Finite Group Databasebeta

← #13176487

Subgroups — #13176487

Order 1728.

1472 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723929C211yesmaximal, characteristic
2C3×#64098C211yesmaximal, characteristic
3C32×#889C211yesmaximal, characteristic
4#11723409C211yesmaximal, characteristic
5#11723406C211yesmaximal, characteristic
6#11723405C211yesmaximal, characteristic
7#11723402C211yesmaximal, characteristic
8#11723400C211yesmaximal, characteristic
9#11723399C211yesmaximal, characteristic
10#11723395C211yesmaximal, characteristic
11#11723395C211yesmaximal, characteristic
12#11723391C211yesmaximal, characteristic
13#11723391C211yesmaximal, characteristic
14#11723390C211yesmaximal, characteristic
15#11723389C211yesmaximal, characteristic
16#10596156—31nomaximal
17#10594315—34nomaximal
18C2×#6308C2211yescharacteristic
19C2×#6308C2211yescharacteristic
20C2×#6293C2211yescharacteristic
21C2×#6293C2211yescharacteristic
22C33×C4⋊C4C2211yescharacteristic
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24#88857C2211yescharacteristic
25#88857C2211yescharacteristic
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
27C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
28C3×#3750C2211yescharacteristic
29C3×#3750C2211yescharacteristic
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#88806C2211yescharacteristic
42#88806C2211yescharacteristic
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#64098D311yescharacteristic
59#64097—31no—
60#64097—31no—
61#64097—31no—
62#64097—34no—
63#64097—64no—
64#64096—31no—
65#64096—31no—
66#64096—31no—
67#64096—31no—
68C4×#634—31no—
69C4×#634—31no—
70C3×#889—34no—
71C3×#889D314yes—
72#63847—34no—
73#63847—34no—
74#63845—34no—
75#63845—34no—
76#63844—34no—
77#63844—34no—
78#63840—34no—
79#63840—34no—
80#63838—34no—
81#63838—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
91#6308—22no—
92#6308—22no—
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C32×Dic3C2311yescharacteristic
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101#6293—22no—
102#6293Q814yescomplemented
103#5685—94no—
104#5685—91no—
105C2×#634—31no—
106C2×#634—31no—
107C2×#634—31no—
108C2×#634—34no—
109C2×#634—34no—
110C2×#634—31no—
111C2×#634—64no—
112C2×#634—64no—
113C2×#627—34no—
114C2×#627—34no—
115C32×C4⋊C4—24no—
116C32×C4⋊C4—31no—
117C32×C4⋊C4—31no—
118C32×C4⋊C4—31no—
119C32×C4⋊C4—31no—
120C32×C4⋊C4—31no—
121C32×C4⋊C4C2×D314yes—
122C32×C4⋊C4C2×D311yescharacteristic
123C32×C42—31no—
124C4⋊(C3⋊Dic3)—31no—
125C4⋊(C3⋊Dic3)—34no—
126C4⋊(C3⋊Dic3)—31no—
127C4⋊(C3⋊Dic3)—31no—
128C4⋊(C3⋊Dic3)—64no—
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—64no—
141#3750—64no—
142#3750C2×D311yescharacteristic
143#3750C2×D311yescharacteristic
144#3750C2×D311yescharacteristic
145#3750C2×D311yescharacteristic
146C4×C3⋊Dic3—31no—
147C4×C3⋊Dic3—34no—
148C4×C3⋊Dic3—31no—
149C4×C3⋊Dic3—31no—
150C4×C3⋊Dic3—31no—
151C4×C3⋊Dic3—31no—
152C4×C3⋊Dic3—31no—
153C4×C3⋊Dic3—31no—
154C4×C3⋊Dic3—64no—
155C4×C3⋊Dic3C2×D311yescharacteristic
156C3×C4⋊Dic3—34no—
157C3×C4⋊Dic3C2×D314yes—
158C3×Dic3⋊C4—34no—
159C3×Dic3⋊C4—34no—
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—
170#3724—34no—
171#3724—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)C2×Q812yes—
189C3⋊(C3⋊Dic3)#4112yes—
190C3×C3⋊Dic3—21no—
191C3×C3⋊Dic3—21no—
192C3×C3⋊Dic3—21no—
193C3×C3⋊Dic3C2×Q812yes—
194C32×Dic3—21no—
195C32×Dic3—21no—
196C32×Dic3—21no—
197C32×Dic3C2×Q812yes—
198C3×#125—34no—
199C3×#125—91no—
200#890—94no—
201#890—91no—
202#889—34no—
203#889—94no—
204#889—94no—
205#889—94no—
206#889—91no—
207#889—91no—
208#889—91no—
209#889C3⋊D311yescharacteristic
210#888—94no—
211#888—94no—
212#888—94no—
213#888—91no—
214#888—91no—
215#888—91no—
216#888—184no—
217#887—94no—
218#887—94no—
219#887—94no—
220#887—94no—
221#887—91no—
222#887—91no—
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—34no—
259#634—62no—
260#634—62no—
261#634—62no—
262#634—68no—
263#634—68no—
264#634—128no—
265#634—128no—
266C2×C3×Dic3—34no—
267C2×C3×Dic3—34no—
268C2×C3×Dic3—34no—
269C2×C3×Dic3—34no—
270C2×C3×Dic3C22×D314yes—
271C2×C3×Dic3C22×D314yes—
272C2×C3×Dic3C22×D314yes—
273C2×C3×Dic3C22×D314yes—
274#627—316no—
275#627—68no—
276#557—271noSylow
277C2×C33C22×Q811yescharacteristic
278C2×C33C2×#4111yescharacteristic
279C2×C332−1+411yescharacteristic
280C2×Dic6—94no—
281C2×Dic6—91no—
282C2×Dic6—94no—
283C2×Dic6—94no—
284C2×Dic6—94no—
285C2×Dic6—91no—
286C2×Dic6—91no—
287C2×Dic6—91no—
288C2×Dic6—184no—
289C2×Dic6—184no—
290C3×C4⋊C4—24no—
291C3×C4⋊C4—34no—
292C3×C4⋊C4—34no—
293C3×C4⋊C4—34no—
294C3×C4⋊C4—34no—
295C3×C4⋊C4—34no—
296C3×C4⋊C4—91no—
297C3×C4⋊C4—91no—
298C3×C4⋊C4—91no—
299C3×C4⋊C4—91no—
300C3×C4⋊C4—91no—
301C3×C4⋊C4D3214yes—
302C3×C4⋊C4C2×C3⋊D311yescharacteristic
303C3×C42—34no—
304C3×C42—91no—
305C4⋊Dic3—34no—
306C4⋊Dic3—94no—
307C4⋊Dic3—94no—
308C4⋊Dic3—94no—
309C4⋊Dic3—91no—
310C4⋊Dic3—91no—
311C4⋊Dic3—91no—
312C4⋊Dic3—184no—
313C4⋊Dic3C2×C3⋊D311yescharacteristic
314Dic3⋊C4—34no—
315Dic3⋊C4—34no—
316Dic3⋊C4—34no—
317Dic3⋊C4—34no—
318Dic3⋊C4—94no—
319Dic3⋊C4—94no—
320Dic3⋊C4—94no—
321Dic3⋊C4—94no—
322Dic3⋊C4—94no—
323Dic3⋊C4—94no—
324Dic3⋊C4—94no—
325Dic3⋊C4—94no—
326Dic3⋊C4—91no—
327Dic3⋊C4—91no—
328Dic3⋊C4—91no—
329Dic3⋊C4—91no—
330Dic3⋊C4—91no—
331Dic3⋊C4—91no—
332Dic3⋊C4—91no—
333Dic3⋊C4—91no—
334Dic3⋊C4—184no—
335Dic3⋊C4—184no—
336Dic3⋊C4C2×C3⋊D311yescharacteristic
337Dic3⋊C4C2×C3⋊D311yescharacteristic
338Dic3⋊C4C2×C3⋊D311yescharacteristic
339Dic3⋊C4C2×C3⋊D311yescharacteristic
340C4×Dic3—34no—
341C4×Dic3—94no—
342C4×Dic3—94no—
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—91no—
354C4×Dic3—91no—
355C4×Dic3—184no—
356C4×Dic3C2×C3⋊D311yescharacteristic
357C22×C32—24no—
358C22×C32C23×D314yes—
359C22×C32C23×D311yescharacteristic
360C32×C4—28no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—24no—
364C32×C4—21no—
365C32×C4—24no—
366C32×C4—24no—
367C32×C4—32no—
368C32×C4—44no—
369C32×C4—44no—
370C32×C4—61no—
371C32×C4—61no—
372C32×C4—61no—
373C3⋊Dic3—21no—
374C3⋊Dic3—21no—
375C3⋊Dic3—21no—
376C3⋊Dic3—32no—
377C3⋊Dic3—32no—
378C3⋊Dic3—38no—
379C3⋊Dic3—38no—
380C3⋊Dic3—68no—
381C3⋊Dic3—68no—
382C3⋊Dic3—61no—
383C3⋊Dic3—61no—
384C3⋊Dic3—64no—
385C3⋊Dic3—64no—
386C3⋊Dic3—124no—
387C3⋊Dic3—124no—
388C3⋊Dic3D3×Q812yes—
389C3×Dic3—24no—
390C3×Dic3—24no—
391C3×Dic3—24no—
392C3×Dic3—38no—
393C3×Dic3—64no—
394C3×Dic3—64no—
395C3×Dic3—64no—
396C3×Dic3D3×Q818yes—
397C4⋊Q8—271no—
398C4⋊Q8—271no—
399C4⋊Q8—271no—
400#125—31no—
401#125—91no—
402#125—271no—
403#125—271no—
404#125—271no—
405#125—271no—
406C4×Q8—271no—
407C4×Q8—271no—
408C4×Q8—271no—
409C4×Q8—271no—
410C4×Q8—271no—
411C4×Q8—271no—
412C33#55711yescharacteristic, complemented, Sylow
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—94no—
436C2×Dic3—91no—
437C2×Dic3—91no—
438C2×Dic3—94no—
439C2×Dic3—94no—
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—916no—
451Dic6—94no—
452Dic6—182no—
453Dic6—188no—
454Dic6—182no—
455Dic6—182no—
456Dic6—188no—
457Dic6—188no—
458Dic6—368no—
459Dic6—368no—
460C2×C32—24no—
461C2×C32—24no—
462C2×C32—24no—
463C2×C32C2×D3×Q814yes—
464C2×C32C2×D3×Q811yescharacteristic
465C2×C32D3×#4114yes—
466C2×C32D3×#4111yescharacteristic
467C2×C32#101014yes—
468C2×C32#101011yescharacteristic
469C2×Q8—271no—
470C2×Q8—271no—
471C2×Q8—271no—
472C2×Q8—271no—
473C4⋊C4—31no—
474C4⋊C4—31no—
475C4⋊C4—31no—
476C4⋊C4—31no—
477C4⋊C4—31no—
478C4⋊C4—91no—
479C4⋊C4—91no—
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⋊C4D3×C3⋊D311yescharacteristic
495C42—31no—
496C42—91no—
497C42—271no—
498C42—271no—
499C42—271no—
500C42—271no—
501C42—271no—
502C42—271no—
503C42—271no—
504C22×C3—24no—
505C22×C3C22×D3214yes—
506C22×C3C23×C3⋊D311yescharacteristic
507C3×C4—28no—
508C3×C4—24no—
509C3×C4—24no—
510C3×C4—24no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—38no—
515C3×C4—44no—
516C3×C4—44no—
517C3×C4—64no—
518C3×C4—64no—
519C3×C4—64no—
520C3×C4—92no—
521C3×C4—181no—
522C3×C4—181no—
523C3×C4—181no—
524Dic3—21no—
525Dic3—21no—
526Dic3—21no—
527Dic3—38no—
528Dic3—64no—
529Dic3—64no—
530Dic3—64no—
531Dic3—98no—
532Dic3—98no—
533Dic3—92no—
534Dic3—92no—
535Dic3—188no—
536Dic3—188no—
537Dic3—184no—
538Dic3—184no—
539Dic3—181no—
540Dic3—181no—
541Dic3—364no—
542Dic3—364no—
543Dic3Q8×C3⋊D312yes—
544C32—24no—
545C32#568514yescomplemented
546C32#568511yescharacteristic, complemented
547Q8—274no—
548Q8—542no—
549Q8—542no—
550Q8—542no—
551C2×C4—31no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—91no—
556C2×C4—91no—
557C2×C4—91no—
558C2×C4—91no—
559C2×C4—271no—
560C2×C4—271no—
561C2×C4—271no—
562C2×C4—271no—
563C2×C4C2×D3×C3⋊D311yescharacteristic
564C2×C4C2×D3×C3⋊D311yescharacteristic
565C2×C4C2×D3×C3⋊D311yescharacteristic
566C2×C3—24no—
567C2×C3—24no—
568C2×C3—24no—
569C2×C3#6429814yes—
570C2×C3#6431214yes—
571C2×C3D32×Q814yes—
572C2×C3C2×Q8×C3⋊D311yescharacteristic
573C2×C3#41×C3⋊D311yescharacteristic
574C2×C3#6436711yescharacteristic
575C22C22×D3×C3⋊D311yescharacteristic
576C4—21no—
577C4—21no—
578C4—21no—
579C4—32no—
580C4—61no—
581C4—61no—
582C4—61no—
583C4—92no—
584C4—181no—
585C4—181no—
586C4—181no—
587C4—272no—
588C4—272no—
589C4—541no—
590C4—541no—
591C3—24no—
592C3#1059431514yescomplemented
593C3#1059615611yescharacteristic, complemented
594C2#1172456311yescharacteristic
595C2#1172458611yescharacteristic
596C2D3×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
412. C33#557semidirect
545. C32#5685semidirect
546. C32#5685semidirect
592. C3#10594315semidirect
593. C3#10596156semidirect

Maximal subgroups

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

subgroupindex
#117239292
C3×#640982
C32×#8892
#117234092
#117234062
#117234052
#117234022
#117234002
#117233992
#117233952
#117233912
#117233902
#117233892
#105961563
#105943153