C2×#3756—31no—
25C2×#3754—31no—
26C3×#947—31no—
27C2×C3×#216—31no—
28C3×#936—31no—
29C2×C3×#214—31no—
30C2×C3×Dic12—31no—
31C2×C3×#204—31no—
32#63873—31no—
33#63866—31no—
34#63866—31no—
35#63863D311yescharacteristic
36C2×#3719D311yescharacteristic, complemented
37C2×#3718—31no—
38C2×#3718—31no—
39C2×#3718—61no—
40C2×#3716D311yescharacteristic
41C2×#3715—31no—
42C2×#3715—31no—
43C2×#3715—61no—
44#63642—31no—
45#63554—31no—
46#63554—31no—
47#63554—61no—
48C22×C32×D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D412yes—
53C3×#634—21no—
54C3×#634—21no—
55C3×#634—21no—
56C3×#634—21no—
57C2×C32×Dic3—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3⋊(C3⋊(C3⋊C8))D412yes—
63C32×C3⋊C8D412yes—
64#5190—91no—
65#5167—91no—
66#5060—91no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C2×#634C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×Dic6—21no—
75C2×C3×Dic6—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—61no—
80C2×C3×Dic6—61no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C32×C8—31no—
84#3756—62no—
85#3754—62no—
86C4×C3⋊Dic3C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—31no—
90C2×C3⋊(C3⋊C8)—61no—
91C2×C3⋊(C3⋊C8)—61no—
92C2×C3⋊(C3⋊C8)—61no—
93C2×C3⋊(C3⋊C8)—121no—
94C3×#217—62no—
95C3×#216—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—21no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C3×Dic12—62no—
105C3×#204—62no—
106#3724—62no—
107Dic3⋊Dic3—62no—
108Dic3⋊Dic3—62no—
109(C2×D3)⋊Dic3—22no—
110#3719—22no—
111#3718—62no—
112#3718—62no—
113#3718—64no—
114#3716—22no—
115#3715—62no—
116#3715—62no—
117#3715—64no—
118C22×C33C2×D411yescharacteristic
119C2×C32×D3—41no—
120C2×C32×D3—42no—
121C33×C4C2×D411yescharacteristic
122C33×C4C2×D411yescharacteristic
123C3×C3⋊Dic3—22no—
124C3×C3⋊Dic3—41no—
125C3×C3⋊Dic3—41no—
126C32×Dic3—41no—
127C2×C3×Q16—91no—
128C2×C3×SD16—91no—
129C3×#124—91no—
130#947—31no—
131#944—91no—
132#944—91no—
133C2×#216—31no—
134C2×#216—91no—
135C2×#216—91no—
136C2×#216—91no—
137C2×#216—181no—
138C2×#215—91no—
139#936—31no—
140C2×#214—31no—
141C2×#214—91no—
142C2×#214—91no—
143C2×#214—181no—
144C2×Dic12—31no—
145C2×#204—31no—
146#869—91no—
147#815—91no—
148#815—91no—
149#815—181no—
150#803—91no—
151C23×C32—61no—
152C22×C3×D3—21no—
153C22×C3×D3—21no—
154C22×C3×D3—41no—
155C32×Q8—61no—
156C32×Q8—61no—
157C32×D4—61no—
158C32×D4—61no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—61no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3#6712yes—
170#634—21no—
171#634—21no—
172#634—21no—
173#634—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—41no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—121no—
185C2×C3×Dic3—121no—
186C3×D12—22no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—41no—
192C3×Dic6—22no—
193C3×Dic6—21no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—41no—
198C3×Dic6—62no—
199C3×Dic6—62no—
200C3×Dic6—61no—
201C3×Dic6—61no—
202C3×Dic6—61no—
203C3×Dic6—61no—
204C3×Dic6—61no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—121no—
209C3×Dic6—121no—
210C32×C8—32no—
211C3⋊(C3⋊C8)—32no—
212C3⋊(C3⋊C8)—32no—
213C3⋊(C3⋊C8)—32no—
214C3⋊(C3⋊C8)—62no—
215C3⋊(C3⋊C8)—62no—
216C3⋊(C3⋊C8)—62no—
217C3⋊(C3⋊C8)—122no—
218C3×C3⋊C8—22no—
219C3×C3⋊C8#6712yes—
220C3×C3⋊C8#6712yes—
221#497—271noSylow
222C2×C33#12711yescharacteristic
223C2×C33#13712yes—
224C32×D3—81no—
225C2×C3×Q8—31no—
226C2×C3×Q8—31no—
227C2×C3×Q8—61no—
228C2×C3×Q8—91no—
229C2×C3×Q8—91no—
230C2×C3×D4—31no—
231C2×C3×D4—31no—
232C2×C3×D4—61no—
233C2×D12D3211yescharacteristic
234C2×Dic6—31no—
235C2×Dic6—31no—
236C2×Dic6—31no—
237C2×Dic6—31no—
238C2×Dic6—61no—
239C2×Dic6—61no—
240C2×Dic6D3211yescharacteristic
241C3×Q16—182no—
242C3×SD16—182no—
243C2×C3×C8—31no—
244C2×C3×C8—31no—
245C2×C3×C8—61no—
246C3×C22⋊C4—182no—
247C3×C42—91no—
248#217—62no—
249#216—62no—
250#216—182no—
251#216—182no—
252#216—182no—
253#216—184no—
254#215—182no—
255#214—62no—
256#214—182no—
257#214—182no—
258#214—184no—
259(C2×D3)⋊C4—62no—
260C4⋊Dic3—182no—
261Dic3⋊C4—182no—
262Dic3⋊C4—182no—
263C4×Dic3—31no—
264C4×Dic3—31no—
265C4×Dic3—61no—
266C2×C3⋊C8—91no—
267C2×C3⋊C8—91no—
268C2×C3⋊C8—91no—
269C2×C3⋊C8—181no—
270C2×C3⋊C8—181no—
271C2×C3⋊C8—181no—
272C2×C3⋊C8—361no—
273C2×C3⋊C8D3211yescharacteristic
274Dic12—62no—
275#204—62no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—41no—
280C22×C32—121no—
281C22×C32—122no—
282C22×C32D3×D411yescharacteristic
283C22×C32D3×D411yescharacteristic
284C22×C32C2×#6711yescharacteristic
285C2×C3×D3—42no—
286C2×C3×D3—42no—
287C2×C3×D3—41no—
288C2×C3×D3—41no—
289C2×C3×D3—42no—
290C2×C3×D3—82no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—121no—
300C32×C4D3×D411yescharacteristic
301C32×C4D3×D411yescharacteristic
302C32×C4D3×D411yescharacteristic
303C32×C4D3×D411yescharacteristic
304C32×C4C2×#6711yescharacteristic
305C32×C4C2×#6711yescharacteristic
306C3⋊Dic3—22no—
307C3⋊Dic3—41no—
308C3⋊Dic3—41no—
309C3×Dic3—42no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—122no—
315C3×Dic3—122no—
316C3×Dic3—122no—
317C3×Dic3—121no—
318C3×Dic3—121no—
319C3×Dic3—121no—
320C3×Dic3—121no—
321C2×Q16—91no—
322C2×Q16—271no—
323C2×SD16—91no—
324C2×SD16—271no—
325C4⋊Q8—271no—
326#124—91no—
327#97—271no—
328C33#49711yescharacteristic, complemented, Sylow
329C23×C3—61no—
330C23×C3—61no—
331C23×C3—121no—
332C22×D3—21no—
333C3×Q8—62no—
334C3×Q8—61no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—121no—
339C3×Q8—181no—
340C3×Q8—181no—
341C3×Q8—181no—
342C3×Q8—181no—
343C3×D4—62no—
344C3×D4—61no—
345C3×D4—61no—
346C3×D4—61no—
347C3×D4—61no—
348C3×D4—121no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—92no—
356C2×C3×C4—121no—
357C2×C3×C4—181no—
358C2×C3×C4—181no—
359C2×C3×C4C2×D3211yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—62no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—121no—
371C2×Dic3—121no—
372D12—21no—
373D12—21no—
374Dic6—21no—
375Dic6—21no—
376Dic6—62no—
377Dic6—62no—
378Dic6—61no—
379Dic6—61no—
380Dic6—61no—
381Dic6—61no—
382Dic6—61no—
383Dic6—61no—
384Dic6—61no—
385Dic6—61no—
386Dic6—121no—
387Dic6—121no—
388C3×C8—32no—
389C3×C8—32no—
390C3×C8—62no—
391C3⋊C8—92no—
392C3⋊C8—92no—
393C3⋊C8—92no—
394C3⋊C8—182no—
395C3⋊C8—182no—
396C3⋊C8—182no—
397C3⋊C8—362no—
398C3⋊C8#62512yes—
399C2×C32—22no—
400C2×C32—21no—
401C2×C32—22no—
402C2×C32—22no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—41no—
406C2×C32—42no—
407C2×C32—241no—
408C2×C32#91512yes—
409C2×C32#91812yes—
410C2×C32#94011yescharacteristic
411C2×C32#94011yescharacteristic
412C2×C32#94011yescharacteristic
413C2×C32#95112yes—
414C3×D3—82no—
415C3×D3—81no—
416C3×D3—81no—
417C2×Q8—31no—
418C2×Q8—91no—
419C2×Q8—91no—
420C2×D4—31no—
421Q16—182no—
422Q16—542no—
423SD16—182no—
424SD16—542no—
425C2×C8—31no—
426C2×C8—271no—
427C4⋊C4—542no—
428C22⋊C4—182no—
429C42—91no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—122no—
435C22×C3—122no—
436C22×C3—121no—
437C22×C3—122no—
438C22×C3—121no—
439C22×C3—242no—
440C22×C3#380211yescharacteristic
441C22×C3D3×#6711yescharacteristic
442C22×C3D3×#6711yescharacteristic
443C2×D3—42no—
444C2×D3—41no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—122no—
454C3×C4—121no—
455C3×C4—121no—
456C3×C4—182no—
457C3×C4—361no—
458C3×C4—361no—
459C3×C4#380211yescharacteristic
460C3×C4C2×#62511yescharacteristic
461C3×C4D3×#6711yescharacteristic
462C3×C4D3×#6711yescharacteristic
463C3×C4#381111yescharacteristic
464C3×C4#381111yescharacteristic
465Dic3—41no—
466Dic3—62no—
467Dic3—62no—
468Dic3—122no—
469Dic3—122no—
470Dic3—122no—
471Dic3—121no—
472Dic3—121no—
473Dic3—121no—
474Dic3—121no—
475C32—21no—
476C32—21no—
477C32—21no—
478C32—41no—
479C32#506011yescharacteristic, complemented
480C32#516711yescharacteristic, complemented
481C32#519011yescharacteristic, complemented
482C23—61no—
483Q8—61no—
484Q8—61no—
485Q8—181no—
486Q8—181no—
487Q8—181no—
488Q8—181no—
489D4—61no—
490D4—61no—
491C2×C4—61no—
492C2×C4—92no—
493C2×C4—181no—
494C2×C4—181no—
495C2×C4D3311yescharacteristic
496C8—32no—
497C8—272no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—22no—
502C2×C3—22no—
503C2×C3—22no—
504C2×C3—41no—
505C2×C3—42no—
506C2×C3—242no—
507C2×C3—241no—
508C2×C3—241no—
509C2×C3#6381112yes—
510C2×C3#6391411yescharacteristic
511C2×C3#6392912yes—
512C2×C3#6394312yes—
513C2×C3#6397911yescharacteristic
514C2×C3#6397911yescharacteristic
515D3—81no—
516C22—121no—
517C22—122no—
518C22#8895311yescharacteristic
519C4—121no—
520C4—182no—
521C4—361no—
522C4—361no—
523C4D3×#62511yescharacteristic
524C4#8895411yescharacteristic
525C3—21no—
526C3—21no—
527C3—21no—
528C3—41no—
529C3#1059087011yescharacteristic, complemented
530C3#1059149111yescharacteristic, complemented
531C3#1059162711yescharacteristic, complemented
532C2—241no—
533C2#1172289112yes—
534C2#1172302911yescharacteristic
subgroupG/H
centerC22#88953
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. #11723678C2semidirect
2. C2×#88798C2semidirect
5. C2×C3×#3719C2semidirect
7. #11722020C2semidirect
36. C2×#3719D3semidirect
328. C33#497semidirect
479. C32#5060semidirect
480. C32#5167semidirect
481. C32#5190semidirect
529. C3#10590870semidirect
530. C3#10591491semidirect
531. C3#10591627semidirect

Maximal subgroups

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

subgroupindex
#117236782
C2×#887982
C2×#887922
C3×#638632
C2×C3×#37192
C2×C3×#37162
#117220202
#105916273
#105914913
#105908703