Finite Group Databasebeta

← #13176456

Subgroups — #13176456

Order 1728.

1448 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723929C211yesmaximal, characteristic, complemented
2C3×#64105C211yesmaximal, characteristic, complemented
3C32×#898C211yesmaximal, characteristic, complemented
4#11723528C211yesmaximal, characteristic
5#11723518C211yesmaximal, characteristic
6#11723496C211yesmaximal, characteristic, complemented
7#11723487C211yesmaximal, characteristic, complemented
8#11723480C211yesmaximal, characteristic
9#11723443C211yesmaximal, characteristic, complemented
10#11723438C211yesmaximal, characteristic, complemented
11#11723433C211yesmaximal, characteristic, complemented
12#11723432C211yesmaximal, characteristic, complemented
13#11723431C211yesmaximal, characteristic, complemented
14#11723412C211yesmaximal, characteristic, complemented
15#11723400C211yesmaximal, characteristic, complemented
16#10596183—31nomaximal
17#10594591—34nomaximal
18C2×#6308C2211yescharacteristic
19C2×#6308C2211yescharacteristic
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21C2×C3×#634C2211yescharacteristic
22C2×#6290C2211yescharacteristic
23C2×D3×C3⋊Dic3C2211yescharacteristic
24C33×C4⋊C4C2211yescharacteristic
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
26#88857C2211yescharacteristic
27#88857C2211yescharacteristic
28C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
29C3×#3752C2211yescharacteristic
30C3×#3752C2211yescharacteristic
31C3×#3750C2211yescharacteristic
32C3×#3750C2211yescharacteristic
33C32×(C2×D3)⋊C4C2211yescharacteristic
34C32×(C2×D3)⋊C4C2211yescharacteristic
35C32×(C2×D3)⋊C4C2211yescharacteristic
36C32×C4⋊Dic3C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38C32×C4×Dic3C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88806C2211yescharacteristic
42Dic3⋊(C3⋊Dic3)C2211yescharacteristic
43Dic3⋊(C3⋊Dic3)C2211yescharacteristic
44Dic3⋊(C3⋊Dic3)C2211yescharacteristic
45Dic3⋊(C3⋊Dic3)C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47#88803C2211yescharacteristic
48#88803C2211yescharacteristic
49#88803C2211yescharacteristic
50(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
51(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×#126—31no—
54#64107—31no—
55#64106—31no—
56#64105D311yescharacteristic, complemented
57#64099—31no—
58#64098—31no—
59#64097—31no—
60#64097—34no—
61#64097—64no—
62#64095—31no—
63#64094—31no—
64#64092—31no—
65#64088—31no—
66#64087—31no—
67#64087—31no—
68#64085—31no—
69#64078—31no—
70C3×#898D314yescomplemented
71C3×#896—34no—
72#63913—34no—
73#63905—34no—
74#63896—34no—
75#63891—34no—
76#63887—34no—
77#63866—34no—
78#63862—34no—
79#63862—34no—
80#63859—34no—
81#63857—34no—
82#63846—34no—
83#63845—34no—
84C22×C3×C3⋊D3C2311yescharacteristic
85C22×C32×D3C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C33×C4C2311yescharacteristic
88C2×C33×C4C2311yescharacteristic
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
93#6308—41no—
94#6308—41no—
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C3×C3⋊Dic3C2311yescharacteristic
98C3×C4×C3⋊D3—41no—
99C3×#634—41no—
100C2×C32×Dic3C2311yescharacteristic
101C2×C32×Dic3C2311yescharacteristic
102C2×C32×Dic3C2311yescharacteristic
103#6290—41no—
104D3×C3⋊Dic3—41no—
105#5712—94no—
106#5646—91no—
107C2×#638—31no—
108C22×C3⋊Dic3—31no—
109C2×C4×C3⋊D3C2×D311yescharacteristic
110C2×#634—31no—
111C2×#634—31no—
112C2×#634—34no—
113C2×#634—34no—
114C2×#634—64no—
115C2×#634—64no—
116C2×#634C2×D311yescharacteristic
117C2×C3×C4×D3—34no—
118C2×C3×Dic6—34no—
119C2×#625—34no—
120C2×D3×Dic3—34no—
121C32×C4⋊C4—24no—
122C32×C4⋊C4—31no—
123C32×C4⋊C4—31no—
124C32×C4⋊C4C2×D314yes—
125C32×C4⋊C4C2×D311yescharacteristic
126C32×C22⋊C4—31no—
127C32×C22⋊C4—31no—
128C32×C22⋊C4—31no—
129C32×C42—31no—
130#3757—31no—
131#3757—31no—
132#3752—31no—
133#3752—31no—
134#3752—31no—
135#3752C2×D311yescharacteristic
136#3752C2×D311yescharacteristic
137C4⋊(C3⋊Dic3)—31no—
138C4⋊(C3⋊Dic3)—34no—
139C4⋊(C3⋊Dic3)—31no—
140C4⋊(C3⋊Dic3)—31no—
141C4⋊(C3⋊Dic3)—31no—
142C4⋊(C3⋊Dic3)—31no—
143C4⋊(C3⋊Dic3)—31no—
144C4⋊(C3⋊Dic3)—64no—
145#3750—31no—
146#3750—31no—
147#3750—34no—
148#3750—34no—
149#3750—31no—
150#3750—31no—
151#3750—31no—
152#3750—64no—
153#3750—64no—
154#3750C2×D311yescharacteristic
155#3750C2×D311yescharacteristic
156C4×C3⋊Dic3—31no—
157C4×C3⋊Dic3—34no—
158C4×C3⋊Dic3—31no—
159C4×C3⋊Dic3—64no—
160C3×(C2×D3)⋊C4—34no—
161C3×(C2×D3)⋊C4—34no—
162C3×(C2×D3)⋊C4C2×D314yes—
163C3×(C2×D3)⋊C4C2×D314yes—
164C3×(C2×D3)⋊C4C2×D314yes—
165C3×C4⋊Dic3C2×D314yes—
166C3×Dic3⋊C4—34no—
167C3×Dic3⋊C4—34no—
168C3×Dic3⋊C4C2×D314yes—
169C3×C4×Dic3C2×D314yes—
170#3724—34no—
171Dic3⋊Dic3—34no—
172Dic3⋊Dic3—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178(C2×D3)⋊Dic3—34no—
179(C2×D3)⋊Dic3—34no—
180(C2×D3)⋊Dic3—34no—
181(C2×D3)⋊Dic3—34no—
182(C2×D3)⋊Dic3—34no—
183Dic32—34no—
184C22×C33C2411yescharacteristic
185C2×C3×C3⋊D3—21no—
186C2×C3×C3⋊D3—21no—
187C2×C3×C3⋊D3—21no—
188C2×C32×D3—21no—
189C2×C32×D3—21no—
190C2×C32×D3—21no—
191C33×C4—21no—
192C33×C4—21no—
193C33×C4—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—
200C3×C3⋊Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C32×Dic3—21no—
204C3×#126—34no—
205C3×C22⋊Q8—91no—
206#946—91no—
207#944—91no—
208#929—91no—
209#924—91no—
210#898—94no—
211#898—91no—
212#898C3⋊D311yescharacteristic, complemented
213#897—94no—
214#896—34no—
215#896—91no—
216#890—94no—
217#890—91no—
218#889—94no—
219#889—91no—
220#888—94no—
221#888—91no—
222#888—184no—
223#886—94no—
224#885—94no—
225#885—91no—
226#883—94no—
227#883—91no—
228#879—94no—
229#879—91no—
230#879—91no—
231#878—94no—
232#878—94no—
233#876—94no—
234#869—94no—
235C23×C32—31no—
236C22×C3⋊D3C22×D311yescharacteristic
237C22×C3×D3—34no—
238C22×C3×D3C22×D314yes—
239C2×C32×C4—24no—
240C2×C32×C4—24no—
241C2×C32×C4—24no—
242C2×C32×C4—31no—
243C2×C32×C4—31no—
244C2×C32×C4—31no—
245C2×C32×C4C22×D314yes—
246C2×C32×C4C22×D314yes—
247C2×C32×C4C22×D314yes—
248C2×C32×C4C22×D311yescharacteristic
249C2×C32×C4C22×D311yescharacteristic
250C2×C32×C4C22×D311yescharacteristic
251#638—121no—
252C2×C3⋊Dic3—31no—
253C2×C3⋊Dic3—31no—
254C2×C3⋊Dic3—31no—
255C2×C3⋊Dic3—31no—
256C2×C3⋊Dic3—34no—
257C2×C3⋊Dic3—34no—
258C2×C3⋊Dic3—34no—
259C2×C3⋊Dic3—34no—
260C2×C3⋊Dic3—64no—
261C2×C3⋊Dic3—64no—
262C2×C3⋊Dic3—64no—
263C2×C3⋊Dic3—64no—
264C2×C3⋊Dic3—121no—
265C2×C3⋊Dic3C22×D311yescharacteristic
266C2×C3⋊Dic3C22×D311yescharacteristic
267C2×C3⋊Dic3C22×D311yescharacteristic
268C4×C3⋊D3—41no—
269#634—41no—
270#634—121no—
271#634—121no—
272#634—124no—
273#634—124no—
274#634—128no—
275#634—128no—
276C2×C3×Dic3—34no—
277C2×C3×Dic3—34no—
278C2×C3×Dic3—34no—
279C2×C3×Dic3C22×D314yes—
280C2×C3×Dic3C22×D314yes—
281C2×C3×Dic3C22×D314yes—
282C3×C4×D3—124no—
283C3×Dic6—124no—
284#625—124no—
285D3×Dic3—124no—
286#563—271noSylow
287C2×C33H(5,𝔽2)11yescharacteristic
288C2×C332−1+411yescharacteristic
289C2×C332−1+411yescharacteristic
290C3×C3⋊D3—41no—
291C32×D3—41no—
292C2×C3×Q8—91no—
293C22×C3×C4—91no—
294C2×#67—94no—
295C2×#67—91no—
296C22×Dic3—94no—
297C2×C4×D3—34no—
298C2×C4×D3—91no—
299C2×Dic6—34no—
300C2×Dic6—94no—
301C2×Dic6—94no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C2×Dic6—184no—
305C2×Dic6—184no—
306C3×C4⋊C4—24no—
307C3×C4⋊C4—34no—
308C3×C4⋊C4—34no—
309C3×C4⋊C4—91no—
310C3×C4⋊C4—91no—
311C3×C4⋊C4D3214yes—
312C3×C4⋊C4C2×C3⋊D311yescharacteristic
313C3×C22⋊C4—34no—
314C3×C22⋊C4—34no—
315C3×C22⋊C4—34no—
316C3×C22⋊C4—91no—
317C3×C22⋊C4—91no—
318C3×C42—34no—
319#217—94no—
320#217—94no—
321#217—91no—
322#217—91no—
323#217—91no—
324(C2×D3)⋊C4—34no—
325(C2×D3)⋊C4—34no—
326(C2×D3)⋊C4—94no—
327(C2×D3)⋊C4—94no—
328(C2×D3)⋊C4—94no—
329(C2×D3)⋊C4—91no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
332(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
333(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
334C4⋊Dic3—94no—
335C4⋊Dic3—94no—
336C4⋊Dic3—94no—
337C4⋊Dic3—94no—
338C4⋊Dic3—94no—
339C4⋊Dic3—94no—
340C4⋊Dic3—91no—
341C4⋊Dic3—91no—
342C4⋊Dic3—91no—
343C4⋊Dic3—184no—
344C4⋊Dic3C2×C3⋊D311yescharacteristic
345Dic3⋊C4—34no—
346Dic3⋊C4—34no—
347Dic3⋊C4—94no—
348Dic3⋊C4—94no—
349Dic3⋊C4—94no—
350Dic3⋊C4—94no—
351Dic3⋊C4—94no—
352Dic3⋊C4—91no—
353Dic3⋊C4—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—91no—
360Dic3⋊C4—184no—
361Dic3⋊C4—184no—
362Dic3⋊C4C2×C3⋊D311yescharacteristic
363C4×Dic3—94no—
364C4×Dic3—94no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3—184no—
368C4×Dic3C2×C3⋊D311yescharacteristic
369C22×C32—24no—
370C22×C32—61no—
371C22×C32—61no—
372C22×C32—61no—
373C22×C32C23×D314yes—
374C22×C32C23×D311yescharacteristic
375C2×C3⋊D3—21no—
376C2×C3⋊D3—21no—
377C2×C3⋊D3—21no—
378C2×C3×D3—24no—
379C2×C3×D3—24no—
380C2×C3×D3—24no—
381C2×C3×D3—64no—
382C2×C3×D3—64no—
383C2×C3×D3—64no—
384C32×C4—28no—
385C32×C4—21no—
386C32×C4—21no—
387C32×C4—24no—
388C32×C4—21no—
389C32×C4—24no—
390C32×C4—24no—
391C32×C4—44no—
392C32×C4—44no—
393C32×C4—61no—
394C32×C4—61no—
395C32×C4—61no—
396C3⋊Dic3—21no—
397C3⋊Dic3—21no—
398C3⋊Dic3—21no—
399C3⋊Dic3—68no—
400C3⋊Dic3—68no—
401C3⋊Dic3—61no—
402C3⋊Dic3—61no—
403C3⋊Dic3—61no—
404C3⋊Dic3—61no—
405C3⋊Dic3—64no—
406C3⋊Dic3—64no—
407C3⋊Dic3—64no—
408C3⋊Dic3—64no—
409C3⋊Dic3—124no—
410C3⋊Dic3—124no—
411C3×Dic3—24no—
412C3×Dic3—24no—
413C3×Dic3—24no—
414C3×Dic3—64no—
415C3×Dic3—64no—
416C3×Dic3—64no—
417C4⋊Q8—271no—
418C4⋊Q8—271no—
419#126—31no—
420#126—271no—
421#126—271no—
422#126—271no—
423#125—271no—
424#125—271no—
425#124—271no—
426#123—271no—
427#123—271no—
428C22⋊Q8—91no—
429C22⋊Q8—271no—
430C22⋊Q8—271no—
431C22⋊Q8—271no—
432C33#56311yescharacteristic, complemented, Sylow
433C23×C3—34no—
434C23×C3—91no—
435C22×D3—34no—
436C22×D3C22×C3⋊D311yescharacteristic
437C3×Q8—361no—
438C2×C3×C4—24no—
439C2×C3×C4—24no—
440C2×C3×C4—24no—
441C2×C3×C4—34no—
442C2×C3×C4—34no—
443C2×C3×C4—34no—
444C2×C3×C4—91no—
445C2×C3×C4—91no—
446C2×C3×C4—91no—
447C2×C3×C4—361no—
448C2×C3×C4C2×D3214yes—
449C2×C3×C4C2×D3214yes—
450C2×C3×C4C2×D3214yes—
451C2×C3×C4C22×C3⋊D311yescharacteristic
452C2×C3×C4C22×C3⋊D311yescharacteristic
453C2×C3×C4C22×C3⋊D311yescharacteristic
454#67—361no—
455#67—364no—
456C2×Dic3—34no—
457C2×Dic3—34no—
458C2×Dic3—34no—
459C2×Dic3—94no—
460C2×Dic3—94no—
461C2×Dic3—94no—
462C2×Dic3—94no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—184no—
468C2×Dic3—184no—
469C2×Dic3—184no—
470C2×Dic3—184no—
471C2×Dic3—364no—
472C2×Dic3C22×C3⋊D311yescharacteristic
473C2×Dic3C22×C3⋊D311yescharacteristic
474C2×Dic3C22×C3⋊D311yescharacteristic
475C4×D3—124no—
476C4×D3—361no—
477Dic6—124no—
478Dic6—364no—
479Dic6—364no—
480Dic6—361no—
481Dic6—361no—
482Dic6—368no—
483Dic6—368no—
484C2×C32—24no—
485C2×C32—24no—
486C2×C32—24no—
487C2×C32—121no—
488C2×C32#100411yescharacteristic
489C2×C32#100711yescharacteristic
490C2×C32#100914yes—
491C2×C32#101014yes—
492C2×C32#101014yes—
493C2×C32#101011yescharacteristic
494C3⋊D3—41no—
495C3×D3—44no—
496C3×D3—124no—
497C2×Q8—91no—
498C2×Q8—271no—
499C2×Q8—271no—
500C2×D4—271no—
501C22×C4—91no—
502C22×C4—271no—
503C4⋊C4—31no—
504C4⋊C4—31no—
505C4⋊C4—91no—
506C4⋊C4—91no—
507C4⋊C4—271no—
508C4⋊C4—271no—
509C4⋊C4—271no—
510C4⋊C4—271no—
511C4⋊C4—271no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4—271no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C4⋊C4—271no—
518C4⋊C4D3×C3⋊D311yescharacteristic
519C22⋊C4—31no—
520C22⋊C4—31no—
521C22⋊C4—31no—
522C22⋊C4—91no—
523C22⋊C4—91no—
524C22⋊C4—271no—
525C22⋊C4—271no—
526C22⋊C4—271no—
527C22⋊C4—271no—
528C22⋊C4—271no—
529C42—31no—
530C42—271no—
531C42—271no—
532C22×C3—24no—
533C22×C3—64no—
534C22×C3—64no—
535C22×C3—64no—
536C22×C3—181no—
537C22×C3—181no—
538C22×C3—181no—
539C22×C3C22×D3214yes—
540C22×C3C23×C3⋊D311yescharacteristic
541C2×D3—21no—
542C2×D3—21no—
543C2×D3—21no—
544C2×D3—64no—
545C2×D3—64no—
546C2×D3—64no—
547C3×C4—28no—
548C3×C4—24no—
549C3×C4—24no—
550C3×C4—24no—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—44no—
555C3×C4—44no—
556C3×C4—64no—
557C3×C4—64no—
558C3×C4—64no—
559C3×C4—181no—
560C3×C4—181no—
561C3×C4—181no—
562Dic3—21no—
563Dic3—21no—
564Dic3—21no—
565Dic3—64no—
566Dic3—64no—
567Dic3—64no—
568Dic3—188no—
569Dic3—188no—
570Dic3—184no—
571Dic3—184no—
572Dic3—184no—
573Dic3—184no—
574Dic3—181no—
575Dic3—181no—
576Dic3—181no—
577Dic3—181no—
578Dic3—364no—
579Dic3—364no—
580C32—24no—
581C32#564611yescharacteristic, complemented
582C32#571214yescomplemented
583C23—31no—
584C23—91no—
585Q8—361no—
586Q8—1081no—
587Q8—1081no—
588D4—1081no—
589C2×C4—31no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—91no—
593C2×C4—91no—
594C2×C4—91no—
595C2×C4—271no—
596C2×C4—271no—
597C2×C4—271no—
598C2×C4—271no—
599C2×C4—361no—
600C2×C4—1081no—
601C2×C4C2×D3×C3⋊D311yescharacteristic
602C2×C4C2×D3×C3⋊D311yescharacteristic
603C2×C4C2×D3×C3⋊D311yescharacteristic
604C2×C3—24no—
605C2×C3—24no—
606C2×C3—24no—
607C2×C3—124no—
608C2×C3—361no—
609C2×C3#6429814yes—
610C2×C3#6431414yes—
611C2×C3#6431514yes—
612C2×C3#6436611yescharacteristic
613C2×C3#6436711yescharacteristic
614C2×C3#6436711yescharacteristic
615D3—41no—
616D3—124no—
617C22—61no—
618C22—61no—
619C22—61no—
620C22—181no—
621C22—181no—
622C22—181no—
623C22C22×D3×C3⋊D311yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—61no—
628C4—61no—
629C4—61no—
630C4—181no—
631C4—181no—
632C4—181no—
633C4—541no—
634C4—541no—
635C4—541no—
636C4—541no—
637C3—24no—
638C3#1059459114yescomplemented
639C3#1059618311yescharacteristic, complemented
640C2—121no—
641C2—361no—
642C2#1172456311yescharacteristic
643C2#1172458911yescharacteristic
644C2#1172459011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×#6290C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#563
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723929C2semidirect
2. C3×#64105C2semidirect
3. C32×#898C2semidirect
6. #11723496C2semidirect
7. #11723487C2semidirect
9. #11723443C2semidirect
10. #11723438C2semidirect
11. #11723433C2semidirect
12. #11723432C2semidirect
13. #11723431C2semidirect
14. #11723412C2semidirect
15. #11723400C2semidirect
56. #64105D3semidirect
70. C3×#898D3semidirect
212. #898C3⋊D3semidirect
432. C33#563semidirect
581. C32#5646semidirect
582. C32#5712semidirect
638. C3#10594591semidirect
639. C3#10596183semidirect

Maximal subgroups

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

subgroupindex
#117239292
C3×#641052
C32×#8982
#117235282
#117235182
#117234962
#117234872
#117234802
#117234432
#117234382
#117234332
#117234322
#117234312
#117234122
#117234002
#105961833
#105945913