Finite Group Databasebeta

← #10844920

Subgroups — #10844920

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
1C2×C3×#446C211yesmaximal, characteristic, complemented
2#82480C211yesmaximal, characteristic, complemented
3#82476C211yesmaximal, characteristic
4C2×#4848C211yesmaximal, characteristic
5#70484C211yesmaximal, characteristic, complemented
6#70483C211yesmaximal, characteristic, complemented
7#70482C211yesmaximal, characteristic, complemented
8#70482C211yesmaximal, characteristic, complemented
9#70455C211yesmaximal, characteristic, complemented
10#69826C211yesmaximal, characteristic, complemented
11#69823C211yesmaximal, characteristic
12#69818C211yesmaximal, characteristic, complemented
13#69813C211yesmaximal, characteristic, complemented
14#69774C211yesmaximal, characteristic, complemented
15#69768C211yesmaximal, characteristic, complemented
16#25310—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947—21no—
19C22×#204C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#874C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#446—24no—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30#5230—21no—
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×#817C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41#4991—21no—
42#4991—21no—
43#4988—21no—
44#4985—21no—
45#4978—21no—
46#4978—21no—
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4939C2211yescharacteristic
50#4939C2211yescharacteristic
51#4936C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53#4929C2211yescharacteristic
54#4929C2211yescharacteristic
55#4928C2211yescharacteristic
56#4928C2211yescharacteristic
57#4848—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#460—31no—
63C2×#446D311yescharacteristic, complemented
64#2581—31no—
65#2548—31no—
66#2067—31no—
67#2065—31no—
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1968—31no—
72#1961—31no—
73#1949—31no—
74#1936—31no—
75#1934—31no—
76#1763—31no—
77C22×C3×Q8D411yescharacteristic
78C22×D12—21no—
79C22×D12—21no—
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95#947—44no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×#204—22no—
115C2×#204—22no—
116C2×#204—22no—
117C2×#204—22no—
118C2×#204—41no—
119C2×#204—41no—
120#874—22no—
121#874—22no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C3×C4⋊C8—22no—
125C3×C4⋊C8—22no—
126C3×C4⋊C8D414yes—
127C3×#97—22no—
128C3×#97—22no—
129C3×C4×C8—22no—
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—
139#821—22no—
140#821—22no—
141#821—22no—
142#821—22no—
143#821—41no—
144#821—41no—
145#818—41no—
146#817—41no—
147#816—22no—
148#816—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127—31no—
152C2×#124—61no—
153C2×C4×Q8—31no—
154C2×C4×Q8C2×D311yescharacteristic
155#460—64no—
156#446—24no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×C4⋊C8C2×D311yescharacteristic
161C2×C4⋊C8C2×D311yescharacteristic
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×#97C2×D311yescharacteristic
171C2×#97C2×D311yescharacteristic
172C2×C4×C8C2×D311yescharacteristic
173#400—61no—
174#395—61no—
175#394—61no—
176#394—61no—
177#391—31no—
178#391—31no—
179#390—31no—
180#390—31no—
181#390—31no—
182#390—31no—
183#390—61no—
184#389—31no—
185#389—31no—
186#387—61no—
187#387—61no—
188#382—31no—
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C23×D3—21no—
195C23×D3—21no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—41no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—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—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—42no—
225C2×D12—42no—
226C2×D12—42no—
227C2×D12—42no—
228C2×D12—81no—
229C2×D12—81no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6#4112yes—
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—
244C2×C3×C8—21no—
245C2×C3×C8C2×D412yes—
246C2×C3×C8C2×D412yes—
247C2×C3×C8#4112yes—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—22no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42C2×D412yes—
256C3×C42#4112yes—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264(C2×D3)⋊C4—81no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—42no—
278C4×Dic3—42no—
279C4×Dic3—42no—
280#204—44no—
281#204—44no—
282C22×Q8—31no—
283C22×Q8D1211yescharacteristic
284C22×D4—31no—
285C22×D4—31no—
286C22×D4—61no—
287C22×D4—61no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×SD16—62no—
291C2×SD16—62no—
292C2×SD16—121no—
293C2×SD16—121no—
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298#127—62no—
299#127—62no—
300#124—124no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—62no—
304C4×Q8—62no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4D1211yescharacteristic
314C2×C4⋊C4D1211yescharacteristic
315C2×C4⋊C4D1211yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327C2×C42C22×D311yescharacteristic
328C2×C42C22×D311yescharacteristic
329#107—121no—
330#106—121no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8#6714yes—
334#103—62no—
335#103—62no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—121no—
341#102—121no—
342#97—22no—
343#97—22no—
344C4×C8—22no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3C22×D411yescharacteristic
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—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C3×Q8—44no—
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—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4AGL(1,ℤ/8)12yes—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394C2×Dic3—22no—
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—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412D12—41no—
413D12—41no—
414D12—41no—
415D12—41no—
416D12—41no—
417D12—41no—
418D12—41no—
419D12—41no—
420D12—84no—
421D12—84no—
422Dic6—22no—
423Dic6—22no—
424Dic6—41no—
425Dic6—41no—
426C3×C8—22no—
427C3×C8—22no—
428C3×C8—22no—
429C3×C8—41no—
430C24—61no—
431C24—61no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—32no—
435C2×Q8—41no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C2×D4—241no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4C2×D1211yescharacteristic
466C22×C4C2×D1211yescharacteristic
467C22×C4C2×D1211yescharacteristic
468C22×C4C2×D1211yescharacteristic
469C22×C4C2×D1211yescharacteristic
470C22×C4C2×D1211yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—124no—
473SD16—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8#23912yes—
484C2×C8C2×#6712yes—
485C2×C8C2×#6712yes—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C22⋊C4—241no—
509C22⋊C4—241no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—122no—
515C42—122no—
516C42#23512yes—
517C42C2×#6712yes—
518C22×C3C2×#12111yescharacteristic
519C22×C3#54811yescharacteristic
520C22×C3#55011yescharacteristic
521C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
522C22×C3#57511yescharacteristic
523C22×C3#57711yescharacteristic
524C22×C3#57711yescharacteristic
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C2×D3—81no—
539C3×C4—22no—
540C3×C4—22no—
541C3×C4—22no—
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—41no—
545C3×C4#52911yescharacteristic
546C3×C4#54211yescharacteristic
547C3×C4#54611yescharacteristic
548C3×C4#54811yescharacteristic
549Dic3—42no—
550Dic3—81no—
551Dic3—81no—
552Dic3—81no—
553Dic3—81no—
554Dic3—81no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23C22×D1211yescharacteristic
570Q8—44no—
571Q8—62no—
572Q8—62no—
573Q8—121no—
574Q8—121no—
575D4—121no—
576D4—121no—
577D4—121no—
578D4—121no—
579D4—121no—
580D4—121no—
581D4—121no—
582D4—121no—
583D4—244no—
584D4—244no—
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—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—62no—
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—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4#90812yes—
619C2×C4#93012yes—
620C2×C4#93012yes—
621C2×C4C2×#23511yescharacteristic
622C2×C4C2×#23911yescharacteristic
623C2×C4#100711yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C2×C4#100911yescharacteristic
626C2×C4C22×#6711yescharacteristic
627C8—22no—
628C8—22no—
629C8—22no—
630C8—41no—
631C2×C3#242611yescharacteristic
632C2×C3#305511yescharacteristic
633C2×C3#305711yescharacteristic
634C2×C3#336811yescharacteristic
635C2×C3#337111yescharacteristic
636C2×C3#339711yescharacteristic
637C2×C3#340111yescharacteristic
638D3—161no—
639D3—161no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22#557611yescharacteristic
655C22#557711yescharacteristic
656C22C2×#90811yescharacteristic
657C22#574811yescharacteristic
658C22#575111yescharacteristic
659C22#575111yescharacteristic
660C22C2×#93011yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—122no—
668C4—241no—
669C4—241no—
670C4—241no—
671C4—241no—
672C4—241no—
673C4#558011yescharacteristic
674C4#558211yescharacteristic
675C4#581711yescharacteristic
676C4#582911yescharacteristic
677C3#2531011yescharacteristic, complemented, Sylow
678C2—481no—
679C2—481no—
680C2#7995011yescharacteristic
681C2#7995411yescharacteristic
682C2#7998011yescharacteristic
683C2#7998211yescharacteristic
684C2#8249511yescharacteristic
685C2#8377211yescharacteristic
686C2#8379811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#446C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenterC2×#446D3
hypocenterC3#25310
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
1. C2×C3×#446C2semidirect
2. #82480C2semidirect
5. #70484C2semidirect
6. #70483C2semidirect
7. #70482C2semidirect
8. #70482C2semidirect
9. #70455C2semidirect
10. #69826C2semidirect
12. #69818C2semidirect
13. #69813C2semidirect
14. #69774C2semidirect
15. #69768C2semidirect
63. C2×#446D3semidirect
677. C3#25310semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4462
#824802
#824762
C2×#48482
#704842
#704832
#704822
#704552
#698262
#698232
#698182
#698132
#697742
#697682
#253103