Finite Group Databasebeta

← #13156028

Subgroups — #13156028

Order 1728.

638 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 102 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×#88812C211yesmaximal, characteristic
2#11723451C211yesmaximal, characteristic
3#11723410C211yesmaximal, characteristic
4C2×C3×#3719C211yesmaximal, characteristic
5#11722050C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591580—31nomaximal
9#10590798—31nomaximal
10#10590566—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88812—22no—
19#88807—21no—
20#88806—21no—
21#88806—21no—
22Dic3×C3⋊Dic3—21no—
23C3×#3719—22no—
24#64153—31no—
25#64096—31no—
26C2×C3×#216—31no—
27C2×C3×Dic12—31no—
28#63873—31no—
29#63873—31no—
30#63873—61no—
31#63843—31no—
32#63843—31no—
33#63843—61no—
34C2×#3719—31no—
35C2×#3719—31no—
36C2×#3719D311yescharacteristic
37C2×#3718—31no—
38#63641—31no—
39C3×#835—31no—
40C3×#816—31no—
41C3×#810—31no—
42C3×#810—31no—
43#63572—31no—
44#63572—31no—
45#63572—61no—
46#63567D311yescharacteristic
47#63565D311yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3⋊(C3⋊Dic3)D411yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634—21no—
55C3×#634—21no—
56C3×#634D412yes—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63#5182—91no—
64#4795—91no—
65#4795—91no—
66C2×C32×Q8—31no—
67C2×#634C2×D311yescharacteristic
68C2×#634C2×D311yescharacteristic
69C2×C3×Dic6—21no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—61no—
75C2×C3×Dic6—61no—
76C2×C3×Dic6C2×D311yescharacteristic
77C2×C3×Dic6C2×D311yescharacteristic
78C2×C32×C8—31no—
79C32×C4⋊C4—31no—
80#3750—61no—
81#3750—61no—
82#3750—61no—
83C4×C3⋊Dic3—31no—
84C4×C3⋊Dic3—31no—
85C4×C3⋊Dic3—31no—
86C4×C3⋊Dic3—61no—
87C4×C3⋊Dic3—61no—
88C4×C3⋊Dic3—61no—
89C4×C3⋊Dic3—61no—
90C4×C3⋊Dic3—121no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#216—62no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C2×D311yescharacteristic
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—61no—
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C3×Dic12—62no—
103#3724—61no—
104#3724—61no—
105#3724—61no—
106#3724—61no—
107#3724—62no—
108#3724—62no—
109Dic3⋊Dic3—61no—
110Dic3⋊Dic3—61no—
111Dic3⋊Dic3—62no—
112Dic32—61no—
113Dic32—61no—
114Dic32—62no—
115#3719—22no—
116#3719—62no—
117#3719—62no—
118#3718—62no—
119C22×C33C2×D411yescharacteristic
120C33×C4C2×D411yescharacteristic
121C33×C4#4111yescharacteristic
122C3⋊(C3⋊Dic3)—21no—
123C3⋊(C3⋊Dic3)Q1612yescomplemented
124C3×C3⋊Dic3—22no—
125C3×C3⋊Dic3—41no—
126C32×Dic3—41no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129Q8×Dic3—91no—
130#944—91no—
131#944—91no—
132#944—91no—
133#944—181no—
134C2×#216—31no—
135C2×#216—91no—
136C2×#216—91no—
137C2×Dic12—31no—
138C2×Dic12—91no—
139#887—91no—
140#887—91no—
141#887—181no—
142C3×#103—91no—
143C3×#103—91no—
144#835—31no—
145#816—31no—
146Dic3⋊C8—91no—
147Dic3⋊C8—91no—
148Dic3⋊C8—91no—
149Dic3⋊C8—181no—
150#810—31no—
151#810—31no—
152C32×Q8—61no—
153C32×Q8—61no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—41no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—31no—
166C2×C3⋊Dic3—31no—
167C2×C3⋊Dic3—31no—
168C2×C3⋊Dic3—31no—
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—121no—
178C2×C3⋊Dic3—121no—
179#634—21no—
180#634—21no—
181#634—21no—
182#634#6712yes—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—32no—
200C3×Dic6—32no—
201C3×Dic6—41no—
202C3×Dic6—41no—
203C3×Dic6—62no—
204C3×Dic6—62no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—121no—
212C3×Dic6—121no—
213C32×C8—61no—
214C3⋊(C3⋊C8)—21no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—41no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—121no—
221#462—271noSylow
222C2×C33#12111yescharacteristic
223C2×C33C2×Q1611yescharacteristic
224C2×C33#13711yescharacteristic
225C2×C3×Q8—31no—
226C2×C3×Q8—31no—
227C2×C3×Q8—61no—
228C2×C3×Q8—91no—
229C2×C3×Q8—91no—
230C2×Dic6—31no—
231C2×Dic6—31no—
232C2×Dic6—31no—
233C2×Dic6—31no—
234C2×Dic6—61no—
235C2×Dic6—61no—
236C2×Dic6D3211yescharacteristic
237C3×Q16—182no—
238C2×C3×C8—31no—
239C2×C3×C8—31no—
240C2×C3×C8—61no—
241C2×C3×C8—91no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—61no—
245#216—62no—
246#216—182no—
247#216—182no—
248C4⋊Dic3—181no—
249C4⋊Dic3D3211yescharacteristic
250Dic3⋊C4—181no—
251Dic3⋊C4—181no—
252Dic3⋊C4—181no—
253Dic3⋊C4—181no—
254Dic3⋊C4—181no—
255Dic3⋊C4—181no—
256Dic3⋊C4—181no—
257Dic3⋊C4—181no—
258Dic3⋊C4—182no—
259Dic3⋊C4—182no—
260Dic3⋊C4—182no—
261C4×Dic3—91no—
262C4×Dic3—91no—
263C4×Dic3—91no—
264C4×Dic3—181no—
265C4×Dic3—181no—
266C4×Dic3—181no—
267C4×Dic3—181no—
268C4×Dic3—181no—
269C4×Dic3—181no—
270C4×Dic3—182no—
271C4×Dic3—361no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—61no—
275C2×C3⋊C8D3211yescharacteristic
276Dic12—62no—
277Dic12—182no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—41no—
282C22×C32D3×D411yescharacteristic
283C22×C32D3×D411yescharacteristic
284C22×C32D3×D411yescharacteristic
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—41no—
292C32×C4—41no—
293C32×C4—121no—
294C32×C4—121no—
295C32×C4#23511yescharacteristic
296C32×C4#23511yescharacteristic
297C32×C4D3×D411yescharacteristic
298C32×C4D3×D411yescharacteristic
299C32×C4#23711yescharacteristic
300C32×C4C2×#6711yescharacteristic
301C3⋊Dic3—22no—
302C3⋊Dic3—32no—
303C3⋊Dic3—32no—
304C3⋊Dic3—32no—
305C3⋊Dic3—41no—
306C3⋊Dic3—62no—
307C3⋊Dic3—62no—
308C3⋊Dic3—62no—
309C3⋊Dic3—61no—
310C3⋊Dic3—62no—
311C3⋊Dic3—61no—
312C3⋊Dic3—61no—
313C3⋊Dic3—122no—
314C3⋊Dic3—121no—
315C3⋊Dic3—121no—
316C3⋊Dic3—122no—
317C3×Dic3—42no—
318C3×Dic3—42no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—41no—
323C3×Dic3—64no—
324C3×Dic3—62no—
325C3×Dic3—62no—
326C3×Dic3—122no—
327C3×Dic3—121no—
328C3×Dic3—121no—
329C2×Q16—91no—
330C2×Q16—271no—
331C4⋊Q8—271no—
332C4×Q8—271no—
333C4⋊C8—271no—
334#103—91no—
335#103—91no—
336C33#46211yescharacteristic, complemented, Sylow
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—61no—
341C3×Q8—92no—
342C3×Q8—121no—
343C3×Q8—121no—
344C3×Q8—181no—
345C3×Q8—181no—
346C3×Q8—181no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—62no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
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—21no—
364C2×Dic3—62no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—62no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—91no—
371C2×Dic3—91no—
372C2×Dic3—91no—
373C2×Dic3—91no—
374C2×Dic3—91no—
375C2×Dic3—91no—
376C2×Dic3—181no—
377C2×Dic3—181no—
378C2×Dic3—181no—
379C2×Dic3—181no—
380C2×Dic3—181no—
381C2×Dic3—181no—
382C2×Dic3—361no—
383C2×Dic3—361no—
384Dic6—21no—
385Dic6—21no—
386Dic6—32no—
387Dic6—32no—
388Dic6—62no—
389Dic6—62no—
390Dic6—61no—
391Dic6—61no—
392Dic6—61no—
393Dic6—61no—
394Dic6—61no—
395Dic6—61no—
396Dic6—121no—
397Dic6—121no—
398C3×C8—61no—
399C3×C8—61no—
400C3×C8—121no—
401C3×C8—181no—
402C3⋊C8—21no—
403C3⋊C8—61no—
404C3⋊C8—61no—
405C3⋊C8—121no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32#88411yescharacteristic
419C2×C32#88411yescharacteristic
420C2×C32#91511yescharacteristic
421C2×C32#91511yescharacteristic
422C2×C32D3×Q1611yescharacteristic
423C2×C32D3×Q1611yescharacteristic
424C2×C32D3×Q1611yescharacteristic
425C2×C32#91811yescharacteristic
426C2×C32#93911yescharacteristic
427C2×Q8—31no—
428C2×Q8—91no—
429C2×Q8—91no—
430Q16—182no—
431Q16—542no—
432C2×C8—31no—
433C2×C8—91no—
434C4⋊C4—31no—
435C4⋊C4—541no—
436C4⋊C4—541no—
437C4⋊C4—541no—
438C42—271no—
439C42—541no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—41no—
444C22×C3#380211yescharacteristic
445C22×C3#381111yescharacteristic
446C22×C3#381111yescharacteristic
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—122no—
456C3×C4—122no—
457C3×C4—121no—
458C3×C4—121no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—182no—
462C3×C4—361no—
463C3×C4#379511yescharacteristic
464C3×C4#379811yescharacteristic
465C3×C4#380211yescharacteristic
466C3×C4#380411yescharacteristic
467C3×C4D3×#6711yescharacteristic
468C3×C4D3×#6711yescharacteristic
469Dic3—41no—
470Dic3—41no—
471Dic3—64no—
472Dic3—62no—
473Dic3—62no—
474Dic3—92no—
475Dic3—92no—
476Dic3—92no—
477Dic3—122no—
478Dic3—121no—
479Dic3—121no—
480Dic3—182no—
481Dic3—182no—
482Dic3—182no—
483Dic3—182no—
484Dic3—181no—
485Dic3—181no—
486Dic3—181no—
487Dic3—362no—
488Dic3—361no—
489Dic3—361no—
490Dic3—362no—
491C32—21no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32#479511yescharacteristic, complemented
496C32#479511yescharacteristic, complemented
497C32#518211yescharacteristic, complemented
498Q8—61no—
499Q8—61no—
500Q8—92no—
501Q8—181no—
502Q8—181no—
503Q8—181no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—181no—
507C2×C4—181no—
508C2×C4—271no—
509C2×C4—271no—
510C2×C4D3311yescharacteristic
511C8—61no—
512C8—181no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—41no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3#6380211yescharacteristic
526C2×C3#6381111yescharacteristic
527C2×C3#6388511yescharacteristic
528C2×C3#6391211yescharacteristic
529C2×C3#6393111yescharacteristic
530C2×C3#6394411yescharacteristic
531C2×C3#6394411yescharacteristic
532C2×C3#6395111yescharacteristic
533C2×C3#6397111yescharacteristic
534C22#8895411yescharacteristic
535C4—121no—
536C4—121no—
537C4—182no—
538C4—272no—
539C4—361no—
540C4—541no—
541C4#8895311yescharacteristic
542C4#8896011yescharacteristic
543C3—21no—
544C3—21no—
545C3—21no—
546C3—41no—
547C3#1059056611yescharacteristic, complemented
548C3#1059079811yescharacteristic, complemented
549C3#1059158011yescharacteristic, complemented
550C2#1172284811yescharacteristic
551C2#1172289311yescharacteristic
552C2#1172296711yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#462
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
123. C3⋊(C3⋊Dic3)Q16semidirect
336. C33#462semidirect
495. C32#4795semidirect
496. C32#4795semidirect
497. C32#5182semidirect
547. C3#10590566semidirect
548. C3#10590798semidirect
549. C3#10591580semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
#117234512
#117234102
C2×C3×#37192
#117220502
C3×#635672
C3×#635652
#105915803
#105907983
#105905663