Finite Group Databasebeta

← #13176489

Subgroups — #13176489

Order 1728.

1424 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723930C211yesmaximal, characteristic
2C3×#64097C211yesmaximal, characteristic
3C32×#889C211yesmaximal, characteristic
4#11723412C211yesmaximal, characteristic
5#11723410C211yesmaximal, characteristic
6#11723409C211yesmaximal, characteristic
7#11723402C211yesmaximal, characteristic
8#11723401C211yesmaximal, characteristic
9Dic3×#634C211yesmaximal, characteristic
10#11723394C211yesmaximal, characteristic
11#11723394C211yesmaximal, characteristic
12#11723391C211yesmaximal, characteristic
13#11723390C211yesmaximal, characteristic
14#11723390C211yesmaximal, characteristic
15#11723389C211yesmaximal, characteristic
16#10596156—31nomaximal
17#10594314—34nomaximal
18C2×C3×#634C2211yescharacteristic
19C2×C3×#634C2211yescharacteristic
20C2×#6293C2211yescharacteristic
21C2×#6293C2211yescharacteristic
22C33×C4⋊C4C2211yescharacteristic
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24#88857C2211yescharacteristic
25#88857C2211yescharacteristic
26#88857C2211yescharacteristic
27#88857C2211yescharacteristic
28C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C3×#3750C2211yescharacteristic
31C3×#3750C2211yescharacteristic
32C3×C4×C3⋊Dic3C2211yescharacteristic
33C32×C4⋊Dic3C2211yescharacteristic
34C32×Dic3⋊C4C2211yescharacteristic
35C32×Dic3⋊C4C2211yescharacteristic
36C32×Dic3⋊C4C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38C32×C4×Dic3C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88807C2211yescharacteristic
42#88807C2211yescharacteristic
43#88806C2211yescharacteristic
44#88806C2211yescharacteristic
45Dic3⋊(C3⋊Dic3)C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47Dic3×C3⋊Dic3C2211yescharacteristic
48Dic3×C3⋊Dic3C2211yescharacteristic
49Dic3×C3⋊Dic3C2211yescharacteristic
50Dic3×C3⋊Dic3C2211yescharacteristic
51Dic3×C3⋊Dic3C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×#125—31no—
54#64099—31no—
55#64098—31no—
56#64098—31no—
57#64098—31no—
58#64098—31no—
59#64098—34no—
60#64098—64no—
61#64097—31no—
62#64097—31no—
63#64097D311yescharacteristic
64#64096—31no—
65#64096—31no—
66#64096—31no—
67#64096—31no—
68C4×#634—31no—
69C4×#634—31no—
70C3×#889D314yes—
71C3×#888—34no—
72#63847—34no—
73#63846—34no—
74#63846—34no—
75#63845—34no—
76#63843—34no—
77Dic3×Dic6—34no—
78#63839—34no—
79#63839—34no—
80#63838—34no—
81#63837—34no—
82#63837—34no—
83#63837—34no—
84C2×C33×C4C2311yescharacteristic
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⋊Dic3C2311yescharacteristic
92C2×C3×C3⋊Dic3C2311yescharacteristic
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C3×#634—22no—
96C3×#634—22no—
97C2×C32×Dic3C2311yescharacteristic
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101#6293—22no—
102#6293Q814yescomplemented
103#5721—91no—
104#5685—94no—
105C2×#634—31no—
106C2×#634—31no—
107C2×#634C2×D311yescharacteristic
108C2×#634C2×D311yescharacteristic
109C2×C3×Dic6—34no—
110C2×C3×Dic6—34no—
111C2×#627—34no—
112C2×#627—34no—
113C32×C4⋊C4—24no—
114C32×C4⋊C4—31no—
115C32×C4⋊C4—31no—
116C32×C4⋊C4—31no—
117C32×C4⋊C4—31no—
118C32×C4⋊C4—31no—
119C32×C4⋊C4C2×D314yes—
120C32×C4⋊C4C2×D311yescharacteristic
121C32×C42—31no—
122C4⋊(C3⋊Dic3)—31no—
123C4⋊(C3⋊Dic3)—34no—
124C4⋊(C3⋊Dic3)—31no—
125C4⋊(C3⋊Dic3)—31no—
126C4⋊(C3⋊Dic3)—64no—
127C4⋊(C3⋊Dic3)C2×D311yescharacteristic
128#3750—31no—
129#3750—31no—
130#3750—31no—
131#3750—31no—
132#3750—34no—
133#3750—34no—
134#3750—34no—
135#3750—34no—
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#3750—64no—
145#3750—64no—
146#3750C2×D311yescharacteristic
147#3750C2×D311yescharacteristic
148C4×C3⋊Dic3—31no—
149C4×C3⋊Dic3—34no—
150C4×C3⋊Dic3—31no—
151C4×C3⋊Dic3—31no—
152C4×C3⋊Dic3—31no—
153C4×C3⋊Dic3—31no—
154C4×C3⋊Dic3—31no—
155C4×C3⋊Dic3—31no—
156C4×C3⋊Dic3—64no—
157C4×C3⋊Dic3C2×D311yescharacteristic
158C3×C4⋊Dic3—34no—
159C3×C4⋊Dic3C2×D314yes—
160C3×Dic3⋊C4—34no—
161C3×Dic3⋊C4—34no—
162C3×Dic3⋊C4C2×D314yes—
163C3×Dic3⋊C4C2×D314yes—
164C3×Dic3⋊C4C2×D314yes—
165C3×Dic3⋊C4C2×D314yes—
166C3×C4×Dic3—34no—
167C3×C4×Dic3C2×D314yes—
168#3724—34no—
169#3724—34no—
170Dic3⋊Dic3—34no—
171Dic3⋊Dic3—34no—
172Dic3⋊Dic3—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic32—34no—
177Dic32—34no—
178Dic32—34no—
179Dic32—34no—
180Dic32—34no—
181Dic32—34no—
182C22×C33C2411yescharacteristic
183C33×C4—21no—
184C33×C4—21no—
185C33×C4—21no—
186C3⋊(C3⋊Dic3)—21no—
187C3⋊(C3⋊Dic3)—21no—
188C3⋊(C3⋊Dic3)—21no—
189C3⋊(C3⋊Dic3)C2×Q812yes—
190C3×C3⋊Dic3—21no—
191C3×C3⋊Dic3—21no—
192C3×C3⋊Dic3C2×Q812yes—
193C3×C3⋊Dic3#4112yes—
194C32×Dic3—21no—
195C32×Dic3—21no—
196C32×Dic3—21no—
197C32×Dic3C2×Q812yes—
198C3×C4⋊Q8—91no—
199C3×#125—34no—
200Q8×Dic3—91no—
201Q8×Dic3—91no—
202#944—91no—
203#944—91no—
204#890—94no—
205#890—91no—
206#890—91no—
207#889—94no—
208#889—94no—
209#889—94no—
210#889—94no—
211#889—91no—
212#889—91no—
213#889—91no—
214#889—184no—
215#889C3⋊D311yescharacteristic
216#888—34no—
217#888—94no—
218#888—94no—
219#887—94no—
220#887—94no—
221#887—94no—
222#887—94no—
223#887—91no—
224#887—91no—
225C4×Dic6—94no—
226C4×Dic6—94no—
227C4×Dic6—91no—
228C4×Dic6—91no—
229C2×C32×C4—24no—
230C2×C32×C4—24no—
231C2×C32×C4—24no—
232C2×C32×C4—31no—
233C2×C32×C4—31no—
234C2×C32×C4—31no—
235C2×C32×C4—31no—
236C2×C32×C4C22×D314yes—
237C2×C32×C4C22×D314yes—
238C2×C32×C4C22×D314yes—
239C2×C32×C4C22×D311yescharacteristic
240C2×C32×C4C22×D311yescharacteristic
241C2×C32×C4C22×D311yescharacteristic
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—31no—
244C2×C3⋊Dic3—31no—
245C2×C3⋊Dic3—31no—
246C2×C3⋊Dic3—34no—
247C2×C3⋊Dic3—34no—
248C2×C3⋊Dic3—34no—
249C2×C3⋊Dic3—34no—
250C2×C3⋊Dic3—64no—
251C2×C3⋊Dic3—64no—
252C2×C3⋊Dic3—64no—
253C2×C3⋊Dic3—64no—
254C2×C3⋊Dic3C22×D311yescharacteristic
255C2×C3⋊Dic3C22×D311yescharacteristic
256C2×C3⋊Dic3C22×D311yescharacteristic
257C2×C3⋊Dic3C22×D311yescharacteristic
258#634—22no—
259#634—22no—
260#634—34no—
261#634—62no—
262C2×C3×Dic3—34no—
263C2×C3×Dic3—34no—
264C2×C3×Dic3—34no—
265C2×C3×Dic3—34no—
266C2×C3×Dic3C22×D314yes—
267C2×C3×Dic3C22×D314yes—
268C2×C3×Dic3C22×D314yes—
269C2×C3×Dic3C22×D314yes—
270C3×Dic6—68no—
271C3×Dic6—68no—
272#627—316no—
273#627—68no—
274#557—271noSylow
275C2×C33C22×Q811yescharacteristic
276C2×C33C2×#4111yescharacteristic
277C2×C332−1+411yescharacteristic
278C2×C3×Q8—91no—
279C2×C3×Q8—91no—
280C2×Dic6—34no—
281C2×Dic6—34no—
282C2×Dic6—94no—
283C2×Dic6—91no—
284C2×Dic6—94no—
285C2×Dic6—91no—
286C3×C4⋊C4—24no—
287C3×C4⋊C4—34no—
288C3×C4⋊C4—34no—
289C3×C4⋊C4—34no—
290C3×C4⋊C4—34no—
291C3×C4⋊C4—34no—
292C3×C4⋊C4—91no—
293C3×C4⋊C4—91no—
294C3×C4⋊C4—91no—
295C3×C4⋊C4D3214yes—
296C3×C4⋊C4C2×C3⋊D311yescharacteristic
297C3×C42—34no—
298C3×C42—91no—
299C4⋊Dic3—34no—
300C4⋊Dic3—94no—
301C4⋊Dic3—94no—
302C4⋊Dic3—94no—
303C4⋊Dic3—91no—
304C4⋊Dic3—91no—
305C4⋊Dic3—91no—
306C4⋊Dic3—91no—
307C4⋊Dic3—91no—
308C4⋊Dic3—184no—
309C4⋊Dic3C2×C3⋊D311yescharacteristic
310Dic3⋊C4—34no—
311Dic3⋊C4—34no—
312Dic3⋊C4—94no—
313Dic3⋊C4—94no—
314Dic3⋊C4—94no—
315Dic3⋊C4—94no—
316Dic3⋊C4—94no—
317Dic3⋊C4—94no—
318Dic3⋊C4—94no—
319Dic3⋊C4—94no—
320Dic3⋊C4—94no—
321Dic3⋊C4—94no—
322Dic3⋊C4—91no—
323Dic3⋊C4—91no—
324Dic3⋊C4—91no—
325Dic3⋊C4—91no—
326Dic3⋊C4—91no—
327Dic3⋊C4—91no—
328Dic3⋊C4—91no—
329Dic3⋊C4—91no—
330Dic3⋊C4—184no—
331Dic3⋊C4—184no—
332Dic3⋊C4—184no—
333Dic3⋊C4—184no—
334Dic3⋊C4C2×C3⋊D311yescharacteristic
335Dic3⋊C4C2×C3⋊D311yescharacteristic
336Dic3⋊C4C2×C3⋊D311yescharacteristic
337Dic3⋊C4C2×C3⋊D311yescharacteristic
338C4×Dic3—34no—
339C4×Dic3—94no—
340C4×Dic3—94no—
341C4×Dic3—94no—
342C4×Dic3—94no—
343C4×Dic3—94no—
344C4×Dic3—94no—
345C4×Dic3—94no—
346C4×Dic3—91no—
347C4×Dic3—91no—
348C4×Dic3—91no—
349C4×Dic3—91no—
350C4×Dic3—91no—
351C4×Dic3—91no—
352C4×Dic3—91no—
353C4×Dic3—184no—
354C4×Dic3C2×C3⋊D311yescharacteristic
355C22×C32—24no—
356C22×C32C23×D314yes—
357C22×C32C23×D311yescharacteristic
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—24no—
362C32×C4—24no—
363C32×C4—24no—
364C32×C4—32no—
365C32×C4—44no—
366C32×C4—44no—
367C32×C4—44no—
368C32×C4—61no—
369C32×C4—61no—
370C32×C4—61no—
371C3⋊Dic3—21no—
372C3⋊Dic3—21no—
373C3⋊Dic3—32no—
374C3⋊Dic3—38no—
375C3⋊Dic3—68no—
376C3⋊Dic3—61no—
377C3⋊Dic3—61no—
378C3⋊Dic3—61no—
379C3⋊Dic3—64no—
380C3⋊Dic3—64no—
381C3⋊Dic3—64no—
382C3⋊Dic3—124no—
383C3⋊Dic3—124no—
384C3⋊Dic3—124no—
385C3⋊Dic3#23712yes—
386C3⋊Dic3D3×Q812yes—
387C3×Dic3—24no—
388C3×Dic3—24no—
389C3×Dic3—24no—
390C3×Dic3—38no—
391C3×Dic3—38no—
392C3×Dic3—64no—
393C3×Dic3—64no—
394C3×Dic3D3×Q818yes—
395C4⋊Q8—91no—
396C4⋊Q8—271no—
397C4⋊Q8—271no—
398#125—31no—
399#125—271no—
400#125—271no—
401#125—271no—
402#125—271no—
403#125—271no—
404C4×Q8—271no—
405C4×Q8—271no—
406C4×Q8—271no—
407C4×Q8—271no—
408C4×Q8—271no—
409C4×Q8—271no—
410C33#55711yescharacteristic, complemented, Sylow
411C3×Q8—182no—
412C3×Q8—182no—
413C2×C3×C4—24no—
414C2×C3×C4—24no—
415C2×C3×C4—24no—
416C2×C3×C4—34no—
417C2×C3×C4—34no—
418C2×C3×C4—34no—
419C2×C3×C4—34no—
420C2×C3×C4—91no—
421C2×C3×C4—91no—
422C2×C3×C4—91no—
423C2×C3×C4—91no—
424C2×C3×C4C2×D3214yes—
425C2×C3×C4C2×D3214yes—
426C2×C3×C4C2×D3214yes—
427C2×C3×C4C22×C3⋊D311yescharacteristic
428C2×C3×C4C22×C3⋊D311yescharacteristic
429C2×C3×C4C22×C3⋊D311yescharacteristic
430C2×Dic3—34no—
431C2×Dic3—34no—
432C2×Dic3—34no—
433C2×Dic3—34no—
434C2×Dic3—94no—
435C2×Dic3—91no—
436C2×Dic3—94no—
437C2×Dic3—94no—
438C2×Dic3—94no—
439C2×Dic3—91no—
440C2×Dic3—91no—
441C2×Dic3—91no—
442C2×Dic3—184no—
443C2×Dic3—184no—
444C2×Dic3—184no—
445C2×Dic3—184no—
446C2×Dic3C22×C3⋊D311yescharacteristic
447C2×Dic3C22×C3⋊D311yescharacteristic
448C2×Dic3C22×C3⋊D311yescharacteristic
449C2×Dic3C22×C3⋊D311yescharacteristic
450Dic6—68no—
451Dic6—68no—
452Dic6—916no—
453Dic6—94no—
454Dic6—182no—
455Dic6—188no—
456C2×C32—24no—
457C2×C32—24no—
458C2×C32—24no—
459C2×C32C2×#23711yescharacteristic
460C2×C32C2×D3×Q814yes—
461C2×C32C2×D3×Q811yescharacteristic
462C2×C32#100711yescharacteristic
463C2×C32D3×#4114yes—
464C2×C32#101014yes—
465C2×Q8—91no—
466C2×Q8—91no—
467C2×Q8—271no—
468C2×Q8—271no—
469C4⋊C4—31no—
470C4⋊C4—31no—
471C4⋊C4—31no—
472C4⋊C4—31no—
473C4⋊C4—31no—
474C4⋊C4—91no—
475C4⋊C4—91no—
476C4⋊C4—91no—
477C4⋊C4—271no—
478C4⋊C4—271no—
479C4⋊C4—271no—
480C4⋊C4—271no—
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⋊C4D3×C3⋊D311yescharacteristic
491C42—31no—
492C42—91no—
493C42—271no—
494C42—271no—
495C42—271no—
496C42—271no—
497C42—271no—
498C42—271no—
499C42—271no—
500C22×C3—24no—
501C22×C3C22×D3214yes—
502C22×C3C23×C3⋊D311yescharacteristic
503C3×C4—24no—
504C3×C4—24no—
505C3×C4—24no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—38no—
510C3×C4—44no—
511C3×C4—44no—
512C3×C4—44no—
513C3×C4—64no—
514C3×C4—64no—
515C3×C4—64no—
516C3×C4—92no—
517C3×C4—92no—
518C3×C4—181no—
519C3×C4—181no—
520Dic3—21no—
521Dic3—21no—
522Dic3—21no—
523Dic3—38no—
524Dic3—38no—
525Dic3—64no—
526Dic3—64no—
527Dic3—98no—
528Dic3—92no—
529Dic3—188no—
530Dic3—184no—
531Dic3—184no—
532Dic3—184no—
533Dic3—181no—
534Dic3—181no—
535Dic3—181no—
536Dic3—364no—
537Dic3—364no—
538Dic3—364no—
539Dic3Q8×C3⋊D312yes—
540C32—24no—
541C32#568514yescomplemented
542C32#572111yescharacteristic, complemented
543Q8—182no—
544Q8—182no—
545Q8—274no—
546Q8—542no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—31no—
550C2×C4—31no—
551C2×C4—91no—
552C2×C4—91no—
553C2×C4—91no—
554C2×C4—91no—
555C2×C4—271no—
556C2×C4—271no—
557C2×C4—271no—
558C2×C4—271no—
559C2×C4C2×D3×C3⋊D311yescharacteristic
560C2×C4C2×D3×C3⋊D311yescharacteristic
561C2×C4C2×D3×C3⋊D311yescharacteristic
562C2×C3—24no—
563C2×C3—24no—
564C2×C3—24no—
565C2×C3#6429714yes—
566C2×C3D3×#23714yes—
567C2×C3D32×Q814yes—
568C2×C3C2×Q8×C3⋊D311yescharacteristic
569C2×C3#41×C3⋊D311yescharacteristic
570C2×C3#6436711yescharacteristic
571C22C22×D3×C3⋊D311yescharacteristic
572C4—21no—
573C4—21no—
574C4—21no—
575C4—32no—
576C4—61no—
577C4—61no—
578C4—61no—
579C4—92no—
580C4—92no—
581C4—181no—
582C4—181no—
583C4—272no—
584C4—541no—
585C4—541no—
586C4—541no—
587C3—24no—
588C3#1059431414yescomplemented
589C3#1059615611yescharacteristic, complemented
590C2#1172456211yescharacteristic
591C2C3⋊D3×#23711yescharacteristic
592C2D3×Q8×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, prime radical, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#557
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
102. #6293Q8semidirect
410. C33#557semidirect
541. C32#5685semidirect
542. C32#5721semidirect
588. C3#10594314semidirect
589. C3#10596156semidirect

Maximal subgroups

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

subgroupindex
#117239302
C3×#640972
C32×#8892
#117234122
#117234102
#117234092
#117234022
#117234012
Dic3×#6342
#117233942
#117233912
#117233902
#117233892
#105961563
#105943143