Finite Group Databasebeta

← #13161996

Subgroups — #13161996

Order 1728.

640 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64153C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic
3#11723668C211yesmaximal, characteristic
4C2×#88812C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic
6#11722078C211yesmaximal, characteristic
7#11722076C211yesmaximal, characteristic
8#10591580—31nomaximal
9#10591575—31nomaximal
10#10591574—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C3×#3756—22no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×#3750—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#88815—21no—
22#88813—21no—
23#88812—22no—
24#64153D311yescharacteristic
25C2×#3756D311yescharacteristic
26C3×#944—31no—
27C3×#944—31no—
28C3×#944—61no—
29C2×C3×#216—31no—
30C2×C3×#216—31no—
31C2×C3×#216—61no—
32#63844—31no—
33#63843—31no—
34Dic3×Dic6—31no—
35C2×#3719—31no—
36C2×#3719—31no—
37C2×#3718—31no—
38#63662D311yescharacteristic
39C3×#835—31no—
40C3×#835—31no—
41C3×#835—61no—
42#63572—31no—
43Dic3⋊(C3⋊C8)—31no—
44Dic3⋊(C3⋊C8)—31no—
45#63567—31no—
46#63566—31no—
47#63565—31no—
48C33×Q8—21no—
49C33×Q8—21no—
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×#634—21no—
58C3×#634—21no—
59C3×#634—21no—
60C3×#634D412yes—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63#5182—91no—
64#5090—91no—
65#4795—91no—
66C2×C32×Q8—21no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—41no—
70C2×C32×Q8C2×D311yescharacteristic
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×#634C2×D311yescharacteristic
74C2×#634C2×D311yescharacteristic
75C2×C3×Dic6—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—61no—
80C2×C3×Dic6—61no—
81#3756—22no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83#3750—21no—
84#3750—21no—
85C4×C3⋊Dic3C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C3×#216—62no—
89C3×#216—62no—
90C3×#216—64no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—61no—
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—62no—
99C3×Dic3⋊C4—62no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—61no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8—61no—
109#3724—61no—
110Dic3⋊Dic3—61no—
111Dic3⋊Dic3—61no—
112Dic32—61no—
113Dic32—61no—
114Dic32—61no—
115#3719—62no—
116#3719—62no—
117#3718—62no—
118C22×C33C2×D411yescharacteristic
119C33×C4—41no—
120C33×C4C2×D411yescharacteristic
121C33×C4#4111yescharacteristic
122C3×C3⋊Dic3—22no—
123C3×C3⋊Dic3—21no—
124C3×C3⋊Dic3—41no—
125C3×C3⋊Dic3—41no—
126C3×C3⋊Dic3Q1612yescomplemented
127C2×C3×Q16—91no—
128C3×C4⋊Q8—91no—
129Q8×Dic3—91no—
130#944—31no—
131#944—31no—
132#944—61no—
133C2×#216—31no—
134C2×#216—31no—
135C2×#216—61no—
136C2×#216—91no—
137C2×#216—91no—
138C2×Dic12—91no—
139#887—91no—
140C4×Dic6—91no—
141C3×#103—91no—
142#835—31no—
143#835—31no—
144#835—61no—
145#835—91no—
146#816—91no—
147Dic3⋊C8—91no—
148Dic3⋊C8—91no—
149#810—91no—
150C4⋊(C3⋊C8)—91no—
151C32×Q8—21no—
152C32×Q8—21no—
153C32×Q8—22no—
154C32×Q8—22no—
155C32×Q8—21no—
156C32×Q8—21no—
157C32×Q8—21no—
158C32×Q8—21no—
159C32×Q8—41no—
160C32×Q8—41no—
161C32×Q8—41no—
162C32×Q8—41no—
163C32×Q8—42no—
164C32×Q8—42no—
165C2×C32×C4—21no—
166C2×C32×C4—22no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—42no—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3#6711yescharacteristic
183C2×C3⋊Dic3#6711yescharacteristic
184#634—21no—
185#634—21no—
186#634—21no—
187#634#6712yes—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—121no—
202C2×C3×Dic3—121no—
203C3×Dic6—32no—
204C3×Dic6—32no—
205C3×Dic6—62no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—121no—
213C3×Dic6—121no—
214C3×Dic6—121no—
215C3⋊(C3⋊C8)—21no—
216C3⋊(C3⋊C8)—21no—
217C3×C3⋊C8—62no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—121no—
223#462—271noSylow
224C2×C33#12111yescharacteristic
225C2×C33C2×Q1611yescharacteristic
226C2×C33#13711yescharacteristic
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8—41no—
231C2×C3×Q8—91no—
232C2×C3×Q8—91no—
233C2×C3×Q8D3211yescharacteristic
234C2×C3×Q8D3211yescharacteristic
235C2×C3×Q8D3211yescharacteristic
236C2×Dic6—31no—
237C2×Dic6—31no—
238C2×Dic6—31no—
239C2×Dic6—31no—
240C2×Dic6—61no—
241C2×Dic6—61no—
242C3×Q16—182no—
243C2×C3×C8—91no—
244C2×C3×C8—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—181no—
247C3×C4⋊C4—181no—
248C3×C42—91no—
249#216—62no—
250#216—62no—
251#216—64no—
252#216—182no—
253#216—182no—
254C4⋊Dic3—31no—
255C4⋊Dic3—31no—
256C4⋊Dic3—61no—
257C4⋊Dic3—181no—
258Dic3⋊C4—61no—
259Dic3⋊C4—61no—
260Dic3⋊C4—61no—
261Dic3⋊C4—61no—
262Dic3⋊C4—62no—
263Dic3⋊C4—62no—
264Dic3⋊C4—181no—
265Dic3⋊C4—181no—
266C4×Dic3—31no—
267C4×Dic3—31no—
268C4×Dic3—61no—
269C4×Dic3—181no—
270C4×Dic3—181no—
271C4×Dic3—181no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—61no—
277C2×C3⋊C8—61no—
278Dic12—182no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32D3×D411yescharacteristic
284C22×C32D3×D411yescharacteristic
285C22×C32C2×#6711yescharacteristic
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—41no—
293C32×C4—41no—
294C32×C4—42no—
295C32×C4—42no—
296C32×C4—42no—
297C32×C4—44no—
298C32×C4—41no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4#23511yescharacteristic
302C32×C4D3×D411yescharacteristic
303C32×C4D3×D411yescharacteristic
304C32×C4#23711yescharacteristic
305C32×C4#23711yescharacteristic
306C32×C4C2×#6711yescharacteristic
307C3⋊Dic3—21no—
308C3⋊Dic3—22no—
309C3⋊Dic3—41no—
310C3⋊Dic3—41no—
311C3⋊Dic3#21612yescomplemented
312C3×Dic3—32no—
313C3×Dic3—32no—
314C3×Dic3—62no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C3×Dic3—61no—
318C3×Dic3—61no—
319C3×Dic3—62no—
320C3×Dic3—122no—
321C3×Dic3—122no—
322C3×Dic3—121no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325C3×Dic3—121no—
326C3×Dic3—241no—
327C2×Q16—91no—
328C2×Q16—271no—
329C4⋊Q8—91no—
330C4×Q8—271no—
331C4⋊C8—271no—
332#103—91no—
333#103—271no—
334C33#46211yescharacteristic, complemented, Sylow
335C3×Q8—22no—
336C3×Q8—22no—
337C3×Q8—21no—
338C3×Q8—21no—
339C3×Q8—21no—
340C3×Q8—21no—
341C3×Q8—21no—
342C3×Q8—21no—
343C3×Q8—41no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×Q8—41no—
347C3×Q8—42no—
348C3×Q8—42no—
349C3×Q8—92no—
350C3×Q8—181no—
351C3×Q8—181no—
352C3×Q8—181no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4—181no—
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—61no—
377C2×Dic3—61no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—62no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—121no—
386C2×Dic3—121no—
387Dic6—32no—
388Dic6—32no—
389Dic6—62no—
390Dic6—61no—
391Dic6—61no—
392Dic6—61no—
393Dic6—61no—
394Dic6—61no—
395Dic6—61no—
396Dic6—121no—
397Dic6—121no—
398Dic6—121no—
399C3×C8—181no—
400C3×C8—181no—
401C3⋊C8—62no—
402C3⋊C8—61no—
403C3⋊C8—61no—
404C3⋊C8—61no—
405C3⋊C8—61no—
406C3⋊C8—121no—
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—21no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32#88411yescharacteristic
420C2×C32#91511yescharacteristic
421C2×C32D3×Q1611yescharacteristic
422C2×C32D3×Q1611yescharacteristic
423C2×C32#91811yescharacteristic
424C2×C32#93811yescharacteristic
425C2×C32#93911yescharacteristic
426C2×C32#94211yescharacteristic
427C2×C32C2×#21611yescharacteristic
428C2×Q8—91no—
429C2×Q8—91no—
430C2×Q8#111311yescharacteristic
431Q16—182no—
432Q16—542no—
433C2×C8—91no—
434C2×C8—91no—
435C4⋊C4—91no—
436C4⋊C4—181no—
437C4⋊C4—181no—
438C4⋊C4—541no—
439C42—91no—
440C42—541no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—41no—
445C22×C3D3×#6711yescharacteristic
446C22×C3D3×#6711yescharacteristic
447C22×C3#381111yescharacteristic
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—44no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—41no—
461C3×C4—41no—
462C3×C4—41no—
463C3×C4—92no—
464C3×C4—181no—
465C3×C4—182no—
466C3×C4—361no—
467C3×C4—361no—
468C3×C4#380411yescharacteristic
469C3×C4#380411yescharacteristic
470C3×C4#380511yescharacteristic
471C3×C4D3×#6711yescharacteristic
472C3×C4D3×#6711yescharacteristic
473C3×C4#381111yescharacteristic
474Dic3—32no—
475Dic3—32no—
476Dic3—62no—
477Dic3—62no—
478Dic3—61no—
479Dic3—61no—
480Dic3—62no—
481Dic3—62no—
482Dic3—122no—
483Dic3—122no—
484Dic3—121no—
485Dic3—121no—
486Dic3—121no—
487Dic3—121no—
488Dic3—241no—
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#479511yescharacteristic, complemented
494C32#509011yescharacteristic, complemented
495C32#518211yescharacteristic, complemented
496Q8—21no—
497Q8—21no—
498Q8—92no—
499Q8—181no—
500Q8—181no—
501Q8—181no—
502C2×C4—21no—
503C2×C4—91no—
504C2×C4—91no—
505C2×C4—181no—
506C2×C4—181no—
507C2×C4—181no—
508C2×C4C2×#111311yescharacteristic
509C8—181no—
510C8—181no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—41no—
523C2×C3D3×#21611yescharacteristic
524C2×C3D3×#21611yescharacteristic
525C2×C3#6394411yescharacteristic
526C2×C3#6394611yescharacteristic
527C2×C3#6394811yescharacteristic
528C2×C3#6395111yescharacteristic
529C2×C3#6395811yescharacteristic
530C2×C3#6397011yescharacteristic
531C2×C3#6397111yescharacteristic
532C22#8904911yescharacteristic
533C4—41no—
534C4—92no—
535C4—181no—
536C4—182no—
537C4—361no—
538C4—361no—
539C4#8904611yescharacteristic
540C4#8904911yescharacteristic
541C3—21no—
542C3—21no—
543C3—21no—
544C3—41no—
545C3#1059157411yescharacteristic, complemented
546C3#1059157511yescharacteristic, complemented
547C3#1059158011yescharacteristic, complemented
548C2#1172371011yescharacteristic
549C2#1172371211yescharacteristic
550C2#1172372511yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8#1113
hypocenterC33#462
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
126. C3×C3⋊Dic3Q16semidirect
311. C3⋊Dic3#216semidirect
334. C33#462semidirect
493. C32#4795semidirect
494. C32#5090semidirect
495. C32#5182semidirect
545. C3#10591574semidirect
546. C3#10591575semidirect
547. C3#10591580semidirect

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
#117236682
C2×#888122
C3×#636622
#117220782
#117220762
#105915803
#105915753
#105915743