Finite Group Databasebeta

← #10844739

Subgroups — #10844739

Order 768.

848 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×#4787C211yesmaximal, characteristic
2C2×C8×Dic6C211yesmaximal, characteristic, complemented
3#76365C211yesmaximal, characteristic, complemented
4#76075C211yesmaximal, characteristic, complemented
5C3×#1934C211yesmaximal, characteristic, complemented
6#70190C211yesmaximal, characteristic
7#70179C211yesmaximal, characteristic, complemented
8#70158C211yesmaximal, characteristic, complemented
9#69774C211yesmaximal, characteristic, complemented
10#69650C211yesmaximal, characteristic
11#69635C211yesmaximal, characteristic, complemented
12#68863C211yesmaximal, characteristic, complemented
13#68850C211yesmaximal, characteristic, complemented
14#68797C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#25296—31nomaximal, Sylow
17C2×#936C2211yescharacteristic
18C22×#213C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#389C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4993—21no—
37#4989—21no—
38#4988C2211yescharacteristic
39#4977C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4952—21no—
48#4951—21no—
49#4947—21no—
50#4944—21no—
51#4939C2211yescharacteristic
52#4929C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54#4787—24no—
55C8×Dic6—24no—
56#4673—21no—
57#4668—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#463—31no—
63C2×C8×Q8—31no—
64#2592—31no—
65#2580—31no—
66#2068—31no—
67#2065—31no—
68#2055—31no—
69#2054—31no—
70#1970—31no—
71#1968—31no—
72#1958—31no—
73#1949—31no—
74#1934D311yescharacteristic, complemented
75#1933—31no—
76C4×#102—31no—
77C22×C3×D4C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6D411yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95#936—22no—
96#936—22no—
97C2×#213—22no—
98C2×#213—22no—
99C2×#213—41no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×#204—22no—
117C2×#204—22no—
118C2×#204—41no—
119#893—22no—
120#893—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#102—22no—
124C3×#102—22no—
125C3×C4×C8—22no—
126C3×#95—21no—
127#832—41no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#821—22no—
137#817—41no—
138#816—22no—
139Dic3⋊C8—22no—
140Dic3⋊C8D414yes—
141C8×Dic3—22no—
142C8×Dic3—22no—
143#810—22no—
144#810—41no—
145#809—22no—
146#807—41no—
147C4⋊(C3⋊C8)—22no—
148C22×SD16—31no—
149C22×D8—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×D4—31no—
153#463—64no—
154C8×Q8—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170C2×C4×C8C2×D311yescharacteristic
171#399—61no—
172#397—61no—
173#393—61no—
174#393—61no—
175#391—31no—
176#391—31no—
177#390—31no—
178#390—31no—
179#390—31no—
180#390C2×D311yescharacteristic
181#389—31no—
182#389C2×D311yescharacteristic
183#388—61no—
184#388—61no—
185#385—61no—
186#385—61no—
187#382—31no—
188C4×C4⋊C4—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
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—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×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12#4112yes—
220C2×C4×D3—42no—
221C2×C4×D3—81no—
222C2×C4×D3—81no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8C2×D412yes—
237C2×C3×C8#4112yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—81no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—81no—
247C3×C42—21no—
248C3×C42—21no—
249#217—42no—
250#217—42no—
251#217—42no—
252#217—81no—
253#213—44no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259C4⋊Dic3—22no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4—81no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—81no—
270C4×Dic3C2×D412yes—
271C4×Dic3#4112yes—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8C2×D412yes—
278#204—44no—
279C22×Q8—31no—
280C22×Q8—31no—
281C22×D4—31no—
282C22×D4C22×D311yescharacteristic
283C23×C4—61no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—121no—
287C2×D8—62no—
288C2×D8—62no—
289C2×D8—121no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294#124—62no—
295#124—62no—
296C4×Q8—62no—
297C4×Q8—62no—
298C4×D4—62no—
299C4×D4—62no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
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×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325#107—121no—
326#106—121no—
327C4⋊C8—34no—
328C4⋊C8—62no—
329C4⋊C8—62no—
330#103—62no—
331#103—62no—
332#103—121no—
333#102—22no—
334#102—22no—
335#102—62no—
336#102—62no—
337#102—121no—
338C4×C8—22no—
339C4×C8—62no—
340C4×C8—62no—
341#95—21no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C3×D4—41no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—41no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
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#13512yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3#12112yes—
409C2×Dic3#12112yes—
410C2×Dic3#13512yes—
411D12—22no—
412D12—22no—
413D12—41no—
414D12—41no—
415C4×D3—84no—
416Dic6—28no—
417Dic6—41no—
418Dic6—41no—
419Dic6—41no—
420Dic6—41no—
421C3×C8—22no—
422C3×C8—22no—
423C3⋊C8—22no—
424C3⋊C8—41no—
425C24—21no—
426C24—61no—
427C2×Q8—62no—
428C2×Q8—62no—
429C2×Q8—62no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—32no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—122no—
459C22×C4—241no—
460C22×C4—241no—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—124no—
465D8—124no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—32no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8#23512yes—
477C2×C8D3×D412yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—62no—
485C4⋊C4—62no—
486C4⋊C4—62no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—81no—
492C4⋊C4—241no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C22⋊C4—241no—
504C42—21no—
505C42—21no—
506C42—32no—
507C42—32no—
508C42—61no—
509C42—61no—
510C42—241no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3C2×#12111yescharacteristic
519C22×C3#54811yescharacteristic
520C22×C3#55011yescharacteristic
521C22×C3C2×#13511yescharacteristic
522C22×C3C2×#13511yescharacteristic
523C22×C3#57611yescharacteristic
524C22×C3#57711yescharacteristic
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—22no—
533C3×C4—41no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4#52911yescharacteristic
538C3×C4#54211yescharacteristic
539C3×C4#54611yescharacteristic
540C3×C4#54811yescharacteristic
541Dic3—22no—
542Dic3—22no—
543Dic3—22no—
544Dic3—41no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—62no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C2×D3×D411yescharacteristic
563Q8—68no—
564Q8—121no—
565Q8—121no—
566Q8—121no—
567Q8—121no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—62no—
573D4—62no—
574D4—121no—
575D4—121no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—32no—
582C2×C4—32no—
583C2×C4—32no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
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—244no—
612C2×C4#91612yes—
613C2×C4C2×#23511yescharacteristic
614C2×C4C2×D3×D411yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C2×C4#100911yescharacteristic
618C2×C4#101011yescharacteristic
619C8—22no—
620C8—22no—
621C8—62no—
622C8—121no—
623C2×C3—161no—
624C2×C3#242611yescharacteristic
625C2×C3#305011yescharacteristic
626C2×C3#306211yescharacteristic
627C2×C3#336011yescharacteristic
628C2×C3#336111yescharacteristic
629C2×C3#339411yescharacteristic
630C2×C3#339611yescharacteristic
631D3—82no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22#555311yescharacteristic
647C22D3×#12111yescharacteristic
648C22#567711yescharacteristic
649C22#575711yescharacteristic
650C22C2×#91611yescharacteristic
651C22D3×#13511yescharacteristic
652C22#577011yescharacteristic
653C4—22no—
654C4—41no—
655C4—62no—
656C4—62no—
657C4—62no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—121no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4#552811yescharacteristic
666C4#555211yescharacteristic
667C4#565411yescharacteristic
668C4#571011yescharacteristic
669C3#2529611yescharacteristic, complemented, Sylow
670C2—161no—
671C2—242no—
672C2#7549911yescharacteristic
673C2#7958011yescharacteristic
674C2#7963811yescharacteristic
675C2#8065611yescharacteristic
676C2#8071811yescharacteristic
677C2#8158511yescharacteristic
678C2#8163211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1934C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1934D3
hypocenterC3#25296
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. C2×C8×Dic6C2semidirect
3. #76365C2semidirect
4. #76075C2semidirect
5. C3×#1934C2semidirect
7. #70179C2semidirect
8. #70158C2semidirect
9. #69774C2semidirect
11. #69635C2semidirect
12. #68863C2semidirect
13. #68850C2semidirect
14. #68797C2semidirect
15. #68780C2semidirect
74. #1934D3semidirect
669. C3#25296semidirect

Maximal subgroups

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

subgroupindex
C2×#47872
C2×C8×Dic62
#763652
#760752
C3×#19342
#701902
#701792
#701582
#697742
#696502
#696352
#688632
#688502
#687972
#687802
#252963