Finite Group Databasebeta

← #10844671

Subgroups — #10844671

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4829C211yesmaximal, characteristic, complemented
2C2×#4713C211yesmaximal, characteristic
3#73712C211yesmaximal, characteristic
4#73659C211yesmaximal, characteristic, complemented
5C3×#1940C211yesmaximal, characteristic, complemented
6#70484C211yesmaximal, characteristic, complemented
7#70483C211yesmaximal, characteristic, complemented
8#70458C211yesmaximal, characteristic, complemented
9#70458C211yesmaximal, characteristic, complemented
10#70457C211yesmaximal, characteristic
11#69813C211yesmaximal, characteristic, complemented
12#68630C211yesmaximal, characteristic, complemented
13#68630C211yesmaximal, characteristic, complemented
14#68626C211yesmaximal, characteristic, complemented
15#68625C211yesmaximal, characteristic, complemented
16#25291—31nomaximal, Sylow
17C2×#875C2211yescharacteristic
18C2×#875C2211yescharacteristic
19C2×#874C2211yescharacteristic
20C2×#870C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41#4988—21no—
42#4988—21no—
43#4978—21no—
44#4976—21no—
45#4939C2211yescharacteristic
46#4939C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52#4829—24no—
53#4713—24no—
54#4672—21no—
55#4672—21no—
56#4652—21no—
57#4649—21no—
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#490—31no—
63C2×#479—31no—
64#2548—31no—
65#2370—31no—
66#2067—31no—
67#2065—31no—
68#2042—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1982—31no—
73#1940D311yescharacteristic, complemented
74#1926—31no—
75#1925—31no—
76#1763—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
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×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113#875—22no—
114#875—22no—
115#875—41no—
116#875—41no—
117#874—22no—
118#874—22no—
119#870—22no—
120#870D414yes—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×C4⋊C8—22no—
124C3×C4⋊C8—41no—
125C3×C4⋊C8—41no—
126C3×#97—22no—
127C3×#95D411yescharacteristic
128C3×#95D411yescharacteristic
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#821—22no—
135#821—22no—
136#821—22no—
137#821—22no—
138#821—22no—
139#821—22no—
140#821—41no—
141#821—41no—
142#818—22no—
143#817—22no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148C2×#127—31no—
149C2×#125—31no—
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153#490—64no—
154#479—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8C2×D311yescharacteristic
158C2×C4⋊C8C2×D311yescharacteristic
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#97C2×D311yescharacteristic
173#394—61no—
174#394—61no—
175#390—31no—
176#390—31no—
177#390—61no—
178#390—61no—
179#389—31no—
180#389—31no—
181#389—61no—
182#387—61no—
183#387—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—61no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#336C2×D311yescharacteristic
192#336C2×D311yescharacteristic
193C23×D3—21no—
194C23×D3—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—42no—
221C2×D12—42no—
222C2×D12—42no—
223C2×D12—42no—
224C2×D12—81no—
225C2×D12—81no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—41no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42C2×D412yes—
251C3×C42#4112yes—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3C2×D412yes—
268C4⋊Dic3C2×D412yes—
269C4⋊Dic3#4112yes—
270Dic3⋊C4—24no—
271Dic3⋊C4—24no—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274Dic3⋊C4—81no—
275C4×Dic3—42no—
276C4×Dic3—42no—
277C4×Dic3—81no—
278C4×Dic3—81no—
279C22×Q8—31no—
280C22×D4—31no—
281C22×D4—31no—
282C22×D4—61no—
283C22×D4—61no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288#127—62no—
289#127—62no—
290#125—34no—
291#125—62no—
292#124—62no—
293#124—62no—
294#124—121no—
295#124—121no—
296C4×Q8—62no—
297C4×Q8—62no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4D1211yescharacteristic
309C2×C4⋊C4D1211yescharacteristic
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323C2×C42C22×D311yescharacteristic
324C2×C42C22×D311yescharacteristic
325#107—62no—
326#106—62no—
327C4⋊C8—22no—
328C4⋊C8—41no—
329C4⋊C8—41no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—62no—
334#102—62no—
335#102—62no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—121no—
341#102—121no—
342#97—22no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95D1211yescharacteristic
349#95D1211yescharacteristic
350C23×C3C22×D411yescharacteristic
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4#12112yes—
382C2×C3×C4#12112yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388C2×C3×C42−1+411yescharacteristic
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407D12—41no—
408D12—41no—
409D12—41no—
410D12—41no—
411D12—41no—
412D12—41no—
413D12—41no—
414D12—41no—
415D12—84no—
416D12—84no—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421C3×C8—41no—
422C3×C8—41no—
423C3×C8—41no—
424C3×C8—41no—
425C24—61no—
426C24—61no—
427C2×Q8—32no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—122no—
446C2×D4—122no—
447C2×D4—122no—
448C2×D4—122no—
449C2×D4—241no—
450C2×D4—241no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4C2×D1211yescharacteristic
458C22×C4C2×D1211yescharacteristic
459C22×C4C2×D1211yescharacteristic
460C22×C4C2×D1211yescharacteristic
461C22×C4C2×D1211yescharacteristic
462C22×C4C2×D1211yescharacteristic
463C22×C4C23×D311yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C4⋊C4—32no—
477C4⋊C4—32no—
478C4⋊C4—32no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—64no—
482C4⋊C4—64no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—122no—
491C4⋊C4—122no—
492C4⋊C4—241no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C22⋊C4—241no—
500C22⋊C4—241no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—122no—
506C42—122no—
507C42—241no—
508C42—241no—
509C42D3×D412yes—
510C42#23912yes—
511C22×C3C2×#12111yescharacteristic
512C22×C3#54611yescharacteristic
513C22×C3#54811yescharacteristic
514C22×C3#57511yescharacteristic
515C22×C3#57511yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57711yescharacteristic
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4#52911yescharacteristic
539C3×C4#54211yescharacteristic
540C3×C4#54811yescharacteristic
541C3×C4#55011yescharacteristic
542Dic3—42no—
543Dic3—42no—
544Dic3—42no—
545Dic3—42no—
546Dic3—81no—
547Dic3—81no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C22×D1211yescharacteristic
563Q8—62no—
564Q8—62no—
565Q8—121no—
566Q8—121no—
567D4—121no—
568D4—121no—
569D4—121no—
570D4—121no—
571D4—121no—
572D4—121no—
573D4—121no—
574D4—121no—
575D4—244no—
576D4—244no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
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—121no—
610C2×C4—121no—
611C2×C4#89512yes—
612C2×C4#89512yes—
613C2×C4C2×D3×D411yescharacteristic
614C2×C4#100411yescharacteristic
615C2×C4C2×#23911yescharacteristic
616C2×C4#100711yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C8—41no—
620C8—41no—
621C8—41no—
622C8—41no—
623C2×C3#242611yescharacteristic
624C2×C3#304011yescharacteristic
625C2×C3#304511yescharacteristic
626C2×C3#330911yescharacteristic
627C2×C3#331711yescharacteristic
628C2×C3#337011yescharacteristic
629C2×C3#337111yescharacteristic
630D3—161no—
631D3—161no—
632C22—241no—
633C22—241no—
634C22—241no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22C2×#89511yescharacteristic
647C22#555411yescharacteristic
648C22#557611yescharacteristic
649C22#574811yescharacteristic
650C22#574811yescharacteristic
651C22#575111yescharacteristic
652C22#575211yescharacteristic
653C4—22no—
654C4—22no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—122no—
660C4—122no—
661C4—122no—
662C4—122no—
663C4—241no—
664C4—241no—
665C4#559911yescharacteristic
666C4#563311yescharacteristic
667C4#569111yescharacteristic
668C4#573311yescharacteristic
669C3#2529111yescharacteristic, complemented, Sylow
670C2—481no—
671C2—481no—
672C2#7621811yescharacteristic
673C2#7905811yescharacteristic
674C2#7908811yescharacteristic
675C2#7951011yescharacteristic
676C2#7951411yescharacteristic
677C2#7997111yescharacteristic
678C2#7998211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1940C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1940D3
hypocenterC3#25291
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
1. C2×#4829C2semidirect
4. #73659C2semidirect
5. C3×#1940C2semidirect
6. #70484C2semidirect
7. #70483C2semidirect
8. #70458C2semidirect
9. #70458C2semidirect
11. #69813C2semidirect
12. #68630C2semidirect
13. #68630C2semidirect
14. #68626C2semidirect
15. #68625C2semidirect
73. #1940D3semidirect
669. C3#25291semidirect

Maximal subgroups

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

subgroupindex
C2×#48292
C2×#47132
#737122
#736592
C3×#19402
#704842
#704832
#704582
#704572
#698132
#686302
#686262
#686252
#252913