Finite Group Databasebeta

← #13162028

Subgroups — #13162028

Order 1728.

620 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C3×#64151C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3#11723661C211yesmaximal, characteristic
4#11723657C211yesmaximal, characteristic, complemented
5C3×#63663C211yesmaximal, characteristic, complemented
6#11722081C211yesmaximal, characteristic, complemented
7#11722073C211yesmaximal, characteristic, complemented
8#10591643—31nomaximal
9#10591642—31nomaximal
10#10591542—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×#3756—21no—
15C3×#3755—21no—
16C3×#3755—21no—
17C3×#3755—21no—
18C3×#3748C2211yescharacteristic
19C3×#3748C2211yescharacteristic
20C3×#3748C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88811—21no—
23#88810—21no—
24#88809—21no—
25#88808—21no—
26#64151D311yescharacteristic, complemented
27C2×#3755D311yescharacteristic, complemented
28C3×#942—31no—
29C3×#942—31no—
30C3×#942—61no—
31C2×C3×#215—31no—
32C2×C3×#215—31no—
33C2×C3×#215—61no—
34#63830—31no—
35#63825—31no—
36#63823—31no—
37#63817—31no—
38#63815—31no—
39#63814—31no—
40#63663D311yescharacteristic, complemented
41C3×#836—31no—
42C3×#836—31no—
43C3×#836—61no—
44#63578—31no—
45#63577—31no—
46#63577—31no—
47#63559—31no—
48#63558—31no—
49#63558—31no—
50C22×C3×C3⋊D3—21no—
51C33×Q8—21no—
52C33×Q8—21no—
53C2×C33×C4—21no—
54C2×C33×C4C2311yescharacteristic
55C3×#638—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)D411yescharacteristic
59C3×C4×C3⋊D3—21no—
60C3×#634D411yescharacteristic
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C3×C3⋊(C3⋊C8)D411yescharacteristic
65C3×C3⋊(C3⋊C8)D411yescharacteristic
66#5203—91no—
67#5177—91no—
68#4754—91no—
69C2×C32×Q8—21no—
70C2×C32×Q8—21no—
71C2×C32×Q8—21no—
72C2×C32×Q8—41no—
73C2×C32×Q8C2×D311yescharacteristic
74C2×C32×Q8C2×D311yescharacteristic
75C2×C32×Q8C2×D311yescharacteristic
76#3828C2×D311yescharacteristic
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C3×#235—31no—
79C3×#235—31no—
80C3×#235—61no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—61no—
84#3756—21no—
85#3755—21no—
86#3755—21no—
87#3755—21no—
88#3748C2×D311yescharacteristic
89#3748C2×D311yescharacteristic
90#3748C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#216—61no—
93C3×#216—61no—
94C3×#216—62no—
95C3×#215—61no—
96C3×#215—61no—
97C3×#215—61no—
98C3×#215—61no—
99C3×#215—61no—
100C3×#215—61no—
101C3×#215—62no—
102C3×#215—62no—
103C3×#215—62no—
104C3×#208—31no—
105C3×#208—31no—
106C3×#208—31no—
107C3×#208—31no—
108C3×#208—31no—
109C3×#208—31no—
110C3×#208—61no—
111C3×#208—61no—
112C3×#208—61no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—61no—
116#3717—61no—
117#3716—61no—
118#3715—61no—
119#3714—61no—
120#3714—61no—
121#3713—61no—
122#3712—61no—
123#3711—61no—
124#3711—61no—
125#3710—61no—
126D3×C3⋊C8—61no—
127D3×C3⋊C8—61no—
128C22×C33#4111yescharacteristic
129C2×C3×C3⋊D3—21no—
130C2×C3×C3⋊D3—41no—
131C2×C3×C3⋊D3—81no—
132C33×C4—41no—
133C33×C4C2×D411yescharacteristic
134C33×C4C2×D411yescharacteristic
135C3×C3⋊Dic3—21no—
136C3×#137—91no—
137C2×C3×SD16—91no—
138#949—91no—
139#948—91no—
140#942—31no—
141#942—31no—
142#942—61no—
143C2×#215—31no—
144C2×#215—31no—
145C2×#215—61no—
146#932—91no—
147#908—91no—
148#907—91no—
149#901—91no—
150C3×#101—91no—
151#836—31no—
152#836—31no—
153#836—61no—
154#833—91no—
155#823—91no—
156#823—91no—
157#822—91no—
158#822—91no—
159#819—91no—
160C22×C3⋊D3—21no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—121no—
164C32×Q8—21no—
165C32×Q8—21no—
166C32×Q8—22no—
167C32×Q8—22no—
168C32×Q8—21no—
169C32×Q8—21no—
170C32×Q8—21no—
171C32×Q8—21no—
172C32×Q8—41no—
173C32×Q8—41no—
174C32×Q8—41no—
175C32×Q8—41no—
176C32×Q8—42no—
177C32×Q8—42no—
178C2×C32×C4—21no—
179C2×C32×C4—22no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—41no—
186C2×C32×C4—41no—
187C2×C32×C4—41no—
188C2×C32×C4—42no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192#638—21no—
193C4⋊(C3⋊D3)—21no—
194C4⋊(C3⋊D3)—21no—
195C4⋊(C3⋊D3)#6711yescharacteristic
196C4×C3⋊D3—21no—
197#634#6711yescharacteristic
198C3×#67—61no—
199C3×#67—61no—
200C3×#67—121no—
201C3×D12—31no—
202C3×D12—31no—
203C3×D12—61no—
204C3×D12—62no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—121no—
210C3×C4×D3—61no—
211C3×C4×D3—61no—
212C3×C4×D3—121no—
213C3×Dic6—31no—
214C3×Dic6—31no—
215C3×Dic6—61no—
216C3⋊(C3⋊C8)—21no—
217C3⋊(C3⋊C8)—21no—
218C3⋊(C3⋊C8)—21no—
219C3⋊(C3⋊C8)#6711yescharacteristic
220C3⋊(C3⋊C8)#6711yescharacteristic
221C3×C3⋊C8—31no—
222C3×C3⋊C8—31no—
223C3×C3⋊C8—31no—
224C3×C3⋊C8—31no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—62no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—121no—
235C3×C3⋊C8—121no—
236#471—271noSylow
237C2×C33—21no—
238C2×C33#12111yescharacteristic
239C3×C3⋊D3—41no—
240C3×C3⋊D3—81no—
241C3×#41—91no—
242C2×C3×Q8—21no—
243C2×C3×Q8—21no—
244C2×C3×Q8—21no—
245C2×C3×Q8—41no—
246C2×C3×Q8D3211yescharacteristic
247C2×C3×Q8D3211yescharacteristic
248C2×C3×Q8D3211yescharacteristic
249C2×C3×D4—91no—
250#235—31no—
251#235—31no—
252#235—61no—
253C2×D12—31no—
254C2×D12—31no—
255C2×D12—61no—
256C3×Q16—181no—
257C3×SD16—181no—
258C3×SD16—181no—
259C3×SD16—181no—
260C3×M16—91no—
261C3×M16—91no—
262C3×M16—91no—
263C2×C3×C8—91no—
264#216—61no—
265#216—61no—
266#216—62no—
267#215—61no—
268#215—61no—
269#215—61no—
270#215—61no—
271#215—61no—
272#215—61no—
273#215—62no—
274#215—62no—
275#215—62no—
276#215—181no—
277#214—181no—
278#213—181no—
279#213—181no—
280#208—31no—
281#208—31no—
282#208—31no—
283#208—31no—
284#208—31no—
285#208—31no—
286#208—61no—
287#208—61no—
288#208—61no—
289#208—181no—
290C2×C3⋊C8—31no—
291C2×C3⋊C8—31no—
292C2×C3⋊C8—61no—
293C2×C3⋊C8—181no—
294D24—181no—
295#204—181no—
296ΓL(1,𝔽25)—181no—
297ΓL(1,𝔽25)—181no—
298D3×C8—181no—
299D3×C8—181no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—41no—
304C22×C32#23511yescharacteristic
305C22×C32#23711yescharacteristic
306C22×C32#23711yescharacteristic
307C2×C3⋊D3—21no—
308C2×C3⋊D3—41no—
309C2×C3⋊D3—81no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C2×C3×D3—121no—
313C2×C3×D3—122no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C2×C3×D3—242no—
317C2×C3×D3—241no—
318C2×C3×D3—241no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—41no—
326C32×C4—41no—
327C32×C4—42no—
328C32×C4—42no—
329C32×C4—42no—
330C32×C4—44no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4—41no—
334C32×C4D3×D411yescharacteristic
335C32×C4D3×D411yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4D3×D411yescharacteristic
338C32×C4C2×#6711yescharacteristic
339C32×C4C2×#6711yescharacteristic
340C3⋊Dic3—21no—
341C3×Dic3—61no—
342C3×Dic3—61no—
343C3×Dic3—121no—
344#137—91no—
345AGL(1,ℤ/8)—271no—
346C2×SD16—91no—
347#131—271no—
348#108—271no—
349#101—91no—
350#100—271no—
351C33#47111yescharacteristic, complemented, Sylow
352C23×C3—181no—
353C22×D3—61no—
354C22×D3—61no—
355C22×D3—121no—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—21no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—21no—
362C3×Q8—21no—
363C3×Q8—21no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—42no—
369C3×Q8—42no—
370C3×Q8—91no—
371C3×D4—91no—
372C3×D4—181no—
373C3×D4—181no—
374C3×D4—181no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—42no—
385C2×C3×C4—41no—
386C2×C3×C4—181no—
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390#67—61no—
391#67—61no—
392#67—121no—
393D12—31no—
394D12—31no—
395D12—61no—
396D12—62no—
397D12—61no—
398D12—61no—
399D12—61no—
400D12—61no—
401D12—121no—
402C4×D3—61no—
403C4×D3—61no—
404C4×D3—121no—
405Dic6—31no—
406Dic6—31no—
407Dic6—61no—
408C3×C8—91no—
409C3×C8—91no—
410C3×C8—181no—
411C3×C8—181no—
412C3×C8—181no—
413C3⋊C8—31no—
414C3⋊C8—31no—
415C3⋊C8—31no—
416C3⋊C8—31no—
417C3⋊C8—61no—
418C3⋊C8—61no—
419C3⋊C8—62no—
420C3⋊C8—61no—
421C3⋊C8—61no—
422C3⋊C8—61no—
423C3⋊C8—61no—
424C3⋊C8—61no—
425C3⋊C8—61no—
426C3⋊C8—121no—
427C3⋊C8—121no—
428C2×C32—21no—
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32—41no—
438C2×C32—42no—
439C2×C32#88411yescharacteristic
440C2×C32#93811yescharacteristic
441C2×C32#93911yescharacteristic
442C3⋊D3—41no—
443C3⋊D3—81no—
444C3×D3—121no—
445C3×D3—121no—
446C3×D3—241no—
447C3×D3—242no—
448C3×D3—241no—
449C3×D3—241no—
450#41—91no—
451C2×Q8#111311yescharacteristic
452C2×D4—91no—
453Q16—181no—
454SD16—181no—
455SD16—181no—
456SD16—181no—
457SD16—541no—
458D8—541no—
459M16—91no—
460M16—91no—
461M16—91no—
462M16—541no—
463C2×C8—91no—
464C2×C8—541no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—41no—
469C22×C3—181no—
470C22×C3—361no—
471C22×C3—721no—
472C22×C3#380411yescharacteristic
473C22×C3#380411yescharacteristic
474C22×C3#380511yescharacteristic
475C2×D3—61no—
476C2×D3—61no—
477C2×D3—121no—
478C2×D3—122no—
479C2×D3—121no—
480C2×D3—121no—
481C2×D3—242no—
482C2×D3—241no—
483C2×D3—241no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—44no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—181no—
500C3×C4D3×#6711yescharacteristic
501C3×C4D3×#6711yescharacteristic
502C3×C4D3×#6711yescharacteristic
503C3×C4D3×#6711yescharacteristic
504C3×C4#381111yescharacteristic
505C3×C4#381111yescharacteristic
506Dic3—61no—
507Dic3—61no—
508Dic3—121no—
509C32—21no—
510C32—21no—
511C32—21no—
512C32—41no—
513C32#475411yescharacteristic, complemented
514C32#517711yescharacteristic, complemented
515C32#520311yescharacteristic, complemented
516C23—181no—
517Q8—21no—
518Q8—21no—
519Q8—91no—
520D4—91no—
521D4—181no—
522D4—181no—
523D4—181no—
524C2×C4—21no—
525C2×C4—181no—
526C2×C4C2×#111311yescharacteristic
527C8—91no—
528C8—91no—
529C8—181no—
530C8—181no—
531C8—181no—
532C2×C3—22no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—41no—
540C2×C3—41no—
541C2×C3—42no—
542C2×C3—41no—
543C2×C3—361no—
544C2×C3—721no—
545C2×C3#6395811yescharacteristic
546C2×C3#6397011yescharacteristic
547C2×C3#6397111yescharacteristic
548D3—121no—
549D3—121no—
550D3—241no—
551D3—242no—
552D3—241no—
553D3—241no—
554C22—181no—
555C22—361no—
556C22—721no—
557C22#8904611yescharacteristic
558C4—41no—
559C4—181no—
560C4#8904911yescharacteristic
561C4#8904911yescharacteristic
562C3—21no—
563C3—21no—
564C3—21no—
565C3—41no—
566C3#1059154211yescharacteristic, complemented
567C3#1059164211yescharacteristic, complemented
568C3#1059164311yescharacteristic, complemented
569C2—21no—
570C2—361no—
571C2—721no—
572C2#1172372511yescharacteristic
subgroupG/H
centerC2#11723725
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC2×C33#121
prime radical#11723661C2
hypercenterC2×Q8#1113
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#64151C2semidirect
2. C2×C3×#3755C2semidirect
4. #11723657C2semidirect
5. C3×#63663C2semidirect
6. #11722081C2semidirect
7. #11722073C2semidirect
26. #64151D3semidirect
27. C2×#3755D3semidirect
40. #63663D3semidirect
351. C33#471semidirect
513. C32#4754semidirect
514. C32#5177semidirect
515. C32#5203semidirect
566. C3#10591542semidirect
567. C3#10591642semidirect
568. C3#10591643semidirect

Maximal subgroups

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

subgroupindex
C3×#641512
C2×C3×#37552
#117236612
#117236572
C3×#636632
#117220812
#117220732
#105916433
#105916423
#105915423