Finite Group Databasebeta

← #13177542

Subgroups — #13177542

Order 1728.

2140 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×#89020C211yesmaximal, characteristic, complemented
3#11723979C211yesmaximal, characteristic, complemented
4C32×#938C211yesmaximal, characteristic, complemented
5#11723641C212yesmaximal, complemented
6#11723635C212yesmaximal, complemented
7#11723631C212yesmaximal, complemented
8#11723624C212yesmaximal, complemented
9C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#11723449C211yesmaximal, characteristic, complemented
11#11723425C211yesmaximal, characteristic, complemented
12#10596079—31nomaximal
13#10595232—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#638C2212yes—
18C22×C3×C3⋊Dic3C2212yes—
19C3×#3830—24no—
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21C2×C3×#634C2211yescharacteristic
22C2×C32×#67C2212yes—
23C2×C32×C4×D3C2211yescharacteristic
24C2×#6293C2211yescharacteristic
25C2×#6292C2211yescharacteristic
26C2×#6291C2211yescharacteristic
27C2×#6290C2211yescharacteristic
28C2×Dic3×C3⋊D3C2212yes—
29#89020—24no—
30#88864C2212yes—
31C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
32C32×#217C2212yes—
33C32×C4⋊Dic3C2211yescharacteristic
34#88807C2212yes—
35#88806C2212yes—
36Dic3⋊(C3⋊Dic3)C2212yes—
37#88804C2212yes—
38(C2×D3)⋊(C3⋊Dic3)C2212yes—
39C2×#3830D311yescharacteristic, complemented
40C2×#3828—31no—
41C2×C3×#237—34no—
42C2×#3798—34no—
43C32×#121—31no—
44#64148—32no—
45#64147—31no—
46#64147—32no—
47#64147—34no—
48#64147—64no—
49#64133—31no—
50#64106—31no—
51C4⋊C4×C3⋊D3—31no—
52#64093—32no—
53#64087—32no—
54C3×#938D314yescomplemented
55#63978—38no—
56#63971—38no—
57#63970—38no—
58#63964—38no—
59D3×C4⋊Dic3—34no—
60#63871—34no—
61#63856—34no—
62C23×C33C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C32×D3C2311yescharacteristic
66C33×D4—22no—
67C2×C33×C4C2311yescharacteristic
68#6312—42no—
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C4×C3⋊(C3⋊D3)—22no—
72C3×#638—24no—
73C2×C3×C3⋊Dic3—24no—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C3×C4×C3⋊D3D414yescomplemented
77C3×#634—22no—
78C32×#67—42no—
79C2×C32×Dic3C2311yescharacteristic
80C2×C32×Dic3C2312yes—
81C32×C4×D3—22no—
82#6293—22no—
83#6292—22no—
84#6291—22no—
85#6290—22no—
86Dic3×C3⋊D3—42no—
87#5833—91no—
88#5608—94no—
89C2×C32×D4—24no—
90C2×C32×D4—32no—
91C2×C32×D4C2×D314yes—
92C2×C32×D4C2×D311yescharacteristic
93C22×C32×C4—31no—
94C2×#638—31no—
95C2×#638—32no—
96C2×#638—38no—
97C2×#638—31no—
98C2×#638—68no—
99C2×#638C2×D312yes—
100C22×C3⋊Dic3C2×D312yes—
101#3830—24no—
102#3828—64no—
103C2×C4⋊(C3⋊D3)—31no—
104C2×C4×C3⋊D3—31no—
105C2×C4×C3⋊D3—34no—
106C2×C4×C3⋊D3—32no—
107C2×C4×C3⋊D3—64no—
108C2×C4×C3⋊D3C2×D311yescharacteristic
109C2×#634—31no—
110C2×#634C2×D311yescharacteristic
111C2×C3×#67—38no—
112C2×C3×#67C2×D318yes—
113C22×C3×Dic3—38no—
114C3×#237—616no—
115C2×C3×C4×D3—34no—
116C2×C3×C4×D3C2×D314yes—
117C2×C3×Dic6—34no—
118C2×#627—34no—
119C2×#626—34no—
120C2×#626—34no—
121C2×#625—34no—
122C2×D3×Dic3—38no—
123#3798—616no—
124C32×C4⋊C4—31no—
125C32×C22⋊C4—32no—
126#3757—32no—
127#3757—38no—
128#3757—32no—
129#3757—68no—
130#3752—32no—
131C4⋊(C3⋊Dic3)—31no—
132C4⋊(C3⋊Dic3)—34no—
133C4⋊(C3⋊Dic3)—32no—
134C4⋊(C3⋊Dic3)—64no—
135#3750—32no—
136#3750—32no—
137C3×#217C2×D318yes—
138C3×C4⋊Dic3C2×D314yes—
139#3724—38no—
140Dic3⋊Dic3—38no—
141Dic3⋊Dic3—38no—
142#3722—38no—
143(C2×D3)⋊Dic3—38no—
144C22×C33—22no—
145C22×C33—22no—
146C22×C33—22no—
147C22×C33C2411yescharacteristic
148C2×C3⋊(C3⋊D3)—21no—
149C2×C3⋊(C3⋊D3)—21no—
150C2×C3⋊(C3⋊D3)—21no—
151C2×C3×C3⋊D3—21no—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3C2×D412yes—
154C2×C32×D3—21no—
155C2×C32×D3—21no—
156C2×C32×D3—21no—
157C33×C4C2×D412yes—
158C3⋊(C3⋊Dic3)—21no—
159C3⋊(C3⋊Dic3)—22no—
160C3×C3⋊Dic3—22no—
161C3×C3⋊Dic3C2×D412yes—
162C32×Dic3—21no—
163C32×Dic3—22no—
164C2×C3×#41—91no—
165C2×#237—34no—
166C2×#235—94no—
167C2×#235—91no—
168C3×#121—34no—
169#946—91no—
170#946—91no—
171#939—98no—
172#939—92no—
173#939—92no—
174#938—98no—
175#938—94no—
176#938—91no—
177#938—184no—
178#938C3⋊D311yescharacteristic, complemented
179#930—92no—
180C2×C4⋊Dic3—91no—
181#924—94no—
182#924—92no—
183#897—94no—
184D3×C4⋊C4—94no—
185#884—98no—
186#878—98no—
187C23×C32—28no—
188C23×C32—31no—
189C23×C32C22×D318yes—
190C23×C32C22×D312yes—
191C22×C3⋊D3—31no—
192C22×C3⋊D3—34no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3C22×D311yescharacteristic
195C22×C3×D3—34no—
196C22×C3×D3C22×D314yes—
197C32×D4—22no—
198C32×D4—28no—
199C32×D4—48no—
200C32×D4—122no—
201C2×C32×C4—24no—
202C2×C32×C4—31no—
203C2×C32×C4—32no—
204C2×C32×C4—62no—
205C2×C32×C4C22×D314yes—
206C2×C32×C4C22×D311yescharacteristic
207#638—24no—
208#638—62no—
209#638—62no—
210#638—122no—
211#638—128no—
212#638—1216no—
213C2×C3⋊Dic3—24no—
214C2×C3⋊Dic3—32no—
215C2×C3⋊Dic3—31no—
216C2×C3⋊Dic3—38no—
217C2×C3⋊Dic3—34no—
218C2×C3⋊Dic3—68no—
219C2×C3⋊Dic3—64no—
220C2×C3⋊Dic3C22×D312yes—
221C2×C3⋊Dic3C22×D311yescharacteristic
222C4⋊(C3⋊D3)—62no—
223C4×C3⋊D3—62no—
224C4×C3⋊D3—68no—
225C4×C3⋊D3—128no—
226C4×C3⋊D3—122no—
227C4×C3⋊D3#6714yescomplemented
228#634—22no—
229#634—62no—
230C3×#67—48no—
231C3×#67—616no—
232C2×C3×Dic3—34no—
233C2×C3×Dic3—38no—
234C2×C3×Dic3—616no—
235C2×C3×Dic3C22×D318yes—
236C2×C3×Dic3C22×D314yes—
237C3×C4×D3—28no—
238C3×C4×D3—316no—
239C3×Dic6—68no—
240#627—68no—
241#626—68no—
242#626—68no—
243#625—68no—
244D3×Dic3—128no—
245#537—271noSylow
246C2×C33—42no—
247C2×C33C22×D411yescharacteristic
248C2×C33H(5,𝔽2)11yescharacteristic
249C2×C332−1+411yescharacteristic
250C3⋊(C3⋊D3)—41no—
251C3×C3⋊D3—22no—
252C32×D3—41no—
253C3×#41—184no—
254C2×C3×Q8—91no—
255C2×C3×D4—24no—
256C2×C3×D4—38no—
257C2×C3×D4—92no—
258C2×C3×D4D3214yes—
259C2×C3×D4C2×C3⋊D311yescharacteristic
260C22×C3×C4—34no—
261C22×C3×C4—91no—
262C22×C3×C4—92no—
263C2×#67—38no—
264C2×#67—94no—
265C2×#67—94no—
266C2×#67—91no—
267C2×#67—98no—
268C2×#67—92no—
269C2×#67—91no—
270C2×#67—188no—
271C2×#67C2×C3⋊D312yes—
272C22×Dic3—38no—
273C22×Dic3—92no—
274#237—616no—
275#235—184no—
276#235—1816no—
277C2×D12—94no—
278C2×D12—91no—
279C2×C4×D3—34no—
280C2×C4×D3—94no—
281C2×C4×D3—91no—
282C2×C4×D3—98no—
283C2×C4×D3—184no—
284C2×C4×D3C2×C3⋊D311yescharacteristic
285C2×Dic6—34no—
286C2×Dic6—94no—
287C2×Dic6—91no—
288C3×C4⋊C4—34no—
289C3×C22⋊C4—38no—
290#217—98no—
291#217—98no—
292#217—92no—
293#217—188no—
294#217C2×C3⋊D312yes—
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298C4⋊Dic3—98no—
299C4⋊Dic3—94no—
300C4⋊Dic3—91no—
301C4⋊Dic3—92no—
302C4⋊Dic3—184no—
303C4⋊Dic3C2×C3⋊D311yescharacteristic
304Dic3⋊C4—98no—
305Dic3⋊C4—98no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—24no—
312C22×C32—28no—
313C22×C32—28no—
314C22×C32—28no—
315C22×C32—48no—
316C22×C32—48no—
317C22×C32—48no—
318C22×C32—61no—
319C22×C32—61no—
320C22×C32—61no—
321C22×C32C23×D314yes—
322C22×C32C23×D311yescharacteristic
323C2×C3⋊D3—21no—
324C2×C3⋊D3—21no—
325C2×C3⋊D3—68no—
326C2×C3⋊D3—64no—
327C2×C3⋊D3—61no—
328C2×C3⋊D3—61no—
329C2×C3⋊D3—61no—
330C2×C3⋊D3—64no—
331C2×C3⋊D3—64no—
332C2×C3⋊D3—124no—
333C2×C3⋊D3—124no—
334C2×C3⋊D3C2×#6712yes—
335C2×C3×D3—24no—
336C2×C3×D3—24no—
337C2×C3×D3—24no—
338C2×C3×D3—38no—
339C2×C3×D3—64no—
340C2×C3×D3—64no—
341C32×C4—28no—
342C32×C4—62no—
343C32×C4—61no—
344C32×C4D3×D418yes—
345C32×C4C2×#6712yes—
346C3⋊Dic3—22no—
347C3⋊Dic3—62no—
348C3⋊Dic3—61no—
349C3⋊Dic3—64no—
350C3⋊Dic3—68no—
351C3⋊Dic3—124no—
352C3⋊Dic3—128no—
353C3⋊Dic3C2×#6712yes—
354C3×Dic3—24no—
355C3×Dic3—28no—
356C3×Dic3—38no—
357C3×Dic3—68no—
358C2×#41—91no—
359C2×#41—271no—
360C22⋊Q8—272no—
361C22⋊Q8—271no—
362C22⋊Q8—271no—
363#121—31no—
364#121—272no—
365#121—272no—
366#121—272no—
367#121—271no—
368C2×C4⋊C4—271no—
369C33#53711yescharacteristic, complemented, Sylow
370C23×C3—28no—
371C23×C3—34no—
372C23×C3—91no—
373C23×C3C2×D3218yes—
374C23×C3C22×C3⋊D312yes—
375C22×D3—34no—
376C22×D3—94no—
377C22×D3—91no—
378C22×D3—184no—
379C22×D3C22×C3⋊D311yescharacteristic
380C3×Q8—182no—
381C3×D4—28no—
382C3×D4—22no—
383C3×D4—48no—
384C3×D4—128no—
385C3×D4—184no—
386C2×C3×C4—24no—
387C2×C3×C4—34no—
388C2×C3×C4—38no—
389C2×C3×C4—68no—
390C2×C3×C4—91no—
391C2×C3×C4—94no—
392C2×C3×C4—92no—
393C2×C3×C4—184no—
394C2×C3×C4C2×D3214yes—
395C2×C3×C4C22×C3⋊D311yescharacteristic
396#67—42no—
397#67—616no—
398#67—182no—
399#67—188no—
400#67—182no—
401#67—188no—
402#67—368no—
403#67—362no—
404#67—3616no—
405C2×Dic3—34no—
406C2×Dic3—38no—
407C2×Dic3—616no—
408C2×Dic3—98no—
409C2×Dic3—94no—
410C2×Dic3—91no—
411C2×Dic3—92no—
412C2×Dic3—188no—
413C2×Dic3—184no—
414C2×Dic3—362no—
415C2×Dic3C22×C3⋊D311yescharacteristic
416C2×Dic3C22×C3⋊D312yes—
417D12—188no—
418D12—182no—
419C4×D3—22no—
420C4×D3—316no—
421C4×D3—188no—
422C4×D3—182no—
423C4×D3—368no—
424C4×D3—368no—
425Dic6—68no—
426Dic6—182no—
427Dic6—188no—
428C2×C32—24no—
429C2×C32—24no—
430C2×C32—24no—
431C2×C32—42no—
432C2×C32—48no—
433C2×C32—88no—
434C2×C32—121no—
435C2×C32C2×D3×D414yes—
436C2×C32#100414yes—
437C2×C32#100911yescharacteristic
438C2×C32#101014yes—
439C2×C32#101011yescharacteristic
440C2×C32C22×#6711yescharacteristic
441C3⋊D3—22no—
442C3⋊D3—128no—
443C3⋊D3—121no—
444C3⋊D3—124no—
445C3×D3—44no—
446C3×D3—68no—
447#41—184no—
448#41—544no—
449C2×Q8—91no—
450C2×Q8—271no—
451C2×D4—32no—
452C2×D4—92no—
453C2×D4—271no—
454C2×D4—271no—
455C2×D4—271no—
456C2×D4—272no—
457C2×D4D3×C3⋊D311yescharacteristic
458C22×C4—31no—
459C22×C4—91no—
460C22×C4—92no—
461C22×C4—272no—
462C22×C4—271no—
463C4⋊C4—31no—
464C4⋊C4—272no—
465C4⋊C4—272no—
466C4⋊C4—272no—
467C4⋊C4—271no—
468C22⋊C4—32no—
469C22⋊C4—272no—
470C22⋊C4—272no—
471C22⋊C4—272no—
472C22×C3—28no—
473C22×C3—28no—
474C22×C3—28no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—24no—
479C22×C3—48no—
480C22×C3—48no—
481C22×C3—48no—
482C22×C3—64no—
483C22×C3—64no—
484C22×C3—64no—
485C22×C3—92no—
486C22×C3—181no—
487C22×C3—181no—
488C22×C3C22×D3214yes—
489C22×C3C23×C3⋊D311yescharacteristic
490C2×D3—21no—
491C2×D3—21no—
492C2×D3—21no—
493C2×D3—38no—
494C2×D3—64no—
495C2×D3—64no—
496C2×D3—188no—
497C2×D3—184no—
498C2×D3—184no—
499C2×D3—184no—
500C2×D3—181no—
501C2×D3—181no—
502C2×D3—181no—
503C2×D3—364no—
504C2×D3—364no—
505C3×C4—28no—
506C3×C4—68no—
507C3×C4—64no—
508C3×C4—92no—
509C3×C4—182no—
510C3×C4D3×#6718yes—
511C3×C4D4×C3⋊D312yes—
512Dic3—21no—
513Dic3—22no—
514Dic3—38no—
515Dic3—68no—
516Dic3—188no—
517Dic3—184no—
518Dic3—181no—
519Dic3—182no—
520Dic3—364no—
521Dic3—368no—
522C32—24no—
523C32#560814yescomplemented
524C32#583311yescharacteristic, complemented
525C23—31no—
526C23—91no—
527C23—271no—
528C23C2×D3×C3⋊D312yes—
529Q8—182no—
530Q8—542no—
531D4—22no—
532D4—122no—
533D4—184no—
534D4—542no—
535D4—542no—
536D4—542no—
537D4—1082no—
538C2×C4—31no—
539C2×C4—32no—
540C2×C4—62no—
541C2×C4—91no—
542C2×C4—94no—
543C2×C4—92no—
544C2×C4—184no—
545C2×C4—272no—
546C2×C4—271no—
547C2×C4—542no—
548C2×C4—1082no—
549C2×C4C2×D3×C3⋊D311yescharacteristic
550C2×C3—24no—
551C2×C3—24no—
552C2×C3—24no—
553C2×C3—48no—
554C2×C3—42no—
555C2×C3—88no—
556C2×C3—124no—
557C2×C3—182no—
558C2×C3#6430914yes—
559C2×C3#6431414yes—
560C2×C3C2×D3×#6714yes—
561C2×C3C2×D4×C3⋊D311yescharacteristic
562C2×C3#6436111yescharacteristic
563C2×C3#6436711yescharacteristic
564D3—41no—
565D3—68no—
566D3—368no—
567D3—364no—
568D3—361no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—61no—
573C22—61no—
574C22—61no—
575C22—92no—
576C22—181no—
577C22—181no—
578C22—541no—
579C22—541no—
580C22—541no—
581C22C22×D3×C3⋊D311yescharacteristic
582C4—62no—
583C4—61no—
584C4—92no—
585C4—182no—
586C4—542no—
587C4—541no—
588C4C3⋊D3×#6712yes—
589C3—24no—
590C3#1059523214yescomplemented
591C3#1059607911yescharacteristic, complemented
592C2—42no—
593C2—121no—
594C2—182no—
595C2—1081no—
596C2#1172458211yescharacteristic
597C2#1172458811yescharacteristic
598C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#537
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C2×#89020C2semidirect
3. #11723979C2semidirect
4. C32×#938C2semidirect
5. #11723641C2semidirect
6. #11723635C2semidirect
7. #11723631C2semidirect
8. #11723624C2semidirect
9. C3⋊D3×C4⋊Dic3C2semidirect
10. #11723449C2semidirect
11. #11723425C2semidirect
39. C2×#3830D3semidirect
54. C3×#938D3semidirect
76. C3×C4×C3⋊D3D4semidirect
178. #938C3⋊D3semidirect
227. C4×C3⋊D3#67semidirect
369. C33#537semidirect
523. C32#5608semidirect
524. C32#5833semidirect
590. C3#10595232semidirect
591. C3#10596079semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#890202
#117239792
C32×#9382
#117236412
#117236352
#117236312
#117236242
C3⋊D3×C4⋊Dic32
#117234492
#117234252
#105960793
#105952323