Finite Group Databasebeta

← #13148986

Subgroups — #13148986

Order 1728.

970 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88642C211yesmaximal, characteristic, complemented
2C2×#88635C211yesmaximal, characteristic, complemented
3#11721749C211yesmaximal, characteristic, complemented
4C9×#897C211yesmaximal, characteristic, complemented
5C3×#63459C211yesmaximal, characteristic, complemented
6#11721041C212yesmaximal, complemented
7C4×#6194C211yesmaximal, characteristic, complemented
8#11721000C212yesmaximal, complemented
9#11720958C212yesmaximal, complemented
10#11720942C212yesmaximal, complemented
11#11720930C211yesmaximal, characteristic, complemented
12#10594530—31nomaximal
13#10589733—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C4×D3×C9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×#6195C2212yes—
21C2×#6194C2211yescharacteristic
22C2×#6192C2212yes—
23C2×(C3⋊D9)⋊C4C2212yes—
24C2×(C3⋊D9)⋊C4C2212yes—
25#88642—24no—
26#88635—24no—
27C3×C9×C4⋊C4C2211yescharacteristic
28#88536C2212yes—
29C4×C3⋊Dic9C2211yescharacteristic
30C9×(C2×D3)⋊C4C2212yes—
31C3×(C2×D9)⋊C4C2212yes—
32C9×C4⋊Dic3C2212yes—
33C3×C4⋊Dic9C2212yes—
34(C2×D3)⋊Dic9C2211yescharacteristic
35(C2×D9)⋊Dic3C2211yescharacteristic
36#88442C2211yescharacteristic
37#88442C2212yes—
38#88442C2212yes—
39C2×#3799—31no—
40C2×#3799—31no—
41#64102—31no—
42C3×#897—31no—
43C3×#897D311yescharacteristic
44#63912—32no—
45C4×#625—31no—
46#63900—32no—
47#63873—31no—
48#63854—32no—
49#63854—32no—
50C2×#3701—31no—
51C2×#3697—31no—
52C9×C22⋊Q8—31no—
53#63506—31no—
54C4×#611—31no—
55#63459D311yescharacteristic, complemented
56#63456—32no—
57#63455—31no—
58#63455—61no—
59C4⋊C4×D9—32no—
60#63446—32no—
61#63445—32no—
62C22×C3⋊D9C2312yes—
63C22×D3×C9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C4⋊(C3⋊D9)D414yescomplemented
70C4×C3⋊D9—24no—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2312yes—
75C4×D3×C9—22no—
76C3×C4×D9—22no—
77C9×Dic6—22no—
78C3×Dic18—22no—
79#6195—24no—
80#6194—22no—
81#6192—24no—
82(C3⋊D9)⋊C4—24no—
83(C3⋊D9)⋊C4—24no—
84#5631—91no—
85#5631—91no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×C4×C3⋊D3—32no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3C2×D311yescharacteristic
90C2×C3×Dic6—31no—
91C2×C3×Dic6C2×D311yescharacteristic
92C2×#626—32no—
93C2×#626—32no—
94C2×#625—31no—
95C2×#624—32no—
96C2×#624—32no—
97#3799—64no—
98#3799—64no—
99C32×C4⋊C4C2×D311yescharacteristic
100#3752—32no—
101C4×C3⋊Dic3—31no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4C2×D312yes—
104C3×C4⋊Dic3—32no—
105C3×C4⋊Dic3C2×D312yes—
106#3724—32no—
107#3724—32no—
108#3724—31no—
109(C2×D3)⋊Dic3—31no—
110(C2×D3)⋊Dic3—31no—
111C2×Q8×C9—31no—
112C22×C4×C9—31no—
113C2×#611—31no—
114C2×#611—32no—
115#3701—64no—
116#3697—64no—
117C2×D36—31no—
118C2×D36—32no—
119C2×D36—61no—
120C2×C4×D9—32no—
121C2×C4×D9—32no—
122C2×C4×D9—32no—
123C2×C4×D9—62no—
124C2×C4×D9C2×D311yescharacteristic
125C2×Dic18C2×D311yescharacteristic
126C9×C4⋊C4—21no—
127C9×C4⋊C4—32no—
128C9×C4⋊C4C2×D311yescharacteristic
129C9×C22⋊C4—32no—
130#3676—31no—
131(C2×D9)⋊C4—31no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—62no—
134(C2×D9)⋊C4C2×D312yes—
135C4⋊Dic9C2×D312yes—
136Dic9⋊C4—31no—
137Dic9⋊C4—32no—
138Dic9⋊C4—32no—
139C4×Dic9—31no—
140C4×Dic9—61no—
141C22×C3×C9C2411yescharacteristic
142C2×C3⋊D9—22no—
143C2×C3⋊D9—22no—
144C2×C3⋊D9C2×D414yes—
145C2×D3×C9—21no—
146C2×D3×C9—21no—
147C2×D3×C9—21no—
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9—21no—
151C3×C4×C9—22no—
152C3×C4×C9C2×D412yes—
153C3⋊Dic9—21no—
154C3⋊Dic9#4112yescomplemented
155C9×Dic3—21no—
156C9×Dic3—22no—
157C3×Dic9—21no—
158C3×Dic9—22no—
159C2×#239—91no—
160C2×#239—91no—
161C2×#235—91no—
162C2×#235—91no—
163C3×C22⋊Q8—31no—
164C3×C22⋊Q8—91no—
165#944—91no—
166#944—91no—
167C4×#67—91no—
168C4×#67—91no—
169#897—31no—
170#897D911yescharacteristic, complemented
171#894—92no—
172#894—92no—
173#893—91no—
174#893—91no—
175#893—181no—
176D3×C4⋊C4—92no—
177D3×C4⋊C4—92no—
178#884—92no—
179#884—92no—
180#883—92no—
181#883—92no—
182C22×C3⋊D3—32no—
183C22×C3×D3—31no—
184C22×C3×D3C22×D311yescharacteristic
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—31no—
189C4⋊(C3⋊D3)—34no—
190C4×C3⋊D3—64no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3C22×D311yescharacteristic
194C2×C3×Dic3C22×D312yes—
195C3×C4×D3—22no—
196C3×C4×D3—62no—
197C3×Dic6—22no—
198C3×Dic6—62no—
199#626—64no—
200#626—64no—
201#625—62no—
202#624—64no—
203#624—64no—
204C23×C9—31no—
205C22×D9—32no—
206C22×D9—62no—
207C22×D9C22×D311yescharacteristic
208Q8×C9—62no—
209C2×C4×C9—21no—
210C2×C4×C9—22no—
211C2×C4×C9—32no—
212C2×C4×C9—31no—
213C2×C4×C9—62no—
214C2×C4×C9C22×D312yes—
215C2×C4×C9C22×D311yescharacteristic
216#611—64no—
217#611—62no—
218C2×Dic9—31no—
219C2×Dic9—31no—
220C2×Dic9—61no—
221C2×Dic9—61no—
222C2×Dic9C22×D312yes—
223C2×Dic9C22×D311yescharacteristic
224D36—34no—
225D36—64no—
226D36—64no—
227C4×D9—22no—
228C4×D9—64no—
229C4×D9—64no—
230C4×D9—64no—
231C4×D9—124no—
232Dic18—22no—
233#547—271noSylow
234C2×C3×C9C22×D411yescharacteristic
235C2×C3×C9C2×#4111yescharacteristic
236C2×C3×C92−1+411yescharacteristic
237C3⋊D9—24no—
238D3×C9—41no—
239C3×D9—41no—
240C2×C3×Q8—31no—
241C2×C3×Q8—91no—
242C22×C3×C4—31no—
243C22×C3×C4—91no—
244C2×#67—92no—
245C2×#67—91no—
246C2×#67—91no—
247C2×#67—92no—
248#239—184no—
249#239—184no—
250#235—184no—
251#235—184no—
252C2×D12—91no—
253C2×D12—91no—
254C2×D12—92no—
255C2×D12—92no—
256C2×D12—181no—
257C2×C4×D3—31no—
258C2×C4×D3—92no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×C4×D3—92no—
262C2×C4×D3—92no—
263C2×C4×D3—92no—
264C2×C4×D3—182no—
265C2×C4×D3C2×D911yescharacteristic
266C2×Dic6—31no—
267C2×Dic6C2×D911yescharacteristic
268C3×C4⋊C4—21no—
269C3×C4⋊C4—32no—
270C3×C4⋊C4—92no—
271C3×C4⋊C4C2×D911yescharacteristic
272C3×C4⋊C4D3211yescharacteristic
273C3×C22⋊C4—32no—
274C3×C22⋊C4—92no—
275#217—91no—
276#217—91no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—91no—
279(C2×D3)⋊C4—91no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—182no—
283(C2×D3)⋊C4C2×D912yes—
284C4⋊Dic3—32no—
285C4⋊Dic3C2×D912yes—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—91no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—91no—
292C4×Dic3—91no—
293C4×Dic3—91no—
294C4×Dic3—181no—
295C22×C32C23×D311yescharacteristic
296C2×C3⋊D3—34no—
297C2×C3⋊D3—62no—
298C2×C3⋊D3—62no—
299C2×C3×D3—21no—
300C2×C3×D3—21no—
301C2×C3×D3—21no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C32×C4—22no—
306C32×C4D3×D412yes—
307C3⋊Dic3—32no—
308C3⋊Dic3—61no—
309C3×Dic3—21no—
310C3×Dic3—22no—
311C3×Dic3—61no—
312C3×Dic3—62no—
313C22×C9—21no—
314C22×C9—61no—
315C22×C9—61no—
316C22×C9—61no—
317C22×C9C23×D311yescharacteristic
318C2×D9—21no—
319C2×D9—21no—
320C2×D9—21no—
321C2×D9—34no—
322C2×D9—64no—
323C2×D9—62no—
324C2×D9—62no—
325C2×D9—122no—
326C2×D9—122no—
327C4×C9—22no—
328C4×C9—22no—
329C4×C9—42no—
330C4×C9—61no—
331C4×C9—62no—
332C4×C9D3×D412yes—
333Dic9—22no—
334Dic9—21no—
335Dic9—32no—
336Dic9—62no—
337Dic9—61no—
338Dic9—121no—
339C2×#41—271no—
340C2×#41—271no—
341C4⋊Q8—271no—
342#123—272no—
343#123—272no—
344C22⋊Q8—31no—
345C22⋊Q8—91no—
346#121—272no—
347C4×D4—271no—
348C4×D4—271no—
349C2×C4⋊C4—272no—
350C3×C9#54711yescharacteristic, complemented, Sylow
351C23×C3—31no—
352C23×C3—91no—
353C22×D3—31no—
354C22×D3—92no—
355C22×D3—92no—
356C22×D3—182no—
357C22×D3C22×D911yescharacteristic
358C3×Q8—62no—
359C3×Q8—182no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—32no—
363C2×C3×C4—31no—
364C2×C3×C4—62no—
365C2×C3×C4—92no—
366C2×C3×C4—91no—
367C2×C3×C4—182no—
368C2×C3×C4C22×D912yes—
369C2×C3×C4C22×D911yescharacteristic
370C2×C3×C4C2×D3212yes—
371C2×C3×C4C2×D3211yescharacteristic
372#67—184no—
373#67—184no—
374#67—182no—
375#67—182no—
376C2×Dic3—32no—
377C2×Dic3—31no—
378C2×Dic3—91no—
379C2×Dic3—91no—
380C2×Dic3—91no—
381C2×Dic3—91no—
382C2×Dic3—181no—
383C2×Dic3—181no—
384C2×Dic3C22×D912yes—
385C2×Dic3C22×D911yescharacteristic
386D12—94no—
387D12—94no—
388D12—184no—
389D12—184no—
390D12—184no—
391C4×D3—22no—
392C4×D3—62no—
393C4×D3—184no—
394C4×D3—184no—
395C4×D3—184no—
396C4×D3—184no—
397C4×D3—184no—
398C4×D3—184no—
399C4×D3—364no—
400Dic6—22no—
401Dic6—62no—
402C2×C32C2×D3×D411yescharacteristic
403C2×C32#100711yescharacteristic
404C2×C32D3×#4111yescharacteristic
405C3⋊D3—64no—
406C3×D3—41no—
407C3×D3—121no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—121no—
412C2×C9C2×D3×D411yescharacteristic
413C2×C9#100711yescharacteristic
414C2×C9D3×#4111yescharacteristic
415D9—41no—
416D9—64no—
417D9—124no—
418#41—544no—
419#41—544no—
420C2×Q8—31no—
421C2×Q8—91no—
422C2×D4—272no—
423C2×D4—272no—
424C2×D4—271no—
425C2×D4—271no—
426C22×C4—31no—
427C22×C4—91no—
428C22×C4—272no—
429C22×C4—272no—
430C22×C4—272no—
431C4⋊C4—32no—
432C4⋊C4—92no—
433C4⋊C4—271no—
434C4⋊C4—272no—
435C4⋊C4—272no—
436C4⋊C4D3×D911yescharacteristic
437C22⋊C4—32no—
438C22⋊C4—92no—
439C22⋊C4—271no—
440C22⋊C4—271no—
441C22⋊C4—272no—
442C42—271no—
443C22×C3—21no—
444C22×C3—61no—
445C22×C3—61no—
446C22×C3—61no—
447C22×C3—181no—
448C22×C3—181no—
449C22×C3—181no—
450C22×C3C23×D911yescharacteristic
451C22×C3C22×D3211yescharacteristic
452C2×D3—21no—
453C2×D3—21no—
454C2×D3—21no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—94no—
459C2×D3—94no—
460C2×D3—184no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3—182no—
464C2×D3—182no—
465C2×D3—362no—
466C2×D3—362no—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—42no—
471C3×C4—61no—
472C3×C4—62no—
473C3×C4—181no—
474C3×C4—182no—
475C3×C4D4×D912yes—
476C3×C4#380212yes—
477Dic3—22no—
478Dic3—21no—
479Dic3—62no—
480Dic3—61no—
481Dic3—92no—
482Dic3—92no—
483Dic3—182no—
484Dic3—181no—
485Dic3—181no—
486Dic3—361no—
487C32#563111yescharacteristic
488C9—21no—
489C9#563111yescharacteristic, complemented
490C23—31no—
491C23—91no—
492C23—272no—
493Q8—62no—
494Q8—182no—
495D4—274no—
496D4—544no—
497D4—544no—
498D4—542no—
499C2×C4—32no—
500C2×C4—31no—
501C2×C4—62no—
502C2×C4—92no—
503C2×C4—91no—
504C2×C4—182no—
505C2×C4—271no—
506C2×C4—271no—
507C2×C4—544no—
508C2×C4—544no—
509C2×C4—544no—
510C2×C4C2×D3×D912yes—
511C2×C4C2×D3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—121no—
516C2×C3—361no—
517C2×C3C2×D4×D911yescharacteristic
518C2×C3#6371311yescharacteristic
519C2×C3#41×D911yescharacteristic
520C2×C3C2×#380211yescharacteristic
521C2×C3#6431211yescharacteristic
522C2×C3#6431611yescharacteristic
523D3—41no—
524D3—121no—
525D3—184no—
526D3—184no—
527D3—364no—
528C22—61no—
529C22—61no—
530C22—61no—
531C22—181no—
532C22—181no—
533C22—181no—
534C22—274no—
535C22—542no—
536C22—542no—
537C22C22×D3×D911yescharacteristic
538C4—22no—
539C4—61no—
540C4—62no—
541C4—181no—
542C4—182no—
543C4—272no—
544C4—541no—
545C4#8864612yes—
546C3—21no—
547C3#1058973311yescharacteristic, complemented
548C3#1059453011yescharacteristic
549C2—121no—
550C2—361no—
551C2—544no—
552C2C2×#8864611yescharacteristic
553C2#1172249811yescharacteristic
554C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#547
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88642C2semidirect
2. C2×#88635C2semidirect
3. #11721749C2semidirect
4. C9×#897C2semidirect
5. C3×#63459C2semidirect
6. #11721041C2semidirect
7. C4×#6194C2semidirect
8. #11721000C2semidirect
9. #11720958C2semidirect
10. #11720942C2semidirect
11. #11720930C2semidirect
55. #63459D3semidirect
69. C4⋊(C3⋊D9)D4semidirect
154. C3⋊Dic9#41semidirect
170. #897D9semidirect
350. C3×C9#547semidirect
489. C9#5631semidirect
547. C3#10589733semidirect

Maximal subgroups

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

subgroupindex
C2×#886422
C2×#886352
#117217492
C9×#8972
C3×#634592
#117210412
C4×#61942
#117210002
#117209582
#117209422
#117209302
#105945303
#105897333