Finite Group Databasebeta

← #10875981

Subgroups — #10875981

Order 768.

704 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
1Q8×Dic3⋊C4C211yesmaximal, characteristic, complemented
2#74172C211yesmaximal, characteristic
3C3×#1805C211yesmaximal, characteristic, complemented
4#71633C211yesmaximal, characteristic
5#70484C211yesmaximal, characteristic
6#70448C211yesmaximal, characteristic, complemented
7#70240C211yesmaximal, characteristic
8#70240C211yesmaximal, characteristic
9#70236C211yesmaximal, characteristic, complemented
10#70220C211yesmaximal, characteristic, complemented
11#69542C211yesmaximal, characteristic, complemented
12#68902C211yesmaximal, characteristic
13#68856C211yesmaximal, characteristic
14#68843C211yesmaximal, characteristic, complemented
15#68814C211yesmaximal, characteristic, complemented
16#27572—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C2×#835C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#815C2211yescharacteristic
30#4988C2211yescharacteristic
31#4988C2211yescharacteristic
32#4984C2211yescharacteristic
33#4975—21no—
34#4975C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4972—21no—
37#4972C2211yescharacteristic
38C2×#809C2211yescharacteristic
39#4939C2211yescharacteristic
40C4×Dic3⋊C4—21no—
41C4×Dic3⋊C4C2211yescharacteristic
42#4928—21no—
43#4928C2211yescharacteristic
44#4675—21no—
45#4674—21no—
46#4670—21no—
47#4669—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4550C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61Q8×C4⋊C4—31no—
62#2101—31no—
63#2100—31no—
64#2071—31no—
65#2067—31no—
66#1936—31no—
67#1918—31no—
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1863—31no—
73#1805D311yescharacteristic, complemented
74#1763—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C4×Q8—22no—
80C3×C4×Q8—41no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87Q8×Dic3—22no—
88Q8×Dic3—22no—
89Q8×Dic3—22no—
90Q8×Dic3—41no—
91C2×#215—22no—
92C2×#215—22no—
93C2×#215—22no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4D411yescharacteristic
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C3×C4⋊C8—41no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#95—21no—
123C3×#95—21no—
124#835—22no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#821—22no—
132#821—41no—
133#818—22no—
134#815—41no—
135#809—41no—
136C22×SD16—31no—
137C2×C4×Q8—31no—
138C2×C4×Q8—31no—
139C2×C4×Q8C2×D311yescharacteristic
140C2×#107—31no—
141C2×C4⋊C8C2×D311yescharacteristic
142C2×#103—31no—
143C2×#103C2×D311yescharacteristic
144C2×#103C2×D311yescharacteristic
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#97—31no—
149#397—61no—
150#397—61no—
151#390—31no—
152#390—31no—
153#390—31no—
154#389—31no—
155#389—31no—
156#389—31no—
157#389—61no—
158#389—61no—
159#388—61no—
160#388—61no—
161#384—31no—
162#384—61no—
163#382C2×D311yescharacteristic
164C4×C4⋊C4—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179C23×D3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8#4112yes—
184C2×C3×Q8#4112yes—
185C2×C3×Q8#4112yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—42no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—42no—
222#215—44no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—81no—
226(C2×D3)⋊C4—81no—
227(C2×D3)⋊C4—81no—
228(C2×D3)⋊C4—81no—
229C4⋊Dic3—24no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—42no—
236C4⋊Dic3—81no—
237C4⋊Dic3#4112yes—
238Dic3⋊C4—22no—
239Dic3⋊C4—41no—
240Dic3⋊C4—42no—
241Dic3⋊C4—81no—
242Dic3⋊C4—81no—
243Dic3⋊C4—81no—
244C4×Dic3—24no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—42no—
250C4×Dic3—81no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C22×Q8C22×D311yescharacteristic
258C22×D4—31no—
259C2×SD16—62no—
260C2×SD16—62no—
261C2×SD16—62no—
262C22×C8—31no—
263C22×C8—31no—
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C4×Q8—22no—
267C4×Q8—41no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8—62no—
271C4×Q8—121no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C42—21no—
295C2×C42—31no—
296C2×C42—31no—
297C2×C42—61no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301#107—62no—
302C4⋊C8—41no—
303#103—22no—
304#103—22no—
305#103—62no—
306#102—62no—
307#102—121no—
308#102—121no—
309#97—121no—
310#95—21no—
311#95—21no—
312#95—31no—
313#95—31no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318C23×C3C22×D411yescharacteristic
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—22no—
329C3×Q8#12614yes—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
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—41no—
340C2×C3×C4—41no—
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—41no—
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4H(5,𝔽2)11yescharacteristic
353C2×C3×C42−1+411yescharacteristic
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—22no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
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#13712yes—
375D12—41no—
376D12—41no—
377D12—41no—
378D12—41no—
379C3×C8—41no—
380C3×C8—41no—
381C3⋊C8—41no—
382C3⋊C8—41no—
383C24—61no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8#23512yes—
388C2×Q8#23712yes—
389C2×Q8#23912yes—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4D3×D411yescharacteristic
408C22×C4D3×D411yescharacteristic
409C22×C4C23×D311yescharacteristic
410SD16—124no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—32no—
430C4⋊C4—42no—
431C4⋊C4—62no—
432C4⋊C4—64no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—121no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C4⋊C4—241no—
442C4⋊C4—241no—
443C4⋊C4—241no—
444C4⋊C4—241no—
445C22⋊C4—122no—
446C22⋊C4—122no—
447C22⋊C4—241no—
448C22⋊C4—241no—
449C22⋊C4—241no—
450C22⋊C4—241no—
451C42—21no—
452C42—21no—
453C42—42no—
454C42—64no—
455C42—61no—
456C42—61no—
457C42—61no—
458C42—61no—
459C42—122no—
460C42—241no—
461C22×C3#52511yescharacteristic
462C22×C3#54611yescharacteristic
463C22×C3#54811yescharacteristic
464C22×C3C2×#13711yescharacteristic
465C22×C3#57511yescharacteristic
466C22×C3#57611yescharacteristic
467C22×C3#57711yescharacteristic
468C2×D3—81no—
469C2×D3—81no—
470C2×D3—81no—
471C2×D3—81no—
472C2×D3—81no—
473C2×D3—81no—
474C2×D3—81no—
475C3×C4—24no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4C2×#12611yescharacteristic
482C3×C4#53911yescharacteristic
483C3×C4#55211yescharacteristic
484C3×C4#55311yescharacteristic
485Dic3—22no—
486Dic3—41no—
487Dic3—41no—
488Dic3—42no—
489Dic3—41no—
490Dic3—81no—
491Dic3—81no—
492C23—121no—
493C23—121no—
494C23—121no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23C2×D3×D411yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—22no—
503Q8#89814yes—
504D4—121no—
505D4—121no—
506D4—121no—
507D4—121no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—32no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—62no—
538C2×C4—61no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4C2×#23511yescharacteristic
548C2×C4C2×#23711yescharacteristic
549C2×C4#100411yescharacteristic
550C2×C4C2×#23911yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4#101011yescharacteristic
553C8—41no—
554C8—41no—
555C8—121no—
556C8—121no—
557C2×C3#248111yescharacteristic
558C2×C3#311711yescharacteristic
559C2×C3#311911yescharacteristic
560C2×C3#330911yescharacteristic
561C2×C3#332911yescharacteristic
562C2×C3#337111yescharacteristic
563C2×C3#337511yescharacteristic
564D3—161no—
565C22—241no—
566C22—241no—
567C22—241no—
568C22—241no—
569C22—241no—
570C22—241no—
571C22—241no—
572C22#549011yescharacteristic
573C22#559811yescharacteristic
574C22#562711yescharacteristic
575C22#576811yescharacteristic
576C22#576911yescharacteristic
577C22D3×#13711yescharacteristic
578C22#577811yescharacteristic
579C4—24no—
580C4—41no—
581C4—41no—
582C4—62no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—121no—
587C4—121no—
588C4—122no—
589C4—121no—
590C4—241no—
591C4—241no—
592C4C2×#89811yescharacteristic
593C4#553111yescharacteristic
594C4#560111yescharacteristic
595C4#564511yescharacteristic
596C3#2757211yescharacteristic, complemented, Sylow
597C2—481no—
598C2#7391011yescharacteristic
599C2#7885411yescharacteristic
600C2#7897311yescharacteristic
601C2#8058311yescharacteristic
602C2#8058511yescharacteristic
603C2#8120711yescharacteristic
604C2#8120811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1805D3
hypocenterC3#27572
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. Q8×Dic3⋊C4C2semidirect
3. C3×#1805C2semidirect
6. #70448C2semidirect
9. #70236C2semidirect
10. #70220C2semidirect
11. #69542C2semidirect
14. #68843C2semidirect
15. #68814C2semidirect
73. #1805D3semidirect
596. C3#27572semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3⋊C42
#741722
C3×#18052
#716332
#704842
#704482
#702402
#702362
#702202
#695422
#689022
#688562
#688432
#688142
#275723