Finite Group Databasebeta

← #13161531

Subgroups — #13161531

Order 1728.

810 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#11724618C211yesmaximal, characteristic
2C3×#64081C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4C3×#63632C212yesmaximal, complemented
5#11722077C212yesmaximal, complemented
6#10590394—31nomaximal
7#10590392—32nomaximal
8C3×#3828C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10#89049—21no—
11#89046—21no—
12C4⋊#1113C411yescharacteristic, complemented
13C4×#1113—21no—
14#89042—21no—
15#89041C412yescomplemented
16#89040C411yescharacteristic, complemented
17C33×C42C2211yescharacteristic, complemented
18C3×#3752—21no—
19C3×#3750—21no—
20C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
21C3×#3748C2212yescomplemented
22#88809—21no—
23#88808—21no—
24#64306—31no—
25D3×#235—32no—
26#64081D311yescharacteristic, complemented
27C3×#872—32no—
28C3×#872—61no—
29#63818—31no—
30D3×#208—32no—
31#63632D312yescomplemented
32C3×#805—32no—
33C3×#805—32no—
34C3×#805—62no—
35#63570—32no—
36#63569—32no—
37#63568—32no—
38C22×C3×C3⋊D3D411yescharacteristic
39C2×#1113—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C3×#638—22no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4⋊(C3⋊D3)C2×C412yes—
46C3×C4×C3⋊D3—22no—
47C3×C4×C3⋊D3C2×C411yescharacteristic
48C3×C4×C3⋊D3C2×C411yescharacteristic
49C3×C4×C3⋊D3D411yescharacteristic
50C3×C4×C3⋊D3D411yescharacteristic
51C3×#634C2×C412yes—
52#6296—21no—
53#6295—22no—
54#6295—21no—
55#6294—21no—
56#6294—22no—
57C3×C3⋊(C3⋊C8)—22no—
58#5005—91no—
59#4821—92no—
60#3828C2×D312yes—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C3×#235—32no—
63C3×#235—32no—
64C3×#235—62no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—61no—
67#3811—61no—
68D3×#67—62no—
69#3805—61no—
70#3804—62no—
71#3802—31no—
72D3×D12—32no—
73C4×D32—62no—
74C4×D32—61no—
75#3799—32no—
76#3798—62no—
77#3798—61no—
78#3797—31no—
79#3796—32no—
80#3795—32no—
81D3×Dic6—32no—
82C32×C42—21no—
83C32×C42—22no—
84C32×C42—41no—
85C32×C42C2×D311yescharacteristic, complemented
86C32×C42C2×D312yescomplemented
87#3752—21no—
88#3750—21no—
89C4×C3⋊Dic3C2×D311yescharacteristic, complemented
90#3748C2×D312yescomplemented
91C3×(C2×D3)⋊C4—62no—
92C3×(C2×D3)⋊C4—62no—
93C3×Dic3⋊C4—62no—
94C3×Dic3⋊C4—62no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—61no—
97C3×#208—32no—
98C3×#208—32no—
99C3×#208—62no—
100#3712—61no—
101#3711—62no—
102#3710—61no—
103D3×C3⋊C8—62no—
104C22×C33C2×D411yescharacteristic
105C2×C3×C3⋊D3—22no—
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3C22⋊C411yescharacteristic
108C2×C3×C3⋊D3C22⋊C411yescharacteristic
109#1113—42no—
110C33×C4—41no—
111C33×C4C22×C411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3×C3⋊Dic3—22no—
114C3×C3⋊Dic3—41no—
115C3×C3⋊Dic3C22⋊C411yescharacteristic
116C3×C3⋊Dic3C22⋊C411yescharacteristic
117D3×#41—92no—
118C2×#235—91no—
119C3×#117—91no—
120C2×#208—91no—
121D3×M16—92no—
122#872—32no—
123#872—61no—
124C3×#104—92no—
125#837—92no—
126#825—92no—
127#805—32no—
128#805—32no—
129#805—62no—
130#805—92no—
131C22×C3⋊D3#6711yescharacteristic
132C22×C3×D3—32no—
133C22×C3×D3—61no—
134C2×D32—62no—
135C2×D32—61no—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4—42no—
143C2×C32×C4—42no—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D312yes—
146#638—22no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3#6711yescharacteristic
149C4⋊(C3⋊D3)C4×D312yes—
150C4×C3⋊D3—22no—
151C4×C3⋊D3C4×D311yescharacteristic
152C4×C3⋊D3C4×D311yescharacteristic
153C4×C3⋊D3D1211yescharacteristic
154C4×C3⋊D3D1211yescharacteristic
155#634C4×D312yes—
156C3×#67—62no—
157C3×#67—62no—
158C3×#67—122no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C3×D12—32no—
164C3×D12—32no—
165C3×D12—62no—
166C3×C4×D3—32no—
167C3×C4×D3—32no—
168C3×C4×D3—32no—
169C3×C4×D3—32no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×C4×D3—61no—
173C3×C4×D3—61no—
174C3×C4×D3—62no—
175C3×C4×D3—62no—
176C3×C4×D3—122no—
177C3×Dic6—32no—
178C3×Dic6—32no—
179C3×Dic6—62no—
180#627—62no—
181#627—61no—
182#626—62no—
183#626—62no—
184#626—62no—
185#625—62no—
186#625—61no—
187#624—62no—
188#624—61no—
189D3×Dic3—62no—
190D3×Dic3—62no—
191D3×Dic3—62no—
192C3⋊(C3⋊C8)—22no—
193C3×C3⋊C8—62no—
194C3×C3⋊C8—62no—
195C3×C3⋊C8—122no—
196#421—271noSylow
197C2×C33—21no—
198C2×C33C2×C22⋊C411yescharacteristic
199C3×C3⋊D3—21no—
200C3×C3⋊D3—21no—
201C3×C3⋊D3—42no—
202C3×#41—92no—
203C22×C3×C4—91no—
204C2×#67—181no—
205#239—92no—
206D3×Q8—92no—
207#237—92no—
208#237—182no—
209D3×D4—92no—
210D3×D4—182no—
211#235—32no—
212#235—32no—
213#235—62no—
214#235—92no—
215#235—182no—
216#235—181no—
217#235—181no—
218C2×D12—91no—
219C2×C4×D3—32no—
220C2×C4×D3—61no—
221C2×C4×D3—182no—
222C2×C4×D3—181no—
223C2×Dic6—91no—
224C3×M16—92no—
225C3×C4⋊C4—181no—
226C3×C22⋊C4—181no—
227C3×C42—21no—
228C3×C42—22no—
229C3×C42—41no—
230C3×C42—91no—
231C3×C42D3211yescharacteristic, complemented
232C3×C42D3212yescomplemented
233(C2×D3)⋊C4—62no—
234(C2×D3)⋊C4—62no—
235Dic3⋊C4—62no—
236Dic3⋊C4—62no—
237C4×Dic3—32no—
238C4×Dic3—61no—
239#208—32no—
240#208—32no—
241#208—62no—
242#208—181no—
243C2×C3⋊C8—181no—
244ΓL(1,𝔽25)—182no—
245D3×C8—182no—
246C22×C32—22no—
247C22×C32—21no—
248C22×C32—41no—
249C22×C32D3×D412yes—
250C22×C32C2×#6711yescharacteristic
251C2×C3⋊D3—22no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
254C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
255C2×C3×D3—32no—
256C2×C3×D3—32no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—62no—
260C2×C3×D3—62no—
261C2×C3×D3—122no—
262C2×C3×D3—122no—
263C2×C3×D3—122no—
264D32—124no—
265D32—122no—
266C32×C4—22no—
267C32×C4—21no—
268C32×C4—22no—
269C32×C4—21no—
270C32×C4—41no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—42no—
274C32×C4—42no—
275C32×C4—44no—
276C32×C4—44no—
277C32×C4C2×C4×D312yes—
278C32×C4C2×C4×D311yescharacteristic
279C32×C4C2×D1211yescharacteristic
280C32×C4D3×D412yes—
281C3⋊Dic3—22no—
282C3⋊Dic3—41no—
283C3⋊Dic3(C2×D3)⋊C411yescharacteristic
284C3⋊Dic3(C2×D3)⋊C411yescharacteristic
285C3×Dic3—32no—
286C3×Dic3—32no—
287C3×Dic3—61no—
288C3×Dic3—61no—
289C3×Dic3—62no—
290C3×Dic3—62no—
291C3×Dic3—122no—
292C3×Dic3—122no—
293C3×Dic3—122no—
294C2×#41—271no—
295C2×M16—271no—
296#117—91no—
297#104—92no—
298#104—272no—
299C33#42111yescharacteristic, complemented, Sylow
300C23×C3—91no—
301C22×D3—32no—
302C22×D3—61no—
303C22×D3—182no—
304C22×D3—181no—
305C3×Q8—92no—
306C3×D4—92no—
307C3×D4—182no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—91no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—91no—
320C2×C3×C4—91no—
321C2×C3×C4—182no—
322C2×C3×C4—181no—
323C2×C3×C4C2×D3212yes—
324C2×C3×C4C2×D3211yescharacteristic
325#67—62no—
326#67—62no—
327#67—122no—
328#67—182no—
329#67—182no—
330#67—182no—
331C2×Dic3—32no—
332C2×Dic3—61no—
333C2×Dic3—62no—
334C2×Dic3—62no—
335C2×Dic3—182no—
336C2×Dic3—181no—
337D12—32no—
338D12—32no—
339D12—62no—
340D12—181no—
341D12—182no—
342C4×D3—32no—
343C4×D3—32no—
344C4×D3—32no—
345C4×D3—32no—
346C4×D3—61no—
347C4×D3—61no—
348C4×D3—61no—
349C4×D3—61no—
350C4×D3—62no—
351C4×D3—62no—
352C4×D3—122no—
353C4×D3—182no—
354C4×D3—182no—
355C4×D3—182no—
356Dic6—32no—
357Dic6—32no—
358Dic6—62no—
359Dic6—182no—
360Dic6—181no—
361C3×C8—182no—
362C3⋊C8—62no—
363C3⋊C8—62no—
364C3⋊C8—122no—
365C2×C32—21no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—22no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—42no—
372C2×C32—42no—
373C2×C32D3×C22⋊C412yes—
374C2×C32C2×(C2×D3)⋊C411yescharacteristic
375C3⋊D3—21no—
376C3⋊D3—21no—
377C3⋊D3—42no—
378C3×D3—62no—
379C3×D3—62no—
380C3×D3—121no—
381C3×D3—121no—
382C3×D3—122no—
383C3×D3—122no—
384C3×D3—242no—
385#41—92no—
386#41—272no—
387#41—541no—
388#41—541no—
389C2×Q8—271no—
390C2×D4—271no—
391C2×D4—541no—
392C22×C4—91no—
393C22×C4—541no—
394M16—92no—
395M16—541no—
396C2×C8—541no—
397C4⋊C4—181no—
398C22⋊C4—181no—
399C42—91no—
400C42#111311yescharacteristic, complemented
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—91no—
405C22×C3—91no—
406C22×C3—182no—
407C22×C3—361no—
408C22×C3D3×#6712yes—
409C22×C3#381111yescharacteristic
410C2×D3—32no—
411C2×D3—32no—
412C2×D3—61no—
413C2×D3—61no—
414C2×D3—62no—
415C2×D3—62no—
416C2×D3—122no—
417C2×D3—122no—
418C2×D3—122no—
419C2×D3—364no—
420C2×D3—362no—
421C3×C4—22no—
422C3×C4—21no—
423C3×C4—22no—
424C3×C4—21no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—41no—
428C3×C4—44no—
429C3×C4—44no—
430C3×C4—41no—
431C3×C4—41no—
432C3×C4—91no—
433C3×C4—91no—
434C3×C4—182no—
435C3×C4—361no—
436C3×C4C4×D3212yes—
437C3×C4C4×D3211yescharacteristic
438C3×C4D3×D1212yes—
439C3×C4#380211yescharacteristic
440Dic3—32no—
441Dic3—32no—
442Dic3—61no—
443Dic3—61no—
444Dic3—62no—
445Dic3—62no—
446Dic3—122no—
447Dic3—122no—
448Dic3—122no—
449C32—22no—
450C32—21no—
451C32—41no—
452C32#482112yescomplemented
453C32#500511yescharacteristic, complemented
454C23—91no—
455C23—541no—
456Q8—92no—
457Q8—541no—
458D4—92no—
459D4—182no—
460D4—541no—
461D4—542no—
462C2×C4—21no—
463C2×C4—91no—
464C2×C4—91no—
465C2×C4—91no—
466C2×C4—91no—
467C2×C4—91no—
468C2×C4—182no—
469C2×C4—181no—
470C2×C4—542no—
471C2×C4—541no—
472C2×C4C2×#111311yescharacteristic
473C8—182no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—22no—
479C2×C3—42no—
480C2×C3—42no—
481C2×C3—41no—
482C2×C3—181no—
483C2×C3—181no—
484C2×C3—362no—
485C2×C3D3×(C2×D3)⋊C412yes—
486C2×C3#6392111yescharacteristic
487D3—62no—
488D3—62no—
489D3—121no—
490D3—121no—
491D3—122no—
492D3—122no—
493D3—242no—
494C22—91no—
495C22—91no—
496C22—182no—
497C22—361no—
498C22—1082no—
499C22#8904911yescharacteristic
500C4—41no—
501C4—91no—
502C4—91no—
503C4—182no—
504C4—361no—
505C4C4×#111311yescharacteristic
506C4C4⋊#111311yescharacteristic
507C3—21no—
508C3—22no—
509C3—41no—
510C3#1059039212yescomplemented
511C3#1059039411yescharacteristic, complemented
512C2—21no—
513C2—181no—
514C2—181no—
515C2—362no—
516C2#1172370011yescharacteristic
subgroupG/H
centerC4C4⋊#1113
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC2×C33×C4C23
hypercenterC42#1113
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#64081C2semidirect
3. #11723656C2semidirect
4. C3×#63632C2semidirect
5. #11722077C2semidirect
12. C4⋊#1113C4semidirect
15. #89041C4semidirect
16. #89040C4semidirect
17. C33×C42C22semidirect
20. C3×C4×C3⋊Dic3C22semidirect
21. C3×#3748C22semidirect
26. #64081D3semidirect
31. #63632D3semidirect
85. C32×C42C2×D3semidirect
86. C32×C42C2×D3semidirect
89. C4×C3⋊Dic3C2×D3semidirect
90. #3748C2×D3semidirect
231. C3×C42D32semidirect
232. C3×C42D32semidirect
299. C33#421semidirect
400. C42#1113semidirect
452. C32#4821semidirect
453. C32#5005semidirect
510. C3#10590392semidirect
511. C3#10590394semidirect

Maximal subgroups

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

subgroupindex
#117246182
C3×#640812
#117236562
C3×#636322
#117220772
#105903943
#105903923