Finite Group Databasebeta

← #13150610

Subgroups — #13150610

Order 1728.

843 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6235C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721301C212yesmaximal
5#11721296C212yesmaximal, complemented
6#11721095C211yesmaximal, characteristic, complemented
7#11721094C211yesmaximal, characteristic, complemented
8#11721089C211yesmaximal, characteristic, complemented
9#11721088C212yesmaximal, complemented
10#11721087C212yesmaximal, complemented
11#11721084C211yesmaximal, characteristic, complemented
12#10594853—31nomaximal
13#10594849—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6235C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2212yes—
18C2×#634⋊C3C2211yescharacteristic
19#88552C2211yescharacteristic
20#88542C2211yescharacteristic
21#88540C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23#3757⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2212yes—
29#88451C2212yes—
30#88451C2212yes—
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#88447C2212yes—
37C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
38#64133D311yescharacteristic, complemented
39C3×C4×#67—31no—
40C3×C4×#67—31no—
41C3×C4×#67—61no—
42C2×C3×C4⋊Dic3—31no—
43C3×#924—31no—
44#63962—32no—
45#63961—32no—
46#63957—32no—
47#63955—32no—
48#63869—31no—
49#63869—31no—
50#63861—32no—
51#63858—32no—
52#63857—31no—
53#63856—31no—
54#63847—32no—
55#63845—32no—
56#63844—31no—
57Dic3×Dic6—31no—
58#63841—31no—
59#63841—31no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×#272C2311yescharacteristic
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65#6235—22no—
66C2×#1084C2311yescharacteristic
67C2×#1084C2311yescharacteristic
68C2×#1084C2311yescharacteristic
69C4×#272—22no—
70C2×(C3⋊Dic3)⋊C3—24no—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2312yes—
75#634⋊C3—22no—
76#5642—91no—
77#5541—91no—
78#5513—91no—
79C22×C32×C4—21no—
80C22×C32×C4C2×D311yescharacteristic
81C22×C32×C4C2×D311yescharacteristic
82C22×C3⋊Dic3C2×D312yes—
83C2×#634C2×D311yescharacteristic
84C2×C3×#67—31no—
85C2×C3×#67—31no—
86C2×C3×#67—61no—
87C22×C3×Dic3—32no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3—61no—
91C2×C3×Dic6—31no—
92#3757C2×D312yes—
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94C4⋊(C3⋊Dic3)C2×D311yescharacteristic
95C4⋊(C3⋊Dic3)C2×D312yes—
96C4⋊(C3⋊Dic3)C2×D311yescharacteristic
97#3750C2×D312yes—
98C3×#217—32no—
99C3×#217—31no—
100C3×#217—31no—
101C3×#217—61no—
102C3×(C2×D3)⋊C4—31no—
103C3×(C2×D3)⋊C4—31no—
104C3×(C2×D3)⋊C4—61no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—32no—
108C3×C4⋊Dic3—31no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—31no—
111C3×Dic3⋊C4—31no—
112C3×Dic3⋊C4—61no—
113C3×C4×Dic3—31no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—61no—
116#3724—32no—
117#3724—32no—
118#3724—32no—
119#3724—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126(C2×D3)⋊Dic3—32no—
127(C2×D3)⋊Dic3—32no—
128Dic32—32no—
129Dic32—32no—
130Dic32—32no—
131Dic32—32no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)#4112yes—
135C22×H(3,𝔽3)C2411yescharacteristic
136C2×#272—21no—
137C2×#272—21no—
138C2×#272—21no—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)#4112yes—
141#1084—21no—
142#1084—21no—
143#1084—21no—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3—22no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3—22no—
148C3×C22⋊Q8—91no—
149C3×C4×D4—91no—
150C2×C3×C4⋊C4—91no—
151#946—91no—
152Q8×Dic3—91no—
153#935—92no—
154#929—92no—
155C4×#67—31no—
156C4×#67—31no—
157C4×#67—61no—
158#926—91no—
159#926—92no—
160C2×C4⋊Dic3—31no—
161C2×C4⋊Dic3—91no—
162#924—31no—
163#924—91no—
164#924—91no—
165#898—92no—
166#898—92no—
167#897—91no—
168#892—91no—
169#892—91no—
170#890—91no—
171#890—91no—
172#890—92no—
173#889—92no—
174#889—92no—
175#887—91no—
176#886—92no—
177#879—92no—
178#877—92no—
179#877—92no—
180#870—91no—
181C4×Dic6—91no—
182C23×C32—21no—
183C23×C32C22×D311yescharacteristic
184C23×C32C22×D311yescharacteristic
185C22×C3×D3—31no—
186C22×C3×D3—31no—
187C22×C3×D3—61no—
188C2×C32×C4—22no—
189C2×C32×C4—22no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—42no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3—24no—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D312yes—
201C2×C3⋊Dic3C22×D312yes—
202#634—22no—
203C3×#67—64no—
204C3×#67—62no—
205C3×#67—62no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—31no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—64no—
220C3×C4×D3—62no—
221C3×C4×D3—62no—
222C3×C4×D3—122no—
223C3×Dic6—62no—
224#552—271noSylow
225C2×H(3,𝔽3)—22no—
226C2×H(3,𝔽3)C2×#4111yescharacteristic
227C2×H(3,𝔽3)C2×#4111yescharacteristic
228C2×H(3,𝔽3)2−1+411yescharacteristic
229#272—41no—
230C2×C3×Q8—91no—
231C2×C3×D4—91no—
232C22×C3×C4—31no—
233C22×C3×C4—31no—
234C22×C3×C4—61no—
235C22×C3×C4—92no—
236C22×C3×C4—91no—
237C22×C3×C4D3211yescharacteristic
238C2×#67—31no—
239C2×#67—31no—
240C2×#67—61no—
241C22×Dic3—32no—
242C22×Dic3—92no—
243C2×C4×D3—31no—
244C2×C4×D3—31no—
245C2×C4×D3—61no—
246C2×Dic6—31no—
247C2×Dic6—91no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4—91no—
254C3×C22⋊C4—92no—
255C3×C22⋊C4—91no—
256C3×C22⋊C4—91no—
257C3×C42—91no—
258#217—31no—
259#217—31no—
260#217—32no—
261#217—61no—
262#217—92no—
263#217—92no—
264#217—92no—
265(C2×D3)⋊C4—31no—
266(C2×D3)⋊C4—31no—
267(C2×D3)⋊C4—61no—
268(C2×D3)⋊C4—92no—
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272C4⋊Dic3—31no—
273C4⋊Dic3—31no—
274C4⋊Dic3—31no—
275C4⋊Dic3—32no—
276C4⋊Dic3—91no—
277C4⋊Dic3—92no—
278C4⋊Dic3—91no—
279C4⋊Dic3—91no—
280C4⋊Dic3—92no—
281C4⋊Dic3—92no—
282C4⋊Dic3—92no—
283C4⋊Dic3—92no—
284Dic3⋊C4—31no—
285Dic3⋊C4—31no—
286Dic3⋊C4—32no—
287Dic3⋊C4—61no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297C4×Dic3—31no—
298C4×Dic3—31no—
299C4×Dic3—61no—
300C4×Dic3—92no—
301C4×Dic3—92no—
302C4×Dic3—92no—
303C4×Dic3—92no—
304C4×Dic3—92no—
305C4×Dic3—92no—
306C22×C32—21no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—22no—
312C22×C32—41no—
313C22×C32—41no—
314C22×C32#23712yes—
315C22×C32#23912yes—
316C22×C32C23×D311yescharacteristic
317C22×C32C23×D311yescharacteristic
318C2×C3×D3—62no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321C2×C3×D3—61no—
322C2×C3×D3—61no—
323C2×C3×D3—61no—
324C2×C3×D3—61no—
325C2×C3×D3—121no—
326C2×C3×D3—121no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—22no—
330C32×C4—41no—
331C32×C4#23512yes—
332C32×C4#23712yes—
333C3⋊Dic3—22no—
334C3⋊Dic3—22no—
335C3⋊Dic3—22no—
336C3⋊Dic3—22no—
337C3×Dic3—62no—
338C3×Dic3—62no—
339C3×Dic3—61no—
340C3×Dic3—61no—
341C3×Dic3—61no—
342C3×Dic3—62no—
343C3×Dic3—62no—
344C3×Dic3—61no—
345C3×Dic3—62no—
346C3×Dic3—61no—
347C3×Dic3—61no—
348C3×Dic3—121no—
349C3×Dic3—121no—
350#126—272no—
351#125—271no—
352#125—272no—
353#123—272no—
354C22⋊Q8—91no—
355C22⋊Q8—271no—
356C4×Q8—271no—
357C4×D4—91no—
358#117—272no—
359#117—271no—
360C2×C4⋊C4—91no—
361H(3,𝔽3)#55211yescharacteristic, complemented, Sylow
362C23×C3—31no—
363C23×C3—31no—
364C23×C3—61no—
365C23×C3—91no—
366C23×C3C2×D3211yescharacteristic
367C22×D3—31no—
368C22×D3—31no—
369C22×D3—61no—
370C3×Q8—182no—
371C3×D4—182no—
372C2×C3×C4—22no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—92no—
384C2×C3×C4—92no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—122no—
389C2×C3×C4—184no—
390C2×C3×C4—182no—
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C2×D3211yescharacteristic
393#67—62no—
394#67—62no—
395#67—64no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—31no—
399C2×Dic3—31no—
400C2×Dic3—31no—
401C2×Dic3—31no—
402C2×Dic3—32no—
403C2×Dic3—32no—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—61no—
407C2×Dic3—61no—
408C2×Dic3—61no—
409C2×Dic3—64no—
410C2×Dic3—92no—
411C2×Dic3—92no—
412C2×Dic3—92no—
413C2×Dic3—92no—
414C2×Dic3—184no—
415C4×D3—62no—
416C4×D3—62no—
417C4×D3—122no—
418Dic6—62no—
419Dic6—182no—
420C2×C32—22no—
421C2×C32—22no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—42no—
426C2×C32C2×#23511yescharacteristic
427C2×C32C2×#23711yescharacteristic
428C2×C32C2×#23711yescharacteristic
429C2×C32C2×#23911yescharacteristic
430C2×C32#100711yescharacteristic
431C2×C32#101011yescharacteristic
432C3×D3—122no—
433C3×D3—121no—
434C3×D3—121no—
435C2×Q8—91no—
436C2×D4—91no—
437C22×C4—92no—
438C22×C4—91no—
439C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
440C4⋊C4—91no—
441C4⋊C4—91no—
442C4⋊C4—92no—
443C4⋊C4—91no—
444C4⋊C4—91no—
445C4⋊C4—92no—
446C4⋊C4—272no—
447C4⋊C4—272no—
448C4⋊C4—272no—
449C4⋊C4—272no—
450C22⋊C4—91no—
451C22⋊C4—92no—
452C22⋊C4—91no—
453C22⋊C4—272no—
454C22⋊C4—272no—
455C42—91no—
456C42—272no—
457C42—272no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—31no—
461C22×C3—31no—
462C22×C3—32no—
463C22×C3—32no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—62no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3—181no—
473C22×C3—181no—
474C22×C3—181no—
475C22×C3#379612yes—
476C22×C3C22×D3211yescharacteristic
477C2×D3—61no—
478C2×D3—61no—
479C2×D3—61no—
480C2×D3—61no—
481C2×D3—61no—
482C2×D3—61no—
483C2×D3—62no—
484C2×D3—121no—
485C2×D3—121no—
486C3×C4—21no—
487C3×C4—32no—
488C3×C4—32no—
489C3×C4—61no—
490C3×C4—61no—
491C3×C4—62no—
492C3×C4—121no—
493C3×C4—182no—
494C3×C4—182no—
495C3×C4—182no—
496C3×C4—182no—
497C3×C4—181no—
498C3×C4—181no—
499C3×C4—181no—
500C3×C4#380412yes—
501Dic3—61no—
502Dic3—61no—
503Dic3—61no—
504Dic3—61no—
505Dic3—61no—
506Dic3—61no—
507Dic3—62no—
508Dic3—62no—
509Dic3—62no—
510Dic3—62no—
511Dic3—62no—
512Dic3—121no—
513Dic3—121no—
514Dic3—182no—
515Dic3—182no—
516Dic3—182no—
517Dic3—182no—
518C32—21no—
519C32#551311yescharacteristic, complemented
520C32#564211yescharacteristic, complemented
521C23—91no—
522C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
523Q8—182no—
524D4—182no—
525C2×C4—22no—
526C2×C4—91no—
527C2×C4—91no—
528C2×C4—91no—
529C2×C4—92no—
530C2×C4—92no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—184no—
534C2×C4—182no—
535C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
536C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
537C2×C3—22no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—62no—
545C2×C3—62no—
546C2×C3—61no—
547C2×C3—61no—
548C2×C3—61no—
549C2×C3—122no—
550C2×C3—361no—
551C2×C3C2×#379611yescharacteristic
552C2×C3#6429711yescharacteristic
553C2×C3C2×#380411yescharacteristic
554D3—121no—
555D3—121no—
556D3—122no—
557C22—21no—
558C22—21no—
559C22—181no—
560C22—181no—
561C22—181no—
562C22#8864912yes—
563C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
564C4—21no—
565C4—181no—
566C4—182no—
567C4—182no—
568C4—182no—
569C4—181no—
570C4—181no—
571C4—182no—
572C4#8867112yes—
573C3—31no—
574C3—31no—
575C3—61no—
576C3#1059482511yescharacteristic
577C2—22no—
578C2—361no—
579C2C2×#8864911yescharacteristic
580C2#1172248211yescharacteristic
581C2C2×#8867111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC2×#6235C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#552
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
3. #64133⋊C3C2semidirect
5. #11721296C2semidirect
6. #11721095C2semidirect
7. #11721094C2semidirect
8. #11721089C2semidirect
9. #11721088C2semidirect
10. #11721087C2semidirect
11. #11721084C2semidirect
37. C2×C4⋊(C3⋊Dic3)D3semidirect
38. #64133D3semidirect
361. H(3,𝔽3)#552semidirect
519. C32#5513semidirect
520. C32#5642semidirect

Maximal subgroups

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

subgroupindex
C4×#62352
C2×C4⋊((C3⋊Dic3)⋊C3)2
#64133⋊C32
#117213012
#117212962
#117210952
#117210942
#117210892
#117210882
#117210872
#117210842
#105948533
#105948493