Finite Group Databasebeta

← #13148837

Subgroups — #13148837

Order 1728.

974 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88640C211yesmaximal, characteristic, complemented
2C2×#88634C211yesmaximal, characteristic, complemented
3C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
4C9×#897C211yesmaximal, characteristic, complemented
5C3×#63457C211yesmaximal, characteristic, complemented
6#11721041C212yesmaximal, complemented
7#11721037C211yesmaximal, characteristic, complemented
8#11721007C212yesmaximal, complemented
9#11720999C212yesmaximal, complemented
10#11720982C212yesmaximal, complemented
11#11720944C211yesmaximal, characteristic, complemented
12#10594528—31nomaximal
13#10589731—31nomaximal
14C2×C4×C3⋊D9C2212yes—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C4×D3×C9C2211yescharacteristic
17C2×C3×D36C2211yescharacteristic
18C2×C3×D36C2212yes—
19C2×C3×C4×D9C2211yescharacteristic
20C2×C9×Dic6C2211yescharacteristic
21C2×#6195C2211yescharacteristic
22C2×#6194C2211yescharacteristic
23C2×#6194C2212yes—
24C2×#6192C2211yescharacteristic
25C2×#6192C2212yes—
26C2×Dic3×D9C2212yes—
27C2×#6189C2211yescharacteristic
28#88640—24no—
29#88634—24no—
30C3×C9×C4⋊C4C2211yescharacteristic
31C4⋊(C3⋊Dic9)C2211yescharacteristic
32#88534C2212yes—
33C9×(C2×D3)⋊C4C2212yes—
34C3×(C2×D9)⋊C4C2212yes—
35C9×C4⋊Dic3C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37(C2×D9)⋊Dic3C2212yes—
38#88442C2212yes—
39C2×#3798—31no—
40C2×#3796—31no—
41C4⋊C4×C3⋊D3—31no—
42C3×#897D311yescharacteristic
43C3×#895—31no—
44#63916—32no—
45#63912—32no—
46#63908—31no—
47#63899—32no—
48#63885—32no—
49#63856—31no—
50C2×#3701—31no—
51C2×#3697—31no—
52C9×C22⋊Q8—31no—
53#63508—31no—
54#63492—31no—
55#63458—32no—
56#63457—32no—
57#63457D311yescharacteristic, complemented
58C4⋊C4×D9—31no—
59C4⋊C4×D9—61no—
60#63446—32no—
61#63446—32no—
62C22×C3⋊D9C2311yescharacteristic
63C22×D3×C9C2311yescharacteristic
64C22×C3×D9C2312yes—
65C22×C3×D9C2311yescharacteristic
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2311yescharacteristic
70C4×C3⋊D9—42no—
71C4×C3⋊D9D414yescomplemented
72C2×C9×Dic3C2311yescharacteristic
73C2×C9×Dic3C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C4×D3×C9—22no—
76C3×D36—22no—
77C3×D36—42no—
78C3×C4×D9—22no—
79C9×Dic6—22no—
80#6195—22no—
81#6194—22no—
82#6194—42no—
83#6192—24no—
84#6192—22no—
85Dic3×D9—24no—
86#6189—22no—
87#5629—91no—
88#5608—91no—
89C2×C4×C3⋊D3—31no—
90C2×C4×C3⋊D3—32no—
91C2×C3×D12—31no—
92C2×C3×D12—32no—
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3C2×D311yescharacteristic
95C2×C3×Dic6C2×D311yescharacteristic
96C2×#627—31no—
97C2×#626—31no—
98C2×#626—32no—
99C2×#626—31no—
100C2×#625—31no—
101C2×#625—32no—
102C2×D3×Dic3—32no—
103#3798—64no—
104#3796—64no—
105C32×C4⋊C4C2×D311yescharacteristic
106C4⋊(C3⋊Dic3)—31no—
107#3750—32no—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4C2×D312yes—
110C3×C4⋊Dic3C2×D312yes—
111#3724—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114C2×Q8×C9—31no—
115C22×C4×C9—31no—
116C2×#611—31no—
117C2×#611—32no—
118C2×#611—31no—
119#3701—64no—
120#3697—64no—
121C2×D36—32no—
122C2×D36—31no—
123C2×D36C2×D311yescharacteristic
124C2×D36C2×D312yes—
125C2×C4×D9—31no—
126C2×C4×D9—32no—
127C2×C4×D9—32no—
128C2×C4×D9—61no—
129C2×C4×D9—62no—
130C2×C4×D9C2×D311yescharacteristic
131C2×Dic18—31no—
132C9×C4⋊C4—21no—
133C9×C4⋊C4—32no—
134C9×C4⋊C4C2×D311yescharacteristic
135C9×C22⋊C4—32no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4C2×D312yes—
139C4⋊Dic9—31no—
140C4⋊Dic9—61no—
141Dic9⋊C4—32no—
142Dic9⋊C4—32no—
143Dic9⋊C4—62no—
144C22×C3×C9C2411yescharacteristic
145C2×C3⋊D9—21no—
146C2×C3⋊D9—21no—
147C2×C3⋊D9C2×D412yes—
148C2×D3×C9—21no—
149C2×D3×C9—21no—
150C2×D3×C9—21no—
151C2×C3×D9—22no—
152C2×C3×D9—21no—
153C2×C3×D9—22no—
154C2×C3×D9—21no—
155C2×C3×D9—22no—
156C2×C3×D9—21no—
157C3×C4×C9—22no—
158C3×C4×C9C2×D412yes—
159C3⋊Dic9—22no—
160C3⋊Dic9C2×D412yes—
161C9×Dic3—21no—
162C9×Dic3—22no—
163C3×Dic9—21no—
164C2×#239—91no—
165C2×#237—91no—
166C2×#235—91no—
167C2×#235—91no—
168C3×C22⋊Q8—31no—
169C3×#121—91no—
170#946—91no—
171#939—92no—
172#938—91no—
173#938—92no—
174#930—91no—
175#924—91no—
176#897D911yescharacteristic, complemented
177#896—92no—
178#895—31no—
179#895—92no—
180D3×C4⋊C4—91no—
181D3×C4⋊C4—91no—
182D3×C4⋊C4—181no—
183#884—92no—
184#884—92no—
185#884—92no—
186#878—92no—
187C22×C3⋊D3—31no—
188C22×C3×D3—31no—
189C22×C3×D3—32no—
190C22×C3×D3C22×D311yescharacteristic
191C2×C32×C4C22×D312yes—
192C2×C32×C4C22×D311yescharacteristic
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—32no—
195C4×C3⋊D3—34no—
196C4×C3⋊D3—122no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3C22×D311yescharacteristic
199C2×C3×Dic3C22×D312yes—
200C3×D12—62no—
201C3×D12—122no—
202C3×C4×D3—22no—
203C3×C4×D3—62no—
204C3×Dic6—22no—
205#627—62no—
206#626—64no—
207#626—62no—
208#626—62no—
209#625—62no—
210#625—122no—
211D3×Dic3—64no—
212C23×C9—31no—
213C22×D9—31no—
214C22×D9—61no—
215C22×D9C22×D311yescharacteristic
216C22×D9C22×D312yes—
217Q8×C9—62no—
218C2×C4×C9—21no—
219C2×C4×C9—22no—
220C2×C4×C9—32no—
221C2×C4×C9—31no—
222C2×C4×C9—62no—
223C2×C4×C9C22×D312yes—
224C2×C4×C9C22×D311yescharacteristic
225#611—62no—
226#611—62no—
227#611—122no—
228C2×Dic9—31no—
229C2×Dic9—32no—
230C2×Dic9—61no—
231C2×Dic9—62no—
232C2×Dic9C22×D311yescharacteristic
233D36—22no—
234D36—42no—
235D36—62no—
236D36—64no—
237C4×D9—22no—
238C4×D9—34no—
239C4×D9—64no—
240C4×D9—64no—
241C4×D9—122no—
242C4×D9—124no—
243Dic18—62no—
244#537—271noSylow
245C2×C3×C9C22×D411yescharacteristic
246C2×C3×C9H(5,𝔽2)11yescharacteristic
247C2×C3×C92−1+411yescharacteristic
248C3⋊D9—22no—
249D3×C9—41no—
250C3×D9—42no—
251C3×D9—41no—
252C2×C3×Q8—31no—
253C2×C3×D4—91no—
254C2×C3×D4—92no—
255C22×C3×C4—31no—
256C22×C3×C4—91no—
257C2×#67—91no—
258C2×#67—91no—
259C2×#67—91no—
260C2×#67—92no—
261C2×#67—92no—
262C2×#67—92no—
263C2×#67—91no—
264C22×Dic3—92no—
265#239—184no—
266#237—184no—
267#235—184no—
268#235—184no—
269C2×D12—31no—
270C2×D12—32no—
271C2×D12—92no—
272C2×D12—91no—
273C2×D12—91no—
274C2×C4×D3—31no—
275C2×C4×D3—91no—
276C2×C4×D3—91no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—92no—
280C2×C4×D3—181no—
281C2×C4×D3—182no—
282C2×C4×D3C2×D911yescharacteristic
283C2×Dic6—91no—
284C2×Dic6—91no—
285C2×Dic6C2×D911yescharacteristic
286C3×C4⋊C4—21no—
287C3×C4⋊C4—32no—
288C3×C4⋊C4C2×D911yescharacteristic
289C3×C4⋊C4D3211yescharacteristic
290C3×C22⋊C4—32no—
291C3×C22⋊C4—92no—
292#217—92no—
293#217—92no—
294(C2×D3)⋊C4—32no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4C2×D912yes—
298C4⋊Dic3—91no—
299C4⋊Dic3—91no—
300C4⋊Dic3—181no—
301C4⋊Dic3C2×D912yes—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—182no—
307C22×C32C23×D311yescharacteristic
308C2×C3⋊D3—32no—
309C2×C3⋊D3—61no—
310C2×C3⋊D3—61no—
311C2×C3×D3—21no—
312C2×C3×D3—21no—
313C2×C3×D3—21no—
314C2×C3×D3—62no—
315C2×C3×D3—61no—
316C2×C3×D3—61no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C2×C3×D3—61no—
320C32×C4—22no—
321C32×C4D3×D412yes—
322C3⋊Dic3—32no—
323C3⋊Dic3—62no—
324C3×Dic3—22no—
325C3×Dic3—21no—
326C3×Dic3—61no—
327C22×C9—21no—
328C22×C9—61no—
329C22×C9—61no—
330C22×C9—61no—
331C22×C9C23×D311yescharacteristic
332C2×D9—21no—
333C2×D9—22no—
334C2×D9—21no—
335C2×D9—22no—
336C2×D9—21no—
337C2×D9—22no—
338C2×D9—32no—
339C2×D9—62no—
340C2×D9—62no—
341C2×D9—62no—
342C2×D9—61no—
343C2×D9—61no—
344C4×C9—22no—
345C4×C9—22no—
346C4×C9—42no—
347C4×C9—61no—
348C4×C9—62no—
349C4×C9D3×D412yes—
350Dic9—21no—
351Dic9—32no—
352Dic9—62no—
353Dic9—62no—
354Dic9—122no—
355C2×#41—271no—
356C2×#41—271no—
357C22⋊Q8—31no—
358C22⋊Q8—272no—
359C22⋊Q8—271no—
360#121—91no—
361#121—272no—
362#121—272no—
363#121—272no—
364#121—271no—
365C2×C4⋊C4—271no—
366C3×C9#53711yescharacteristic, complemented, Sylow
367C23×C3—31no—
368C23×C3—91no—
369C23×C3—92no—
370C22×D3—32no—
371C22×D3—31no—
372C22×D3—91no—
373C22×D3—91no—
374C22×D3—181no—
375C22×D3C22×D911yescharacteristic
376C3×Q8—62no—
377C3×D4—182no—
378C3×D4—362no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—32no—
382C2×C3×C4—31no—
383C2×C3×C4—62no—
384C2×C3×C4—91no—
385C2×C3×C4—182no—
386C2×C3×C4C22×D912yes—
387C2×C3×C4C22×D911yescharacteristic
388C2×C3×C4C2×D3212yes—
389C2×C3×C4C2×D3211yescharacteristic
390#67—182no—
391#67—184no—
392#67—182no—
393#67—182no—
394#67—182no—
395#67—362no—
396#67—362no—
397C2×Dic3—31no—
398C2×Dic3—91no—
399C2×Dic3—91no—
400C2×Dic3—92no—
401C2×Dic3—92no—
402C2×Dic3—181no—
403C2×Dic3—182no—
404C2×Dic3—184no—
405C2×Dic3C22×D912yes—
406C2×Dic3C22×D911yescharacteristic
407D12—62no—
408D12—122no—
409D12—182no—
410D12—184no—
411D12—182no—
412C4×D3—22no—
413C4×D3—62no—
414C4×D3—94no—
415C4×D3—94no—
416C4×D3—184no—
417C4×D3—184no—
418C4×D3—362no—
419C4×D3—362no—
420C4×D3—364no—
421Dic6—22no—
422Dic6—182no—
423Dic6—182no—
424C2×C32C2×D3×D411yescharacteristic
425C2×C32#100711yescharacteristic
426C2×C32#100911yescharacteristic
427C3⋊D3—62no—
428C3×D3—41no—
429C3×D3—122no—
430C3×D3—121no—
431C2×C9—21no—
432C2×C9—21no—
433C2×C9—21no—
434C2×C9—121no—
435C2×C9C2×D3×D411yescharacteristic
436C2×C9#100411yescharacteristic
437C2×C9#101011yescharacteristic
438D9—41no—
439D9—42no—
440D9—64no—
441D9—62no—
442#41—544no—
443#41—544no—
444C2×Q8—31no—
445C2×Q8—271no—
446C2×D4—91no—
447C2×D4—92no—
448C2×D4—272no—
449C2×D4—271no—
450C2×D4—271no—
451C2×D4—271no—
452C2×D4—272no—
453C22×C4—31no—
454C22×C4—91no—
455C22×C4—272no—
456C22×C4—271no—
457C22×C4—272no—
458C4⋊C4—32no—
459C4⋊C4—272no—
460C4⋊C4—272no—
461C4⋊C4—271no—
462C4⋊C4D3×D911yescharacteristic
463C22⋊C4—32no—
464C22⋊C4—92no—
465C22⋊C4—272no—
466C22⋊C4—272no—
467C22×C3—21no—
468C22×C3—61no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—182no—
472C22×C3—181no—
473C22×C3—181no—
474C22×C3—182no—
475C22×C3—182no—
476C22×C3—181no—
477C22×C3C23×D911yescharacteristic
478C22×C3C22×D3211yescharacteristic
479C2×D3—21no—
480C2×D3—21no—
481C2×D3—21no—
482C2×D3—61no—
483C2×D3—62no—
484C2×D3—61no—
485C2×D3—62no—
486C2×D3—62no—
487C2×D3—61no—
488C2×D3—92no—
489C2×D3—92no—
490C2×D3—182no—
491C2×D3—182no—
492C2×D3—182no—
493C2×D3—181no—
494C2×D3—181no—
495C2×D3—181no—
496C2×D3—181no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—22no—
500C3×C4—42no—
501C3×C4—61no—
502C3×C4—62no—
503C3×C4—181no—
504C3×C4D4×D912yes—
505C3×C4#380212yes—
506Dic3—22no—
507Dic3—21no—
508Dic3—61no—
509Dic3—92no—
510Dic3—92no—
511Dic3—182no—
512Dic3—182no—
513Dic3—182no—
514Dic3—362no—
515C32#562911yescharacteristic
516C9—21no—
517C9#560811yescharacteristic, complemented
518C23—31no—
519C23—92no—
520C23—91no—
521C23—271no—
522Q8—62no—
523Q8—542no—
524D4—182no—
525D4—362no—
526D4—542no—
527D4—544no—
528D4—542no—
529D4—542no—
530D4—1082no—
531C2×C4—32no—
532C2×C4—31no—
533C2×C4—62no—
534C2×C4—91no—
535C2×C4—182no—
536C2×C4—271no—
537C2×C4—274no—
538C2×C4—272no—
539C2×C4—544no—
540C2×C4—1082no—
541C2×C4C2×D3×D912yes—
542C2×C4C2×D3×D911yescharacteristic
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—121no—
547C2×C3—362no—
548C2×C3—361no—
549C2×C3C2×D4×D911yescharacteristic
550C2×C3#6371311yescharacteristic
551C2×C3#6371511yescharacteristic
552C2×C3C2×#380211yescharacteristic
553C2×C3#6431411yescharacteristic
554C2×C3#6431511yescharacteristic
555D3—41no—
556D3—121no—
557D3—122no—
558D3—184no—
559D3—182no—
560D3—182no—
561C22—61no—
562C22—61no—
563C22—61no—
564C22—181no—
565C22—182no—
566C22—182no—
567C22—181no—
568C22—181no—
569C22—182no—
570C22—272no—
571C22—541no—
572C22—541no—
573C22C22×D3×D911yescharacteristic
574C4—22no—
575C4—61no—
576C4—62no—
577C4—181no—
578C4—272no—
579C4—542no—
580C4#8864612yes—
581C3—21no—
582C3#1058973111yescharacteristic, complemented
583C3#1059452811yescharacteristic
584C2—121no—
585C2—361no—
586C2—362no—
587C2—542no—
588C2C2×#8864611yescharacteristic
589C2#1172250111yescharacteristic
590C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#537
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×#88640C2semidirect
2. C2×#88634C2semidirect
3. C4⋊C4×C3⋊D9C2semidirect
4. C9×#897C2semidirect
5. C3×#63457C2semidirect
6. #11721041C2semidirect
7. #11721037C2semidirect
8. #11721007C2semidirect
9. #11720999C2semidirect
10. #11720982C2semidirect
11. #11720944C2semidirect
57. #63457D3semidirect
71. C4×C3⋊D9D4semidirect
176. #897D9semidirect
366. C3×C9#537semidirect
517. C9#5608semidirect
582. C3#10589731semidirect

Maximal subgroups

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

subgroupindex
C2×#886402
C2×#886342
C4⋊C4×C3⋊D92
C9×#8972
C3×#634572
#117210412
#117210372
#117210072
#117209992
#117209822
#117209442
#105945283
#105897313