Finite Group Databasebeta

← #13162030

Subgroups — #13162030

Order 1728.

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

normal subgroups only

This group has 596 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
1C2×C3×#3756C211yesmaximal, characteristic, complemented
2C3×#64151C211yesmaximal, characteristic
3#11723663C211yesmaximal, characteristic
4#11723657C211yesmaximal, characteristic
5C3×#63663C211yesmaximal, characteristic, complemented
6#11722081C211yesmaximal, characteristic, complemented
7#11722074C211yesmaximal, characteristic, complemented
8#10591644—31nomaximal
9#10591641—31nomaximal
10#10591543—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C3×#3756—21no—
15C3×#3756—21no—
16C3×#3756—21no—
17C3×#3755—21no—
18C3×#3748C2211yescharacteristic
19C3×#3748C2211yescharacteristic
20C3×#3748C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88812—21no—
23#88811—21no—
24#88809—21no—
25#88808—21no—
26C2×#3756D311yescharacteristic, complemented
27#64151D311yescharacteristic
28C2×C3×#216—31no—
29C2×C3×#216—31no—
30C2×C3×#216—61no—
31C3×#942—31no—
32C3×#942—31no—
33C3×#942—61no—
34#63833—31no—
35#63831—31no—
36#63827—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#63561—31no—
48#63560—31no—
49#63560—31no—
50C33×Q8—21no—
51C33×Q8—21no—
52C2×C33×C4—21no—
53C2×C33×C4C2311yescharacteristic
54C3×#638—21no—
55C2×C3×C3⋊Dic3—21no—
56C3×C4⋊(C3⋊D3)D411yescharacteristic
57C3×C4×C3⋊D3—21no—
58C3×#634—21no—
59C3×#634—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#5204—91no—
67#5192—91no—
68#4755—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×#634C2×D311yescharacteristic
78C3×#235—31no—
79C3×#235—31no—
80C3×#235—61no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84#3756—21no—
85#3756—21no—
86#3756—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—61no—
95C3×#216—61no—
96C3×#216—61no—
97C3×#216—61no—
98C3×#216—62no—
99C3×#216—62no—
100C3×#216—62no—
101C3×#215—61no—
102C3×#215—61no—
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#3719—61no—
117#3719—61no—
118#3718—61no—
119#3717—61no—
120#3716—61no—
121#3715—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—
130C33×C4—41no—
131C33×C4C2×D411yescharacteristic
132C33×C4C2×D411yescharacteristic
133C3×C3⋊Dic3—21no—
134C3×C3⋊Dic3—41no—
135C3×#137—91no—
136C2×C3×Q16—91no—
137#951—91no—
138#948—91no—
139C2×#216—31no—
140C2×#216—31no—
141C2×#216—61no—
142#942—31no—
143#942—31no—
144#942—61no—
145#942—91no—
146#909—91no—
147#907—91no—
148#901—91no—
149C3×#101—91no—
150#836—31no—
151#836—31no—
152#836—61no—
153#836—91no—
154#824—91no—
155#824—91no—
156#822—91no—
157#822—91no—
158#819—91no—
159C32×Q8—21no—
160C32×Q8—21no—
161C32×Q8—22no—
162C32×Q8—22no—
163C32×Q8—21no—
164C32×Q8—21no—
165C32×Q8—21no—
166C32×Q8—21no—
167C32×Q8—41no—
168C32×Q8—41no—
169C32×Q8—41no—
170C32×Q8—41no—
171C32×Q8—42no—
172C32×Q8—42no—
173C2×C32×C4—21no—
174C2×C32×C4—22no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—41no—
181C2×C32×C4—41no—
182C2×C32×C4—41no—
183C2×C32×C4—42no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187#638—21no—
188C2×C3⋊Dic3—21no—
189C4⋊(C3⋊D3)#6711yescharacteristic
190C4×C3⋊D3—21no—
191#634—21no—
192#634—21no—
193#634#6711yescharacteristic
194C3×#67—61no—
195C3×#67—61no—
196C3×#67—121no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—121no—
200C3×D12—31no—
201C3×D12—31no—
202C3×D12—61no—
203C3×C4×D3—61no—
204C3×C4×D3—61no—
205C3×C4×D3—121no—
206C3×Dic6—31no—
207C3×Dic6—31no—
208C3×Dic6—61no—
209C3×Dic6—62no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—121no—
215C3⋊(C3⋊C8)—21no—
216C3⋊(C3⋊C8)—21no—
217C3⋊(C3⋊C8)—21no—
218C3⋊(C3⋊C8)#6711yescharacteristic
219C3⋊(C3⋊C8)#6711yescharacteristic
220C3×C3⋊C8—31no—
221C3×C3⋊C8—31no—
222C3×C3⋊C8—31no—
223C3×C3⋊C8—31no—
224C3×C3⋊C8—61no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—62no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—121no—
234C3×C3⋊C8—121no—
235#473—271noSylow
236C2×C33—21no—
237C2×C33#12111yescharacteristic
238C3×C3⋊D3—41no—
239C3×#41—91no—
240C2×C3×Q8—21no—
241C2×C3×Q8—21no—
242C2×C3×Q8—21no—
243C2×C3×Q8—41no—
244C2×C3×Q8—91no—
245C2×C3×Q8D3211yescharacteristic
246C2×C3×Q8D3211yescharacteristic
247C2×C3×Q8D3211yescharacteristic
248#235—31no—
249#235—31no—
250#235—61no—
251C2×Dic6—31no—
252C2×Dic6—31no—
253C2×Dic6—61no—
254C3×Q16—181no—
255C3×Q16—181no—
256C3×Q16—181no—
257C3×SD16—181no—
258C3×M16—91no—
259C3×M16—91no—
260C3×M16—91no—
261C2×C3×C8—91no—
262#216—61no—
263#216—61no—
264#216—61no—
265#216—61no—
266#216—61no—
267#216—61no—
268#216—62no—
269#216—62no—
270#216—62no—
271#216—181no—
272#216—181no—
273#215—61no—
274#215—61no—
275#215—62no—
276#215—181no—
277#214—181no—
278#208—31no—
279#208—31no—
280#208—31no—
281#208—31no—
282#208—31no—
283#208—31no—
284#208—61no—
285#208—61no—
286#208—61no—
287#208—181no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—31no—
290C2×C3⋊C8—61no—
291C2×C3⋊C8—181no—
292Dic12—181no—
293#204—181no—
294ΓL(1,𝔽25)—181no—
295ΓL(1,𝔽25)—181no—
296D3×C8—181no—
297D3×C8—181no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—41no—
302C22×C32#23511yescharacteristic
303C22×C32#23711yescharacteristic
304C22×C32#23711yescharacteristic
305C2×C3⋊D3—21no—
306C2×C3×D3—61no—
307C2×C3×D3—61no—
308C2×C3×D3—121no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4—42no—
318C32×C4—42no—
319C32×C4—42no—
320C32×C4—44no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4D3×D411yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4D3×D411yescharacteristic
327C32×C4D3×D411yescharacteristic
328C32×C4C2×#6711yescharacteristic
329C32×C4C2×#6711yescharacteristic
330C3⋊Dic3—21no—
331C3⋊Dic3—41no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—121no—
335C3×Dic3—122no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338#137—91no—
339#137—271no—
340C2×Q16—91no—
341#131—271no—
342#108—271no—
343#101—91no—
344#101—271no—
345C33#47311yescharacteristic, complemented, Sylow
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—21no—
349C3×Q8—21no—
350C3×Q8—21no—
351C3×Q8—21no—
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—42no—
359C3×Q8—42no—
360C3×Q8—91no—
361C3×Q8—181no—
362C3×Q8—181no—
363C3×D4—91no—
364C3×D4—181no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—42no—
375C2×C3×C4—41no—
376C2×C3×C4—181no—
377C2×C3×C4—181no—
378C2×C3×C4C2×D3211yescharacteristic
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381#67—61no—
382#67—61no—
383#67—121no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—121no—
387D12—31no—
388D12—31no—
389D12—61no—
390C4×D3—61no—
391C4×D3—61no—
392C4×D3—121no—
393Dic6—31no—
394Dic6—31no—
395Dic6—61no—
396Dic6—62no—
397Dic6—61no—
398Dic6—61no—
399Dic6—61no—
400Dic6—61no—
401Dic6—121no—
402C3×C8—91no—
403C3×C8—91no—
404C3×C8—181no—
405C3×C8—181no—
406C3×C8—181no—
407C3⋊C8—31no—
408C3⋊C8—31no—
409C3⋊C8—31no—
410C3⋊C8—31no—
411C3⋊C8—61no—
412C3⋊C8—61no—
413C3⋊C8—62no—
414C3⋊C8—61no—
415C3⋊C8—61no—
416C3⋊C8—61no—
417C3⋊C8—61no—
418C3⋊C8—61no—
419C3⋊C8—61no—
420C3⋊C8—121no—
421C3⋊C8—121no—
422C2×C32—21no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32—41no—
432C2×C32—42no—
433C2×C32#88411yescharacteristic
434C2×C32#93811yescharacteristic
435C2×C32#93911yescharacteristic
436C3⋊D3—41no—
437C3×D3—121no—
438C3×D3—121no—
439C3×D3—241no—
440#41—91no—
441C2×Q8—91no—
442C2×Q8#111311yescharacteristic
443Q16—181no—
444Q16—181no—
445Q16—181no—
446Q16—541no—
447SD16—181no—
448SD16—541no—
449M16—91no—
450M16—91no—
451M16—91no—
452M16—541no—
453C2×C8—91no—
454C2×C8—541no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—41no—
459C22×C3—181no—
460C22×C3#380411yescharacteristic
461C22×C3#380411yescharacteristic
462C22×C3#380511yescharacteristic
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—121no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4—44no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—181no—
482C3×C4—361no—
483C3×C4D3×#6711yescharacteristic
484C3×C4D3×#6711yescharacteristic
485C3×C4D3×#6711yescharacteristic
486C3×C4D3×#6711yescharacteristic
487C3×C4#381111yescharacteristic
488C3×C4#381111yescharacteristic
489Dic3—61no—
490Dic3—61no—
491Dic3—121no—
492Dic3—122no—
493Dic3—121no—
494Dic3—121no—
495C32—21no—
496C32—21no—
497C32—21no—
498C32—41no—
499C32#475511yescharacteristic, complemented
500C32#519211yescharacteristic, complemented
501C32#520411yescharacteristic, complemented
502Q8—21no—
503Q8—21no—
504Q8—91no—
505Q8—181no—
506Q8—181no—
507D4—91no—
508D4—181no—
509C2×C4—21no—
510C2×C4—181no—
511C2×C4—181no—
512C2×C4C2×#111311yescharacteristic
513C8—91no—
514C8—91no—
515C8—181no—
516C8—181no—
517C8—181no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—41no—
526C2×C3—41no—
527C2×C3—42no—
528C2×C3—41no—
529C2×C3—361no—
530C2×C3#6395811yescharacteristic
531C2×C3#6397011yescharacteristic
532C2×C3#6397111yescharacteristic
533D3—121no—
534D3—121no—
535D3—241no—
536C22—181no—
537C22#8904611yescharacteristic
538C4—41no—
539C4—181no—
540C4—361no—
541C4#8904911yescharacteristic
542C4#8904911yescharacteristic
543C3—21no—
544C3—21no—
545C3—21no—
546C3—41no—
547C3#1059154311yescharacteristic, complemented
548C3#1059164111yescharacteristic, complemented
549C3#1059164411yescharacteristic, complemented
550C2—21no—
551C2—361no—
552C2#1172372511yescharacteristic
subgroupG/H
centerC2#11723725
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC2×C33#121
prime radicalC3×#3828C22
hypercenterC2×Q8#1113
hypocenterC33#473
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. C2×C3×#3756C2semidirect
5. C3×#63663C2semidirect
6. #11722081C2semidirect
7. #11722074C2semidirect
26. C2×#3756D3semidirect
40. #63663D3semidirect
345. C33#473semidirect
499. C32#4755semidirect
500. C32#5192semidirect
501. C32#5204semidirect
547. C3#10591543semidirect
548. C3#10591641semidirect
549. C3#10591644semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37562
C3×#641512
#117236632
#117236572
C3×#636632
#117220812
#117220742
#105916443
#105916413
#105915433