Finite Group Databasebeta

← #10701553

Subgroups — #10701553

Order 768.

930 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5006C211yesmaximal, characteristic
2C2×#5005C211yesmaximal, characteristic
3C2×#5002C211yesmaximal, characteristic
4C2×#4997C211yesmaximal, characteristic, complemented
5C3×#1740C211yesmaximal, characteristic, complemented
6#71580C211yesmaximal, characteristic, complemented
7#71574C211yesmaximal, characteristic, complemented
8#10944—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C23×C3⋊C8C2211yescharacteristic
16#5006—22no—
17#5006C414yescomplemented
18#5005—22no—
19#5005C414yescomplemented
20#5002—21no—
21#5002—21no—
22#5002—22no—
23#4997—22no—
24#4997—22no—
25C2×#810C412yescomplemented
26C2×#809C412yescomplemented
27C2×#806—22no—
28C2×#805—22no—
29C2×#803—21no—
30C2×C4×C3⋊C8—21no—
31C3×#342—22no—
32#4532—22no—
33#4529—22no—
34C2×#421—31no—
35C2×#416—31no—
36C2×#410—31no—
37C2×#405—31no—
38#1740D311yescharacteristic, complemented
39#1735—31no—
40#1733—31no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235—21no—
53C2×#235C2×C412yes—
54C2×#235C2×C412yes—
55C22×D12C2×C411yescharacteristic
56C22×C4×D3—21no—
57C22×Dic6C2×C411yescharacteristic
58C3×#117—21no—
59C3×#117—21no—
60C3×#117—21no—
61C3×#117—21no—
62C3×#117—21no—
63C3×#117—21no—
64C3×#117—21no—
65C3×#117—21no—
66C3×#117—21no—
67C3×#117C2×C412yes—
68C3×#117C2×C412yes—
69C3×#117D412yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2×C412yes—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4D412yes—
76C2×C3×C42—21no—
77C2×C3×C42—21no—
78C2×C3×C42—21no—
79C2×#217—21no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926C2×C412yes—
84C2×C4⋊Dic3—21no—
85C2×C4×Dic3—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208C2×C412yes—
90C2×#208D412yes—
91C2×#208D412yes—
92C22×C3⋊C8—21no—
93C22×C3⋊C8—21no—
94C22×C3⋊C8—21no—
95C22×C3⋊C8—21no—
96C22×C3⋊C8—21no—
97C22×C3⋊C8C2×C412yes—
98C22×C3⋊C8D411yescharacteristic
99C22×C3⋊C8D411yescharacteristic
100C22×C3⋊C8D412yes—
101#810—22no—
102#810—22no—
103#809—22no—
104#809—22no—
105#806—28no—
106#805—28no—
107#803—24no—
108C4×C3⋊C8—24no—
109C22×#41—31no—
110C22×M16—31no—
111C23×C8—31no—
112C2×#117—31no—
113C2×#117C2×D311yescharacteristic
114C2×#117C2×D311yescharacteristic
115C2×#117C2×D311yescharacteristic
116#421—34no—
117#421—62no—
118C2×#104—62no—
119#416—62no—
120#416—61no—
121#416—61no—
122C2×#103—32no—
123C2×#102—32no—
124#410—34no—
125#410—62no—
126C2×#99—62no—
127#405—62no—
128#405—62no—
129C2×#97—61no—
130C2×C4×C8—61no—
131#342—22no—
132#340—62no—
133#339—62no—
134C24×C3C2×D411yescharacteristic
135C23×D3—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—41no—
139C22×C3×C4C22×C411yescharacteristic
140C22×C3×C4C22×C411yescharacteristic
141C22×C3×C4C22×C411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C22×C3×C4#4111yescharacteristic
151C22×C3×C4#4111yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C2×#67—22no—
154C2×#67—22no—
155C2×#67—41no—
156C2×#67—41no—
157C2×#67—41no—
158C2×#67—41no—
159C22×Dic3—21no—
160C22×Dic3—41no—
161#235—22no—
162#235—22no—
163#235—22no—
164#235—22no—
165#235—22no—
166#235—22no—
167#235—22no—
168#235—22no—
169#235—41no—
170#235—41no—
171#235—41no—
172#235—41no—
173#235—41no—
174#235—41no—
175#235—41no—
176#235—41no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12C4212yes—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—41no—
186C2×C4×D3—41no—
187C2×C4×D3—41no—
188C2×C4×D3—41no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6C4212yes—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—41no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—41no—
204C3×C22⋊C4—41no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4C22⋊C414yes—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42—41no—
211C3×C42—41no—
212C3×C42—41no—
213#217—22no—
214#217—41no—
215C4⋊Dic3—22no—
216C4⋊Dic3—41no—
217C4×Dic3—22no—
218C4×Dic3—41no—
219#208—22no—
220#208—22no—
221#208—22no—
222#208C22⋊C414yes—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8C22⋊C412yes—
235C2×C3⋊C8C22⋊C412yes—
236C2×#41—32no—
237C2×#41—32no—
238C2×#41—61no—
239C2×#41—61no—
240C2×#41—61no—
241C2×#41—61no—
242C2×#41—61no—
243C2×#41—61no—
244C2×#41—61no—
245C2×#41—61no—
246C2×#41—61no—
247C2×#41—61no—
248C22×Q8—31no—
249C22×D4—31no—
250C22×D4—61no—
251C23×C4—61no—
252C23×C4C22×D311yescharacteristic
253C2×M16—32no—
254C2×M16—32no—
255C2×M16—32no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16—61no—
259C22×C8—32no—
260C22×C8—32no—
261C22×C8—31no—
262C22×C8—31no—
263C22×C8—61no—
264C22×C8—61no—
265C22×C8—61no—
266C22×C8—61no—
267C22×C8—61no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—32no—
278#117—61no—
279#117—61no—
280#117—61no—
281#117C4×D312yes—
282#117C4×D312yes—
283#117D1212yes—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4C4×D312yes—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4#6712yes—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—61no—
296#104—68no—
297#103—62no—
298#103—62no—
299#102—62no—
300#102—62no—
301#99—68no—
302#97—64no—
303C4×C8—64no—
304C23×C3—42no—
305C23×C3C2×C22⋊C411yescharacteristic
306C23×C3C4×D411yescharacteristic
307C23×C3C4×D411yescharacteristic
308C23×C3#12111yescharacteristic
309C23×C3#12111yescharacteristic
310C23×C3#12411yescharacteristic
311C23×C3#12711yescharacteristic
312C22×D3—22no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—82no—
316C2×C3×C4—24no—
317C2×C3×C4—44no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—82no—
321C2×C3×C4—82no—
322C2×C3×C4C2×C4211yescharacteristic
323C2×C3×C4C2×C22⋊C411yescharacteristic
324C2×C3×C4C2×C22⋊C411yescharacteristic
325C2×C3×C4#11711yescharacteristic
326C2×C3×C4#11711yescharacteristic
327C2×C3×C4#11711yescharacteristic
328C2×C3×C4C4×D411yescharacteristic
329C2×C3×C4C4×D411yescharacteristic
330C2×C3×C4C4×D411yescharacteristic
331C2×C3×C4C4×D411yescharacteristic
332C2×C3×C4C4×D411yescharacteristic
333C2×C3×C4C4×D411yescharacteristic
334C2×C3×C4C4×D411yescharacteristic
335C2×C3×C4C4×D411yescharacteristic
336C2×C3×C4C4×D411yescharacteristic
337C2×C3×C4C4×D411yescharacteristic
338C2×C3×C4#12011yescharacteristic
339C2×C3×C4#12111yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12311yescharacteristic
344C2×C3×C4#12311yescharacteristic
345C2×C3×C4#12311yescharacteristic
346C2×C3×C4#12411yescharacteristic
347C2×C3×C4#12411yescharacteristic
348C2×C3×C4#12611yescharacteristic
349C2×C3×C4#12611yescharacteristic
350#67—42no—
351#67—42no—
352#67—42no—
353#67—42no—
354C2×Dic3—22no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—42no—
358C2×Dic3—82no—
359D12—22no—
360D12—22no—
361D12—41no—
362D12—41no—
363C4×D3—42no—
364C4×D3—42no—
365C4×D3—42no—
366C4×D3—42no—
367Dic6—22no—
368Dic6—22no—
369Dic6—41no—
370Dic6—41no—
371C3⋊C8—22no—
372C3⋊C8—22no—
373C3⋊C8—24no—
374C24—61no—
375C24C2×#6711yescharacteristic
376#41—62no—
377#41—62no—
378#41—62no—
379#41—62no—
380#41—62no—
381#41—62no—
382#41—62no—
383#41—62no—
384#41—121no—
385#41—121no—
386#41—121no—
387#41—121no—
388#41—121no—
389#41—121no—
390#41—121no—
391#41—121no—
392C2×Q8—32no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×D4—32no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—121no—
407C2×D4—121no—
408C2×D4—121no—
409C2×D4—121no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—61no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—121no—
417C22×C4—121no—
418C22×C4—121no—
419C22×C4—121no—
420C22×C4—121no—
421C22×C4C2×C4×D311yescharacteristic
422C22×C4C2×C4×D311yescharacteristic
423C22×C4C2×C4×D311yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4#23511yescharacteristic
426C22×C4#23511yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4#23711yescharacteristic
432C22×C4#23711yescharacteristic
433C22×C4#23911yescharacteristic
434C22×C4#23911yescharacteristic
435M16—34no—
436M16—62no—
437M16—62no—
438M16—62no—
439C2×C8—32no—
440C2×C8—32no—
441C2×C8—62no—
442C2×C8—61no—
443C2×C8—62no—
444C2×C8—61no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—62no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—41no—
458C4⋊C4—62no—
459C4⋊C4—121no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—62no—
466C22⋊C4—121no—
467C22⋊C4(C2×D3)⋊C414yes—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—62no—
475C42—121no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3C4×C22⋊C411yescharacteristic
484C22×C3#38611yescharacteristic
485C22×C3#38811yescharacteristic
486C22×C3#38811yescharacteristic
487C22×C3#39011yescharacteristic
488C22×C3#39011yescharacteristic
489C22×C3#39011yescharacteristic
490C2×D3—42no—
491C2×D3—82no—
492C2×D3—82no—
493C2×D3—82no—
494C3×C4—44no—
495C3×C4—82no—
496C3×C4—82no—
497C3×C4C4×C22⋊C411yescharacteristic
498C3×C4C4×C22⋊C411yescharacteristic
499C3×C4#38611yescharacteristic
500C3×C4#38611yescharacteristic
501C3×C4#38811yescharacteristic
502C3×C4#38811yescharacteristic
503C3×C4#38811yescharacteristic
504C3×C4#38811yescharacteristic
505Dic3—42no—
506Dic3—82no—
507C23—42no—
508C23—62no—
509C23—121no—
510C23—121no—
511C23—242no—
512C23C2×(C2×D3)⋊C411yescharacteristic
513C23C4×#6711yescharacteristic
514C23C4×#6711yescharacteristic
515C23#93911yescharacteristic
516C23#93911yescharacteristic
517C23#94011yescharacteristic
518C23#94711yescharacteristic
519Q8—62no—
520Q8—62no—
521Q8—121no—
522Q8—121no—
523D4—62no—
524D4—62no—
525D4—122no—
526D4—122no—
527D4—122no—
528D4—122no—
529D4—121no—
530D4—121no—
531C2×C4—24no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—62no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—242no—
546C2×C4C42×D311yescharacteristic
547C2×C4#87211yescharacteristic
548C2×C4C4×D1211yescharacteristic
549C2×C4C4×D1211yescharacteristic
550C2×C4D3×C22⋊C411yescharacteristic
551C2×C4D3×C22⋊C411yescharacteristic
552C2×C4#67⋊C411yescharacteristic
553C2×C4#67⋊C411yescharacteristic
554C2×C4#67⋊C411yescharacteristic
555C2×C4#67⋊C411yescharacteristic
556C2×C4#88211yescharacteristic
557C2×C4#88411yescharacteristic
558C2×C4#88411yescharacteristic
559C2×C4#88511yescharacteristic
560C2×C4#88511yescharacteristic
561C2×C4#88611yescharacteristic
562C2×C4#89211yescharacteristic
563C2×C4#89211yescharacteristic
564C2×C4#89311yescharacteristic
565C2×C4#89311yescharacteristic
566C2×C4#89311yescharacteristic
567C2×C4#89311yescharacteristic
568C2×C4#89411yescharacteristic
569C2×C4#89411yescharacteristic
570C2×C4#89511yescharacteristic
571C2×C4#89511yescharacteristic
572C2×C4#89811yescharacteristic
573C2×C4#89811yescharacteristic
574C8—62no—
575C8—62no—
576C8—64no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3#143711yescharacteristic
584C2×C3#197612yes—
585D3—82no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—122no—
594C22—242no—
595C22—242no—
596C22—242no—
597C22C4×(C2×D3)⋊C411yescharacteristic
598C22#495711yescharacteristic
599C22#495711yescharacteristic
600C22#495811yescharacteristic
601C22#498811yescharacteristic
602C22#499011yescharacteristic
603C22#499011yescharacteristic
604C4—44no—
605C4—82no—
606C4—82no—
607C4—122no—
608C4—242no—
609C4#466511yescharacteristic
610C4#466511yescharacteristic
611C4#466611yescharacteristic
612C4#466611yescharacteristic
613C4#466911yescharacteristic
614C4#466911yescharacteristic
615C4#467011yescharacteristic
616C4#467011yescharacteristic
617C3#1094411yescharacteristic, complemented, Sylow
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—242no—
625C2#6940811yescharacteristic
626C2#7004212yes—
subgroupG/H
centerC2×C4#884
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC3×#1740C2
socleC22×C3#390
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1740D3
hypocenterC3#10944
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
4. C2×#4997C2semidirect
5. C3×#1740C2semidirect
6. #71580C2semidirect
7. #71574C2semidirect
17. #5006C4semidirect
19. #5005C4semidirect
25. C2×#810C4semidirect
26. C2×#809C4semidirect
38. #1740D3semidirect
617. C3#10944semidirect

Maximal subgroups

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

subgroupindex
C2×#50062
C2×#50052
C2×#50022
C2×#49972
C3×#17402
#715802
#715742
#109443