Finite Group Databasebeta

← #13156636

Subgroups — #13156636

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#11723677C211yesmaximal, characteristic
2C2×#88798C211yesmaximal, characteristic, complemented
3C2×#88793C211yesmaximal, characteristic
4C3×#63873C211yesmaximal, characteristic, complemented
5C2×C3×#3719C211yesmaximal, characteristic, complemented
6C2×C3×#3717C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic, complemented
8#10591608—31nomaximal
9#10591605—31nomaximal
10#10590870—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88814—22no—
19#88798—22no—
20#88793—22no—
21C3×#3724—22no—
22C3×#3719—22no—
23C3×#3717—22no—
24C2×#3756—31no—
25C2×#3755—31no—
26C3×#944—31no—
27C3×#944—31no—
28C2×C3×#216—31no—
29C2×C3×#215—31no—
30C2×C3×Dic12—31no—
31C2×C3×#204—31no—
32#63873D311yescharacteristic, complemented
33#63864—31no—
34#63863—31no—
35#63860—31no—
36C2×#3719D311yescharacteristic, complemented
37C2×#3718—31no—
38C2×#3718—31no—
39C2×#3718—61no—
40C2×#3717D311yescharacteristic
41C2×#3715—31no—
42C2×#3715—31no—
43C2×#3715—61no—
44#63642—31no—
45#63554—31no—
46#63554—31no—
47#63554—61no—
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D412yes—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3⋊(C3⋊(C3⋊C8))D412yes—
63C32×C3⋊C8D412yes—
64#5190—91no—
65#5167—91no—
66#5086—91no—
67C2×C32×Q8—31no—
68C2×C32×Q8—31no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C2×D311yescharacteristic
81C2×C3×Dic6C2×D311yescharacteristic
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C32×C8—31no—
84#3756—62no—
85#3755—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×#216—62no—
95C3×#215—62no—
96C3×Dic3⋊C4—62no—
97C3×Dic3⋊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—22no—
107#3722—62no—
108(C2×D3)⋊Dic3—62no—
109(C2×D3)⋊Dic3—62no—
110#3719—22no—
111#3718—62no—
112#3718—62no—
113#3718—64no—
114#3717—22no—
115#3715—62no—
116#3715—62no—
117#3715—64no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—42no—
121C33×C4C2×D411yescharacteristic
122C33×C4C2×D411yescharacteristic
123C3×C3⋊Dic3—22no—
124C3×C3⋊Dic3—41no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C2×C3×Q16—91no—
128C2×C3×SD16—91no—
129C3×C4⋊Q8—91no—
130#947—91no—
131#944—31no—
132#944—31no—
133C2×#216—31no—
134C2×#216—91no—
135C2×#216—91no—
136C2×#216—91no—
137C2×#216—181no—
138C2×#215—31no—
139C2×#215—91no—
140C2×#215—91no—
141C2×#215—181no—
142#936—91no—
143C2×#214—91no—
144C2×Dic12—31no—
145C2×#204—31no—
146#875—91no—
147#815—91no—
148#815—91no—
149#815—181no—
150#803—91no—
151C22×C3⋊D3—21no—
152C22×C3×D3—61no—
153C22×C3×D3—61no—
154C22×C3×D3—121no—
155C32×Q8—61no—
156C32×Q8—61no—
157C32×Q8—61no—
158C32×Q8—61no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—61no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3#6712yes—
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)—21no—
172#634—21no—
173#634—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—41no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—121no—
186C3×D12—62no—
187C3×D12—61no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—121no—
192C3×Dic6—22no—
193C3×Dic6—22no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—41no—
203C3×Dic6—41no—
204C3×Dic6—62no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—61no—
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—
224C3×C3⋊D3—81no—
225C2×C3×Q8—31no—
226C2×C3×Q8—31no—
227C2×C3×Q8—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—61no—
230C2×C3×Q8—61no—
231C2×C3×Q8—91no—
232C2×C3×D4—91no—
233C2×D12—31no—
234C2×D12—31no—
235C2×D12—61no—
236C2×Dic6—31no—
237C2×Dic6—31no—
238C2×Dic6—61no—
239C2×Dic6D3211yescharacteristic
240C2×Dic6D3211yescharacteristic
241C3×Q16—182no—
242C3×SD16—182no—
243C2×C3×C8—31no—
244C2×C3×C8—31no—
245C2×C3×C8—61no—
246C3×C4⋊C4—182no—
247C3×C42—91no—
248#217—182no—
249#216—62no—
250#216—182no—
251#216—182no—
252#216—182no—
253#216—184no—
254#215—62no—
255#215—182no—
256#215—182no—
257#215—184no—
258#214—182no—
259(C2×D3)⋊C4—182no—
260(C2×D3)⋊C4—182no—
261Dic3⋊C4—62no—
262Dic3⋊C4—62no—
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×C32D3×D411yescharacteristic
281C22×C32D3×D411yescharacteristic
282C22×C32C2×#6711yescharacteristic
283C2×C3⋊D3—41no—
284C2×C3⋊D3—42no—
285C2×C3×D3—122no—
286C2×C3×D3—122no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C2×C3×D3—122no—
290C2×C3×D3—242no—
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×C4—121no—
301C32×C4D3×D411yescharacteristic
302C32×C4D3×D411yescharacteristic
303C32×C4D3×D411yescharacteristic
304C32×C4D3×D411yescharacteristic
305C32×C4C2×#6711yescharacteristic
306C32×C4C2×#6711yescharacteristic
307C3⋊Dic3—22no—
308C3⋊Dic3—41no—
309C3×Dic3—42no—
310C3×Dic3—42no—
311C3×Dic3—41no—
312C3×Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—41no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C3×Dic3—122no—
318C3×Dic3—122no—
319C3×Dic3—121no—
320C3×Dic3—121no—
321C2×Q16—91no—
322C2×Q16—271no—
323C2×SD16—91no—
324C2×SD16—271no—
325C4⋊Q8—91no—
326#124—271no—
327#97—271no—
328C33#49711yescharacteristic, complemented, Sylow
329C23×C3—181no—
330C22×D3—61no—
331C22×D3—61no—
332C22×D3—121no—
333C3×Q8—62no—
334C3×Q8—62no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—61no—
341C3×Q8—61no—
342C3×Q8—61no—
343C3×Q8—121no—
344C3×Q8—121no—
345C3×Q8—181no—
346C3×Q8—181no—
347C3×D4—181no—
348C3×D4—181no—
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—61no—
356C2×C3×C4—61no—
357C2×C3×C4—92no—
358C2×C3×C4—121no—
359C2×C3×C4—121no—
360C2×C3×C4—181no—
361C2×C3×C4C2×D3211yescharacteristic
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—62no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—121no—
372D12—62no—
373D12—61no—
374D12—61no—
375D12—61no—
376D12—61no—
377D12—121no—
378Dic6—21no—
379Dic6—21no—
380Dic6—21no—
381Dic6—21no—
382Dic6—62no—
383Dic6—61no—
384Dic6—61no—
385Dic6—61no—
386Dic6—61no—
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#91512yes—
408C2×C32#91812yes—
409C2×C32#94011yescharacteristic
410C2×C32#94011yescharacteristic
411C2×C32#94011yescharacteristic
412C2×C32#94212yes—
413C3⋊D3—81no—
414C3×D3—242no—
415C3×D3—241no—
416C3×D3—241no—
417C2×Q8—31no—
418C2×Q8—31no—
419C2×Q8—91no—
420C2×D4—91no—
421Q16—182no—
422Q16—542no—
423SD16—182no—
424SD16—542no—
425C2×C8—31no—
426C2×C8—271no—
427C4⋊C4—182no—
428C22⋊C4—542no—
429C42—91no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—362no—
435C22×C3—361no—
436C22×C3#380211yescharacteristic
437C22×C3D3×#6711yescharacteristic
438C22×C3D3×#6711yescharacteristic
439C2×D3—122no—
440C2×D3—122no—
441C2×D3—121no—
442C2×D3—122no—
443C2×D3—121no—
444C2×D3—242no—
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—122no—
455C3×C4—121no—
456C3×C4—121no—
457C3×C4—121no—
458C3×C4—121no—
459C3×C4—182no—
460C3×C4—361no—
461C3×C4#380211yescharacteristic
462C3×C4C2×#62511yescharacteristic
463C3×C4D3×#6711yescharacteristic
464C3×C4D3×#6711yescharacteristic
465C3×C4#381111yescharacteristic
466C3×C4#381111yescharacteristic
467Dic3—41no—
468Dic3—41no—
469Dic3—62no—
470Dic3—62no—
471Dic3—122no—
472Dic3—122no—
473Dic3—121no—
474Dic3—121no—
475C32—21no—
476C32—21no—
477C32—21no—
478C32—41no—
479C32#508611yescharacteristic, complemented
480C32#516711yescharacteristic, complemented
481C32#519011yescharacteristic, complemented
482C23—181no—
483Q8—61no—
484Q8—61no—
485Q8—61no—
486Q8—61no—
487Q8—181no—
488Q8—181no—
489D4—181no—
490D4—181no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—92no—
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—721no—
507C2×C3#6381112yes—
508C2×C3#6391411yescharacteristic
509C2×C3#6394612yes—
510C2×C3#6394812yes—
511C2×C3#6397911yescharacteristic
512C2×C3#6397911yescharacteristic
513D3—242no—
514D3—241no—
515D3—241no—
516C22—361no—
517C22—362no—
518C22#8895311yescharacteristic
519C4—121no—
520C4—121no—
521C4—182no—
522C4—361no—
523C4D3×#62511yescharacteristic
524C4#8895411yescharacteristic
525C3—21no—
526C3—21no—
527C3—21no—
528C3—41no—
529C3#1059087011yescharacteristic, complemented
530C3#1059160511yescharacteristic, complemented
531C3#1059160811yescharacteristic, complemented
532C2—721no—
533C2#1172289212yes—
534C2#1172302911yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
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
2. C2×#88798C2semidirect
4. C3×#63873C2semidirect
5. C2×C3×#3719C2semidirect
7. #11722020C2semidirect
32. #63873D3semidirect
36. C2×#3719D3semidirect
328. C33#497semidirect
479. C32#5086semidirect
480. C32#5167semidirect
481. C32#5190semidirect
529. C3#10590870semidirect
530. C3#10591605semidirect
531. C3#10591608semidirect

Maximal subgroups

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

subgroupindex
#117236772
C2×#887982
C2×#887932
C3×#638732
C2×C3×#37192
C2×C3×#37172
#117220202
#105916083
#105916053
#105908703