Finite Group Databasebeta

← #10835914

Subgroups — #10835914

Order 768.

814 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4849C211yesmaximal, characteristic, complemented
2C2×#4840C211yesmaximal, characteristic
3#74193C211yesmaximal, characteristic
4#73762C211yesmaximal, characteristic
5C3×#1949C211yesmaximal, characteristic, complemented
6#70635C211yesmaximal, characteristic
7#70627C211yesmaximal, characteristic, complemented
8#70582C211yesmaximal, characteristic
9#70575C211yesmaximal, characteristic, complemented
10#69559C211yesmaximal, characteristic, complemented
11#69486C211yesmaximal, characteristic, complemented
12#68888C211yesmaximal, characteristic
13#68888C211yesmaximal, characteristic
14#68855C211yesmaximal, characteristic, complemented
15#68851C211yesmaximal, characteristic, complemented
16#24664—31nomaximal, Sylow
17C22×Dic12C2211yescharacteristic
18C22×Dic12C2211yescharacteristic
19C2×#897C2211yescharacteristic
20C2×#897C2211yescharacteristic
21C2×#897C2211yescharacteristic
22C2×#872C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C3×#382C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#815C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4987C2211yescharacteristic
37#4984C2211yescharacteristic
38#4973C2211yescharacteristic
39#4973C2211yescharacteristic
40#4973C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47#4940—21no—
48#4928—21no—
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4849—24no—
51#4840—24no—
52#4674—21no—
53#4674—21no—
54#4670—21no—
55#4661—21no—
56#4661—21no—
57#4653—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#462—31no—
63C2×#433—31no—
64#2621—31no—
65#2323—31no—
66#2036—31no—
67#2035—31no—
68#2001—31no—
69#2001—31no—
70#1981—31no—
71#1971—31no—
72#1969—31no—
73#1949D311yescharacteristic, complemented
74#1935—31no—
75#1933—31no—
76#1805—31no—
77C22×C4×D3C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4D411yescharacteristic
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4D411yescharacteristic
104C2×Dic3⋊C4D411yescharacteristic
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C2×Dic12—22no—
110C2×Dic12—22no—
111C2×Dic12—22no—
112C2×Dic12—22no—
113C2×Dic12—41no—
114C2×Dic12—41no—
115#897—22no—
116#897—22no—
117#897—41no—
118#897—41no—
119#897—41no—
120#897—41no—
121#872—22no—
122#872—22no—
123#869—22no—
124#869—22no—
125C4×Dic6—22no—
126C4×Dic6—22no—
127C3×#107—41no—
128C3×#106—41no—
129C3×C4⋊C8D414yes—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#816—22no—
144#816—22no—
145#815—22no—
146C8×Dic3—22no—
147#810—22no—
148#810—22no—
149#810—41no—
150#810—41no—
151#810—41no—
152#810—41no—
153C4⋊(C3⋊C8)—22no—
154C22×Q16—31no—
155C22×Q16—31no—
156C2×C4⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C22⋊Q8—31no—
159C2×C22⋊Q8—31no—
160C2×C4×Q8—31no—
161C2×#117—31no—
162#462—64no—
163#433—64no—
164C2×#107C2×D311yescharacteristic
165C2×#106C2×D311yescharacteristic
166C2×C4⋊C8—31no—
167C2×C4⋊C8C2×D311yescharacteristic
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#103—31no—
174C2×#103—31no—
175C2×#103—31no—
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178C2×#97—31no—
179C2×C4×C8—31no—
180#398—61no—
181#398—61no—
182#397—61no—
183#397—61no—
184#391—31no—
185#391—31no—
186#391—31no—
187#389—31no—
188#389—61no—
189#388—61no—
190#388—61no—
191#387—61no—
192#382C2×D311yescharacteristic
193#380—31no—
194C4×C4⋊C4C2×D311yescharacteristic
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199C23×D3C2×D411yescharacteristic
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6#4112yes—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8C2×D412yes—
243C2×C3×C8C2×D412yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4#4112yes—
252C3×C42—81no—
253C3×C42C2×D412yes—
254C3×C42#4112yes—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—81no—
261C4⋊Dic3—24no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—42no—
265C4⋊Dic3—42no—
266C4⋊Dic3—42no—
267C4⋊Dic3—42no—
268C4⋊Dic3—81no—
269Dic3⋊C4—22no—
270Dic3⋊C4—22no—
271Dic3⋊C4—42no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—42no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281Dic12—44no—
282Dic12—44no—
283C22×Q8—31no—
284C22×Q8—31no—
285C22×Q8—31no—
286C23×C4—31no—
287C2×Q16—62no—
288C2×Q16—62no—
289C2×Q16—62no—
290C2×Q16—62no—
291C2×Q16—121no—
292C2×Q16—121no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299C22⋊Q8—62no—
300C22⋊Q8—62no—
301C22⋊Q8—121no—
302C22⋊Q8—121no—
303C22⋊Q8—121no—
304C22⋊Q8—121no—
305C4×Q8—62no—
306C4×Q8—62no—
307#117—62no—
308#117—62no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—31no—
324C2×C22⋊C4—31no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—21no—
330C2×C42—31no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—41no—
334#106—41no—
335C4⋊C8—62no—
336C4⋊C8D1214yes—
337#103—62no—
338#103—62no—
339#103—62no—
340#103—62no—
341#103—121no—
342#103—121no—
343#103—121no—
344#103—121no—
345C22⋊C8—62no—
346C22⋊C8—62no—
347#97—62no—
348C4×C8—62no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3C22×D411yescharacteristic
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4#12112yes—
385C2×C3×C4#12112yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4H(5,𝔽2)11yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C4×D3—41no—
414C4×D3—41no—
415C4×D3—41no—
416C4×D3—41no—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421Dic6—41no—
422Dic6—41no—
423Dic6—41no—
424Dic6—41no—
425Dic6—41no—
426Dic6—41no—
427Dic6—41no—
428Dic6—41no—
429C3×C8—22no—
430C3×C8—22no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—31no—
434C2×Q8—32no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×Q8—61no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×Q8—61no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—61no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472Q16—124no—
473Q16—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8C2×D1212yes—
485C2×C8C2×D1212yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—64no—
494C4⋊C4—62no—
495C4⋊C4—62no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—122no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4—122no—
503C4⋊C4—241no—
504C4⋊C4#23912yes—
505C22⋊C4—64no—
506C22⋊C4—62no—
507C22⋊C4—62no—
508C22⋊C4—122no—
509C22⋊C4—122no—
510C22⋊C4—241no—
511C42—61no—
512C42—61no—
513C42—81no—
514C42—122no—
515C42C2×D1212yes—
516C42#23912yes—
517C22×C3C2×#12111yescharacteristic
518C22×C3#52511yescharacteristic
519C22×C3#53511yescharacteristic
520C22×C3#57711yescharacteristic
521C22×C3#57811yescharacteristic
522C22×C3#57811yescharacteristic
523C22×C3#57811yescharacteristic
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—42no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4#54611yescharacteristic
538C3×C4#55011yescharacteristic
539C3×C4#55311yescharacteristic
540C3×C4#55511yescharacteristic
541Dic3—42no—
542Dic3—41no—
543Dic3—41no—
544Dic3—41no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23C2×D3×D411yescharacteristic
556Q8—62no—
557Q8—62no—
558Q8—121no—
559Q8—121no—
560Q8—121no—
561Q8—121no—
562Q8—121no—
563Q8—121no—
564Q8—121no—
565Q8—121no—
566Q8—121no—
567Q8—121no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—62no—
593C2×C4—61no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4#89512yes—
610C2×C4#89512yes—
611C2×C4C22×D1211yescharacteristic
612C2×C4#100411yescharacteristic
613C2×C4C2×#23911yescharacteristic
614C2×C4C2×#23911yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4#100911yescharacteristic
617C8—22no—
618C8—22no—
619C8—121no—
620C8—121no—
621C2×C3#241411yescharacteristic
622C2×C3#305911yescharacteristic
623C2×C3#306311yescharacteristic
624C2×C3#312011yescharacteristic
625C2×C3#312711yescharacteristic
626C2×C3#322311yescharacteristic
627C2×C3#322511yescharacteristic
628D3—81no—
629C22—121no—
630C22—121no—
631C22—121no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22C2×#89511yescharacteristic
637C22#560711yescharacteristic
638C22#562711yescharacteristic
639C22#577111yescharacteristic
640C22#577111yescharacteristic
641C22#577511yescharacteristic
642C22#577911yescharacteristic
643C4—22no—
644C4—22no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—121no—
650C4—121no—
651C4—121no—
652C4—122no—
653C4—241no—
654C4—241no—
655C4—241no—
656C4#555411yescharacteristic
657C4#557711yescharacteristic
658C4#569511yescharacteristic
659C4#573411yescharacteristic
660C3#2466411yescharacteristic, complemented, Sylow
661C2—241no—
662C2#7620911yescharacteristic
663C2#7918411yescharacteristic
664C2#7921411yescharacteristic
665C2#8054711yescharacteristic
666C2#8060911yescharacteristic
667C2#8114511yescharacteristic
668C2#8121111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1949C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1949D3
hypocenterC3#24664
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
1. C2×#4849C2semidirect
5. C3×#1949C2semidirect
7. #70627C2semidirect
9. #70575C2semidirect
10. #69559C2semidirect
11. #69486C2semidirect
14. #68855C2semidirect
15. #68851C2semidirect
73. #1949D3semidirect
660. C3#24664semidirect

Maximal subgroups

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

subgroupindex
C2×#48492
C2×#48402
#741932
#737622
C3×#19492
#706352
#706272
#705822
#705752
#695592
#694862
#688882
#688552
#688512
#246643