Finite Group Databasebeta

← #10845326

Subgroups — #10845326

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4854C211yesmaximal, characteristic, complemented
2C2×#67⋊C8C211yesmaximal, characteristic
3#73937C211yesmaximal, characteristic
4#73936C211yesmaximal, characteristic
5C3×#2077C211yesmaximal, characteristic, complemented
6#70577C211yesmaximal, characteristic, complemented
7#69936C211yesmaximal, characteristic, complemented
8#69514C211yesmaximal, characteristic, complemented
9#69511C211yesmaximal, characteristic, complemented
10#68882C211yesmaximal, characteristic, complemented
11#68833C211yesmaximal, characteristic, complemented
12#68824C211yesmaximal, characteristic, complemented
13#68803C211yesmaximal, characteristic, complemented
14#68796C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16#25341—31nomaximal, Sylow
17C2×C4×#67C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C2×#897C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×#885—21no—
25C2×C3×#106C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×#380C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5212C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4975C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45#4957—21no—
46#4952—21no—
47#4948—21no—
48#4854—24no—
49#67⋊C8—24no—
50#4676—21no—
51#4675—21no—
52#4647—21no—
53#4646—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#475—31no—
63C2×C8×D4—31no—
64#2547—31no—
65#2538—31no—
66#2077—31no—
67#2077D311yescharacteristic, complemented
68#2035—31no—
69#2035—31no—
70#1990—31no—
71#1948—31no—
72#1892—31no—
73#1892—31no—
74#1807—31no—
75#1807—31no—
76C4×#103—31no—
77C23×C3×C4C2311yescharacteristic
78C22×#67D411yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C22×Dic6—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217D411yescharacteristic
93C4×#67—22no—
94C4×#67—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104#924—22no—
105#924—41no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8—22no—
116C22×C3⋊C8—22no—
117C22×C3⋊C8—41no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×D3×C8—22no—
121C2×D3×C8—22no—
122C2×D3×C8—41no—
123#897—22no—
124#897—41no—
125#888—22no—
126#888—22no—
127#887—22no—
128#887—22no—
129#885—44no—
130C3×#106—22no—
131C3×C22⋊C8—22no—
132C3×#95—21no—
133C3×#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139(C2×D3)⋊C8—22no—
140#817—41no—
141#816—22no—
142#816—22no—
143Dic3⋊C8D414yes—
144C8×Dic3—22no—
145#810—22no—
146#810—22no—
147#808—22no—
148#808—41no—
149C23×C8—31no—
150C23×C8—31no—
151C2×C4⋊Q8—31no—
152C2×#124—61no—
153C2×C22⋊Q8—31no—
154C2×C22⋊Q8—31no—
155C2×C4×Q8—31no—
156C2×C4×D4—31no—
157#475—64no—
158C8×D4—64no—
159C2×#106—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×C4⋊C8—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8C2×D311yescharacteristic
170C2×C4×C8—31no—
171#397—61no—
172#397—61no—
173#396—61no—
174#396—61no—
175#391—61no—
176#390—61no—
177#389—61no—
178#384—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380C2×D311yescharacteristic
184#380C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194C24×C3C2×D411yescharacteristic
195C23×D3C2×D411yescharacteristic
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C2×#67—22no—
209C2×#67—22no—
210C2×#67SD1614yes—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—42no—
230C2×Dic6—42no—
231C2×Dic6—42no—
232C2×Dic6#4112yes—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8C2×D412yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C22⋊C4—42no—
248C3×C22⋊C4—81no—
249#217—22no—
250#217—42no—
251#217—42no—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—81no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—42no—
264Dic3⋊C4—22no—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269Dic3⋊C4—81no—
270C4×Dic3—42no—
271C4×Dic3—42no—
272C4×Dic3C2×D412yes—
273C4×Dic3#4112yes—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—44no—
279C2×C3⋊C8C2×D412yes—
280C2×C3⋊C8#4112yes—
281D3×C8—44no—
282C22×Q8—31no—
283C22×Q8—61no—
284C22×D4—31no—
285C23×C4—31no—
286C23×C4C22×D311yescharacteristic
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—62no—
290C22×C8—62no—
291C22×C8—62no—
292C22×C8—62no—
293C22×C8—121no—
294C22×C8—121no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299#124—124no—
300C22⋊Q8—62no—
301C22⋊Q8—62no—
302C22⋊Q8—121no—
303C22⋊Q8—121no—
304C4×Q8—62no—
305C4×Q8—62no—
306C4×D4—62no—
307C4×D4—62no—
308C2×C4⋊C4—31no—
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⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—31no—
325C2×C22⋊C4—31no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C42—31no—
333C2×C42—61no—
334C2×C42—61no—
335#106—22no—
336#106—62no—
337#106—121no—
338#106—121no—
339C4⋊C8—34no—
340#103—62no—
341#103—62no—
342#103—62no—
343#103—62no—
344C22⋊C8—22no—
345C22⋊C8—62no—
346C4×C8—62no—
347#95—21no—
348#95—21no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3C2×SD1612yes—
361C23×C3C22×D411yescharacteristic
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3C2×SD1612yes—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
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—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
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
394#67—28no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—21no—
400C2×Dic3—21no—
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—
412C2×Dic3#12112yes—
413C2×Dic3#12112yes—
414C2×Dic3C2×SD1612yes—
415C2×Dic3#13512yes—
416C4×D3—41no—
417C4×D3—41no—
418C4×D3—41no—
419C4×D3—41no—
420Dic6—22no—
421Dic6—22no—
422Dic6—41no—
423Dic6—41no—
424Dic6—84no—
425C3×C8—22no—
426C3×C8—41no—
427C3⋊C8—22no—
428C3⋊C8—22no—
429C24—31no—
430C24D3×D411yescharacteristic
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—122no—
438C2×Q8—122no—
439C2×Q8—122no—
440C2×D4—34no—
441C2×D4—62no—
442C2×D4—62no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4D3×D411yescharacteristic
467C22×C4C23×D311yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—32no—
474C2×C8—32no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—124no—
480C2×C8—124no—
481C2×C8D3×D412yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—62no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4—241no—
503C22⋊C4—42no—
504C22⋊C4—62no—
505C22⋊C4—62no—
506C22⋊C4—81no—
507C22⋊C4—122no—
508C22⋊C4—122no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—241no—
513C42—32no—
514C42—32no—
515C42—122no—
516C42—122no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#54611yescharacteristic
526C22×C3#54711yescharacteristic
527C22×C3C22×SD1611yescharacteristic
528C22×C3C2×#13511yescharacteristic
529C22×C3#57711yescharacteristic
530C22×C3#57811yescharacteristic
531C2×D3—22no—
532C2×D3—22no—
533C2×D3—22no—
534C2×D3—41no—
535C2×D3—41no—
536C2×D3—41no—
537C2×D3—41no—
538C3×C4—41no—
539C3×C4—81no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4#52911yescharacteristic
544C3×C4#54211yescharacteristic
545C3×C4#54611yescharacteristic
546C3×C4#54811yescharacteristic
547Dic3—22no—
548Dic3—42no—
549Dic3—41no—
550Dic3—81no—
551Dic3—81no—
552Dic3—81no—
553Dic3#46514yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—32no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23—61no—
567C23D3×SD1612yes—
568C23C2×D3×D411yescharacteristic
569Q8—62no—
570Q8—62no—
571Q8—121no—
572Q8—121no—
573Q8—244no—
574D4—68no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—32no—
581C2×C4—32no—
582C2×C4—41no—
583C2×C4—41no—
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—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—62no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
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—121no—
619C2×C4C2×D3×D411yescharacteristic
620C2×C4D3×#4111yescharacteristic
621C2×C4D3×#4111yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C2×C4#100911yescharacteristic
624C2×C4#101011yescharacteristic
625C8—22no—
626C8—41no—
627C8—62no—
628C8—62no—
629C2×C3—42no—
630C2×C3#242711yescharacteristic
631C2×C3C2×#46511yescharacteristic
632C2×C3#306311yescharacteristic
633C2×C3#329811yescharacteristic
634C2×C3#330411yescharacteristic
635C2×C3#333511yescharacteristic
636C2×C3#334211yescharacteristic
637D3—42no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—62no—
646C22—62no—
647C22—62no—
648C22—121no—
649C22—121no—
650C22—121no—
651C22—121no—
652C22D3×#12111yescharacteristic
653C22#565411yescharacteristic
654C22#565511yescharacteristic
655C22C2×D3×SD1611yescharacteristic
656C22D3×#13511yescharacteristic
657C22#577511yescharacteristic
658C22#577911yescharacteristic
659C4—34no—
660C4—41no—
661C4—62no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—121no—
667C4—122no—
668C4—241no—
669C4—241no—
670C4—241no—
671C4#565411yescharacteristic
672C4#567711yescharacteristic
673C4#570611yescharacteristic
674C4#571011yescharacteristic
675C3#2534111yescharacteristic, complemented, Sylow
676C2—42no—
677C2—122no—
678C2#7746611yescharacteristic
679C2D3×#46511yescharacteristic
680C2#8081011yescharacteristic
681C2#8132011yescharacteristic
682C2#8140611yescharacteristic
683C2#8145011yescharacteristic
684C2#8153311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2077C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#2077D3
hypocenterC3#25341
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2077

Complements

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

NH ≅ G/Ntype
1. C2×#4854C2semidirect
5. C3×#2077C2semidirect
6. #70577C2semidirect
7. #69936C2semidirect
8. #69514C2semidirect
9. #69511C2semidirect
10. #68882C2semidirect
11. #68833C2semidirect
12. #68824C2semidirect
13. #68803C2semidirect
14. #68796C2semidirect
15. #68781C2semidirect
67. #2077D3semidirect
675. C3#25341semidirect

Maximal subgroups

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

subgroupindex
C2×#48542
C2×#67⋊C82
#739372
#739362
C3×#20772
#705772
#699362
#695142
#695112
#688822
#688332
#688242
#688032
#687962
#687812
#253413