Finite Group Databasebeta

← #10698067

Subgroups — #10698067

Order 768.

910 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C3×#2848C211yesmaximal, characteristic
2#84031C211yesmaximal, characteristic
3#82741C211yesmaximal, characteristic, complemented
4#72126C211yesmaximal, characteristic
5#71690C211yesmaximal, characteristic, complemented
6#69510C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#10324—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C2×C3×C22⋊Q8—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#924—21no—
13C2×#924C2211yescharacteristic
14C3×#385—21no—
15C3×#385—22no—
16C3×#380—21no—
17C3×#380C2211yescharacteristic
18C3×#379—21no—
19#5247—21no—
20#5228—21no—
21#5212—21no—
22#5212C2211yescharacteristic
23#5210—21no—
24#5210—22no—
25C2×#830—22no—
26C2×#830C2211yescharacteristic
27C2×#830C2211yescharacteristic
28#5051—22no—
29#5050—21no—
30#5044—21no—
31#4970—21no—
32#4538—22no—
33#4452—22no—
34#3189—31no—
35#2848—31no—
36#2848D311yescharacteristic
37#1779—31no—
38#1520—31no—
39#1520—31no—
40#1289—31no—
41C25×C3D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C23×C3×C4—21no—
44C23×C3×C4—21no—
45C23×C3×C4—21no—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4—21no—
49C23×C3×C4—44no—
50C23×C3×C4D411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C22×Dic6D411yescharacteristic
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—44no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×#217—22no—
67C2×#217—22no—
68C2×#217—21no—
69C2×#217—21no—
70C2×#216—22no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3D411yescharacteristic
73#924—21no—
74#924—21no—
75#924—21no—
76#924—21no—
77#924—44no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—21no—
80C22×C3⋊C8—21no—
81C22×C3⋊C8D412yes—
82C3×#95—22no—
83C3×#95D412yes—
84#835—22no—
85C3⋊#95—22no—
86C3⋊#95D412yes—
87#830—21no—
88#830—21no—
89#830—21no—
90#830—21no—
91#830—44no—
92#810—22no—
93#807—22no—
94C24×C4C2×D311yescharacteristic
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—31no—
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—31no—
106C2×C22⋊C8—62no—
107#385—21no—
108#385—22no—
109#385—62no—
110#385—61no—
111#380—21no—
112#380—31no—
113#380—61no—
114#380C2×D311yescharacteristic
115#379—21no—
116#379—61no—
117#328—62no—
118#328—62no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—21no—
125C24×C3—42no—
126C24×C3—42no—
127C24×C3Q1612yescomplemented
128C24×C3C2×D411yescharacteristic
129C2×C3×Q8—21no—
130C2×C3×Q8—21no—
131C2×C3×Q8—42no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4—42no—
144C22×C3×C4—42no—
145C22×C3×C4—42no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4#4111yescharacteristic
162C22×Dic3—21no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C2×Dic6—21no—
166C2×Dic6—21no—
167C2×Dic6—42no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—44no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—41no—
174C3×C22⋊C4—44no—
175C3×C22⋊C4—44no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—41no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—41no—
180#217—44no—
181#217—44no—
182#217—41no—
183#217—41no—
184#217—41no—
185#217—41no—
186#216—82no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—42no—
190Dic3⋊C4—44no—
191Dic3⋊C4—41no—
192Dic3⋊C4—41no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—42no—
195C2×C3⋊C8—42no—
196C2×C3⋊C8—41no—
197C25#6711yescharacteristic
198C22×Q8—31no—
199C22×Q8#6711yescharacteristic
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—44no—
207C23×C4#6711yescharacteristic
208C23×C4C22×D311yescharacteristic
209C2×Q16—62no—
210C22×C8—32no—
211C22×C8—61no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—44no—
217C22⋊Q8—61no—
218C22⋊Q8—61no—
219C22⋊Q8—61no—
220C22⋊Q8—61no—
221C22⋊Q8—124no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4#6711yescharacteristic
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
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—61no—
242C22⋊C8—61no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245C22⋊C8—124no—
246#95—22no—
247#95—32no—
248#95—62no—
249#95#6712yes—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—41no—
260C23×C3—44no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—44no—
269C23×C3—44no—
270C23×C3—44no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3#12011yescharacteristic
276C23×C3#12011yescharacteristic
277C23×C3#12111yescharacteristic
278C23×C3C2×Q1611yescharacteristic
279C23×C3#13511yescharacteristic
280C23×C3#13711yescharacteristic
281C23×C3#13711yescharacteristic
282C3×Q8—42no—
283C3×Q8—81no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—44no—
294C2×C3×C4—44no—
295C2×C3×C4—44no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—44no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4#12111yescharacteristic
310C2×C3×C4#12111yescharacteristic
311C2×C3×C4#12711yescharacteristic
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—42no—
319Dic6—42no—
320Dic6—81no—
321C3⋊C8—42no—
322C3⋊C8—81no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—42no—
330C24—42no—
331C24#21612yescomplemented
332C24C2×#6711yescharacteristic
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—42no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×Q8—122no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—62no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4D3×D411yescharacteristic
367C22×C4D3×D411yescharacteristic
368C22×C4D3×D411yescharacteristic
369C22×C4#23711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372Q16—242no—
373C2×C8—62no—
374C2×C8—122no—
375C2×C8—122no—
376C2×C8—121no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—61no—
384C4⋊C4—61no—
385C4⋊C4—124no—
386C4⋊C4—122no—
387C4⋊C4—121no—
388C4⋊C4—121no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
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—24no—
408C22×C3—24no—
409C22×C3—24no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3#39211yescharacteristic
427C22×C3#45111yescharacteristic
428C22×C3#45111yescharacteristic
429C22×C3#45211yescharacteristic
430C22×C3#45211yescharacteristic
431C22×C3#46711yescharacteristic
432C22×C3#47011yescharacteristic
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—42no—
437C3×C4—82no—
438C3×C4—81no—
439C3×C4—81no—
440C3×C4B(4,𝔽2)12yes—
441Dic3—82no—
442Dic3—81no—
443Dic3—81no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—41no—
454C23—44no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23#93711yescharacteristic
470C23#93911yescharacteristic
471C23#94211yescharacteristic
472C23C2×#21611yescharacteristic
473C23#95011yescharacteristic
474C23#95111yescharacteristic
475C23#95311yescharacteristic
476Q8—42no—
477Q8—81no—
478Q8—122no—
479Q8—241no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—122no—
511C2×C4#93711yescharacteristic
512C2×C4#93811yescharacteristic
513C2×C4#93911yescharacteristic
514C2×C4#94011yescharacteristic
515C8—122no—
516C8—241no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—44no—
524C2×C3#200111yescharacteristic
525C2×C3#201011yescharacteristic
526C2×C3#202111yescharacteristic
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—24no—
534C22—24no—
535C22—24no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22#504711yescharacteristic
553C22#505011yescharacteristic
554C22#505111yescharacteristic
555C22#505211yescharacteristic
556C22#522311yescharacteristic
557C22#522811yescharacteristic
558C22#523911yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—42no—
563C4—82no—
564C4—81no—
565C4—81no—
566C4—242no—
567C4—241no—
568C4—241no—
569C4#503212yes—
570C3#1032411yescharacteristic, complemented, Sylow
571C2—21no—
572C2—21no—
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2—44no—
578C2#7167111yescharacteristic
579C2#7170011yescharacteristic
580C2#7171511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2848C2
socleC22×C3#392
prime radicalC25×C3D4
hypercenter#2848D3
hypocenterC3#10324
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
3. #82741C2semidirect
5. #71690C2semidirect
6. #69510C2semidirect
7. #67625C2semidirect
127. C24×C3Q16semidirect
331. C24#216semidirect
570. C3#10324semidirect

Maximal subgroups

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

subgroupindex
C3×#28482
#840312
#827412
#721262
#716902
#695102
#676252
#103243