Finite Group Databasebeta

← #13175888

Subgroups — #13175888

Order 1728.

1456 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723918C211yesmaximal, characteristic
2C3×#64088C211yesmaximal, characteristic
3C32×#879C211yesmaximal, characteristic
4#11723642C211yesmaximal, characteristic
5#11723642C211yesmaximal, characteristic
6#11723597C211yesmaximal, characteristic
7#11723596C211yesmaximal, characteristic
8#11723536C211yesmaximal, characteristic, complemented
9C4×#6293C211yesmaximal, characteristic, complemented
10#11723448C211yesmaximal, characteristic, complemented
11#11723445C211yesmaximal, characteristic, complemented
12#11723444C211yesmaximal, characteristic, complemented
13#11723412C211yesmaximal, characteristic, complemented
14#11723401C211yesmaximal, characteristic, complemented
15#11723391C211yesmaximal, characteristic, complemented
16#10596172—31nomaximal
17#10594249—34nomaximal
18C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×#6293C2211yescharacteristic
20C2×#6293C2211yescharacteristic
21C33×C22⋊C4C2211yescharacteristic
22#88864C2211yescharacteristic
23#88857C2211yescharacteristic
24#88857C2211yescharacteristic
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
27C3×#3757C2211yescharacteristic
28C3×#3757C2211yescharacteristic
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C3×#3750C2211yescharacteristic
31C3×#3750C2211yescharacteristic
32C3×C4×C3⋊Dic3C2211yescharacteristic
33C32×#217C2211yescharacteristic
34C32×#217C2211yescharacteristic
35C32×C4⋊Dic3C2211yescharacteristic
36C32×Dic3⋊C4C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38C32×C4×Dic3C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88807C2211yescharacteristic
42#88807C2211yescharacteristic
43#88806C2211yescharacteristic
44#88806C2211yescharacteristic
45#88806C2211yescharacteristic
46#88806C2211yescharacteristic
47Dic3⋊(C3⋊Dic3)C2211yescharacteristic
48Dic3⋊(C3⋊Dic3)C2211yescharacteristic
49Dic3⋊(C3⋊Dic3)C2211yescharacteristic
50Dic3⋊(C3⋊Dic3)C2211yescharacteristic
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#64088D311yescharacteristic
63#64087—31no—
64#64087—31no—
65#64086—31no—
66#64086—34no—
67#64086—64no—
68#64079—31no—
69C4×#634—31no—
70C3×#879—34no—
71C3×#879D314yes—
72#63982—34no—
73#63982—34no—
74#63956—34no—
75#63956—34no—
76#63919—34no—
77C4×#627—34no—
78#63870—34no—
79#63867—34no—
80#63867—34no—
81#63846—34no—
82#63846—34no—
83#63838—34no—
84C23×C33C2311yescharacteristic
85C2×C33×C4C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C3⋊(C3⋊Dic3)—22no—
88C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92C2×C3×C3⋊Dic3C2311yescharacteristic
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C32×Dic3C2311yescharacteristic
97C2×C32×Dic3C2311yescharacteristic
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100#6293—22no—
101#6293—22no—
102#5701—94no—
103#5701—91no—
104C22×C3⋊Dic3—31no—
105C22×C3⋊Dic3—34no—
106C22×C3⋊Dic3—64no—
107C2×#634—31no—
108C2×#634—31no—
109C2×#627—34no—
110C2×#627—34no—
111C32×C4⋊C4—31no—
112C32×C4⋊C4—31no—
113C32×C4⋊C4—31no—
114C32×C22⋊C4—24no—
115C32×C22⋊C4—31no—
116C32×C22⋊C4—31no—
117C32×C22⋊C4C2×D314yes—
118C32×C22⋊C4C2×D311yescharacteristic
119C32×C42—31no—
120#3757—31no—
121#3757—34no—
122#3757—64no—
123#3757C2×D311yescharacteristic
124#3757C2×D311yescharacteristic
125C4⋊(C3⋊Dic3)—31no—
126C4⋊(C3⋊Dic3)—31no—
127C4⋊(C3⋊Dic3)—31no—
128C4⋊(C3⋊Dic3)—31no—
129C4⋊(C3⋊Dic3)C2×D311yescharacteristic
130#3750—31no—
131#3750—31no—
132#3750—34no—
133#3750—34no—
134#3750—31no—
135#3750—31no—
136#3750—31no—
137#3750—31no—
138#3750—31no—
139#3750—31no—
140#3750—31no—
141#3750—31no—
142#3750—64no—
143#3750—64no—
144#3750C2×D311yescharacteristic
145#3750C2×D311yescharacteristic
146C4×C3⋊Dic3—31no—
147C4×C3⋊Dic3—31no—
148C4×C3⋊Dic3—34no—
149C4×C3⋊Dic3—34no—
150C4×C3⋊Dic3—31no—
151C4×C3⋊Dic3—31no—
152C4×C3⋊Dic3—64no—
153C4×C3⋊Dic3—64no—
154C4×C3⋊Dic3C2×D311yescharacteristic
155C3×#217—34no—
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×Dic3—34no—
166C3×C4×Dic3C2×D314yes—
167#3724—34no—
168#3724—34no—
169#3724—34no—
170#3724—34no—
171Dic3⋊Dic3—34no—
172Dic3⋊Dic3—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178Dic3⋊Dic3—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)#4112yes—
191C3×C3⋊Dic3—21no—
192C3×C3⋊Dic3—21no—
193C3×C3⋊Dic3—21no—
194C3×C3⋊Dic3—21no—
195C32×Dic3—21no—
196C32×Dic3—21no—
197C32×Dic3—21no—
198C32×Dic3—21no—
199C3×#126—34no—
200C3×#126—91no—
201#890—94no—
202#890—91no—
203#889—94no—
204#889—94no—
205#889—94no—
206#889—91no—
207#889—91no—
208#889—91no—
209#888—94no—
210#888—91no—
211#887—94no—
212#887—91no—
213#879—34no—
214#879—94no—
215#879—94no—
216#879—91no—
217#879—91no—
218#879C3⋊D311yescharacteristic
219#878—94no—
220#878—94no—
221#878—91no—
222#878—91no—
223#877—94no—
224#877—91no—
225#877—184no—
226#870—94no—
227#870—91no—
228C4×Dic6—94no—
229C4×Dic6—91no—
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—31no—
244C2×C3⋊Dic3—31no—
245C2×C3⋊Dic3—31no—
246C2×C3⋊Dic3—31no—
247C2×C3⋊Dic3—34no—
248C2×C3⋊Dic3—34no—
249C2×C3⋊Dic3—34no—
250C2×C3⋊Dic3—34no—
251C2×C3⋊Dic3—64no—
252C2×C3⋊Dic3—64no—
253C2×C3⋊Dic3—64no—
254C2×C3⋊Dic3—64no—
255C2×C3⋊Dic3—62no—
256C2×C3⋊Dic3—68no—
257C2×C3⋊Dic3—616no—
258C2×C3⋊Dic3C22×D311yescharacteristic
259C2×C3⋊Dic3C22×D311yescharacteristic
260C2×C3⋊Dic3C22×D311yescharacteristic
261C2×C3⋊Dic3C22×D311yescharacteristic
262#634—62no—
263#634—62no—
264C2×C3×Dic3—34no—
265C2×C3×Dic3—34no—
266C2×C3×Dic3—34no—
267C2×C3×Dic3—34no—
268C2×C3×Dic3C22×D314yes—
269C2×C3×Dic3C22×D314yes—
270C2×C3×Dic3C22×D314yes—
271C2×C3×Dic3C22×D314yes—
272#627—68no—
273#627—68no—
274#541—271noSylow
275C2×C33—41no—
276C2×C33C2×#4111yescharacteristic
277C2×C332−1+411yescharacteristic
278C2×C332−1+411yescharacteristic
279C22×Dic3—94no—
280C22×Dic3—91no—
281C22×Dic3—184no—
282C2×Dic6—94no—
283C2×Dic6—94no—
284C2×Dic6—91no—
285C2×Dic6—91no—
286C3×C4⋊C4—34no—
287C3×C4⋊C4—34no—
288C3×C4⋊C4—34no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4—91no—
291C3×C4⋊C4—91no—
292C3×C22⋊C4—24no—
293C3×C22⋊C4—34no—
294C3×C22⋊C4—34no—
295C3×C22⋊C4—91no—
296C3×C22⋊C4—91no—
297C3×C22⋊C4D3214yes—
298C3×C22⋊C4C2×C3⋊D311yescharacteristic
299C3×C42—34no—
300C3×C42—91no—
301#217—34no—
302#217—34no—
303#217—94no—
304#217—91no—
305#217—184no—
306#217C2×C3⋊D311yescharacteristic
307#217C2×C3⋊D311yescharacteristic
308C4⋊Dic3—34no—
309C4⋊Dic3—94no—
310C4⋊Dic3—94no—
311C4⋊Dic3—94no—
312C4⋊Dic3—94no—
313C4⋊Dic3—91no—
314C4⋊Dic3—91no—
315C4⋊Dic3—91no—
316C4⋊Dic3—91no—
317C4⋊Dic3C2×C3⋊D311yescharacteristic
318Dic3⋊C4—34no—
319Dic3⋊C4—34no—
320Dic3⋊C4—94no—
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—91no—
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—91no—
340Dic3⋊C4—184no—
341Dic3⋊C4—184no—
342Dic3⋊C4C2×C3⋊D311yescharacteristic
343Dic3⋊C4C2×C3⋊D311yescharacteristic
344C4×Dic3—34no—
345C4×Dic3—94no—
346C4×Dic3—94no—
347C4×Dic3—94no—
348C4×Dic3—94no—
349C4×Dic3—91no—
350C4×Dic3—91no—
351C4×Dic3—91no—
352C4×Dic3—91no—
353C4×Dic3—184no—
354C4×Dic3—184no—
355C4×Dic3C2×C3⋊D311yescharacteristic
356C22×C32—28no—
357C22×C32—24no—
358C22×C32—21no—
359C22×C32—21no—
360C22×C32—21no—
361C22×C32—24no—
362C22×C32—24no—
363C22×C32—24no—
364C22×C32—44no—
365C22×C32—44no—
366C22×C32C23×D314yes—
367C22×C32C23×D311yescharacteristic
368C32×C4—21no—
369C32×C4—21no—
370C32×C4—24no—
371C32×C4—24no—
372C32×C4—44no—
373C32×C4—44no—
374C32×C4—61no—
375C32×C4—61no—
376C32×C4—61no—
377C32×C4—61no—
378C3⋊Dic3—21no—
379C3⋊Dic3—21no—
380C3⋊Dic3—21no—
381C3⋊Dic3—21no—
382C3⋊Dic3—32no—
383C3⋊Dic3—38no—
384C3⋊Dic3—68no—
385C3⋊Dic3—68no—
386C3⋊Dic3—64no—
387C3⋊Dic3—61no—
388C3⋊Dic3—61no—
389C3⋊Dic3—61no—
390C3⋊Dic3—64no—
391C3⋊Dic3—64no—
392C3⋊Dic3—124no—
393C3⋊Dic3—124no—
394C3×Dic3—24no—
395C3×Dic3—24no—
396C3×Dic3—24no—
397C3×Dic3—24no—
398C3×Dic3—64no—
399C3×Dic3—64no—
400C3×Dic3—64no—
401C3×Dic3—64no—
402C4⋊Q8—271no—
403#126—31no—
404#126—91no—
405#126—271no—
406#126—271no—
407#125—271no—
408#125—271no—
409#125—271no—
410#125—271no—
411#125—271no—
412C22⋊Q8—271no—
413C22⋊Q8—271no—
414C4×Q8—271no—
415C4×Q8—271no—
416#117—271no—
417C33#54111yescharacteristic, complemented, Sylow
418C23×C3—24no—
419C23×C3C2×D3214yes—
420C23×C3C22×C3⋊D311yescharacteristic
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×C4C2×D3214yes—
432C2×C3×C4C2×D3214yes—
433C2×C3×C4C22×C3⋊D311yescharacteristic
434C2×C3×C4C22×C3⋊D311yescharacteristic
435C2×Dic3—34no—
436C2×Dic3—34no—
437C2×Dic3—34no—
438C2×Dic3—34no—
439C2×Dic3—94no—
440C2×Dic3—91no—
441C2×Dic3—94no—
442C2×Dic3—94no—
443C2×Dic3—94no—
444C2×Dic3—91no—
445C2×Dic3—91no—
446C2×Dic3—91no—
447C2×Dic3—184no—
448C2×Dic3—184no—
449C2×Dic3—184no—
450C2×Dic3—184no—
451C2×Dic3—188no—
452C2×Dic3—182no—
453C2×Dic3—1816no—
454C2×Dic3C22×C3⋊D311yescharacteristic
455C2×Dic3C22×C3⋊D311yescharacteristic
456C2×Dic3C22×C3⋊D311yescharacteristic
457C2×Dic3C22×C3⋊D311yescharacteristic
458Dic6—182no—
459Dic6—182no—
460Dic6—188no—
461Dic6—188no—
462C2×C32—24no—
463C2×C32—24no—
464C2×C32—24no—
465C2×C32—48no—
466C2×C32—44no—
467C2×C32—41no—
468C2×C32D3×#4114yes—
469C2×C32D3×#4111yescharacteristic
470C2×C32#101014yes—
471C2×C32#101014yes—
472C2×C32#101011yescharacteristic
473C2×C32#101011yescharacteristic
474C2×Q8—271no—
475C2×Q8—271no—
476C22×C4—271no—
477C4⋊C4—31no—
478C4⋊C4—31no—
479C4⋊C4—31no—
480C4⋊C4—91no—
481C4⋊C4—91no—
482C4⋊C4—91no—
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—
495C4⋊C4—271no—
496C4⋊C4—271no—
497C22⋊C4—31no—
498C22⋊C4—31no—
499C22⋊C4—91no—
500C22⋊C4—91no—
501C22⋊C4—271no—
502C22⋊C4D3×C3⋊D311yescharacteristic
503C42—31no—
504C42—91no—
505C42—271no—
506C42—271no—
507C42—271no—
508C42—271no—
509C22×C3—28no—
510C22×C3—24no—
511C22×C3—24no—
512C22×C3—24no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—24no—
517C22×C3—44no—
518C22×C3—44no—
519C22×C3C22×D3214yes—
520C22×C3C23×C3⋊D311yescharacteristic
521C3×C4—24no—
522C3×C4—24no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—44no—
526C3×C4—44no—
527C3×C4—64no—
528C3×C4—64no—
529C3×C4—64no—
530C3×C4—64no—
531C3×C4—181no—
532C3×C4—181no—
533C3×C4—181no—
534C3×C4—181no—
535Dic3—21no—
536Dic3—21no—
537Dic3—21no—
538Dic3—21no—
539Dic3—64no—
540Dic3—64no—
541Dic3—64no—
542Dic3—64no—
543Dic3—98no—
544Dic3—92no—
545Dic3—188no—
546Dic3—188no—
547Dic3—184no—
548Dic3—184no—
549Dic3—184no—
550Dic3—181no—
551Dic3—181no—
552Dic3—181no—
553Dic3—364no—
554Dic3—364no—
555C32—24no—
556C32#570114yescomplemented
557C32#570111yescharacteristic, complemented
558C23C2×D3×C3⋊D311yescharacteristic
559Q8—542no—
560Q8—542no—
561C2×C4—31no—
562C2×C4—31no—
563C2×C4—31no—
564C2×C4—31no—
565C2×C4—91no—
566C2×C4—91no—
567C2×C4—91no—
568C2×C4—91no—
569C2×C4—271no—
570C2×C4—271no—
571C2×C4—271no—
572C2×C4—271no—
573C2×C4—542no—
574C2×C4C2×D3×C3⋊D311yescharacteristic
575C2×C4C2×D3×C3⋊D311yescharacteristic
576C2×C3—24no—
577C2×C3—24no—
578C2×C3—24no—
579C2×C3—48no—
580C2×C3—41no—
581C2×C3—44no—
582C2×C3#6430614yes—
583C2×C3#6430914yes—
584C2×C3#6430914yes—
585C2×C3#41×C3⋊D311yescharacteristic
586C2×C3#6436711yescharacteristic
587C2×C3#6436711yescharacteristic
588C22—21no—
589C22—21no—
590C22—21no—
591C22C22×D3×C3⋊D311yescharacteristic
592C4—21no—
593C4—21no—
594C4—61no—
595C4—61no—
596C4—61no—
597C4—61no—
598C4—181no—
599C4—181no—
600C4—181no—
601C4—181no—
602C4—272no—
603C4—541no—
604C4—541no—
605C4—541no—
606C3—24no—
607C3#1059424914yescomplemented
608C3#1059617211yescharacteristic, complemented
609C2—41no—
610C2#1172457811yescharacteristic
611C2#1172458211yescharacteristic
612C2#1172458211yescharacteristic
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. #11723536C2semidirect
9. C4×#6293C2semidirect
10. #11723448C2semidirect
11. #11723445C2semidirect
12. #11723444C2semidirect
13. #11723412C2semidirect
14. #11723401C2semidirect
15. #11723391C2semidirect
417. C33#541semidirect
556. C32#5701semidirect
557. C32#5701semidirect
607. C3#10594249semidirect
608. C3#10596172semidirect

Maximal subgroups

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

subgroupindex
#117239182
C3×#640882
C32×#8792
#117236422
#117235972
#117235962
#117235362
C4×#62932
#117234482
#117234452
#117234442
#117234122
#117234012
#117233912
#105961723
#105942493