Finite Group Databasebeta

← #10817110

Subgroups — #10817110

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×C3×#478C211yesmaximal, characteristic
2#83025C211yesmaximal, characteristic, complemented
3#82333C211yesmaximal, characteristic
4#71740C211yesmaximal, characteristic, complemented
5C4×#832C211yesmaximal, characteristic
6#70613C211yesmaximal, characteristic, complemented
7#70609C211yesmaximal, characteristic, complemented
8#70562C211yesmaximal, characteristic, complemented
9#70557C211yesmaximal, characteristic, complemented
10#70466C211yesmaximal, characteristic, complemented
11#70455C211yesmaximal, characteristic, complemented
12#70170C211yesmaximal, characteristic
13#70168C211yesmaximal, characteristic
14#70152C211yesmaximal, characteristic
15#70152C211yesmaximal, characteristic
16#22972—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C3×#478C418yescomplemented
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29#5212—21no—
30C2×C3⋊#95—21no—
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×#815C2211yescharacteristic
35#4984—21no—
36#4984C2211yescharacteristic
37#4984C2211yescharacteristic
38#4984C2211yescharacteristic
39#4984C2211yescharacteristic
40#4980—21no—
41#4980—21no—
42Dic3×C4⋊C4—21no—
43Dic3×C4⋊C4C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45#4950—21no—
46#4950—21no—
47#4936C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50C2×C4×C3⋊C8C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#478D311yescharacteristic
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#1770—31no—
73#1762—31no—
74#1762—31no—
75C4×#102—31no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×Dic3—21no—
79C23×Dic3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#125—22no—
83C3×#125C2×C414yescomplemented
84C3×C4×D4—22no—
85C3×C4×D4C2×C414yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95D4×Dic3—22no—
96D4×Dic3—22no—
97D4×Dic3—41no—
98D4×Dic3—41no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C3×#107C2×C414yescomplemented
117C3×#106C2×C414yescomplemented
118C3×C4⋊C8C2×C414yescomplemented
119C3×#102C2×C414yes—
120C3×#102C2×C414yes—
121#832—22no—
122#832Q814yes—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134#818—41no—
135#817—41no—
136#815—41no—
137#815—41no—
138C4×C3⋊C8—22no—
139C2×#125C2×D311yescharacteristic
140C2×C4×D4—31no—
141C2×C4×D4—31no—
142C2×C4×D4C2×D311yescharacteristic
143#478Dic318yescomplemented
144C2×#107—31no—
145C2×#107C2×D311yescharacteristic
146C2×#106—31no—
147C2×#106C2×D311yescharacteristic
148C2×C4⋊C8C2×D311yescharacteristic
149C2×#102—31no—
150C2×#102—31no—
151C2×#102C2×D311yescharacteristic
152C2×#102C2×D311yescharacteristic
153C2×#97—31no—
154C2×#97—31no—
155C2×C4×C8—31no—
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—31no—
171C4×C4⋊C4—61no—
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#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—21no—
192C22×C3×C4—21no—
193C22×C3×C4—24no—
194C22×C3×C4—81no—
195C22×C3×C4—81no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—42no—
204C22×Dic3—42no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C22×Dic3—81no—
208C22×Dic3—81no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8C22×C412yes—
218C2×C3×C8C22×C412yes—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4C22×C412yes—
225C3×C4⋊C4C22×C412yes—
226C3×C4⋊C4C22×C412yes—
227C3×C22⋊C4—24no—
228C3×C42C22×C412yes—
229C3×C42#4112yes—
230#217—42no—
231#217—42no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—41no—
240C4⋊Dic3—41no—
241C4⋊Dic3—42no—
242C4⋊Dic3—81no—
243C4⋊Dic3—81no—
244C4⋊Dic3—81no—
245C4⋊Dic3C2×Q812yes—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—42no—
251C4×Dic3—81no—
252C4×Dic3—81no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8C2×Q812yes—
259C22×D4C22×D311yescharacteristic
260C23×C4—21no—
261C23×C4—61no—
262C23×C4—61no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#125—22no—
268#125C2×Dic314yescomplemented
269C4×D4—22no—
270C4×D4—62no—
271C4×D4—62no—
272C4×D4—121no—
273C4×D4—121no—
274C4×D4C2×Dic314yes—
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—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C42—31no—
294C2×C42—31no—
295C2×C42—61no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#107—121no—
300#107C2×Dic314yescomplemented
301#106—121no—
302#106C2×Dic314yescomplemented
303C4⋊C8C2×Dic314yescomplemented
304#102—34no—
305#102—62no—
306#102C2×Dic314yes—
307#102C2×Dic314yes—
308#97—121no—
309#97—121no—
310C4×C8—62no—
311#95—31no—
312#95—31no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
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—22no—
340C2×C3×C4—22no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—84no—
348C2×C3×C4C4×D412yes—
349C2×C3×C4C4×D412yes—
350C2×C3×C4C23×C411yescharacteristic
351C2×C3×C4C22×Q811yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4H(5,𝔽2)11yescharacteristic
355C2×C3×C42−1+411yescharacteristic
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—84no—
381C2×Dic3—84no—
382C3×C8—22no—
383C3×C8—22no—
384C3⋊C8—22no—
385C3⋊C8—41no—
386C24—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4D3×Q812yes—
391C2×D4#23912yes—
392C2×D4C22×Dic312yes—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—122no—
408C22×C4—122no—
409C22×C4—122no—
410C22×C4—122no—
411C22×C4—241no—
412C22×C4—241no—
413C22×C4D3×D411yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4C23×D311yescharacteristic
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—32no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8C22×Dic312yes—
427C2×C8C22×Dic312yes—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—32no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—121no—
442C4⋊C4—121no—
443C4⋊C4—122no—
444C4⋊C4—241no—
445C4⋊C4—241no—
446C4⋊C4—241no—
447C4⋊C4C22×Dic312yes—
448C4⋊C4C22×Dic312yes—
449C4⋊C4C22×Dic312yes—
450C22⋊C4—24no—
451C22⋊C4—122no—
452C22⋊C4—122no—
453C42—61no—
454C42—61no—
455C42—61no—
456C42—61no—
457C42—122no—
458C42—241no—
459C42—241no—
460C42#23712yes—
461C42C22×Dic312yes—
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—42no—
480C3×C4—42no—
481C3×C4C2×C4×Q811yescharacteristic
482C3×C4#52011yescharacteristic
483C3×C4#55211yescharacteristic
484C3×C4#55411yescharacteristic
485Dic3—42no—
486Dic3—41no—
487Dic3—41no—
488Dic3—41no—
489Dic3—41no—
490Dic3—81no—
491Dic3—81no—
492Dic3—81no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23C2×D3×D411yescharacteristic
501D4—22no—
502D4—22no—
503D4—22no—
504D4Q8×Dic314yes—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—62no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—84no—
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—121no—
543C2×C4—244no—
544C2×C4—244no—
545C2×C4D4×Dic312yes—
546C2×C4D4×Dic312yes—
547C2×C4C2×#23711yescharacteristic
548C2×C4C2×D3×Q811yescharacteristic
549C2×C4C2×#23911yescharacteristic
550C2×C4#100911yescharacteristic
551C2×C4#101011yescharacteristic
552C2×C4C23×Dic311yescharacteristic
553C8—22no—
554C8—22no—
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—
572C22#563011yescharacteristic
573C22#563211yescharacteristic
574C22#576811yescharacteristic
575C22#576811yescharacteristic
576C22#577411yescharacteristic
577C22#577511yescharacteristic
578C22C2×D4×Dic311yescharacteristic
579C4—22no—
580C4—22no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—122no—
585C4—121no—
586C4—121no—
587C4—121no—
588C4—121no—
589C4—241no—
590C4—241no—
591C4—241no—
592C4#563511yescharacteristic
593C4#564211yescharacteristic
594C4C2×Q8×Dic311yescharacteristic
595C4#582711yescharacteristic
596C3#2297211yescharacteristic, complemented, Sylow
597C2—44no—
598C2#8116011yescharacteristic
599C2#8116211yescharacteristic
600C2#8119211yescharacteristic
601C2#8119311yescharacteristic
602C2#8274711yescharacteristic
603C2#8389711yescharacteristic
604C2#8393011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#478C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenterC2×#478D3
hypocenterC3#22972
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
2. #83025C2semidirect
4. #71740C2semidirect
6. #70613C2semidirect
7. #70609C2semidirect
8. #70562C2semidirect
9. #70557C2semidirect
10. #70466C2semidirect
11. #70455C2semidirect
21. C3×#478C4semidirect
83. C3×#125C2×C4semidirect
116. C3×#107C2×C4semidirect
117. C3×#106C2×C4semidirect
118. C3×C4⋊C8C2×C4semidirect
143. #478Dic3semidirect
268. #125C2×Dic3semidirect
300. #107C2×Dic3semidirect
302. #106C2×Dic3semidirect
303. C4⋊C8C2×Dic3semidirect
596. C3#22972semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4782
#830252
#823332
#717402
C4×#8322
#706132
#706092
#705622
#705572
#704662
#704552
#701702
#701682
#701522
#229723