Finite Group Databasebeta

← #13181080

Subgroups — #13181080

Order 1728.

914 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 123 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×#89040C211yesmaximal, characteristic, complemented
2C3×#64306C211yesmaximal, characteristic
3#11724511C211yesmaximal, characteristic, complemented
4#11724328C214yesmaximal, complemented
5#11724311C212yesmaximal, complemented
6#11724310C212yesmaximal, complemented
7#11724309C214yesmaximal, complemented
8#10596505—31nomaximal
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#634—21no—
11C32×#235—21no—
12C2×#6296C2212yescomplemented
13C2×#6294—21no—
14#89040—21no—
15#89040—21no—
16#89040C2212yes—
17#89040C2212yescomplemented
18C3×#3811—21no—
19C3×#3805C2212yes—
20C3×#3802C2211yescharacteristic
21C3×C4×D32—21no—
22C3×#3798C2212yes—
23C3×#3797C2211yescharacteristic
24C3×#3796—21no—
25#88993C2212yescomplemented
26#88989C2212yescomplemented
27#88988C2212yes—
28#88938C2218yescomplemented
29#88937C2218yescomplemented
30C3×D3×#41—61no—
31#64306D311yescharacteristic
32C2×#3797—31no—
33C2×D3×Dic6—61no—
34#64283—31no—
35#64239—34no—
36#64228—32no—
37#64227—32no—
38#64226—34no—
39C22×C3×C3⋊D3D411yescharacteristic
40C2×C3×D32—21no—
41C2×C33×C4D411yescharacteristic
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3C2311yescharacteristic
44C3×C4×C3⋊D3D411yescharacteristic
45C3×C4×C3⋊D3D411yescharacteristic
46C3×C4×C3⋊D3C2311yescharacteristic
47C3×C4×C3⋊D3C2311yescharacteristic
48C3×#634—22no—
49C3×#634—22no—
50C32×#67—22no—
51C32×D12—21no—
52C32×C4×D3—22no—
53C32×Dic6—21no—
54#6296—22no—
55#6296C2314yes—
56#6294—22no—
57#6294—22no—
58C3×#627C2312yes—
59C3×#626—22no—
60C3×#625C2312yes—
61C3×#624—21no—
62C3×D3×Dic3—22no—
63C3⋊#622C2314yes—
64C2×#951—91nomaximal
65C32×#41—61no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×#634—21no—
68C3×#239—61no—
69C3×D3×Q8—61no—
70C3×#237—61no—
71C3×#237—62no—
72C3×D3×D4—61no—
73C3×D3×D4—62no—
74C3×#235—21no—
75C3×#235—21no—
76C3×#235—41no—
77C3×#235—62no—
78C2×C3×C4×D3—62no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—61no—
82C2×C3×Dic6—61no—
83C2×C3×Dic6—121no—
84#3811—21no—
85C2×#627—32no—
86C2×#627—62no—
87C2×#624—61no—
88#3805C2×D312yes—
89C2×D3×Dic3—62no—
90#3802C2×D311yescharacteristic
91C4×D32—21no—
92#3798C2×D312yes—
93#3797—32no—
94#3797—32no—
95#3797—61no—
96#3797—61no—
97#3797C2×D311yescharacteristic
98#3796—21no—
99D3×Dic6—62no—
100D3×Dic6—62no—
101D3×Dic6—62no—
102D3×Dic6—62no—
103#3792—32no—
104#3788—32no—
105#3787—32no—
106#3776—38no—
107#3775—38no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3C2×D411yescharacteristic
111C2×C3×C3⋊D3C2×D411yescharacteristic
112C2×C32×D3—22no—
113C3×D32—42no—
114C33×C4C2×D411yescharacteristic
115C33×C4C2×D411yescharacteristic
116C3×C3⋊Dic3—24no—
117C3×C3⋊Dic3C2×D411yescharacteristic
118C3×C3⋊Dic3C2411yescharacteristic
119C32×Dic3—22no—
120C2×C3×#41—91no—
121D3×#41—61no—
122C2×D3×Q8—181no—
123C22×Dic6—91no—
124#951—94no—
125#951—94no—
126C2×#216—92no—
127C2×#214—92no—
128C2×#208—91no—
129C22×C3⋊D3#6711yescharacteristic
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—62no—
133C2×D32—21no—
134C32×Q8—61no—
135C32×D4—61no—
136C32×D4—62no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4—41no—
141C2×C32×C4—62no—
142C2×C32×C4#6711yescharacteristic
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3C22×D311yescharacteristic
145C4×C3⋊D3#6711yescharacteristic
146C4×C3⋊D3#6711yescharacteristic
147C4×C3⋊D3C22×D311yescharacteristic
148C4×C3⋊D3C22×D311yescharacteristic
149#634—22no—
150#634—22no—
151C3×#67—22no—
152C3×#67—22no—
153C3×#67—42no—
154C3×#67—62no—
155C3×#67—62no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—122no—
162C3×D12—21no—
163C3×D12—21no—
164C3×D12—41no—
165C3×D12—62no—
166C3×C4×D3—22no—
167C3×C4×D3—22no—
168C3×C4×D3—42no—
169C3×C4×D3—61no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×C4×D3—61no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—62no—
178C3×C4×D3—62no—
179C3×Dic6—21no—
180C3×Dic6—21no—
181C3×Dic6—41no—
182C3×Dic6—62no—
183C3×Dic6—62no—
184C3×Dic6—62no—
185C3×Dic6—62no—
186C3×Dic6—62no—
187C3×Dic6—122no—
188C3×Dic6—122no—
189#627—34no—
190#627—64no—
191#627—64no—
192#627—62no—
193#627C22×D312yes—
194#626—22no—
195#625C22×D312yes—
196#624—21no—
197#624—62no—
198#624—62no—
199D3×Dic3—22no—
200D3×Dic3—64no—
201D3×Dic3—64no—
202#622—34no—
203C2×#137—271noSylow
204C2×C33—21no—
205C2×C33C22×D411yescharacteristic
206C3×C3⋊D3—21no—
207C3×C3⋊D3#13712yescomplemented
208C32×D3—42no—
209C3×#41—61no—
210C3×#41—61no—
211C3×#41—92no—
212C3×#41—92no—
213C3×#41—121no—
214C3×#41—181no—
215C2×C3×Q8—91no—
216C2×C3×Q8—181no—
217C2×C3×D4—91no—
218C2×C3×D4—181no—
219C22×C3×C4—91no—
220C22×C3×C4—181no—
221C22×Dic3—181no—
222#239—61no—
223D3×Q8—61no—
224D3×Q8—182no—
225D3×Q8—182no—
226D3×Q8—182no—
227D3×Q8—182no—
228#237—62no—
229#237—61no—
230D3×D4—61no—
231D3×D4—62no—
232#235—21no—
233#235—62no—
234C2×C4×D3—61no—
235C2×C4×D3—61no—
236C2×C4×D3—62no—
237C2×C4×D3—182no—
238C2×Dic6—61no—
239C2×Dic6—61no—
240C2×Dic6—92no—
241C2×Dic6—92no—
242C2×Dic6—92no—
243C2×Dic6—121no—
244C2×Dic6—182no—
245C2×Dic6—181no—
246C2×Dic6—181no—
247#216—98no—
248#214—98no—
249#208—92no—
250#208—92no—
251C2×C3⋊C8—92no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—41no—
255C22×C32—41no—
256C22×C32—62no—
257C22×C32C2×#6711yescharacteristic
258C2×C3⋊D3—22no—
259C2×C3⋊D3C2×#6711yescharacteristic
260C2×C3⋊D3C2×#6711yescharacteristic
261C2×C3×D3—22no—
262C2×C3×D3—22no—
263C2×C3×D3—42no—
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—64no—
269C2×C3×D3—122no—
270C2×C3×D3—124no—
271D32—42no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—41no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—62no—
281C32×C4C2×#6711yescharacteristic
282C32×C4C2×#6711yescharacteristic
283C3⋊Dic3—24no—
284C3⋊Dic3C2×#6711yescharacteristic
285C3⋊Dic3C23×D311yescharacteristic
286C3×Dic3—22no—
287C3×Dic3—22no—
288C3×Dic3—42no—
289C3×Dic3—61no—
290C3×Dic3—61no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3—64no—
294C3×Dic3—64no—
295C3×Dic3—124no—
296C2×#41—91no—
297C22×Q8—271no—
298#137—274no—
299#137—274no—
300C2×Q16—272no—
301C2×SD16—272no—
302C2×M16—271no—
303C33C2×#13711yescharacteristic, complemented, Sylow
304C23×C3—91no—
305C23×C3—181no—
306C22×D3—61no—
307C22×D3—61no—
308C22×D3—62no—
309C3×Q8—61no—
310C3×Q8—61no—
311C3×Q8—92no—
312C3×Q8—121no—
313C3×Q8—182no—
314C3×Q8—182no—
315C3×D4—61no—
316C3×D4—62no—
317C3×D4—62no—
318C3×D4—61no—
319C3×D4—92no—
320C3×D4—121no—
321C3×D4—122no—
322C3×D4—182no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—91no—
333C2×C3×C4—91no—
334C2×C3×C4—122no—
335C2×C3×C4—181no—
336C2×C3×C4—182no—
337C2×C3×C4—182no—
338C2×C3×C4GO+(4,𝔽2)11yescharacteristic
339#67—22no—
340#67—62no—
341#67—62no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—62no—
345C2×Dic3—62no—
346C2×Dic3—62no—
347C2×Dic3—122no—
348C2×Dic3—182no—
349C2×Dic3—182no—
350D12—21no—
351D12—62no—
352C4×D3—22no—
353C4×D3—61no—
354C4×D3—61no—
355C4×D3—61no—
356C4×D3—61no—
357C4×D3—61no—
358C4×D3—61no—
359C4×D3—61no—
360C4×D3—61no—
361C4×D3—62no—
362C4×D3—62no—
363C4×D3—184no—
364C4×D3—184no—
365Dic6—21no—
366Dic6—62no—
367Dic6—62no—
368Dic6—62no—
369Dic6—62no—
370Dic6—62no—
371Dic6—94no—
372Dic6—122no—
373Dic6—122no—
374Dic6—184no—
375Dic6—182no—
376Dic6—184no—
377C3⋊C8—94no—
378C2×C32—22no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—42no—
386C2×C32—81no—
387C2×C32—122no—
388C2×C32C22×#6711yescharacteristic
389C3⋊D3—21no—
390C3⋊D3#95112yescomplemented
391C3×D3—42no—
392C3×D3—42no—
393C3×D3—62no—
394C3×D3—62no—
395C3×D3—62no—
396C3×D3—82no—
397C3×D3—121no—
398#41—61no—
399#41—92no—
400#41—92no—
401#41—181no—
402C2×Q8—91no—
403C2×Q8—181no—
404C2×Q8—272no—
405C2×Q8—272no—
406C2×Q8—272no—
407C2×Q8—541no—
408C2×Q8—541no—
409C2×D4—91no—
410C2×D4—181no—
411C22×C4—91no—
412C22×C4—181no—
413C22×C4—541no—
414Q16—278no—
415SD16—278no—
416M16—272no—
417M16—272no—
418C2×C8—272no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3—91no—
426C22×C3—91no—
427C22×C3—122no—
428C22×C3—182no—
429C22×C3—362no—
430C22×C3C2×GO+(4,𝔽2)11yescharacteristic
431C2×D3—22no—
432C2×D3—61no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—64no—
437C2×D3—122no—
438C2×D3—124no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—62no—
448C3×C4—62no—
449C3×C4—91no—
450C3×C4—91no—
451C3×C4—122no—
452C3×C4—184no—
453C3×C4C2×GO+(4,𝔽2)11yescharacteristic
454C3×C4C2×GO+(4,𝔽2)11yescharacteristic
455Dic3—22no—
456Dic3—61no—
457Dic3—61no—
458Dic3—61no—
459Dic3—61no—
460Dic3—64no—
461Dic3—64no—
462Dic3—124no—
463C32—21no—
464C32—21no—
465C32—41no—
466C32—41no—
467C32C2×#95111yescharacteristic, complemented
468C23—91no—
469C23—181no—
470Q8—61no—
471Q8—92no—
472Q8—182no—
473Q8—182no—
474Q8—274no—
475Q8—542no—
476D4—62no—
477D4—61no—
478D4—92no—
479D4—182no—
480C2×C4—62no—
481C2×C4—91no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—91no—
485C2×C4—91no—
486C2×C4—181no—
487C2×C4—182no—
488C2×C4—182no—
489C2×C4—542no—
490C2×C4—542no—
491C2×C4D32⋊D311yescharacteristic
492C8—274no—
493C2×C3—22no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—41no—
498C2×C3—42no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—81no—
502C2×C3—92no—
503C2×C3—122no—
504C2×C3—122no—
505C2×C3—181no—
506C2×C3—242no—
507C2×C3C22×GO+(4,𝔽2)11yescharacteristic
508D3—42no—
509D3—62no—
510D3—62no—
511D3—62no—
512D3—121no—
513C22—62no—
514C22—91no—
515C22—91no—
516C22—182no—
517C22—362no—
518C22C2×D32⋊D311yescharacteristic
519C4—62no—
520C4—91no—
521C4—91no—
522C4—184no—
523C4C2×D32⋊D311yescharacteristic
524C4C2×D32⋊D311yescharacteristic
525C3—21no—
526C3—21no—
527C3—41no—
528C3—41no—
529C3#1059650511yescharacteristic, complemented
530C2—21no—
531C2—92no—
532C2—122no—
533C2—181no—
534C2C22×D32⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FittingC2×C33×C4D4
socleC2×C33C22×D4
prime radicalC3×#64306C2
hypercenterC2×C4D32⋊D3
hypocenterC33C2×#137
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89040C2semidirect
3. #11724511C2semidirect
4. #11724328C2semidirect
5. #11724311C2semidirect
6. #11724310C2semidirect
7. #11724309C2semidirect
12. C2×#6296C22semidirect
17. #89040C22semidirect
25. #88993C22semidirect
26. #88989C22semidirect
28. #88938C22semidirect
29. #88937C22semidirect
207. C3×C3⋊D3#137semidirect
303. C33C2×#137semidirect
390. C3⋊D3#951semidirect
467. C32C2×#951semidirect
529. C3#10596505semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C3×#643062
#117245112
#117243282
#117243112
#117243102
#117243092
#105965053
C2×#9519