Finite Group Databasebeta

← #10817108

Subgroups — #10817108

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
1C2×#4847C211yesmaximal, characteristic
2#73770C211yesmaximal, characteristic
3#73659C211yesmaximal, characteristic, complemented
4C3×#1770C211yesmaximal, characteristic, complemented
5#70455C211yesmaximal, characteristic, complemented
6#69820C211yesmaximal, characteristic, complemented
7#69819C211yesmaximal, characteristic, complemented
8#69812C211yesmaximal, characteristic
9#69812C211yesmaximal, characteristic
10#69778C211yesmaximal, characteristic, complemented
11C4×#821C211yesmaximal, characteristic
12#68671C211yesmaximal, characteristic
13#68670C211yesmaximal, characteristic
14#68622C211yesmaximal, characteristic, complemented
15#68621C211yesmaximal, characteristic, complemented
16#22972—31nomaximal, Sylow
17C2×C4×D12C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×#870C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#97C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#817C2211yescharacteristic
34#4989—21no—
35#4989—21no—
36#4978—21no—
37#4976—21no—
38#4936C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4847C418yescomplemented
47#4667—21no—
48D3×#95—21no—
49#4652—21no—
50#4649—21no—
51C3×#336C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61C2×#478—31no—
62#2370—31no—
63#2349—31no—
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#1770D311yescharacteristic, complemented
73#1762—31no—
74#1762—31no—
75C4×#102—31no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C4×D12—22no—
109C4×D12—22no—
110C4×D12—22no—
111C4×D12—41no—
112C4×D12—41no—
113C4×D12C2×C414yes—
114#870—22no—
115#870C2×C414yescomplemented
116C3×C4⋊C8C2×C414yescomplemented
117C3×#97—41no—
118C3×#97—41no—
119C3×C4×C8—22no—
120C3×#95D411yescharacteristic
121C3×#95D411yescharacteristic
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#821—22no—
132#821C2×C414yes—
133#821C2×C414yes—
134#821Q814yes—
135#818—41no—
136#818C2×C414yescomplemented
137#817—41no—
138#817C2×C414yescomplemented
139C2×#125—31no—
140C2×C4×D4—31no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143#478—38no—
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8C2×D311yescharacteristic
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#97C2×D311yescharacteristic
154C2×#97C2×D311yescharacteristic
155C2×C4×C8C2×D311yescharacteristic
156#389—31no—
157#389—31no—
158#389—31no—
159#389—31no—
160#389—61no—
161#387—61no—
162#387—61no—
163#385—61no—
164#385—61no—
165#382—31no—
166#380—61no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C4⋊C4—61no—
171C4×C4⋊C4C2×D311yescharacteristic
172C2×#95—61no—
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#336C2×D311yescharacteristic
182#336C2×D311yescharacteristic
183C23×D3—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12C22×C412yes—
202C2×D12C2×Q812yes—
203C2×D12#4112yes—
204C2×C4×D3—24no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×C4×D3—81no—
210C2×C4×D3—81no—
211C2×C4×D3—81no—
212C2×C4×D3—81no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8C22×C412yes—
223C2×C3×C8C22×C412yes—
224C2×C3×C8C2×Q812yes—
225C3×C4⋊C4—41no—
226C3×C4⋊C4—41no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42C22×C412yes—
232C3×C42#4112yes—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3C22×C412yes—
247C4⋊Dic3C22×C412yes—
248C4⋊Dic3C22×C412yes—
249C4⋊Dic3C2×Q812yes—
250Dic3⋊C4—24no—
251Dic3⋊C4—24no—
252Dic3⋊C4—42no—
253Dic3⋊C4—81no—
254Dic3⋊C4—81no—
255Dic3⋊C4—81no—
256C4×Dic3—42no—
257C4×Dic3—81no—
258C4×Dic3—81no—
259C22×D4—31no—
260C23×C4—61no—
261C23×C4—61no—
262C23×C4—61no—
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#125—34no—
268#125—62no—
269C4×D4—34no—
270C4×D4—62no—
271C4×D4—62no—
272C4×D4—62no—
273C4×D4—121no—
274C4×D4—121no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4D1211yescharacteristic
289C2×C4⋊C4D1211yescharacteristic
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
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⋊C8C4×D314yescomplemented
304#102—34no—
305#102—34no—
306#102—34no—
307#102—62no—
308#97—41no—
309#97—41no—
310C4×C8—22no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95D1211yescharacteristic
321#95D1211yescharacteristic
322C23×C3C22×D411yescharacteristic
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
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×C4C4×D412yes—
347C2×C3×C4C4×D412yes—
348C2×C3×C4C23×C411yescharacteristic
349C2×C3×C4C22×Q811yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4H(5,𝔽2)11yescharacteristic
353C2×C3×C42−1+411yescharacteristic
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375D12—22no—
376D12—22no—
377D12—22no—
378D12C4×Q814yes—
379C4×D3—84no—
380C4×D3—84no—
381C4×D3—84no—
382C3×C8—22no—
383C3×C8—22no—
384C3×C8—22no—
385C3×C8—41no—
386C24—61no—
387C2×D4—32no—
388C2×D4—32no—
389C2×D4—32no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—64no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—122no—
402C22×C4—122no—
403C22×C4—122no—
404C22×C4—122no—
405C22×C4—241no—
406C22×C4—241no—
407C22×C4—241no—
408C22×C4—241no—
409C22×C4C2×D1211yescharacteristic
410C22×C4C2×D1211yescharacteristic
411C22×C4C2×D1211yescharacteristic
412C22×C4C2×D1211yescharacteristic
413C22×C4C2×D1211yescharacteristic
414C22×C4C2×D1211yescharacteristic
415C22×C4C23×D311yescharacteristic
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8C2×C4×D312yes—
426C2×C8C2×C4×D312yes—
427C2×C8D3×Q812yes—
428C4⋊C4—32no—
429C4⋊C4—32no—
430C4⋊C4—32no—
431C4⋊C4—32no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—64no—
435C4⋊C4—64no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—122no—
447C4⋊C4—241no—
448C4⋊C4—241no—
449C4⋊C4—241no—
450C22⋊C4—64no—
451C22⋊C4—122no—
452C22⋊C4—122no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—122no—
458C42—241no—
459C42—241no—
460C42C2×C4×D312yes—
461C42#23512yes—
462C22×C3C2×C4×D411yescharacteristic
463C22×C3#54611yescharacteristic
464C22×C3D4×Q811yescharacteristic
465C22×C3#57511yescharacteristic
466C22×C3#57511yescharacteristic
467C22×C3#57611yescharacteristic
468C22×C3#57711yescharacteristic
469C2×D3—24no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—42no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4C2×C4×Q811yescharacteristic
483C3×C4#52011yescharacteristic
484C3×C4#55211yescharacteristic
485C3×C4#55411yescharacteristic
486Dic3—42no—
487Dic3—42no—
488Dic3—42no—
489Dic3—42no—
490Dic3—81no—
491Dic3—81no—
492Dic3—81no—
493C23—62no—
494C23—62no—
495C23—62no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23C22×D1211yescharacteristic
501D4—34no—
502D4—62no—
503D4—62no—
504D4—62no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
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—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—244no—
543C2×C4—244no—
544C2×C4—244no—
545C2×C4C4×D1212yes—
546C2×C4C4×D1212yes—
547C2×C4C22×C4×D311yescharacteristic
548C2×C4C2×#23511yescharacteristic
549C2×C4#100411yescharacteristic
550C2×C4C2×D3×Q811yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4#101011yescharacteristic
553C8—22no—
554C8—22no—
555C8—22no—
556C8—41no—
557C2×C3#236011yescharacteristic
558C2×C3#294311yescharacteristic
559C2×C3#294611yescharacteristic
560C2×C3#332211yescharacteristic
561C2×C3#332311yescharacteristic
562C2×C3#338811yescharacteristic
563C2×C3#338911yescharacteristic
564D3—44no—
565C22—64no—
566C22—122no—
567C22—122no—
568C22—122no—
569C22—122no—
570C22—122no—
571C22—122no—
572C22C2×C4×D1211yescharacteristic
573C22#555411yescharacteristic
574C22Q8×D1211yescharacteristic
575C22#574811yescharacteristic
576C22#574811yescharacteristic
577C22#575111yescharacteristic
578C22#575211yescharacteristic
579C4—22no—
580C4—22no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—122no—
586C4—122no—
587C4—122no—
588C4—122no—
589C4—241no—
590C4—241no—
591C4—241no—
592C4#551111yescharacteristic
593C4#552911yescharacteristic
594C4#554611yescharacteristic
595C4C4×D3×Q811yescharacteristic
596C3#2297211yescharacteristic, complemented, Sylow
597C2—124no—
598C2#7377311yescharacteristic
599C2#7849411yescharacteristic
600C2#7856011yescharacteristic
601C2#7955311yescharacteristic
602C2#7955411yescharacteristic
603C2#7997811yescharacteristic
604C2#7997911yescharacteristic
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1770C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1770D3
hypocenterC3#22972
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1770

Complements

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

NH ≅ G/Ntype
3. #73659C2semidirect
4. C3×#1770C2semidirect
5. #70455C2semidirect
6. #69820C2semidirect
7. #69819C2semidirect
10. #69778C2semidirect
14. #68622C2semidirect
15. #68621C2semidirect
46. #4847C4semidirect
72. #1770D3semidirect
115. #870C2×C4semidirect
116. C3×C4⋊C8C2×C4semidirect
136. #818C2×C4semidirect
138. #817C2×C4semidirect
303. C4⋊C8C4×D3semidirect
596. C3#22972semidirect

Maximal subgroups

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

subgroupindex
C2×#48472
#737702
#736592
C3×#17702
#704552
#698202
#698192
#698122
#697782
C4×#8212
#686712
#686702
#686222
#686212
#229723