Finite Group Databasebeta

← #10698068

Subgroups — #10698068

Order 768.

910 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 101 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#82834C211yesmaximal, characteristic
2#82741C211yesmaximal, characteristic, complemented
3#74049C211yesmaximal, characteristic
4C3×#1520C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic
6#69946C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#10324—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×#946—21no—
12C2×#924C2211yescharacteristic
13C2×#897—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16#5228—21no—
17#5212C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×(C2×D3)⋊C8—22no—
20#5051—22no—
21#5050—21no—
22#5044—21no—
23#4989—22no—
24#4987—21no—
25#4987—21no—
26#4970—21no—
27#4955—21no—
28#4955—21no—
29#4950—21no—
30#4946—21no—
31#4668—22no—
32C3×#328—22no—
33#4467—22no—
34#3189—31no—
35#2848—31no—
36#2848—31no—
37#1779—31no—
38#1520—31no—
39#1520D311yescharacteristic, complemented
40#1289—31no—
41C24×D3D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3D411yescharacteristic
45C22×C4×D3—21no—
46C22×C4×D3—21no—
47C22×C4×D3—21no—
48C22×C4×D3—21no—
49C22×C4×D3—21no—
50C22×C4×D3—21no—
51C22×C4×D3—44no—
52C22×Dic6D411yescharacteristic
53C22×C3×C8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×#217—21no—
64C2×#217—21no—
65#946—44no—
66C2×#216—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3D411yescharacteristic
74#924—21no—
75#924—21no—
76#924—21no—
77#924—21no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—21no—
80C22×C3⋊C8D412yes—
81#897—44no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8—21no—
84C3×#95D412yes—
85#835—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95D412yes—
89#830—21no—
90#830—21no—
91(C2×D3)⋊C8—44no—
92#810—22no—
93#807—22no—
94C24×C4—31no—
95C2×C22⋊Q8—31no—
96C2×C22⋊Q8—61no—
97C2×C22⋊Q8—61no—
98C2×C22⋊Q8C2×D311yescharacteristic
99#484—61no—
100#484—61no—
101#467—62no—
102#451—61no—
103#451—61no—
104C2×C22⋊C8—31no—
105C2×C22⋊C8—62no—
106C2×C22⋊C8C2×D311yescharacteristic
107#385—62no—
108#385—61no—
109#385—61no—
110#385—62no—
111#380—31no—
112#380—61no—
113#380—61no—
114#380C2×D311yescharacteristic
115#379—61no—
116#379—61no—
117#328—22no—
118#328—62no—
119C24×C3C2×D411yescharacteristic
120C23×D3—21no—
121C23×D3—21no—
122C23×D3—21no—
123C23×D3—21no—
124C23×D3—21no—
125C23×D3—21no—
126C23×D3—42no—
127C23×D3—42no—
128C23×D3Q1612yescomplemented
129C2×C3×Q8—21no—
130C2×C3×Q8—21no—
131C2×C3×Q8—42no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—42no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—22no—
147C22×Dic3—21no—
148C22×Dic3—42no—
149C2×C4×D3—24no—
150C2×C4×D3—24no—
151C2×C4×D3—24no—
152C2×C4×D3—42no—
153C2×C4×D3—42no—
154C2×C4×D3—42no—
155C2×C4×D3—42no—
156C2×C4×D3—42no—
157C2×C4×D3—42no—
158C2×C4×D3—42no—
159C2×C4×D3—42no—
160C2×C4×D3—42no—
161C2×C4×D3—41no—
162C2×C4×D3—41no—
163C2×C4×D3—41no—
164C2×C4×D3—41no—
165C2×Dic6—21no—
166C2×Dic6—21no—
167C2×Dic6—42no—
168C2×C3×C8—42no—
169C2×C3×C8—41no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—41no—
174C3×C4⋊C4—41no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—41no—
178C3×C22⋊C4—41no—
179#217—41no—
180#217—41no—
181#217—41no—
182#217—41no—
183#216—82no—
184(C2×D3)⋊C4—44no—
185(C2×D3)⋊C4—44no—
186(C2×D3)⋊C4—44no—
187(C2×D3)⋊C4—44no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—44no—
191C4⋊Dic3—42no—
192Dic3⋊C4—44no—
193Dic3⋊C4—41no—
194Dic3⋊C4—41no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—42no—
197C25—31no—
198C22×Q8—31no—
199C22×Q8D1211yescharacteristic
200C23×C4—31no—
201C23×C4—61no—
202C23×C4—61no—
203C23×C4—61no—
204C23×C4—61no—
205C23×C4—61no—
206C23×C4—61no—
207C23×C4—124no—
208C23×C4C22×D311yescharacteristic
209C2×Q16—62no—
210C22×C8—21no—
211C22×C8—32no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—61no—
217C22⋊Q8—61no—
218C22⋊Q8—61no—
219C22⋊Q8—61no—
220C22⋊Q8—124no—
221C22⋊Q8—124no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4D1211yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4—62no—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—61no—
238#107—62no—
239#103—62no—
240#103—62no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245C22⋊C8—124no—
246#95—32no—
247#95—62no—
248#95—62no—
249#95#6712yes—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3#12011yescharacteristic
253C23×C3#12011yescharacteristic
254C23×C3#12111yescharacteristic
255C23×C3C2×Q1611yescharacteristic
256C23×C3#13511yescharacteristic
257C23×C3#13711yescharacteristic
258C23×C3#13711yescharacteristic
259C22×D3—22no—
260C22×D3—22no—
261C22×D3—22no—
262C22×D3—22no—
263C22×D3—22no—
264C22×D3—22no—
265C22×D3—22no—
266C22×D3—22no—
267C22×D3—22no—
268C22×D3—44no—
269C22×D3—44no—
270C22×D3—44no—
271C22×D3—44no—
272C22×D3—41no—
273C22×D3—41no—
274C22×D3—41no—
275C22×D3—41no—
276C22×D3—41no—
277C22×D3—41no—
278C22×D3—41no—
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—41no—
282C3×Q8—42no—
283C3×Q8—81no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4#12011yescharacteristic
298C2×C3×C4#12111yescharacteristic
299C2×C3×C4#12111yescharacteristic
300C2×C3×C4#12711yescharacteristic
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—42no—
315C4×D3—44no—
316C4×D3—44no—
317C4×D3—44no—
318C4×D3—44no—
319Dic6—42no—
320Dic6—81no—
321C3×C8—81no—
322C3⋊C8—42no—
323C24—32no—
324C24—61no—
325C24—61no—
326C24—61no—
327C24—61no—
328C24—61no—
329C24—61no—
330C24—122no—
331C24—122no—
332C24D3×D411yescharacteristic
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—42no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×Q8—122no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—64no—
344C22×C4—64no—
345C22×C4—64no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—62no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—122no—
353C22×C4—122no—
354C22×C4—122no—
355C22×C4—122no—
356C22×C4—122no—
357C22×C4—122no—
358C22×C4—122no—
359C22×C4—122no—
360C22×C4—122no—
361C22×C4—122no—
362C22×C4—121no—
363C22×C4—121no—
364C22×C4—121no—
365C22×C4—121no—
366C22×C4C2×D1211yescharacteristic
367C22×C4#23511yescharacteristic
368C22×C4D3×D411yescharacteristic
369C22×C4D3×D411yescharacteristic
370C22×C4D3×D411yescharacteristic
371C22×C4C2×#6711yescharacteristic
372Q16—242no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—62no—
376C2×C8—122no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C4⋊C4—122no—
385C4⋊C4—124no—
386C4⋊C4—124no—
387C4⋊C4—121no—
388C4⋊C4—121no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—124no—
394C22⋊C4—124no—
395C22⋊C4—124no—
396C22⋊C4—124no—
397C22⋊C4—121no—
398C22⋊C4—121no—
399C22⋊C4—121no—
400C22⋊C4—121no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3#39211yescharacteristic
412C22×C3#45111yescharacteristic
413C22×C3#45111yescharacteristic
414C22×C3#45211yescharacteristic
415C22×C3#45211yescharacteristic
416C22×C3#46711yescharacteristic
417C22×C3#47011yescharacteristic
418C2×D3—24no—
419C2×D3—24no—
420C2×D3—24no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—42no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—82no—
437C3×C4—81no—
438C3×C4—81no—
439C3×C4B(4,𝔽2)12yes—
440Dic3—42no—
441Dic3—82no—
442Dic3—81no—
443Dic3—81no—
444C23—41no—
445C23—41no—
446C23—62no—
447C23—62no—
448C23—62no—
449C23—62no—
450C23—62no—
451C23—62no—
452C23—62no—
453C23—62no—
454C23—62no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—124no—
462C23—124no—
463C23—124no—
464C23—124no—
465C23—121no—
466C23—121no—
467C23—121no—
468C23—121no—
469C23#88211yescharacteristic
470C23#88411yescharacteristic
471C23#91511yescharacteristic
472C23#91611yescharacteristic
473C23D3×Q1611yescharacteristic
474C23#91811yescharacteristic
475C23#93711yescharacteristic
476Q8—42no—
477Q8—81no—
478Q8—122no—
479Q8—241no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—124no—
497C2×C4—124no—
498C2×C4—124no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—124no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—122no—
511C2×C4#88211yescharacteristic
512C2×C4#88411yescharacteristic
513C2×C4#93011yescharacteristic
514C2×C4#94011yescharacteristic
515C8—81no—
516C8—122no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3#200111yescharacteristic
524C2×C3#201011yescharacteristic
525C2×C3#202111yescharacteristic
526D3—44no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—64no—
538C22—64no—
539C22—64no—
540C22—122no—
541C22—122no—
542C22—122no—
543C22—122no—
544C22—122no—
545C22—122no—
546C22—122no—
547C22—122no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—122no—
552C22#480811yescharacteristic
553C22#480911yescharacteristic
554C22#481311yescharacteristic
555C22#481611yescharacteristic
556C22#496011yescharacteristic
557C22#517311yescharacteristic
558C22#518611yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—82no—
563C4—81no—
564C4—81no—
565C4—122no—
566C4—242no—
567C4—241no—
568C4—241no—
569C4#474112yes—
570C3#1032411yescharacteristic, complemented, Sylow
571C2—21no—
572C2—21no—
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2—124no—
578C2#7003711yescharacteristic
579C2#7027311yescharacteristic
580C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1520C2
socleC22×C3#392
prime radicalC24×D3D4
hypercenter#1520D3
hypocenterC3#10324
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
2. #82741C2semidirect
4. C3×#1520C2semidirect
6. #69946C2semidirect
7. #67625C2semidirect
39. #1520D3semidirect
128. C23×D3Q16semidirect
570. C3#10324semidirect

Maximal subgroups

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

subgroupindex
#828342
#827412
#740492
C3×#15202
#699552
#699462
#676252
#103243