Finite Group Databasebeta

← #13156235

Subgroups — #13156235

Order 1728.

622 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 88 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×#88789C211yesmaximal, characteristic, complemented
2#11723350C211yesmaximal, characteristic, complemented
3C3×#63825C211yesmaximal, characteristic, complemented
4C3×#63823C211yesmaximal, characteristic, complemented
5#11722073C211yesmaximal, characteristic, complemented
6#11722059C211yesmaximal, characteristic, complemented
7C3×#63558C211yesmaximal, characteristic, complemented
8#10591678—31nomaximal
9#10591508—31nomaximal
10#10590982—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C32×#208C2211yescharacteristic
18#88789—21no—
19#88789—21no—
20#88786—21no—
21D3×C3⋊(C3⋊C8)—21no—
22C3×#3717—21no—
23C3×#3715—21no—
24C3×#3714—21no—
25C3×#3713—21no—
26#64157—31no—
27C2×#3753—31no—
28C3×#949—31no—
29C3×#949—31no—
30C3×#932—31no—
31C3×#908—31no—
32#63825D311yescharacteristic, complemented
33#63823D311yescharacteristic, complemented
34C2×#3713—31no—
35C2×#3713—31no—
36C2×#3713—61no—
37#63815—31no—
38#63815—31no—
39#63815—61no—
40#63649—31no—
41C3×#833—31no—
42C3×#823—31no—
43#63578—31no—
44#63578—31no—
45#63578—61no—
46#63559—31no—
47#63558—31no—
48#63558—31no—
49#63558D311yescharacteristic, complemented
50C22×C3×C3⋊D3—21no—
51C22×C32×D3—21no—
52C2×C33×C4C2311yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C32×#67—21no—
56C32×D12—21no—
57C32×D12—21no—
58C32×D12D411yescharacteristic
59C32×C4×D3—21no—
60C32×Dic6D411yescharacteristic
61C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
62C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
63C3×C3⋊(C3⋊C8)—21no—
64C3×C3⋊(C3⋊C8)—21no—
65C32×C3⋊C8—21no—
66#5199—91no—
67#5199—91no—
68#4749—91no—
69C32×#41—31no—
70C2×C32×D4—31no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C3×#235—21no—
73C3×#235C2×D311yescharacteristic
74C3×#235C2×D311yescharacteristic
75C2×C3×D12—21no—
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C2×D311yescharacteristic
81C32×M16—31no—
82#3753—61no—
83#3753—61no—
84#3748—61no—
85#3748C2×D311yescharacteristic
86#3748C2×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)—61no—
94C2×C3⋊(C3⋊C8)—121no—
95C3×#215—61no—
96C3×#215—61no—
97C3×#214—61no—
98C3×#213—61no—
99C3×#213—61no—
100C3×#213—61no—
101C3×#208—21no—
102C3×#208—31no—
103C3×#208—31no—
104C3×#208—31no—
105C3×#208—31no—
106C3×#208—61no—
107C3×#208—61no—
108C3×#208C2×D311yescharacteristic
109C3×#208C2×D311yescharacteristic
110C3×D24—61no—
111C3×#204—61no—
112#3717—21no—
113#3715—21no—
114#3714—21no—
115#3713—21no—
116#3713—61no—
117#3713—61no—
118#3713—61no—
119#3713—61no—
120#3713—62no—
121#3713—62no—
122#3711—61no—
123#3711—61no—
124#3711—62no—
125D3×C3⋊C8—61no—
126D3×C3⋊C8—61no—
127D3×C3⋊C8—62no—
128C22×C33#4111yescharacteristic
129C2×C3×C3⋊D3—41no—
130C2×C3×C3⋊D3—81no—
131C2×C32×D3—21no—
132C2×C32×D3—41no—
133C2×C32×D3—81no—
134C33×C4C2×D411yescharacteristic
135C33×C4C2×D411yescharacteristic
136C32×Dic3—21no—
137C3×AGL(1,ℤ/8)—91no—
138C3×AGL(1,ℤ/8)—91no—
139#949—31no—
140#949—31no—
141#948—91no—
142#948—91no—
143#948—181no—
144#932—31no—
145C2×#213—91no—
146C2×#213—91no—
147C2×#213—91no—
148C2×#213—181no—
149#908—31no—
150#907—91no—
151C3×#100—91no—
152#833—31no—
153#833—91no—
154#823—31no—
155#823—91no—
156#823—91no—
157#822—91no—
158#822—91no—
159#822—91no—
160#822—181no—
161C23×C32—61no—
162C22×C3⋊D3—21no—
163C22×C3×D3—21no—
164C22×C3×D3—21no—
165C22×C3×D3—41no—
166C22×C3×D3—61no—
167C22×C3×D3—61no—
168C22×C3×D3—121no—
169C32×Q8—31no—
170C32×D4—31no—
171C32×D4—61no—
172C32×D4—61no—
173C32×D4—61no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—41no—
178C2×C32×C4—61no—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C4⋊(C3⋊D3)—21no—
183C4⋊(C3⋊D3)—21no—
184C3×#67—22no—
185C3×#67—21no—
186C3×#67—21no—
187C3×D12—22no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—21no—
193C3×D12—41no—
194C3×D12—62no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—121no—
200C3×D12#6711yescharacteristic
201C3×D12#6711yescharacteristic
202C3×C4×D3—22no—
203C3×C4×D3—21no—
204C3×C4×D3—21no—
205C3×Dic6—21no—
206C3×Dic6#6711yescharacteristic
207C3×Dic6#6711yescharacteristic
208C32×C8—61no—
209C3⋊(C3⋊C8)—21no—
210C3⋊(C3⋊C8)—21no—
211C3⋊(C3⋊C8)—31no—
212C3⋊(C3⋊C8)—31no—
213C3⋊(C3⋊C8)—31no—
214C3⋊(C3⋊C8)—31no—
215C3⋊(C3⋊C8)—31no—
216C3⋊(C3⋊C8)—31no—
217C3⋊(C3⋊C8)—61no—
218C3⋊(C3⋊C8)—61no—
219C3⋊(C3⋊C8)—61no—
220C3⋊(C3⋊C8)—61no—
221C3⋊(C3⋊C8)—61no—
222C3⋊(C3⋊C8)—61no—
223C3⋊(C3⋊C8)—121no—
224C3⋊(C3⋊C8)—121no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—41no—
228C3×C3⋊C8—62no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—121no—
234#472—271noSylow
235C2×C33—21no—
236C2×C33#12111yescharacteristic
237C3×C3⋊D3—81no—
238C32×D3—41no—
239C32×D3—81no—
240C3×#41—31no—
241C3×#41—31no—
242C3×#41—61no—
243C2×C3×D4—31no—
244C2×C3×D4—31no—
245C2×C3×D4—61no—
246C2×C3×D4—91no—
247#235D3211yescharacteristic
248C2×D12—31no—
249C2×D12—31no—
250C2×D12—61no—
251C2×D12D3211yescharacteristic
252C3×SD16—181no—
253C3×SD16—181no—
254C3×D8—181no—
255C3×D8—181no—
256C3×M16—31no—
257C3×M16—31no—
258C3×M16—61no—
259C3×M16—91no—
260C3×M16—91no—
261#215—61no—
262#215—61no—
263#214—61no—
264#213—61no—
265#213—61no—
266#213—61no—
267#213—181no—
268#213—181no—
269#213—181no—
270#213—181no—
271#213—181no—
272#213—181no—
273#213—182no—
274#213—182no—
275#208—31no—
276#208—31no—
277#208—31no—
278#208—31no—
279#208—61no—
280#208—61no—
281#208—181no—
282#208—181no—
283#208—182no—
284#208D3211yescharacteristic
285C2×C3⋊C8—91no—
286C2×C3⋊C8—91no—
287C2×C3⋊C8—91no—
288C2×C3⋊C8—181no—
289C2×C3⋊C8—181no—
290C2×C3⋊C8—181no—
291C2×C3⋊C8—181no—
292C2×C3⋊C8—181no—
293C2×C3⋊C8—182no—
294C2×C3⋊C8—361no—
295D24—61no—
296#204—61no—
297ΓL(1,𝔽25)—181no—
298D3×C8—181no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—41no—
303C22×C32—61no—
304C22×C32—121no—
305C22×C32—241no—
306C22×C32#23511yescharacteristic
307C22×C32#23711yescharacteristic
308C22×C32#23711yescharacteristic
309C2×C3⋊D3—41no—
310C2×C3⋊D3—81no—
311C2×C3×D3—22no—
312C2×C3×D3—21no—
313C2×C3×D3—21no—
314C2×C3×D3—42no—
315C2×C3×D3—41no—
316C2×C3×D3—41no—
317C2×C3×D3—82no—
318C2×C3×D3—81no—
319C2×C3×D3—81no—
320C2×C3×D3—122no—
321C2×C3×D3—121no—
322C2×C3×D3—121no—
323C2×C3×D3—242no—
324C2×C3×D3—241no—
325C2×C3×D3—241no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—41no—
333C32×C4—41no—
334C32×C4—61no—
335C32×C4D3×D411yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4D3×D411yescharacteristic
338C32×C4D3×D411yescharacteristic
339C32×C4C2×#6711yescharacteristic
340C32×C4C2×#6711yescharacteristic
341C3×Dic3—22no—
342C3×Dic3—21no—
343C3×Dic3—21no—
344AGL(1,ℤ/8)—91no—
345AGL(1,ℤ/8)—91no—
346C2×D8—271no—
347#131—271no—
348#108—271no—
349#100—91no—
350#100—271no—
351C33#47211yescharacteristic, complemented, Sylow
352C23×C3—61no—
353C23×C3—61no—
354C23×C3—121no—
355C23×C3—181no—
356C22×D3—21no—
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—121no—
360C3×Q8—31no—
361C3×Q8—31no—
362C3×Q8—61no—
363C3×D4—31no—
364C3×D4—31no—
365C3×D4—61no—
366C3×D4—62no—
367C3×D4—62no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—61no—
372C3×D4—61no—
373C3×D4—61no—
374C3×D4—121no—
375C3×D4—181no—
376C3×D4—181no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—62no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4C2×D3211yescharacteristic
385C2×C3×C4C2×D3211yescharacteristic
386C2×C3×C4C2×D3211yescharacteristic
387#67—21no—
388D12—21no—
389D12—21no—
390D12—62no—
391D12—61no—
392D12—61no—
393D12—61no—
394D12—61no—
395D12—121no—
396D12#62511yescharacteristic
397C4×D3—21no—
398Dic6#62511yescharacteristic
399C3×C8—61no—
400C3×C8—61no—
401C3×C8—121no—
402C3×C8—181no—
403C3×C8—181no—
404C3⋊C8—21no—
405C3⋊C8—62no—
406C3⋊C8—61no—
407C3⋊C8—61no—
408C3⋊C8—61no—
409C3⋊C8—61no—
410C3⋊C8—91no—
411C3⋊C8—91no—
412C3⋊C8—91no—
413C3⋊C8—91no—
414C3⋊C8—91no—
415C3⋊C8—91no—
416C3⋊C8—121no—
417C3⋊C8—181no—
418C3⋊C8—181no—
419C3⋊C8—181no—
420C3⋊C8—181no—
421C3⋊C8—181no—
422C3⋊C8—181no—
423C3⋊C8—361no—
424C3⋊C8—361no—
425C2×C32—21no—
426C2×C32—22no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—41no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—42no—
436C2×C32—121no—
437C2×C32—241no—
438C2×C32#88411yescharacteristic
439C2×C32#93911yescharacteristic
440C2×C32#93911yescharacteristic
441C3⋊D3—81no—
442C3×D3—42no—
443C3×D3—41no—
444C3×D3—41no—
445C3×D3—82no—
446C3×D3—81no—
447C3×D3—81no—
448C3×D3—242no—
449C3×D3—241no—
450C3×D3—241no—
451#41—31no—
452C2×D4—31no—
453C2×D4—91no—
454SD16—181no—
455SD16—181no—
456D8—181no—
457D8—181no—
458D8—541no—
459D8—541no—
460M16—31no—
461M16—91no—
462M16—91no—
463M16—541no—
464C2×C8—271no—
465C2×C8—541no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—62no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—122no—
474C22×C3—121no—
475C22×C3—121no—
476C22×C3—241no—
477C22×C3—241no—
478C22×C3—242no—
479C22×C3—361no—
480C22×C3—721no—
481C22×C3#379511yescharacteristic
482C22×C3#380411yescharacteristic
483C22×C3#380511yescharacteristic
484C2×D3—21no—
485C2×D3—41no—
486C2×D3—81no—
487C2×D3—122no—
488C2×D3—121no—
489C2×D3—121no—
490C2×D3—242no—
491C2×D3—241no—
492C2×D3—241no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—62no—
502C3×C4—61no—
503C3×C4—61no—
504C3×C4#380211yescharacteristic
505C3×C4C2×#62511yescharacteristic
506C3×C4D3×#6711yescharacteristic
507C3×C4D3×#6711yescharacteristic
508C3×C4#381111yescharacteristic
509C3×C4#381111yescharacteristic
510Dic3—21no—
511C32—21no—
512C32—21no—
513C32—21no—
514C32—41no—
515C32#474911yescharacteristic, complemented
516C32#519911yescharacteristic, complemented
517C32#519911yescharacteristic, complemented
518C23—61no—
519C23—181no—
520Q8—31no—
521D4—31no—
522D4—61no—
523D4—61no—
524D4—61no—
525D4—181no—
526D4—181no—
527C2×C4—61no—
528C2×C4D3311yescharacteristic
529C8—61no—
530C8—181no—
531C8—181no—
532C8—271no—
533C8—271no—
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—41no—
542C2×C3—41no—
543C2×C3—42no—
544C2×C3—41no—
545C2×C3—122no—
546C2×C3—121no—
547C2×C3—121no—
548C2×C3—242no—
549C2×C3—241no—
550C2×C3—241no—
551C2×C3—721no—
552C2×C3#6388511yescharacteristic
553C2×C3#6397111yescharacteristic
554C2×C3#6398011yescharacteristic
555D3—41no—
556D3—81no—
557D3—242no—
558D3—241no—
559D3—241no—
560C22—61no—
561C22—121no—
562C22—241no—
563C22—361no—
564C22—721no—
565C22#8896111yescharacteristic
566C4—61no—
567C4D3×#62511yescharacteristic
568C4#8895411yescharacteristic
569C3—21no—
570C3—21no—
571C3—21no—
572C3—41no—
573C3#1059098211yescharacteristic, complemented
574C3#1059150811yescharacteristic, complemented
575C3#1059167811yescharacteristic, complemented
576C2—21no—
577C2—121no—
578C2—241no—
579C2—721no—
580C2#1172295411yescharacteristic
subgroupG/H
centerC2#11722954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#121
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88789C2semidirect
2. #11723350C2semidirect
3. C3×#63825C2semidirect
4. C3×#63823C2semidirect
5. #11722073C2semidirect
6. #11722059C2semidirect
7. C3×#63558C2semidirect
32. #63825D3semidirect
33. #63823D3semidirect
49. #63558D3semidirect
351. C33#472semidirect
515. C32#4749semidirect
516. C32#5199semidirect
517. C32#5199semidirect
573. C3#10590982semidirect
574. C3#10591508semidirect
575. C3#10591678semidirect

Maximal subgroups

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

subgroupindex
C2×#887892
#117233502
C3×#638252
C3×#638232
#117220732
#117220592
C3×#635582
#105916783
#105915083
#105909823