Finite Group Databasebeta

← #10693088

Subgroups — #10693088

Order 768.

900 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 93 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#84047C211yesmaximal, characteristic
2#73820C211yesmaximal, characteristic, complemented
3C3×#1781C211yesmaximal, characteristic
4#72130C211yesmaximal, characteristic
5#69495C211yesmaximal, characteristic, complemented
6C3⋊#1304C211yesmaximal, characteristic, complemented
7C3⋊#1304C211yesmaximal, characteristic, complemented
8#9496—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×#217C2211yescharacteristic
11C2×#935—21no—
12C2×C3×#99—22no—
13C3×#379C2211yescharacteristic
14C3×#379C2211yescharacteristic
15#5247C412yescomplemented
16#5247C2211yescharacteristic
17#5247C2211yescharacteristic
18#5247C2211yescharacteristic
19#5222—22no—
20#5220—21no—
21C2×#834—22no—
22#5210C412yescomplemented
23#4945—22no—
24#4944—22no—
25#4942—21no—
26Dic3×C22⋊C4—21no—
27#4527—22no—
28#2405—31no—
29#2312—31no—
30#1819—31no—
31#1781—31no—
32#1781D311yescharacteristic
33#1304—31no—
34#1304—31no—
35C25×C3C2×C411yescharacteristic
36C25×C3C2311yescharacteristic
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—44no—
44C23×C3×C4C2×C411yescharacteristic
45C23×Dic3D411yescharacteristic
46C23×Dic3D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4D412yes—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217—21no—
59C2×#217—22no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#217—21no—
64C2×#217—22no—
65C2×#217—21no—
66C2×#217C2×C412yes—
67C2×#217C2×C412yes—
68C2×#217D412yes—
69C2×#217D412yes—
70#935—44no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×C4×Dic3—22no—
74C3×#99—44no—
75#834—44no—
76C3⋊#95—22no—
77C3⋊#95—22no—
78C3⋊#95—22no—
79C23×D4C2×D311yescharacteristic
80C2×#123—61no—
81C22×C22⋊C4—31no—
82C2×#99—22no—
83C2×#99—62no—
84#394—62no—
85#388—62no—
86#388—62no—
87#386—61no—
88#385—32no—
89#380—61no—
90#379—32no—
91#379—31no—
92#379—31no—
93#379—31no—
94#379C2×D311yescharacteristic
95#379C2×D311yescharacteristic
96C4×C22⋊C4—61no—
97#342—62no—
98C24×C3—21no—
99C24×C3—21no—
100C24×C3—21no—
101C24×C3—42no—
102C24×C3—42no—
103C24×C3—42no—
104C24×C3—42no—
105C24×C3—81no—
106C24×C3—81no—
107C24×C3C22×C411yescharacteristic
108C24×C3C2×D411yescharacteristic
109C24×C3C2×D411yescharacteristic
110C24×C3C2×D411yescharacteristic
111C24×C3C2×D411yescharacteristic
112C24×C3#4111yescharacteristic
113C24×C3#4111yescharacteristic
114C2×C3×D4—24no—
115C2×C3×D4—24no—
116C2×C3×D4—24no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C2×C3×D4—42no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4—42no—
123C2×C3×D4—42no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—42no—
135C22×C3×C4—41no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—42no—
143C22×Dic3—41no—
144C22×Dic3—81no—
145C22×Dic3C22⋊C412yes—
146C22×Dic3C22⋊C412yes—
147C3×C22⋊C4—42no—
148C3×C22⋊C4—42no—
149C3×C22⋊C4—42no—
150C3×C22⋊C4—42no—
151C3×C22⋊C4—81no—
152C3×C22⋊C4—81no—
153#217—22no—
154#217—22no—
155#217—22no—
156#217—22no—
157#217—44no—
158#217—42no—
159#217—42no—
160#217—42no—
161#217—42no—
162#217—42no—
163#217—42no—
164#217—82no—
165#217—81no—
166#217—81no—
167#217—81no—
168Dic3⋊C4—44no—
169Dic3⋊C4—82no—
170C4×Dic3—82no—
171C25C4×D311yescharacteristic
172C25C22×D311yescharacteristic
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—44no—
180C23×C4—31no—
181C23×C4—31no—
182C23×C4C4×D311yescharacteristic
183#123—124no—
184C2×C4⋊C4—62no—
185C2×C4⋊C4—62no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—32no—
192C2×C22⋊C4—32no—
193C2×C22⋊C4—32no—
194C2×C22⋊C4—32no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
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#6712yes—
209C2×C42—62no—
210#99—44no—
211#99—124no—
212#95—62no—
213#95—62no—
214#95—62no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—22no—
223C23×C3—22no—
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—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—42no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—42no—
245C23×C3—81no—
246C23×C3—81no—
247C23×C3—81no—
248C23×C3—81no—
249C23×C3—88no—
250C23×C3—88no—
251C23×C3C2×C22⋊C411yescharacteristic
252C23×C3C4×D411yescharacteristic
253C23×C3C4×D411yescharacteristic
254C23×C3#12011yescharacteristic
255C23×C3#12111yescharacteristic
256C23×C3#12111yescharacteristic
257C23×C3#12311yescharacteristic
258C3×D4—44no—
259C3×D4—44no—
260C3×D4—44no—
261C3×D4—44no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—81no—
273C2×C3×C4—81no—
274C2×C3×C4—81no—
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—81no—
288C2×Dic3—81no—
289C2×Dic3—81no—
290C2×Dic3—81no—
291C2×Dic3—81no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—81no—
300C24—81no—
301C24C2×C4×D311yescharacteristic
302C24#23511yescharacteristic
303C24D3×D411yescharacteristic
304C24D3×D411yescharacteristic
305C24D3×D411yescharacteristic
306C24#23711yescharacteristic
307C24C2×#6711yescharacteristic
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—32no—
329C22×C4—32no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—61no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—122no—
339C22×C4—121no—
340C22×C4—241no—
341C4⋊C4—124no—
342C4⋊C4—242no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—62no—
348C22⋊C4—62no—
349C22⋊C4—62no—
350C22⋊C4—62no—
351C22⋊C4—81no—
352C22⋊C4—81no—
353C22⋊C4—124no—
354C22⋊C4—122no—
355C22⋊C4—122no—
356C22⋊C4—122no—
357C22⋊C4—122no—
358C22⋊C4—122no—
359C22⋊C4—122no—
360C22⋊C4—242no—
361C22⋊C4—241no—
362C22⋊C4—241no—
363C22⋊C4—241no—
364C42—242no—
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—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—44no—
385C22×C3—44no—
386C22×C3—44no—
387C22×C3—42no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3#38611yescharacteristic
401C22×C3#41012yes—
402C22×C3B(4,𝔽2)12yes—
403C3×C4—42no—
404C3×C4—82no—
405C3×C4—161no—
406Dic3—42no—
407Dic3—42no—
408Dic3—82no—
409Dic3—82no—
410Dic3—161no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—88no—
446C23—88no—
447C23D3×C22⋊C411yescharacteristic
448C23#67⋊C411yescharacteristic
449C23#88311yescharacteristic
450C23#88411yescharacteristic
451C23C4×#6711yescharacteristic
452C23#93711yescharacteristic
453C23#93811yescharacteristic
454D4—44no—
455D4—44no—
456D4—44no—
457D4—44no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—241no—
484C2×C4—241no—
485C2×C4—241no—
486C2×C4—241no—
487C2×C4—241no—
488C2×C3—22no—
489C2×C3—22no—
490C2×C3—41no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—82no—
496C2×C3#188911yescharacteristic
497C2×C3#189911yescharacteristic
498C2×C3#191311yescharacteristic
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—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—42no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—81no—
533C22—81no—
534C22#474012yes—
535C22#495611yescharacteristic
536C22#503212yes—
537C4—42no—
538C4—82no—
539C4—122no—
540C4—122no—
541C4—161no—
542C4—242no—
543C4—242no—
544C4—481no—
545C3#949611yescharacteristic, complemented, Sylow
546C2—22no—
547C2—22no—
548C2—41no—
549C2—41no—
550C2—41no—
551C2—41no—
552C2—41no—
553C2—82no—
554C2#6949911yescharacteristic
555C2#7002911yescharacteristic
556C2#7003411yescharacteristic
subgroupG/H
centerC22#4956
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC3×#1781C2
socleC22×C3#386
prime radicalC23×C3×D4C22
cosocleC25×C3C23
hypercenter#1781D3
hypocenterC3#9496
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. #73820C2semidirect
5. #69495C2semidirect
6. C3⋊#1304C2semidirect
7. C3⋊#1304C2semidirect
15. #5247C4semidirect
22. #5210C4semidirect
545. C3#9496semidirect

Maximal subgroups

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

subgroupindex
#840472
#738202
C3×#17812
#721302
#694952
C3⋊#13042
#94963