Finite Group Databasebeta

← #13175904

Subgroups — #13175904

Order 1728.

1392 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723920C211yesmaximal, characteristic
2C3×#64087C211yesmaximal, characteristic
3C32×#879C211yesmaximal, characteristic
4#11723641C211yesmaximal, characteristic
5#11723596C211yesmaximal, characteristic
6#11723596C211yesmaximal, characteristic
7#11723595C211yesmaximal, characteristic
8#11723454C211yesmaximal, characteristic, complemented
9#11723448C211yesmaximal, characteristic, complemented
10#11723412C211yesmaximal, characteristic, complemented
11#11723410C211yesmaximal, characteristic, complemented
12#11723409C211yesmaximal, characteristic, complemented
13#11723401C211yesmaximal, characteristic, complemented
14#11723394C211yesmaximal, characteristic, complemented
15#11723389C211yesmaximal, characteristic, complemented
16#10596172—31nomaximal
17#10594027—34nomaximal
18C22×C3×C3⋊Dic3C2211yescharacteristic
19C2×C3×#634C2211yescharacteristic
20C2×#6293C2211yescharacteristic
21C33×C22⋊C4C2211yescharacteristic
22#88864C2211yescharacteristic
23#88864C2211yescharacteristic
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25#88857C2211yescharacteristic
26#88857C2211yescharacteristic
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
28C3×#3757C2211yescharacteristic
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C3×#3750C2211yescharacteristic
31C3×#3750C2211yescharacteristic
32C32×#217C2211yescharacteristic
33C32×#217C2211yescharacteristic
34C32×C4⋊Dic3C2211yescharacteristic
35C32×Dic3⋊C4C2211yescharacteristic
36C32×Dic3⋊C4C2211yescharacteristic
37C32×C4×Dic3C2211yescharacteristic
38#88807C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88807C2211yescharacteristic
42#88807C2211yescharacteristic
43#88806C2211yescharacteristic
44#88806C2211yescharacteristic
45#88806C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47Dic3⋊(C3⋊Dic3)C2211yescharacteristic
48Dic3⋊(C3⋊Dic3)C2211yescharacteristic
49Dic3×C3⋊Dic3C2211yescharacteristic
50Dic3×C3⋊Dic3C2211yescharacteristic
51Dic3×C3⋊Dic3C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×#126—31no—
54#64099—31no—
55#64098—31no—
56#64098—31no—
57#64098—31no—
58#64097—31no—
59#64096—31no—
60#64088—31no—
61#64088—31no—
62#64088—31no—
63#64088—34no—
64#64088—64no—
65#64087—31no—
66#64087D311yescharacteristic
67#64086—31no—
68#64079—31no—
69C4×#634—31no—
70C3×#879D314yes—
71C3×#878—34no—
72#63964—34no—
73#63956—34no—
74#63956—34no—
75#63955—34no—
76#63870—34no—
77#63869—34no—
78#63846—34no—
79#63846—34no—
80#63845—34no—
81#63843—34no—
82#63839—34no—
83#63837—34no—
84C23×C33C2311yescharacteristic
85C2×C33×C4C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
88C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3×C3⋊Dic3—22no—
92C2×C3×C3⋊Dic3C2311yescharacteristic
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C3×#634—22no—
97C2×C32×Dic3C2311yescharacteristic
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101#6293—22no—
102#5701—94no—
103#5625—91no—
104C22×C3⋊Dic3C2×D311yescharacteristic
105C2×#634—31no—
106C2×#634C2×D311yescharacteristic
107C22×C3×Dic3—34no—
108C2×C3×Dic6—34no—
109C2×#627—34no—
110C32×C4⋊C4—31no—
111C32×C4⋊C4—31no—
112C32×C4⋊C4—31no—
113C32×C22⋊C4—24no—
114C32×C22⋊C4—31no—
115C32×C22⋊C4—31no—
116C32×C22⋊C4C2×D314yes—
117C32×C22⋊C4C2×D311yescharacteristic
118C32×C42—31no—
119#3757—31no—
120#3757—31no—
121#3757—34no—
122#3757—34no—
123#3757—64no—
124#3757—64no—
125#3757C2×D311yescharacteristic
126C4⋊(C3⋊Dic3)—31no—
127C4⋊(C3⋊Dic3)—34no—
128C4⋊(C3⋊Dic3)—31no—
129C4⋊(C3⋊Dic3)—31no—
130C4⋊(C3⋊Dic3)—31no—
131C4⋊(C3⋊Dic3)—64no—
132C4⋊(C3⋊Dic3)C2×D311yescharacteristic
133#3750—31no—
134#3750—31no—
135#3750—34no—
136#3750—34no—
137#3750—31no—
138#3750—31no—
139#3750—31no—
140#3750—31no—
141#3750—31no—
142#3750—31no—
143#3750—31no—
144#3750—31no—
145#3750—64no—
146#3750—64no—
147#3750C2×D311yescharacteristic
148#3750C2×D311yescharacteristic
149C4×C3⋊Dic3—31no—
150C4×C3⋊Dic3—34no—
151C4×C3⋊Dic3—31no—
152C4×C3⋊Dic3—31no—
153C4×C3⋊Dic3—31no—
154C4×C3⋊Dic3—31no—
155C4×C3⋊Dic3—64no—
156C3×#217—34no—
157C3×#217C2×D314yes—
158C3×#217C2×D314yes—
159C3×C4⋊Dic3—34no—
160C3×C4⋊Dic3C2×D314yes—
161C3×Dic3⋊C4—34no—
162C3×Dic3⋊C4—34no—
163C3×Dic3⋊C4C2×D314yes—
164C3×Dic3⋊C4C2×D314yes—
165C3×C4×Dic3C2×D314yes—
166#3724—34no—
167#3724—34no—
168#3724—34no—
169Dic3⋊Dic3—34no—
170Dic3⋊Dic3—34no—
171Dic3⋊Dic3—34no—
172Dic3⋊Dic3—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic32—34no—
178Dic32—34no—
179Dic32—34no—
180Dic32—34no—
181C22×C33—21no—
182C22×C33—21no—
183C22×C33—21no—
184C22×C33C2411yescharacteristic
185C33×C4—21no—
186C33×C4—21no—
187C3⋊(C3⋊Dic3)—21no—
188C3⋊(C3⋊Dic3)—21no—
189C3⋊(C3⋊Dic3)—21no—
190C3⋊(C3⋊Dic3)—21no—
191C3×C3⋊Dic3—21no—
192C3×C3⋊Dic3—21no—
193C3×C3⋊Dic3—21no—
194C3×C3⋊Dic3#4112yes—
195C32×Dic3—21no—
196C32×Dic3—21no—
197C32×Dic3—21no—
198C32×Dic3—21no—
199C3×#126—34no—
200C3×C22⋊Q8—91no—
201Q8×Dic3—91no—
202#944—91no—
203#926—91no—
204#924—91no—
205#890—94no—
206#890—91no—
207#890—91no—
208#890—91no—
209#889—94no—
210#889—94no—
211#889—94no—
212#889—91no—
213#889—91no—
214#888—94no—
215#887—94no—
216#887—91no—
217#879—94no—
218#879—94no—
219#879—94no—
220#879—91no—
221#879—91no—
222#879—91no—
223#879—184no—
224#879C3⋊D311yescharacteristic
225#878—34no—
226#878—94no—
227#877—94no—
228#870—94no—
229C4×Dic6—94no—
230C23×C32—24no—
231C23×C32C22×D314yes—
232C23×C32C22×D311yescharacteristic
233C2×C32×C4—24no—
234C2×C32×C4—24no—
235C2×C32×C4—31no—
236C2×C32×C4—31no—
237C2×C32×C4—31no—
238C2×C32×C4—31no—
239C2×C32×C4C22×D314yes—
240C2×C32×C4C22×D314yes—
241C2×C32×C4C22×D311yescharacteristic
242C2×C32×C4C22×D311yescharacteristic
243C2×C3⋊Dic3—22no—
244C2×C3⋊Dic3—31no—
245C2×C3⋊Dic3—31no—
246C2×C3⋊Dic3—31no—
247C2×C3⋊Dic3—31no—
248C2×C3⋊Dic3—34no—
249C2×C3⋊Dic3—34no—
250C2×C3⋊Dic3—34no—
251C2×C3⋊Dic3—34no—
252C2×C3⋊Dic3—64no—
253C2×C3⋊Dic3—64no—
254C2×C3⋊Dic3—64no—
255C2×C3⋊Dic3—64no—
256C2×C3⋊Dic3C22×D311yescharacteristic
257C2×C3⋊Dic3C22×D311yescharacteristic
258C2×C3⋊Dic3C22×D311yescharacteristic
259C2×C3⋊Dic3C22×D311yescharacteristic
260#634—22no—
261#634—62no—
262C2×C3×Dic3—34no—
263C2×C3×Dic3—34no—
264C2×C3×Dic3—34no—
265C2×C3×Dic3—34no—
266C2×C3×Dic3—68no—
267C2×C3×Dic3C22×D314yes—
268C2×C3×Dic3C22×D314yes—
269C2×C3×Dic3C22×D314yes—
270C2×C3×Dic3C22×D314yes—
271C3×Dic6—68no—
272#627—68no—
273#541—271noSylow
274C2×C33—41no—
275C2×C33C2×#4111yescharacteristic
276C2×C332−1+411yescharacteristic
277C2×C332−1+411yescharacteristic
278C2×C3×Q8—91no—
279C22×C3×C4—91no—
280C22×Dic3—34no—
281C2×Dic6—34no—
282C2×Dic6—94no—
283C2×Dic6—91no—
284C3×C4⋊C4—34no—
285C3×C4⋊C4—34no—
286C3×C4⋊C4—34no—
287C3×C4⋊C4—91no—
288C3×C4⋊C4—91no—
289C3×C4⋊C4—91no—
290C3×C22⋊C4—24no—
291C3×C22⋊C4—34no—
292C3×C22⋊C4—34no—
293C3×C22⋊C4—91no—
294C3×C22⋊C4D3214yes—
295C3×C22⋊C4C2×C3⋊D311yescharacteristic
296C3×C42—34no—
297#217—34no—
298#217—94no—
299#217—94no—
300#217—91no—
301#217—91no—
302#217—184no—
303#217—184no—
304#217C2×C3⋊D311yescharacteristic
305#217C2×C3⋊D311yescharacteristic
306C4⋊Dic3—34no—
307C4⋊Dic3—94no—
308C4⋊Dic3—94no—
309C4⋊Dic3—94no—
310C4⋊Dic3—94no—
311C4⋊Dic3—91no—
312C4⋊Dic3—91no—
313C4⋊Dic3—91no—
314C4⋊Dic3—91no—
315C4⋊Dic3—91no—
316C4⋊Dic3—91no—
317C4⋊Dic3—184no—
318C4⋊Dic3C2×C3⋊D311yescharacteristic
319Dic3⋊C4—34no—
320Dic3⋊C4—34no—
321Dic3⋊C4—94no—
322Dic3⋊C4—94no—
323Dic3⋊C4—94no—
324Dic3⋊C4—94no—
325Dic3⋊C4—94no—
326Dic3⋊C4—94no—
327Dic3⋊C4—94no—
328Dic3⋊C4—94no—
329Dic3⋊C4—94no—
330Dic3⋊C4—94no—
331Dic3⋊C4—91no—
332Dic3⋊C4—91no—
333Dic3⋊C4—91no—
334Dic3⋊C4—91no—
335Dic3⋊C4—91no—
336Dic3⋊C4—91no—
337Dic3⋊C4—91no—
338Dic3⋊C4—91no—
339Dic3⋊C4—184no—
340Dic3⋊C4—184no—
341Dic3⋊C4C2×C3⋊D311yescharacteristic
342Dic3⋊C4C2×C3⋊D311yescharacteristic
343C4×Dic3—94no—
344C4×Dic3—94no—
345C4×Dic3—94no—
346C4×Dic3—94no—
347C4×Dic3—94no—
348C4×Dic3—91no—
349C4×Dic3—91no—
350C4×Dic3—91no—
351C4×Dic3—91no—
352C4×Dic3—91no—
353C4×Dic3—184no—
354C4×Dic3C2×C3⋊D311yescharacteristic
355C22×C32—21no—
356C22×C32—21no—
357C22×C32—21no—
358C22×C32—24no—
359C22×C32—24no—
360C22×C32—24no—
361C22×C32—24no—
362C22×C32—44no—
363C22×C32—44no—
364C22×C32—44no—
365C22×C32C23×D314yes—
366C22×C32C23×D311yescharacteristic
367C32×C4—21no—
368C32×C4—21no—
369C32×C4—24no—
370C32×C4—24no—
371C32×C4—44no—
372C32×C4—44no—
373C32×C4—61no—
374C32×C4—61no—
375C32×C4—61no—
376C32×C4—61no—
377C3⋊Dic3—21no—
378C3⋊Dic3—21no—
379C3⋊Dic3—21no—
380C3⋊Dic3—61no—
381C3⋊Dic3—61no—
382C3⋊Dic3—61no—
383C3⋊Dic3—61no—
384C3⋊Dic3—64no—
385C3⋊Dic3—64no—
386C3⋊Dic3—64no—
387C3⋊Dic3—64no—
388C3⋊Dic3—124no—
389C3⋊Dic3—124no—
390C3⋊Dic3—124no—
391C3⋊Dic3—124no—
392C3⋊Dic3#23712yes—
393C3×Dic3—24no—
394C3×Dic3—24no—
395C3×Dic3—24no—
396C3×Dic3—24no—
397C3×Dic3—38no—
398C3×Dic3—64no—
399C3×Dic3—64no—
400C3×Dic3—64no—
401C4⋊Q8—271no—
402#126—31no—
403#126—271no—
404#126—271no—
405#126—271no—
406#125—271no—
407#125—271no—
408#125—271no—
409#125—271no—
410#125—271no—
411C22⋊Q8—91no—
412C22⋊Q8—271no—
413C4×Q8—271no—
414C4×Q8—271no—
415#117—271no—
416C33#54111yescharacteristic, complemented, Sylow
417C23×C3—24no—
418C23×C3C2×D3214yes—
419C23×C3C22×C3⋊D311yescharacteristic
420C3×Q8—182no—
421C2×C3×C4—24no—
422C2×C3×C4—24no—
423C2×C3×C4—34no—
424C2×C3×C4—34no—
425C2×C3×C4—34no—
426C2×C3×C4—34no—
427C2×C3×C4—91no—
428C2×C3×C4—91no—
429C2×C3×C4—91no—
430C2×C3×C4—91no—
431C2×C3×C4—182no—
432C2×C3×C4C2×D3214yes—
433C2×C3×C4C2×D3214yes—
434C2×C3×C4C22×C3⋊D311yescharacteristic
435C2×C3×C4C22×C3⋊D311yescharacteristic
436C2×Dic3—34no—
437C2×Dic3—34no—
438C2×Dic3—34no—
439C2×Dic3—34no—
440C2×Dic3—68no—
441C2×Dic3—94no—
442C2×Dic3—94no—
443C2×Dic3—94no—
444C2×Dic3—94no—
445C2×Dic3—91no—
446C2×Dic3—91no—
447C2×Dic3—91no—
448C2×Dic3—91no—
449C2×Dic3—184no—
450C2×Dic3—184no—
451C2×Dic3—184no—
452C2×Dic3—184no—
453C2×Dic3C22×C3⋊D311yescharacteristic
454C2×Dic3C22×C3⋊D311yescharacteristic
455C2×Dic3C22×C3⋊D311yescharacteristic
456C2×Dic3C22×C3⋊D311yescharacteristic
457Dic6—68no—
458Dic6—182no—
459Dic6—188no—
460C2×C32—24no—
461C2×C32—24no—
462C2×C32—24no—
463C2×C32—41no—
464C2×C32—44no—
465C2×C32—84no—
466C2×C32C2×#23711yescharacteristic
467C2×C32#100711yescharacteristic
468C2×C32D3×#4114yes—
469C2×C32#101014yes—
470C2×C32#101014yes—
471C2×C32#101011yescharacteristic
472C2×Q8—91no—
473C2×Q8—271no—
474C22×C4—91no—
475C4⋊C4—31no—
476C4⋊C4—31no—
477C4⋊C4—31no—
478C4⋊C4—91no—
479C4⋊C4—91no—
480C4⋊C4—91no—
481C4⋊C4—271no—
482C4⋊C4—271no—
483C4⋊C4—271no—
484C4⋊C4—271no—
485C4⋊C4—271no—
486C4⋊C4—271no—
487C4⋊C4—271no—
488C4⋊C4—271no—
489C4⋊C4—271no—
490C4⋊C4—271no—
491C4⋊C4—271no—
492C4⋊C4—271no—
493C4⋊C4—271no—
494C4⋊C4—271no—
495C22⋊C4—31no—
496C22⋊C4—31no—
497C22⋊C4—91no—
498C22⋊C4—271no—
499C22⋊C4—271no—
500C22⋊C4D3×C3⋊D311yescharacteristic
501C42—31no—
502C42—271no—
503C42—271no—
504C42—271no—
505C42—271no—
506C42—271no—
507C22×C3—24no—
508C22×C3—24no—
509C22×C3—24no—
510C22×C3—21no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—24no—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3—44no—
517C22×C3C22×D3214yes—
518C22×C3C23×C3⋊D311yescharacteristic
519C3×C4—24no—
520C3×C4—24no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—44no—
524C3×C4—44no—
525C3×C4—64no—
526C3×C4—64no—
527C3×C4—64no—
528C3×C4—64no—
529C3×C4—92no—
530C3×C4—181no—
531C3×C4—181no—
532C3×C4—181no—
533Dic3—21no—
534Dic3—21no—
535Dic3—21no—
536Dic3—21no—
537Dic3—38no—
538Dic3—64no—
539Dic3—64no—
540Dic3—64no—
541Dic3—184no—
542Dic3—184no—
543Dic3—184no—
544Dic3—184no—
545Dic3—181no—
546Dic3—181no—
547Dic3—181no—
548Dic3—181no—
549Dic3—364no—
550Dic3—364no—
551Dic3—364no—
552Dic3—364no—
553C32—24no—
554C32#562511yescharacteristic, complemented
555C32#570114yescomplemented
556C23C2×D3×C3⋊D311yescharacteristic
557Q8—182no—
558Q8—542no—
559C2×C4—31no—
560C2×C4—31no—
561C2×C4—31no—
562C2×C4—31no—
563C2×C4—91no—
564C2×C4—91no—
565C2×C4—91no—
566C2×C4—91no—
567C2×C4—182no—
568C2×C4—271no—
569C2×C4—271no—
570C2×C4—271no—
571C2×C4—271no—
572C2×C4C2×D3×C3⋊D311yescharacteristic
573C2×C4C2×D3×C3⋊D311yescharacteristic
574C2×C3—24no—
575C2×C3—24no—
576C2×C3—24no—
577C2×C3—41no—
578C2×C3—44no—
579C2×C3—84no—
580C2×C3#6429714yes—
581C2×C3#6430914yes—
582C2×C3D3×#23714yes—
583C2×C3#41×C3⋊D311yescharacteristic
584C2×C3#6436711yescharacteristic
585C2×C3#6436711yescharacteristic
586C22—21no—
587C22—21no—
588C22—21no—
589C22C22×D3×C3⋊D311yescharacteristic
590C4—21no—
591C4—21no—
592C4—61no—
593C4—61no—
594C4—61no—
595C4—61no—
596C4—92no—
597C4—181no—
598C4—181no—
599C4—181no—
600C4—541no—
601C4—541no—
602C4—541no—
603C4—541no—
604C3—24no—
605C3#1059402714yescomplemented
606C3#1059617211yescharacteristic, complemented
607C2—41no—
608C2#1172456211yescharacteristic
609C2#1172458211yescharacteristic
610C2C3⋊D3×#23711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC23×C33C23
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#541
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
8. #11723454C2semidirect
9. #11723448C2semidirect
10. #11723412C2semidirect
11. #11723410C2semidirect
12. #11723409C2semidirect
13. #11723401C2semidirect
14. #11723394C2semidirect
15. #11723389C2semidirect
416. C33#541semidirect
554. C32#5625semidirect
555. C32#5701semidirect
605. C3#10594027semidirect
606. C3#10596172semidirect

Maximal subgroups

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

subgroupindex
#117239202
C3×#640872
C32×#8792
#117236412
#117235962
#117235952
#117234542
#117234482
#117234122
#117234102
#117234092
#117234012
#117233942
#117233892
#105961723
#105940273