Finite Group Databasebeta

← #10844583

Subgroups — #10844583

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82381C211yesmaximal, characteristic
2C2×#5007C211yesmaximal, characteristic, complemented
3C2×#4714C211yesmaximal, characteristic, complemented
4#73656C211yesmaximal, characteristic, complemented
5C3×#1762C211yesmaximal, characteristic, complemented
6#70216C211yesmaximal, characteristic, complemented
7#70211C211yesmaximal, characteristic
8#70162C211yesmaximal, characteristic, complemented
9#70161C211yesmaximal, characteristic
10#69836C211yesmaximal, characteristic, complemented
11#69488C211yesmaximal, characteristic, complemented
12#68913C211yesmaximal, characteristic, complemented
13#68901C211yesmaximal, characteristic, complemented
14#68794C211yesmaximal, characteristic, complemented
15#68791C211yesmaximal, characteristic, complemented
16#25285—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×#936C2211yescharacteristic
20C2×#875C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C2×#832C2211yescharacteristic
28#5213—21no—
29#5213—21no—
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#5007—24no—
37#4988C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4973C2211yescharacteristic
41#4972C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4961—21no—
50#4961—21no—
51C4×C4⋊Dic3—21no—
52C4×C4⋊Dic3C2211yescharacteristic
53#4928—21no—
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55#4714—24no—
56#4646—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62C2×#489—31no—
63C2×#474—31no—
64#2544—31no—
65#2369—31no—
66#2068—31no—
67#2067—31no—
68#2042—31no—
69#2041—31no—
70#1983—31no—
71#1983—31no—
72#1982—31no—
73#1940—31no—
74#1928—31no—
75#1925—31no—
76#1762D311yescharacteristic, complemented
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×#67—21no—
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C22×Dic6—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96#940—22no—
97#940—41no—
98#936—22no—
99#936—41no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#875—22no—
118#875—22no—
119#869—22no—
120#869D414yescomplemented
121C3×#102—22no—
122C3×#102—22no—
123C3×#97—22no—
124C3×#95—21no—
125#832—22no—
126#832—22no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#821—22no—
136#821—22no—
137#816—22no—
138#816—22no—
139Dic3⋊C8—41no—
140Dic3⋊C8—41no—
141#810—22no—
142#810—41no—
143#809—22no—
144#809—41no—
145#807—22no—
146#807—22no—
147C4⋊(C3⋊C8)—22no—
148C2×C4⋊Q8—31no—
149C2×#127—31no—
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×D4C2×D311yescharacteristic
153#489—64no—
154#474—64no—
155C2×#107—31no—
156C2×#107—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
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×#102C2×D311yescharacteristic
171C2×#102C2×D311yescharacteristic
172C2×#97C2×D311yescharacteristic
173#394—61no—
174#394—61no—
175#391—31no—
176#391—61no—
177#391—61no—
178#390—31no—
179#389—31no—
180#389—31no—
181#389—61no—
182#386—61no—
183#386—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—61no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C23×D3—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4#4112yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—42no—
205C22×C3×C4—81no—
206C22×C3×C4—81no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C2×#67—42no—
211C2×#67—42no—
212C2×#67—42no—
213C2×#67—42no—
214C2×#67—81no—
215C2×#67—81no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—24no—
236C2×Dic6—81no—
237C2×Dic6—81no—
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—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4#4112yes—
250C3×C22⋊C4—42no—
251C3×C42—42no—
252C3×C42C2×D412yes—
253C3×C42#4112yes—
254#217—42no—
255#217—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—22no—
264C4⋊Dic3—41no—
265C4⋊Dic3C2×D412yes—
266C4⋊Dic3C2×D412yes—
267Dic3⋊C4—81no—
268Dic3⋊C4—81no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—81no—
274C4×Dic3—81no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C22×Q8—31no—
282C22×Q8—61no—
283C22×D4—31no—
284C22×D4—61no—
285C22×D4—61no—
286C22×D4C22×D311yescharacteristic
287C23×C4—21no—
288C22×C8—31no—
289C22×C8—31no—
290C22×C8C22×D311yescharacteristic
291C22×C8C22×D311yescharacteristic
292C4⋊Q8—34no—
293C4⋊Q8—62no—
294#127—62no—
295#127—121no—
296#124—62no—
297#124—62no—
298#124—62no—
299#124—121no—
300C4×D4—22no—
301C4×D4—22no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C42—21no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325#107—62no—
326#107—62no—
327C4⋊C8—62no—
328C4⋊C8—121no—
329C4⋊C8—121no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—121no—
334#102—22no—
335#102—22no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—62no—
341#102—121no—
342#97—22no—
343#95—21no—
344#95—31no—
345#95—31no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3—22no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3C22×D411yescharacteristic
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—41no—
370C3×D4—41no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
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—84no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393#67—84no—
394#67—84no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3#13712yes—
416D12—41no—
417D12—41no—
418D12—41no—
419D12—41no—
420Dic6—41no—
421Dic6—41no—
422Dic6—41no—
423Dic6—41no—
424Dic6—84no—
425C3×C8—41no—
426C3×C8—41no—
427C3⋊C8—41no—
428C3⋊C8—41no—
429C24—21no—
430C24—61no—
431C2×Q8—64no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—241no—
439C2×Q8—241no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—122no—
452C2×D4—122no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—241no—
456C2×D4—241no—
457C2×D4#23512yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—42no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4D3×D411yescharacteristic
472C22×C4D3×D411yescharacteristic
473C22×C4C23×D311yescharacteristic
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—32no—
492C4⋊C4—32no—
493C4⋊C4—62no—
494C4⋊C4—64no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—121no—
498C4⋊C4—241no—
499C4⋊C4—241no—
500C4⋊C4#23512yes—
501C22⋊C4—42no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C42—42no—
509C42—61no—
510C42—61no—
511C42—61no—
512C42—61no—
513C42—241no—
514C42—241no—
515C42#23512yes—
516C42D3×D412yes—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#54611yescharacteristic
526C22×C3#54811yescharacteristic
527C22×C3C2×#13711yescharacteristic
528C22×C3#57511yescharacteristic
529C22×C3#57611yescharacteristic
530C22×C3#57611yescharacteristic
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—22no—
539C3×C4—22no—
540C3×C4—42no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4#52911yescharacteristic
544C3×C4#54211yescharacteristic
545C3×C4#54811yescharacteristic
546C3×C4#55011yescharacteristic
547Dic3—22no—
548Dic3—42no—
549Dic3—42no—
550Dic3—41no—
551Dic3—41no—
552Dic3—41no—
553Dic3—81no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C2×D3×D411yescharacteristic
569Q8—121no—
570Q8—121no—
571Q8—121no—
572Q8—121no—
573Q8—244no—
574D4—22no—
575D4—22no—
576D4—41no—
577D4—41no—
578D4—121no—
579D4—121no—
580D4—121no—
581D4—121no—
582D4—244no—
583D4—244no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—32no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—84no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4#88412yes—
620C2×C4#88412yes—
621C2×C4C2×#23511yescharacteristic
622C2×C4C2×#23511yescharacteristic
623C2×C4C2×#23511yescharacteristic
624C2×C4C2×D3×D411yescharacteristic
625C2×C4#100911yescharacteristic
626C2×C4#101011yescharacteristic
627C8—41no—
628C8—41no—
629C8—121no—
630C8—121no—
631C2×C3—82no—
632C2×C3#242611yescharacteristic
633C2×C3#303711yescharacteristic
634C2×C3#304111yescharacteristic
635C2×C3#331411yescharacteristic
636C2×C3#331911yescharacteristic
637C2×C3#337011yescharacteristic
638C2×C3#337511yescharacteristic
639D3—161no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22C2×#88411yescharacteristic
655C22#555011yescharacteristic
656C22#555211yescharacteristic
657C22#577311yescharacteristic
658C22#577411yescharacteristic
659C22#577411yescharacteristic
660C22D3×#13711yescharacteristic
661C4—22no—
662C4—22no—
663C4—42no—
664C4—62no—
665C4—81no—
666C4—81no—
667C4—121no—
668C4—121no—
669C4—122no—
670C4—122no—
671C4—121no—
672C4—241no—
673C4#547911yescharacteristic
674C4#553411yescharacteristic
675C4#555211yescharacteristic
676C4#555311yescharacteristic
677C3#2528511yescharacteristic, complemented, Sylow
678C2—82no—
679C2—481no—
680C2#7348511yescharacteristic
681C2#7891311yescharacteristic
682C2#7894411yescharacteristic
683C2#7975611yescharacteristic
684C2#7980211yescharacteristic
685C2#7985211yescharacteristic
686C2#7988311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1762C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#1762D3
hypocenterC3#25285
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#5007C2semidirect
3. C2×#4714C2semidirect
4. #73656C2semidirect
5. C3×#1762C2semidirect
6. #70216C2semidirect
8. #70162C2semidirect
10. #69836C2semidirect
11. #69488C2semidirect
12. #68913C2semidirect
13. #68901C2semidirect
14. #68794C2semidirect
15. #68791C2semidirect
76. #1762D3semidirect
120. #869D4semidirect
677. C3#25285semidirect

Maximal subgroups

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

subgroupindex
#823812
C2×#50072
C2×#47142
#736562
C3×#17622
#702162
#702112
#701622
#701612
#698362
#694882
#689132
#689012
#687942
#687912
#252853