Finite Group Databasebeta

← #13175499

Subgroups — #13175499

Order 1728.

2292 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89031C211yesmaximal, characteristic, complemented
2C22×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
3#11723927C211yesmaximal, characteristic, complemented
4C3×#64091C211yesmaximal, characteristic, complemented
5C32×#877C211yesmaximal, characteristic, complemented
6#11723636C211yesmaximal, characteristic, complemented
7#11723609C211yesmaximal, characteristic, complemented
8#11723483C212yesmaximal, complemented
9#11723482C212yesmaximal, complemented
10Dic3×C4⋊(C3⋊D3)C212yesmaximal, complemented
11#11723471C212yesmaximal, complemented
12#10595995—31nomaximal
13#10594078—34nomaximal
14C23×C3×C3⋊D3C2211yescharacteristic
15C2×#6312C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×#638C2211yescharacteristic
18C2×C3×C4⋊(C3⋊D3)C2212yes—
19C22×C32×Dic3C2211yescharacteristic
20C2×#6293C2211yescharacteristic, complemented
21C2×#6292C2212yes—
22C2×#6292C2211yescharacteristic, complemented
23C2×#6289C2211yescharacteristic, complemented
24#89031—24no—
25C2×Dic3×C3⋊D3—24no—
26C2×Dic3×C3⋊D3C2212yes—
27C2×Dic3×C3⋊D3C2212yes—
28C2×Dic3×C3⋊D3C2211yescharacteristic, complemented
29C33×C22⋊C4C2211yescharacteristic
30#88859C2212yescomplemented
31C4⋊(C3⋊(C3⋊Dic3))C2212yescomplemented
32C3×#3752C2212yes—
33C32×#217C2211yescharacteristic
34C32×Dic3⋊C4C2212yescomplemented
35C32×C4×Dic3C2212yescomplemented
36Dic3⋊(C3⋊Dic3)C2212yescomplemented
37#88803C2212yes—
38#88803C2212yescomplemented
39C2×#3828—31no—
40C22×C4×C3⋊D3—31no—
41C2×#3804—34no—
42C22×D3×Dic3—34no—
43C32×#117—31no—
44#64106—32no—
45#64104—32no—
46#64095—31no—
47#64095—31no—
48#64095—34no—
49#64095—64no—
50#64091—31no—
51#64091D311yescharacteristic, complemented
52C4×C4⋊(C3⋊D3)—32no—
53C4×C4⋊(C3⋊D3)—32no—
54C3×#882—34no—
55C3×#877D314yescomplemented
56#63977—34no—
57#63962—34no—
58#63893—38no—
59Dic3×D12—38no—
60#63886—38no—
61#63880—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3—22no—
65C22×C3×C3⋊D3—24no—
66C22×C3×C3⋊D3C2311yescharacteristic
67C22×C3×C3⋊D3C2312yes—
68C2×C33×C4C2312yes—
69#6312—22no—
70C2×C3⋊(C3⋊Dic3)—22no—
71C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
72C2×C3⋊(C3⋊Dic3)C2312yes—
73C3×#638—22no—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C3×C4⋊(C3⋊D3)—24no—
76C2×C32×Dic3D414yescomplemented
77C2×C32×Dic3C2312yes—
78C2×C32×Dic3C2311yescharacteristic
79C2×C32×Dic3C2311yescharacteristic
80#6293—22no—
81#6292—24no—
82#6292—22no—
83#6289—22no—
84Dic3×C3⋊D3—24no—
85Dic3×C3⋊D3—24no—
86Dic3×C3⋊D3—22no—
87#5586—91no—
88#5524—94no—
89C23×C3⋊D3C2×D311yescharacteristic
90C23×C3×D3—34no—
91C22×C32×C4—31no—
92C2×#638—31no—
93C2×#638—34no—
94C2×#638—64no—
95C2×#638C2×D311yescharacteristic
96C22×C3⋊Dic3—31no—
97C22×C3⋊Dic3—34no—
98C22×C3⋊Dic3—64no—
99#3828—64no—
100C2×C4⋊(C3⋊D3)—32no—
101C2×C4⋊(C3⋊D3)—31no—
102C2×C4⋊(C3⋊D3)C2×D312yes—
103C2×C4×C3⋊D3—32no—
104C2×C4×C3⋊D3—31no—
105C2×C4×C3⋊D3—32no—
106C2×C4×C3⋊D3—31no—
107C2×C4×C3⋊D3—64no—
108C2×#634—31no—
109C2×C3×#67—34no—
110C22×C3×Dic3C2×D314yes—
111C2×C3×D12—38no—
112C2×#627—34no—
113C2×#626—34no—
114C2×#626—38no—
115C2×#624—34no—
116#3804—616no—
117C2×D3×Dic3—38no—
118C2×D3×Dic3—38no—
119C2×D3×Dic3—34no—
120C2×D3×Dic3—616no—
121C32×C4⋊C4—32no—
122C32×C22⋊C4—24no—
123C32×C22⋊C4—31no—
124C32×C22⋊C4C2×D314yes—
125C32×C22⋊C4C2×D311yescharacteristic
126C32×C42—32no—
127#3752—32no—
128#3752—32no—
129#3752—32no—
130#3752—38no—
131#3752—68no—
132#3752C2×D312yes—
133C4⋊(C3⋊Dic3)—32no—
134C4⋊(C3⋊Dic3)—38no—
135C4⋊(C3⋊Dic3)—32no—
136C4⋊(C3⋊Dic3)—68no—
137C3×#217C2×D314yes—
138C3×(C2×D3)⋊C4—38no—
139C3×Dic3⋊C4C2×D318yescomplemented
140C3×C4×Dic3C2×D318yescomplemented
141Dic3⋊Dic3—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—38no—
144C22×C33—22no—
145C22×C33C2×D412yes—
146C22×C33C2411yescharacteristic
147C2×C3⋊(C3⋊D3)—21no—
148C2×C3⋊(C3⋊D3)—22no—
149C2×C3×C3⋊D3—28no—
150C2×C3×C3⋊D3—22no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—22no—
154C2×C3×C3⋊D3#4114yescomplemented
155C33×C4—22no—
156C3⋊(C3⋊Dic3)—22no—
157C3⋊(C3⋊Dic3)—21no—
158C3×C3⋊Dic3—21no—
159C32×Dic3—22no—
160C32×Dic3C2×D412yes—
161C32×Dic3C2×D412yes—
162C23×Dic3—91no—
163C2×#237—91no—
164C2×#235—94no—
165C22×C4×D3—94no—
166C3×#120—91no—
167C3×#117—34no—
168#953—91no—
169#939—92no—
170#935—91no—
171D4×Dic3—92no—
172#897—98no—
173#895—98no—
174#886—94no—
175#886—94no—
176#886—91no—
177#886—184no—
178#882—34no—
179#882—94no—
180#67⋊C4—92no—
181#878—92no—
182#877C3⋊D311yescharacteristic, complemented
183C4×D12—98no—
184C4×D12—98no—
185C23×C32—24no—
186C23×C32C22×D314yes—
187C23×C32C22×D311yescharacteristic
188C22×C3⋊D3—22no—
189C22×C3⋊D3—24no—
190C22×C3⋊D3—31no—
191C22×C3⋊D3—34no—
192C22×C3⋊D3—64no—
193C22×C3⋊D3C22×D311yescharacteristic
194C22×C3⋊D3C22×D312yes—
195C22×C3×D3—38no—
196C22×C3×D3—34no—
197C22×C3×D3—616no—
198C22×C3×D3—68no—
199C2×C32×C4—28no—
200C2×C32×C4—34no—
201C2×C32×C4—31no—
202C2×C32×C4—31no—
203C2×C32×C4—32no—
204C2×C32×C4C22×D318yes—
205C2×C32×C4C22×D312yes—
206#638—22no—
207#638—62no—
208#638—68no—
209#638—128no—
210C2×C3⋊Dic3—31no—
211C2×C3⋊Dic3—32no—
212C2×C3⋊Dic3—34no—
213C2×C3⋊Dic3—38no—
214C2×C3⋊Dic3—64no—
215C2×C3⋊Dic3—68no—
216C2×C3⋊Dic3—62no—
217C2×C3⋊Dic3—68no—
218C2×C3⋊Dic3—128no—
219C2×C3⋊Dic3C22×D311yescharacteristic
220C4⋊(C3⋊D3)—24no—
221C4⋊(C3⋊D3)—62no—
222C4⋊(C3⋊D3)—64no—
223C4×C3⋊D3—62no—
224C4×C3⋊D3—64no—
225C4×C3⋊D3—64no—
226C4×C3⋊D3—62no—
227#634—62no—
228C3×#67—68no—
229C2×C3×Dic3—34no—
230C2×C3×Dic3D12116yescomplemented
231C2×C3×Dic3C22×D318yes—
232C2×C3×Dic3C22×D314yes—
233C2×C3×Dic3C22×D314yes—
234C3×D12—616no—
235#627—68no—
236#626—616no—
237#626—68no—
238#624—68no—
239D3×Dic3—616no—
240D3×Dic3—616no—
241D3×Dic3—68no—
242#525—271noSylow
243C2×C33—22no—
244C2×C33C22×D411yescharacteristic
245C2×C33C2×#4112yes—
246C3⋊(C3⋊D3)—41no—
247C3×C3⋊D3—24no—
248C3×C3⋊D3—41no—
249C24×C3—91no—
250C23×D3—34no—
251C2×C3×D4—91no—
252C2×C3×D4—92no—
253C22×C3×C4—34no—
254C2×#67—34no—
255C2×#67—94no—
256C2×#67—91no—
257C2×#67—92no—
258C2×#67—91no—
259C2×#67—184no—
260C22×Dic3—94no—
261C22×Dic3—91no—
262C22×Dic3—92no—
263C22×Dic3—92no—
264C22×Dic3—91no—
265C22×Dic3—184no—
266C22×Dic3—184no—
267C22×Dic3C2×C3⋊D311yescharacteristic
268#237—184no—
269#235—1816no—
270C2×D12—38no—
271C2×D12—98no—
272C2×D12—94no—
273C2×C4×D3—94no—
274C2×C4×D3—98no—
275C2×C4×D3—94no—
276C2×C4×D3—98no—
277C2×C4×D3—91no—
278C2×C4×D3—1816no—
279C2×Dic6—94no—
280C2×Dic6—91no—
281C3×C4⋊C4—38no—
282C3×C22⋊C4—24no—
283C3×C22⋊C4—34no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4D3214yes—
286C3×C22⋊C4C2×C3⋊D311yescharacteristic
287C3×C42—38no—
288#217—92no—
289#217—92no—
290#217C2×C3⋊D311yescharacteristic
291(C2×D3)⋊C4—38no—
292(C2×D3)⋊C4—98no—
293(C2×D3)⋊C4—98no—
294(C2×D3)⋊C4—98no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—188no—
297C4⋊Dic3—98no—
298C4⋊Dic3—98no—
299C4⋊Dic3—92no—
300C4⋊Dic3—188no—
301Dic3⋊C4—92no—
302Dic3⋊C4C2×C3⋊D312yescomplemented
303C4×Dic3C2×C3⋊D312yescomplemented
304C22×C32—22no—
305C22×C32—24no—
306C22×C32—28no—
307C22×C32—28no—
308C22×C32—48no—
309C22×C32C2×D1218yes—
310C22×C32D3×D412yes—
311C22×C32C23×D314yes—
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3—28no—
314C2×C3⋊D3—22no—
315C2×C3⋊D3—22no—
316C2×C3⋊D3—21no—
317C2×C3⋊D3—22no—
318C2×C3⋊D3—61no—
319C2×C3⋊D3—62no—
320C2×C3⋊D3—64no—
321C2×C3⋊D3—68no—
322C2×C3⋊D3—124no—
323C2×C3⋊D3—128no—
324C2×C3⋊D3#23714yescomplemented
325C2×C3×D3—316no—
326C2×C3×D3—632no—
327C2×C3×D3—68no—
328C2×C3×D3—68no—
329C2×C3×D3—64no—
330C2×C3×D3—68no—
331C32×C4—22no—
332C32×C4—28no—
333C32×C4—32no—
334C32×C4—32no—
335C32×C4—48no—
336C32×C4—62no—
337C3⋊Dic3—21no—
338C3⋊Dic3—61no—
339C3⋊Dic3—62no—
340C3⋊Dic3—68no—
341C3⋊Dic3—64no—
342C3⋊Dic3—128no—
343C3⋊Dic3—124no—
344C3×Dic3—28no—
345C3×Dic3—64no—
346C3×Dic3C2×D1218yes—
347C3×Dic3C2×D1218yes—
348C2×#41—271no—
349C23×C4—271no—
350#123—271no—
351#123—271no—
352C22⋊Q8—272no—
353#121—272no—
354#120—91no—
355#120—271no—
356C4×D4—272no—
357C4×D4—272no—
358#117—31no—
359C33#52511yescharacteristic, complemented, Sylow
360C23×C3—24no—
361C23×C3—92no—
362C23×C3—91no—
363C23×C3—184no—
364C23×C3—182no—
365C23×C3C2×D3214yes—
366C23×C3C22×C3⋊D311yescharacteristic
367C22×D3—38no—
368C22×D3—34no—
369C22×D3—616no—
370C22×D3—68no—
371C22×D3—94no—
372C22×D3—91no—
373C22×D3—184no—
374C3×D4—184no—
375C3×D4—182no—
376C2×C3×C4—28no—
377C2×C3×C4—34no—
378C2×C3×C4—34no—
379C2×C3×C4—316no—
380C2×C3×C4—38no—
381C2×C3×C4—91no—
382C2×C3×C4C2×D3218yes—
383C2×C3×C4C22×C3⋊D312yes—
384#67—68no—
385#67—184no—
386#67—188no—
387#67—182no—
388#67—182no—
389#67—368no—
390C2×Dic3—34no—
391C2×Dic3—94no—
392C2×Dic3—98no—
393C2×Dic3—92no—
394C2×Dic3—91no—
395C2×Dic3—184no—
396C2×Dic3—188no—
397C2×Dic3—188no—
398C2×Dic3—182no—
399C2×Dic3—182no—
400C2×Dic3—184no—
401C2×Dic3—184no—
402C2×Dic3—368no—
403C2×Dic3C4⋊(C3⋊D3)14yescomplemented
404C2×Dic3C22×C3⋊D312yes—
405C2×Dic3C22×C3⋊D311yescharacteristic
406C2×Dic3C22×C3⋊D311yescharacteristic
407D12—616no—
408D12—188no—
409D12—1816no—
410C4×D3—188no—
411C4×D3—1816no—
412C4×D3—1816no—
413C4×D3—188no—
414C4×D3—182no—
415Dic6—188no—
416Dic6—182no—
417C2×C32—22no—
418C2×C32—24no—
419C2×C32—28no—
420C2×C32—28no—
421C2×C32—48no—
422C2×C32C22×D1214yes—
423C2×C32C2×D3×D411yescharacteristic
424C2×C32C2×#23712yes—
425C2×C32D3×#4118yes—
426C3⋊D3—24no—
427C3⋊D3—41no—
428C3⋊D3—121no—
429C3⋊D3—124no—
430C3⋊D3—244no—
431C3×D3—616no—
432C3×D3—124no—
433C24—91no—
434#41—544no—
435C2×Q8—271no—
436C2×D4—91no—
437C2×D4—92no—
438C2×D4—272no—
439C2×D4—271no—
440C2×D4—271no—
441C22×C4—31no—
442C22×C4—271no—
443C22×C4—272no—
444C22×C4—271no—
445C22×C4—272no—
446C22×C4—271no—
447C22×C4—544no—
448C4⋊C4—32no—
449C4⋊C4—272no—
450C4⋊C4—272no—
451C22⋊C4—31no—
452C22⋊C4—92no—
453C22⋊C4—272no—
454C22⋊C4—272no—
455C22⋊C4—272no—
456C22⋊C4D3×C3⋊D311yescharacteristic
457C42—32no—
458C22×C3—28no—
459C22×C3—22no—
460C22×C3—24no—
461C22×C3—28no—
462C22×C3—48no—
463C22×C3—94no—
464C22×C3—188no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3—181no—
468C22×C3—182no—
469C22×C3D3×D1218yes—
470C22×C3C2×C4⋊(C3⋊D3)12yes—
471C22×C3C22×D3214yes—
472C22×C3C23×C3⋊D311yescharacteristic
473C2×D3—316no—
474C2×D3—632no—
475C2×D3—68no—
476C2×D3—68no—
477C2×D3—64no—
478C2×D3—68no—
479C2×D3—184no—
480C2×D3—188no—
481C2×D3—181no—
482C2×D3—182no—
483C2×D3—364no—
484C2×D3—368no—
485C3×C4—28no—
486C3×C4—22no—
487C3×C4—38no—
488C3×C4—38no—
489C3×C4—48no—
490C3×C4—68no—
491C3×C4—181no—
492Dic3—22no—
493Dic3—64no—
494Dic3—184no—
495Dic3—188no—
496Dic3—181no—
497Dic3—182no—
498Dic3—368no—
499Dic3—364no—
500Dic3C2×C4⋊(C3⋊D3)12yes—
501Dic3C2×C4⋊(C3⋊D3)12yes—
502C32—24no—
503C32#552414yescomplemented
504C32#558611yescharacteristic, complemented
505C23—92no—
506C23—91no—
507C23—184no—
508C23—182no—
509C23—271no—
510C23C2×D3×C3⋊D311yescharacteristic
511Q8—542no—
512D4—184no—
513D4—182no—
514D4—542no—
515D4—544no—
516D4—542no—
517C2×C4—31no—
518C2×C4—31no—
519C2×C4—34no—
520C2×C4—32no—
521C2×C4—91no—
522C2×C4—272no—
523C2×C4—271no—
524C2×C4—542no—
525C2×C4—544no—
526C2×C4—544no—
527C2×C4—542no—
528C2×C4—542no—
529C2×C4C2×D3×C3⋊D312yes—
530C2×C3—28no—
531C2×C3—22no—
532C2×C3—24no—
533C2×C3—28no—
534C2×C3—48no—
535C2×C3—184no—
536C2×C3—361no—
537C2×C3C2×D3×D1214yes—
538C2×C3D3×#23718yes—
539C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
540C2×C3#41×C3⋊D312yes—
541D3—616no—
542D3—124no—
543D3—364no—
544D3—361no—
545D3—724no—
546C22—22no—
547C22—94no—
548C22—188no—
549C22—182no—
550C22—182no—
551C22—181no—
552C22—182no—
553C22—541no—
554C22—542no—
555C22D3×C4⋊(C3⋊D3)12yes—
556C22C22×D3×C3⋊D311yescharacteristic
557C4—22no—
558C4—32no—
559C4—32no—
560C4—62no—
561C4—181no—
562C4—542no—
563C4—541no—
564C3—24no—
565C3#1059407814yescomplemented
566C3#1059599511yescharacteristic, complemented
567C2—22no—
568C2—184no—
569C2—361no—
570C2—1081no—
571C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
572C2C3⋊D3×#23712yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89031C2semidirect
2. C22×Dic3×C3⋊D3C2semidirect
3. #11723927C2semidirect
4. C3×#64091C2semidirect
5. C32×#877C2semidirect
6. #11723636C2semidirect
7. #11723609C2semidirect
8. #11723483C2semidirect
9. #11723482C2semidirect
10. Dic3×C4⋊(C3⋊D3)C2semidirect
11. #11723471C2semidirect
20. C2×#6293C22semidirect
22. C2×#6292C22semidirect
23. C2×#6289C22semidirect
28. C2×Dic3×C3⋊D3C22semidirect
30. #88859C22semidirect
31. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
34. C32×Dic3⋊C4C22semidirect
35. C32×C4×Dic3C22semidirect
36. Dic3⋊(C3⋊Dic3)C22semidirect
38. #88803C22semidirect
51. #64091D3semidirect
55. C3×#877D3semidirect
76. C2×C32×Dic3D4semidirect
139. C3×Dic3⋊C4C2×D3semidirect
140. C3×C4×Dic3C2×D3semidirect
154. C2×C3×C3⋊D3#41semidirect
182. #877C3⋊D3semidirect
230. C2×C3×Dic3D12semidirect
302. Dic3⋊C4C2×C3⋊D3semidirect
303. C4×Dic3C2×C3⋊D3semidirect
324. C2×C3⋊D3#237semidirect
359. C33#525semidirect
403. C2×Dic3C4⋊(C3⋊D3)semidirect
503. C32#5524semidirect
504. C32#5586semidirect
565. C3#10594078semidirect
566. C3#10595995semidirect

Maximal subgroups

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

subgroupindex
C2×#890312
C22×Dic3×C3⋊D32
#117239272
C3×#640912
C32×#8772
#117236362
#117236092
#117234832
#117234822
Dic3×C4⋊(C3⋊D3)2
#117234712
#105959953
#105940783