Finite Group Databasebeta

← #10693103

Subgroups — #10693103

Order 768.

956 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82126C211yesmaximal, characteristic
2#73849C211yesmaximal, characteristic
3C3×#1819C211yesmaximal, characteristic, complemented
4#71542C211yesmaximal, characteristic, complemented
5#69498C211yesmaximal, characteristic
6C3⋊#1304C211yesmaximal, characteristic, complemented
7C3⋊#1304C211yesmaximal, characteristic, complemented
8#9499—31nomaximal, Sylow
9C23×#67C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C2×#929C412yescomplemented
12C2×D3×C22⋊C4—21no—
13C3×#379C2211yescharacteristic
14C3×#379C2211yescharacteristic
15#5247C2211yescharacteristic
16#5247C2211yescharacteristic
17#5247C2211yescharacteristic
18#5213C412yescomplemented
19#4959—22no—
20#4956—22no—
21#4956—22no—
22#4955—21no—
23C2×#806—22no—
24#4946—21no—
25#4943—21no—
26C3×#342—22no—
27#4526—22no—
28#2405—31no—
29#2308—31no—
30#1819—31no—
31#1819D311yescharacteristic, complemented
32#1781—31no—
33#1304—31no—
34#1304—31no—
35C25×C3C2311yescharacteristic
36C24×D3D411yescharacteristic
37C23×C3×C4C2×C411yescharacteristic, complemented
38C23×C3×C4C2×C411yescharacteristic, complemented
39C22×#67—21no—
40C22×#67—21no—
41C22×#67—21no—
42C22×#67—21no—
43C22×#67—44no—
44C22×#67C2×C412yes—
45C22×#67D412yes—
46C23×Dic3D411yescharacteristic
47C22×C4×D3—22no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4D412yes—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#217—21no—
64C2×#217—21no—
65C2×#217—21no—
66C2×#217—21no—
67C2×#217C2×C412yescomplemented
68C2×#217D412yes—
69#929—44no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×Dic3⋊C4—22no—
74D3×C22⋊C4—44no—
75C3⋊#95—24no—
76C3⋊#95—22no—
77#806—44no—
78C23×D4—31no—
79C2×#123—32no—
80C22×C22⋊C4—61no—
81C22×C22⋊C4C2×D311yescharacteristic
82C2×#99—62no—
83#394—62no—
84#386—32no—
85#386—62no—
86#386—62no—
87#385—61no—
88#381—61no—
89#379—31no—
90#379—31no—
91#379—31no—
92#379—61no—
93#379C2×D311yescharacteristic
94#379C2×D311yescharacteristic
95#342—22no—
96#342—62no—
97C24×C3—42no—
98C24×C3—42no—
99C24×C3—81no—
100C24×C3C22×C411yescharacteristic
101C24×C3C2×D411yescharacteristic
102C24×C3C2×D411yescharacteristic
103C24×C3C2×D411yescharacteristic
104C24×C3C2×D411yescharacteristic
105C24×C3#4111yescharacteristic
106C24×C3#4111yescharacteristic
107C23×D3—21no—
108C23×D3—21no—
109C23×D3—24no—
110C23×D3—42no—
111C23×D3—81no—
112C23×D3C22⋊C412yes—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—42no—
121C22×C3×C4—41no—
122C22×C3×C4—81no—
123C2×#67—22no—
124C2×#67—22no—
125C2×#67—22no—
126C2×#67—22no—
127C2×#67—24no—
128C2×#67—24no—
129C2×#67—42no—
130C2×#67—42no—
131C2×#67—42no—
132C2×#67—42no—
133C2×#67—42no—
134C2×#67—42no—
135C2×#67—44no—
136C2×#67—44no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—42no—
142C22×Dic3—41no—
143C22×Dic3C22⋊C412yes—
144C2×C4×D3—44no—
145C2×C4×D3—82no—
146C2×C4×D3—82no—
147C3×C22⋊C4—42no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—42no—
153C3×C22⋊C4—42no—
154C3×C22⋊C4—42no—
155C3×C22⋊C4—81no—
156C3×C22⋊C4—81no—
157#217—42no—
158#217—42no—
159#217—42no—
160#217—42no—
161#217—42no—
162#217—42no—
163#217—81no—
164#217—81no—
165#217—81no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—44no—
168(C2×D3)⋊C4—82no—
169Dic3⋊C4—44no—
170C25—31no—
171C25C22×D311yescharacteristic
172C22×D4—32no—
173C22×D4—32no—
174C22×D4—61no—
175C22×D4—61no—
176C22×D4—61no—
177C22×D4—61no—
178C22×D4—124no—
179C23×C4—31no—
180C23×C4—62no—
181C23×C4C4×D311yescharacteristic, complemented
182C23×C4C4×D311yescharacteristic, complemented
183#123—124no—
184C2×C4⋊C4—62no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—32no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—124no—
209C2×C22⋊C4#6712yes—
210#99—124no—
211#95—62no—
212#95—64no—
213C23×C3—22no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—88no—
235C23×C3C2×C22⋊C411yescharacteristic
236C23×C3C4×D411yescharacteristic
237C23×C3C4×D411yescharacteristic
238C23×C3#12011yescharacteristic
239C23×C3#12111yescharacteristic
240C23×C3#12111yescharacteristic
241C23×C3#12311yescharacteristic
242C22×D3—22no—
243C22×D3—22no—
244C22×D3—22no—
245C22×D3—24no—
246C22×D3—22no—
247C22×D3—42no—
248C22×D3—42no—
249C22×D3—42no—
250C22×D3—42no—
251C22×D3—88no—
252C22×D3—81no—
253C22×D3—81no—
254C22×D3#9914yes—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—81no—
263C2×C3×C4—81no—
264C2×C3×C4—81no—
265C2×C3×C4—81no—
266C2×C3×C4—81no—
267C2×C3×C4—81no—
268C2×C3×C4—81no—
269#67—44no—
270#67—44no—
271#67—44no—
272#67—44no—
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—81no—
286C2×Dic3—81no—
287C2×Dic3—81no—
288C4×D3—88no—
289C24—32no—
290C24—42no—
291C24—42no—
292C24—64no—
293C24—61no—
294C24—61no—
295C24—81no—
296C24—122no—
297C24—241no—
298C24C2×C4×D311yescharacteristic
299C24#23511yescharacteristic
300C24D3×D411yescharacteristic
301C24D3×D411yescharacteristic
302C24D3×D411yescharacteristic
303C24#23711yescharacteristic
304C24C2×#6711yescharacteristic
305C2×D4—64no—
306C2×D4—64no—
307C2×D4—62no—
308C2×D4—62no—
309C2×D4—62no—
310C2×D4—62no—
311C2×D4—122no—
312C2×D4—122no—
313C2×D4—124no—
314C2×D4—122no—
315C2×D4—122no—
316C2×D4—124no—
317C2×D4—122no—
318C2×D4—122no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—32no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—81no—
334C22×C4—124no—
335C22×C4—122no—
336C22×C4—121no—
337C22×C4—242no—
338C22×C4—242no—
339C4⋊C4—124no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—81no—
349C22⋊C4—81no—
350C22⋊C4—124no—
351C22⋊C4—124no—
352C22⋊C4—122no—
353C22⋊C4—122no—
354C22⋊C4—122no—
355C22⋊C4—122no—
356C22⋊C4—122no—
357C22⋊C4—122no—
358C22⋊C4—242no—
359C22⋊C4—241no—
360C22⋊C4—241no—
361C22⋊C4—241no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—44no—
382C22×C3—44no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3—82no—
386C22×C3—82no—
387C22×C3—82no—
388C22×C3#38611yescharacteristic
389C22×C3C2×#9912yes—
390C22×C3#45812yes—
391C2×D3—24no—
392C2×D3—44no—
393C2×D3—44no—
394C2×D3—42no—
395C2×D3—42no—
396C2×D3—44no—
397C2×D3—82no—
398C2×D3—84no—
399C3×C4—82no—
400C3×C4—81no—
401C3×C4—81no—
402C3×C4—161no—
403Dic3—42no—
404Dic3—82no—
405Dic3—82no—
406Dic3—161no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—34no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—62no—
430C23—62no—
431C23—64no—
432C23—62no—
433C23—62no—
434C23—88no—
435C23—122no—
436C23—122no—
437C23—122no—
438C23—122no—
439C23—241no—
440C23—241no—
441C23—248no—
442C23D3×C22⋊C411yescharacteristic
443C23#67⋊C411yescharacteristic
444C23#88311yescharacteristic
445C23#88411yescharacteristic
446C23C4×#6711yescharacteristic
447C23#93711yescharacteristic
448C23#93811yescharacteristic
449D4—124no—
450D4—124no—
451D4—124no—
452D4—124no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—248no—
480C2×C4—241no—
481C2×C4—241no—
482C2×C4—241no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—41no—
486C2×C3—41no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3—41no—
490C2×C3#189011yescharacteristic
491C2×C3#189811yescharacteristic
492C2×C3#191311yescharacteristic
493D3—44no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—44no—
514C22—44no—
515C22—64no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—124no—
522C22—124no—
523C22—124no—
524C22—122no—
525C22—122no—
526C22—242no—
527C22—244no—
528C22D3×#9912yes—
529C22#495611yescharacteristic
530C22#505312yes—
531C4—82no—
532C4—81no—
533C4—81no—
534C4—122no—
535C4—161no—
536C4—242no—
537C4—242no—
538C4—481no—
539C3#949911yescharacteristic, complemented, Sylow
540C2—22no—
541C2—22no—
542C2—41no—
543C2—41no—
544C2—41no—
545C2—41no—
546C2—41no—
547C2—124no—
548C2#6953811yescharacteristic
549C2#7003411yescharacteristic
550C2#7003611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC3×#1819C2
socleC22×C3#386
prime radicalC23×#67C22
cosocleC25×C3C23
hypercenter#1819D3
hypocenterC3#9499
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. C3×#1819C2semidirect
4. #71542C2semidirect
6. C3⋊#1304C2semidirect
7. C3⋊#1304C2semidirect
11. C2×#929C4semidirect
18. #5213C4semidirect
31. #1819D3semidirect
37. C23×C3×C4C2×C4semidirect
38. C23×C3×C4C2×C4semidirect
67. C2×#217C2×C4semidirect
181. C23×C4C4×D3semidirect
182. C23×C4C4×D3semidirect
539. C3#9499semidirect

Maximal subgroups

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

subgroupindex
#821262
#738492
C3×#18192
#715422
#694982
C3⋊#13042
#94993