Finite Group Databasebeta

← #13175901

Subgroups — #13175901

Order 1728.

1488 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723924C211yesmaximal, characteristic
2C3×#64088C211yesmaximal, characteristic, complemented
3C32×#878C211yesmaximal, characteristic, complemented
4#11723640C211yesmaximal, characteristic, complemented
5#11723612C211yesmaximal, characteristic
6#11723596C211yesmaximal, characteristic, complemented
7#11723592C211yesmaximal, characteristic
8#11723535C211yesmaximal, characteristic, complemented
9#11723456C211yesmaximal, characteristic, complemented
10#11723446C211yesmaximal, characteristic, complemented
11#11723442C211yesmaximal, characteristic, complemented
12#11723411C211yesmaximal, characteristic, complemented
13#11723409C211yesmaximal, characteristic, complemented
14#11723401C211yesmaximal, characteristic, complemented
15#11723393C211yesmaximal, characteristic, complemented
16#10596117—31nomaximal
17#10594251—34nomaximal
18C2×#6312C2211yescharacteristic
19C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
20C22×C32×Dic3C2211yescharacteristic
21C2×C32×Dic6C2211yescharacteristic
22C2×#6293C2211yescharacteristic
23C2×#6293C2211yescharacteristic
24C33×C22⋊C4C2211yescharacteristic
25#88864C2211yescharacteristic
26#88859C2211yescharacteristic
27C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
28#88857C2211yescharacteristic
29C3×#3757C2211yescharacteristic
30C3×#3757C2211yescharacteristic
31C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×#3750C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×C4×C3⋊Dic3C2211yescharacteristic
35C32×#217C2211yescharacteristic
36C32×C4⋊Dic3C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38C32×Dic3⋊C4C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88807C2211yescharacteristic
42#88807C2211yescharacteristic
43#88806C2211yescharacteristic
44#88806C2211yescharacteristic
45Dic3⋊(C3⋊Dic3)C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47#88804C2211yescharacteristic
48#88804C2211yescharacteristic
49#88804C2211yescharacteristic
50#88804C2211yescharacteristic
51Dic3×C3⋊Dic3C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×C22⋊Q8—31no—
54#64155—31no—
55#64153—31no—
56#64138—31no—
57#64133—31no—
58#64107—31no—
59#64107—31no—
60#64105—31no—
61#64099—31no—
62#64098—31no—
63#64097—31no—
64#64094—31no—
65#64092—31no—
66#64092—34no—
67#64092—64no—
68#64088—31no—
69#64088D311yescharacteristic, complemented
70C3×#879—34no—
71C3×#878D314yescomplemented
72#63966—34no—
73#63964—34no—
74#63956—34no—
75#63953—34no—
76#63918—34no—
77#63871—34no—
78#63868—34no—
79#63865—34no—
80#63848—34no—
81#63846—34no—
82#63845—34no—
83#63839—34no—
84C23×C33C2311yescharacteristic
85C22×C3⋊(C3⋊D3)C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C33×C4C2311yescharacteristic
88#6312—41no—
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92C4×C3⋊(C3⋊D3)—41no—
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C32×Dic3—41no—
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101C2×C32×Dic3C2311yescharacteristic
102C32×Dic6—41no—
103#6293—41no—
104#6293—41no—
105#5712—91no—
106#5646—94no—
107C2×C32×Q8—31no—
108C22×C32×C4—31no—
109C2×#638—31no—
110C2×#638—34no—
111C2×#638—64no—
112C2×C4×C3⋊D3—31no—
113C2×C4×C3⋊D3—34no—
114C2×C4×C3⋊D3—64no—
115C2×#634—31no—
116C2×#634—31no—
117C22×C3×Dic3C2×D314yes—
118C2×C3×Dic6C2×D314yes—
119C2×#627—34no—
120C2×#627—34no—
121C32×C4⋊C4—31no—
122C32×C4⋊C4—31no—
123C32×C4⋊C4—31no—
124C32×C22⋊C4—24no—
125C32×C22⋊C4—31no—
126C32×C22⋊C4C2×D314yes—
127C32×C22⋊C4C2×D311yescharacteristic
128#3757—31no—
129#3757—34no—
130#3757—64no—
131#3757C2×D311yescharacteristic
132#3757C2×D311yescharacteristic
133#3752—31no—
134#3752—34no—
135#3752—31no—
136#3752—31no—
137#3752—31no—
138#3752—31no—
139#3752—64no—
140C4⋊(C3⋊Dic3)—31no—
141C4⋊(C3⋊Dic3)—34no—
142C4⋊(C3⋊Dic3)—31no—
143C4⋊(C3⋊Dic3)—31no—
144C4⋊(C3⋊Dic3)—64no—
145C4⋊(C3⋊Dic3)C2×D311yescharacteristic
146#3750—31no—
147#3750—34no—
148#3750—31no—
149#3750—31no—
150#3750—31no—
151#3750—31no—
152#3750—31no—
153#3750—31no—
154#3750—64no—
155#3750C2×D311yescharacteristic
156#3750C2×D311yescharacteristic
157C4×C3⋊Dic3—31no—
158C4×C3⋊Dic3—31no—
159C4×C3⋊Dic3C2×D311yescharacteristic
160C3×#217—34no—
161C3×#217—34no—
162C3×#217C2×D314yes—
163C3×C4⋊Dic3—34no—
164C3×C4⋊Dic3C2×D314yes—
165C3×Dic3⋊C4—34no—
166C3×Dic3⋊C4—34no—
167C3×Dic3⋊C4C2×D314yes—
168C3×Dic3⋊C4C2×D314yes—
169C3×C4×Dic3—34no—
170#3724—34no—
171#3724—34no—
172Dic3⋊Dic3—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178#3722—34no—
179#3722—34no—
180#3722—34no—
181#3722—34no—
182Dic32—34no—
183Dic32—34no—
184C22×C33—21no—
185C22×C33—21no—
186C22×C33—21no—
187C22×C33C2411yescharacteristic
188C2×C3⋊(C3⋊D3)—21no—
189C2×C3⋊(C3⋊D3)—21no—
190C2×C3⋊(C3⋊D3)—21no—
191C33×C4—21no—
192C33×C4—21no—
193C3⋊(C3⋊Dic3)—21no—
194C3⋊(C3⋊Dic3)—21no—
195C3⋊(C3⋊Dic3)—21no—
196C3×C3⋊Dic3—21no—
197C3×C3⋊Dic3—21no—
198C3×C3⋊Dic3—21no—
199C3×C3⋊Dic3—21no—
200C32×Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C32×Dic3—21no—
204C3×#126—91no—
205C3×C22⋊Q8—34no—
206#946—94no—
207#944—94no—
208#929—94no—
209#924—94no—
210#898—94no—
211#898—94no—
212#898—91no—
213#897—91no—
214#896—94no—
215#896—91no—
216#890—94no—
217#890—91no—
218#889—94no—
219#889—91no—
220#888—94no—
221#888—91no—
222#886—91no—
223#885—94no—
224#885—91no—
225#883—94no—
226#883—91no—
227#883—184no—
228#879—34no—
229#879—94no—
230#879—91no—
231#878—91no—
232#878C3⋊D311yescharacteristic, complemented
233#876—91no—
234#869—91no—
235C23×C32—24no—
236C23×C32C22×D314yes—
237C23×C32C22×D311yescharacteristic
238C22×C3⋊D3—31no—
239C22×C3⋊D3—34no—
240C22×C3⋊D3—64no—
241C32×Q8—121no—
242C2×C32×C4—24no—
243C2×C32×C4—24no—
244C2×C32×C4—31no—
245C2×C32×C4—31no—
246C2×C32×C4—31no—
247C2×C32×C4—31no—
248C2×C32×C4—121no—
249C2×C32×C4C22×D314yes—
250C2×C32×C4C22×D314yes—
251C2×C32×C4C22×D311yescharacteristic
252C2×C32×C4C22×D311yescharacteristic
253#638—121no—
254#638—124no—
255#638—128no—
256C2×C3⋊Dic3—31no—
257C2×C3⋊Dic3—31no—
258C2×C3⋊Dic3—31no—
259C2×C3⋊Dic3—34no—
260C2×C3⋊Dic3—34no—
261C2×C3⋊Dic3—34no—
262C2×C3⋊Dic3—64no—
263C2×C3⋊Dic3—64no—
264C2×C3⋊Dic3—64no—
265C2×C3⋊Dic3C22×D311yescharacteristic
266C2×C3⋊Dic3C22×D311yescharacteristic
267C2×C3⋊Dic3C22×D311yescharacteristic
268C2×C3⋊Dic3C22×D311yescharacteristic
269C4×C3⋊D3—121no—
270C4×C3⋊D3—124no—
271C4×C3⋊D3—128no—
272#634—121no—
273#634—121no—
274C2×C3×Dic3—34no—
275C2×C3×Dic3—34no—
276C2×C3×Dic3—34no—
277C2×C3×Dic3—34no—
278C2×C3×Dic3—44no—
279C2×C3×Dic3C22×D314yes—
280C2×C3×Dic3C22×D314yes—
281C2×C3×Dic3C22×D314yes—
282C2×C3×Dic3C22×D314yes—
283C3×Dic6—44no—
284#627—124no—
285#627—124no—
286#563—271noSylow
287C2×C33—41no—
288C2×C33H(5,𝔽2)11yescharacteristic
289C2×C332−1+411yescharacteristic
290C2×C332−1+411yescharacteristic
291C3⋊(C3⋊D3)—41no—
292C2×C3×Q8—34no—
293C22×C3×C4—34no—
294C2×#67—94no—
295C2×#67—91no—
296C2×#67—184no—
297C22×Dic3C2×C3⋊D311yescharacteristic
298C2×C4×D3—94no—
299C2×C4×D3—91no—
300C2×C4×D3—184no—
301C2×Dic6—94no—
302C2×Dic6—94no—
303C2×Dic6—91no—
304C2×Dic6—91no—
305C2×Dic6C2×C3⋊D311yescharacteristic
306C3×C4⋊C4—34no—
307C3×C4⋊C4—34no—
308C3×C4⋊C4—34no—
309C3×C4⋊C4—91no—
310C3×C4⋊C4—91no—
311C3×C4⋊C4—91no—
312C3×C22⋊C4—24no—
313C3×C22⋊C4—34no—
314C3×C22⋊C4—91no—
315C3×C22⋊C4—91no—
316C3×C22⋊C4D3214yes—
317C3×C22⋊C4C2×C3⋊D311yescharacteristic
318C3×C42—91no—
319#217—34no—
320#217—34no—
321#217—94no—
322#217—91no—
323#217—184no—
324#217C2×C3⋊D311yescharacteristic
325(C2×D3)⋊C4—94no—
326(C2×D3)⋊C4—94no—
327(C2×D3)⋊C4—94no—
328(C2×D3)⋊C4—94no—
329(C2×D3)⋊C4—94no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4—91no—
332(C2×D3)⋊C4—91no—
333(C2×D3)⋊C4—91no—
334(C2×D3)⋊C4—91no—
335(C2×D3)⋊C4—184no—
336C4⋊Dic3—34no—
337C4⋊Dic3—94no—
338C4⋊Dic3—94no—
339C4⋊Dic3—94no—
340C4⋊Dic3—91no—
341C4⋊Dic3—91no—
342C4⋊Dic3—91no—
343C4⋊Dic3—91no—
344C4⋊Dic3—91no—
345C4⋊Dic3—184no—
346C4⋊Dic3C2×C3⋊D311yescharacteristic
347Dic3⋊C4—34no—
348Dic3⋊C4—34no—
349Dic3⋊C4—94no—
350Dic3⋊C4—94no—
351Dic3⋊C4—94no—
352Dic3⋊C4—94no—
353Dic3⋊C4—94no—
354Dic3⋊C4—94no—
355Dic3⋊C4—94no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—91no—
360Dic3⋊C4—91no—
361Dic3⋊C4—184no—
362Dic3⋊C4C2×C3⋊D311yescharacteristic
363Dic3⋊C4C2×C3⋊D311yescharacteristic
364C4×Dic3—34no—
365C4×Dic3—94no—
366C4×Dic3—94no—
367C4×Dic3—91no—
368C4×Dic3—91no—
369C22×C32—21no—
370C22×C32—21no—
371C22×C32—21no—
372C22×C32—24no—
373C22×C32—24no—
374C22×C32—24no—
375C22×C32—24no—
376C22×C32—44no—
377C22×C32—44no—
378C22×C32—44no—
379C22×C32C23×D314yes—
380C22×C32C23×D311yescharacteristic
381C2×C3⋊D3—68no—
382C2×C3⋊D3—64no—
383C2×C3⋊D3—61no—
384C2×C3⋊D3—61no—
385C2×C3⋊D3—61no—
386C2×C3⋊D3—64no—
387C2×C3⋊D3—64no—
388C2×C3⋊D3—124no—
389C2×C3⋊D3—124no—
390C32×C4—21no—
391C32×C4—21no—
392C32×C4—24no—
393C32×C4—24no—
394C32×C4—44no—
395C32×C4—44no—
396C32×C4—61no—
397C32×C4—61no—
398C32×C4—61no—
399C32×C4—61no—
400C3⋊Dic3—21no—
401C3⋊Dic3—21no—
402C3⋊Dic3—21no—
403C3⋊Dic3—21no—
404C3⋊Dic3—61no—
405C3⋊Dic3—61no—
406C3⋊Dic3—61no—
407C3⋊Dic3—64no—
408C3⋊Dic3—64no—
409C3⋊Dic3—64no—
410C3⋊Dic3—124no—
411C3⋊Dic3—124no—
412C3⋊Dic3—124no—
413C3×Dic3—24no—
414C3×Dic3—24no—
415C3×Dic3—24no—
416C3×Dic3—24no—
417C3×Dic3—64no—
418C3×Dic3—64no—
419C3×Dic3—64no—
420C3×Dic3—64no—
421C4⋊Q8—271no—
422C4⋊Q8—271no—
423#126—91no—
424#126—271no—
425#126—271no—
426#126—271no—
427#125—271no—
428#125—271no—
429#124—271no—
430#123—271no—
431#123—271no—
432C22⋊Q8—31no—
433C22⋊Q8—271no—
434C22⋊Q8—271no—
435C22⋊Q8—271no—
436C33#56311yescharacteristic, complemented, Sylow
437C23×C3—24no—
438C23×C3C2×D3214yes—
439C23×C3C22×C3⋊D311yescharacteristic
440C22×D3—94no—
441C22×D3—91no—
442C22×D3—184no—
443C3×Q8—124no—
444C2×C3×C4—24no—
445C2×C3×C4—24no—
446C2×C3×C4—34no—
447C2×C3×C4—34no—
448C2×C3×C4—34no—
449C2×C3×C4—34no—
450C2×C3×C4—91no—
451C2×C3×C4—91no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4—124no—
455C2×C3×C4C2×D3214yes—
456C2×C3×C4C2×D3214yes—
457C2×C3×C4C22×C3⋊D311yescharacteristic
458C2×C3×C4C22×C3⋊D311yescharacteristic
459#67—364no—
460#67—361no—
461#67—368no—
462C2×Dic3—34no—
463C2×Dic3—34no—
464C2×Dic3—34no—
465C2×Dic3—34no—
466C2×Dic3—41no—
467C2×Dic3—94no—
468C2×Dic3—94no—
469C2×Dic3—94no—
470C2×Dic3—91no—
471C2×Dic3—91no—
472C2×Dic3—91no—
473C2×Dic3—184no—
474C2×Dic3—184no—
475C2×Dic3—184no—
476C2×Dic3C22×C3⋊D311yescharacteristic
477C2×Dic3C22×C3⋊D311yescharacteristic
478C2×Dic3C22×C3⋊D311yescharacteristic
479C2×Dic3C22×C3⋊D311yescharacteristic
480C4×D3—364no—
481C4×D3—361no—
482C4×D3—368no—
483Dic6—41no—
484Dic6—361no—
485Dic6—361no—
486Dic6—364no—
487Dic6—364no—
488C2×C32—24no—
489C2×C32—24no—
490C2×C32—24no—
491C2×C32—41no—
492C2×C32—44no—
493C2×C32—84no—
494C2×C32#100414yes—
495C2×C32#100714yes—
496C2×C32#100911yescharacteristic
497C2×C32#101014yes—
498C2×C32#101011yescharacteristic
499C2×C32#101011yescharacteristic
500C3⋊D3—128no—
501C3⋊D3—124no—
502C3⋊D3—121no—
503C2×Q8—31no—
504C2×Q8—271no—
505C2×Q8—271no—
506C2×D4—271no—
507C22×C4—31no—
508C22×C4—271no—
509C4⋊C4—31no—
510C4⋊C4—31no—
511C4⋊C4—31no—
512C4⋊C4—91no—
513C4⋊C4—91no—
514C4⋊C4—91no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C4⋊C4—271no—
521C4⋊C4—271no—
522C4⋊C4—271no—
523C4⋊C4—271no—
524C4⋊C4—271no—
525C22⋊C4—31no—
526C22⋊C4—91no—
527C22⋊C4—91no—
528C22⋊C4—271no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4—271no—
532C22⋊C4—271no—
533C22⋊C4—271no—
534C22⋊C4D3×C3⋊D311yescharacteristic
535C42—91no—
536C42—271no—
537C42—271no—
538C22×C3—24no—
539C22×C3—24no—
540C22×C3—24no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—21no—
544C22×C3—24no—
545C22×C3—44no—
546C22×C3—44no—
547C22×C3—44no—
548C22×C3C22×D3214yes—
549C22×C3C23×C3⋊D311yescharacteristic
550C2×D3—188no—
551C2×D3—184no—
552C2×D3—184no—
553C2×D3—184no—
554C2×D3—181no—
555C2×D3—181no—
556C2×D3—181no—
557C2×D3—364no—
558C2×D3—364no—
559C3×C4—24no—
560C3×C4—24no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—44no—
564C3×C4—44no—
565C3×C4—64no—
566C3×C4—64no—
567C3×C4—64no—
568C3×C4—64no—
569C3×C4—181no—
570C3×C4—181no—
571C3×C4—181no—
572C3×C4—181no—
573Dic3—21no—
574Dic3—21no—
575Dic3—21no—
576Dic3—21no—
577Dic3—64no—
578Dic3—64no—
579Dic3—64no—
580Dic3—64no—
581Dic3—184no—
582Dic3—184no—
583Dic3—184no—
584Dic3—181no—
585Dic3—181no—
586Dic3—181no—
587Dic3—364no—
588Dic3—364no—
589Dic3—364no—
590C32—24no—
591C32#564614yescomplemented
592C32#571211yescharacteristic, complemented
593C23—271no—
594C23C2×D3×C3⋊D311yescharacteristic
595Q8—121no—
596Q8—1081no—
597Q8—1081no—
598D4—1081no—
599C2×C4—31no—
600C2×C4—31no—
601C2×C4—31no—
602C2×C4—31no—
603C2×C4—91no—
604C2×C4—91no—
605C2×C4—91no—
606C2×C4—91no—
607C2×C4—121no—
608C2×C4—271no—
609C2×C4—271no—
610C2×C4—271no—
611C2×C4—1081no—
612C2×C4C2×D3×C3⋊D311yescharacteristic
613C2×C4C2×D3×C3⋊D311yescharacteristic
614C2×C3—24no—
615C2×C3—24no—
616C2×C3—24no—
617C2×C3—41no—
618C2×C3—44no—
619C2×C3—84no—
620C2×C3#6429714yes—
621C2×C3#6430914yes—
622C2×C3#6431414yes—
623C2×C3#6436111yescharacteristic
624C2×C3#6436411yescharacteristic
625C2×C3#6436711yescharacteristic
626D3—368no—
627D3—361no—
628D3—364no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—541no—
633C22—541no—
634C22—541no—
635C22C22×D3×C3⋊D311yescharacteristic
636C4—21no—
637C4—21no—
638C4—61no—
639C4—61no—
640C4—61no—
641C4—61no—
642C4—181no—
643C4—181no—
644C4—181no—
645C4—181no—
646C4—541no—
647C4—541no—
648C4—541no—
649C3—24no—
650C3#1059425114yescomplemented
651C3#1059611711yescharacteristic, complemented
652C2—41no—
653C2—1081no—
654C2#1172456611yescharacteristic
655C2#1172458211yescharacteristic
656C2#1172458811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×#6312C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#563
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64088C2semidirect
3. C32×#878C2semidirect
4. #11723640C2semidirect
6. #11723596C2semidirect
8. #11723535C2semidirect
9. #11723456C2semidirect
10. #11723446C2semidirect
11. #11723442C2semidirect
12. #11723411C2semidirect
13. #11723409C2semidirect
14. #11723401C2semidirect
15. #11723393C2semidirect
69. #64088D3semidirect
71. C3×#878D3semidirect
232. #878C3⋊D3semidirect
436. C33#563semidirect
591. C32#5646semidirect
592. C32#5712semidirect
650. C3#10594251semidirect
651. C3#10596117semidirect

Maximal subgroups

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

subgroupindex
#117239242
C3×#640882
C32×#8782
#117236402
#117236122
#117235962
#117235922
#117235352
#117234562
#117234462
#117234422
#117234112
#117234092
#117234012
#117233932
#105961173
#105942513