Finite Group Databasebeta

← #10817106

Subgroups — #10817106

Order 768.

834 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2349C211yesmaximal, characteristic
2C2×#5009C211yesmaximal, characteristic
3#73659C211yesmaximal, characteristic, complemented
4#71609C211yesmaximal, characteristic
5#71609C211yesmaximal, characteristic
6C4×#832C211yesmaximal, characteristic
7#71607C211yesmaximal, characteristic
8#71606C211yesmaximal, characteristic
9#69621C211yesmaximal, characteristic, complemented
10#69620C211yesmaximal, characteristic, complemented
11#69601C211yesmaximal, characteristic, complemented
12#69600C211yesmaximal, characteristic, complemented
13#69554C211yesmaximal, characteristic, complemented
14#69540C211yesmaximal, characteristic, complemented
15#69477C211yesmaximal, characteristic, complemented
16#22972—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#870C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#385—21no—
24C3×#385—21no—
25C3×#382C2211yescharacteristic
26C3×#380—21no—
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×C3×#95—21no—
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5009C418yescomplemented
34#4978—21no—
35#4976—21no—
36C2×#808C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C2×#807C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44#4928C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46C2×#803C2211yescharacteristic
47C2×#803C2211yescharacteristic
48C2×C4×C3⋊C8C2211yescharacteristic
49#4652—21no—
50#4649—21no—
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×#478—31no—
62#2370—31no—
63#2349D311yescharacteristic
64#1991—31no—
65#1949—31no—
66#1944—31no—
67#1943—31no—
68#1919—31no—
69#1918—31no—
70#1864—31no—
71#1862—31no—
72#1770—31no—
73#1762—31no—
74#1762—31no—
75C4×#102—31no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4—21no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—41no—
84C3×C4×D4—41no—
85C3×C4×D4C2×C414yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#870—22no—
115#870C2×C414yescomplemented
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95D411yescharacteristic
121C3×#95D411yescharacteristic
122#832—22no—
123#832C2×C414yes—
124#832C2×C414yes—
125#832Q814yes—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#808—41no—
132#808C2×C414yescomplemented
133#807—41no—
134#807C2×C414yescomplemented
135C4⋊(C3⋊C8)C2×C414yescomplemented
136#803—41no—
137#803—41no—
138C4×C3⋊C8—22no—
139C2×#125—31no—
140C2×C4×D4C2×D311yescharacteristic
141C2×C4×D4C2×D311yescharacteristic
142C2×C4×D4C2×D311yescharacteristic
143#478—38no—
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#97—31no—
154C2×#97—31no—
155C2×C4×C8—31no—
156#389—31no—
157#389—31no—
158#389—61no—
159#389C2×D311yescharacteristic
160#389C2×D311yescharacteristic
161#387—61no—
162#387—61no—
163#385—21no—
164#385—21no—
165#382C2×D311yescharacteristic
166#380—21no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C4⋊C4—61no—
171C4×C4⋊C4C2×D311yescharacteristic
172C2×#95—21no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183C24×C3—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4C22×C412yes—
188C2×C3×D4C2×Q812yes—
189C2×C3×D4#4112yes—
190C22×C3×C4—21no—
191C22×C3×C4—24no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4—81no—
199C22×C3×C4—81no—
200C22×C3×C4—81no—
201C22×C3×C4—81no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—41no—
219C3×C4⋊C4—41no—
220C3×C4⋊C4—81no—
221C3×C4⋊C4—81no—
222C3×C4⋊C4C22×C412yes—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—42no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42C22×C412yes—
231C3×C42#4112yes—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3C22×C412yes—
238C4⋊Dic3C22×C412yes—
239C4⋊Dic3C2×Q812yes—
240Dic3⋊C4—24no—
241Dic3⋊C4—24no—
242Dic3⋊C4—42no—
243Dic3⋊C4—81no—
244C4×Dic3—42no—
245C4×Dic3—81no—
246C4×Dic3—81no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8C22×C412yes—
257C2×C3⋊C8C22×C412yes—
258C2×C3⋊C8C2×Q812yes—
259C22×D4C22×D311yescharacteristic
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267#125—34no—
268#125—62no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4C4×D314yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4#6711yescharacteristic
286C2×C4⋊C4#6711yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C42—61no—
294C2×C42—61no—
295C2×C42—61no—
296C2×C42C22×D311yescharacteristic
297C2×C42C22×D311yescharacteristic
298C2×C42C22×D311yescharacteristic
299#107—34no—
300#107—121no—
301#106—34no—
302#106—121no—
303C4⋊C8—34no—
304#102—34no—
305#102—34no—
306#102—34no—
307#102—62no—
308#97—121no—
309#97—121no—
310C4×C8—62no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95#6711yescharacteristic
321#95#6711yescharacteristic
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C22×D411yescharacteristic
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4C4×Q814yes—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—84no—
360C2×C3×C4—84no—
361C2×C3×C4—84no—
362C2×C3×C4C4×D412yes—
363C2×C3×C4C4×D412yes—
364C2×C3×C4C23×C411yescharacteristic
365C2×C3×C4C22×Q811yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C42−1+411yescharacteristic
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C3⋊C8—22no—
383C3⋊C8—22no—
384C3⋊C8—22no—
385C3⋊C8—41no—
386C24—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4C2×C4×D312yes—
391C2×D4#23712yes—
392C2×D4D3×Q812yes—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4C2×#6711yescharacteristic
410C22×C4C2×#6711yescharacteristic
411C22×C4C2×#6711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6711yescharacteristic
415C22×C4C23×D311yescharacteristic
416C2×C8—32no—
417C2×C8—32no—
418C2×C8—32no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—32no—
434C4⋊C4—32no—
435C4⋊C4—32no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—64no—
439C4⋊C4—64no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—122no—
448C4⋊C4—241no—
449C4⋊C4C2×C4×D312yes—
450C22⋊C4—24no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—122no—
458C42—241no—
459C42—241no—
460C42C2×C4×D312yes—
461C42#23512yes—
462C22×C3—24no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3C2×C4×D411yescharacteristic
470C22×C3#54611yescharacteristic
471C22×C3D4×Q811yescharacteristic
472C22×C3#57511yescharacteristic
473C22×C3#57511yescharacteristic
474C22×C3#57611yescharacteristic
475C22×C3#57711yescharacteristic
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—42no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4C2×C4×Q811yescharacteristic
486C3×C4#52011yescharacteristic
487C3×C4#55211yescharacteristic
488C3×C4#55411yescharacteristic
489Dic3—42no—
490Dic3—42no—
491Dic3—42no—
492Dic3—81no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23C22×#6711yescharacteristic
501D4—22no—
502D4—22no—
503D4—22no—
504D4#88714yes—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4C4×#6712yes—
546C2×C4C4×#6712yes—
547C2×C4C22×C4×D311yescharacteristic
548C2×C4C2×#23511yescharacteristic
549C2×C4C2×#23711yescharacteristic
550C2×C4#100411yescharacteristic
551C2×C4C2×D3×Q811yescharacteristic
552C2×C4#100711yescharacteristic
553C8—62no—
554C8—62no—
555C8—62no—
556C8—121no—
557C2×C3—44no—
558C2×C3#236011yescharacteristic
559C2×C3#294311yescharacteristic
560C2×C3#294611yescharacteristic
561C2×C3#332211yescharacteristic
562C2×C3#332311yescharacteristic
563C2×C3#338811yescharacteristic
564C2×C3#338911yescharacteristic
565C22—24no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22C2×C4×#6711yescharacteristic
573C22#580611yescharacteristic
574C22Q8×#6711yescharacteristic
575C22#582211yescharacteristic
576C22#582211yescharacteristic
577C22#583511yescharacteristic
578C22#583611yescharacteristic
579C4—22no—
580C4—22no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—122no—
589C4—122no—
590C4—122no—
591C4—241no—
592C4C2×#88711yescharacteristic
593C4#550411yescharacteristic
594C4#551111yescharacteristic
595C4#551311yescharacteristic
596C3#2297211yescharacteristic, complemented, Sylow
597C2—44no—
598C2#8216211yescharacteristic
599C2#8250611yescharacteristic
600C2#8253711yescharacteristic
601C2#8350011yescharacteristic
602C2#8350111yescharacteristic
603C2#8363211yescharacteristic
604C2#8363311yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2349C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#2349D3
hypocenterC3#22972
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #73659C2semidirect
9. #69621C2semidirect
10. #69620C2semidirect
11. #69601C2semidirect
12. #69600C2semidirect
13. #69554C2semidirect
14. #69540C2semidirect
15. #69477C2semidirect
33. #5009C4semidirect
115. #870C2×C4semidirect
132. #808C2×C4semidirect
134. #807C2×C4semidirect
135. C4⋊(C3⋊C8)C2×C4semidirect
596. C3#22972semidirect

Maximal subgroups

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

subgroupindex
C3×#23492
C2×#50092
#736592
#716092
C4×#8322
#716072
#716062
#696212
#696202
#696012
#696002
#695542
#695402
#694772
#229723