Finite Group Databasebeta

← #10702279

Subgroups — #10702279

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5139C211yesmaximal, characteristic
2#82270C211yesmaximal, characteristic
3#82232C211yesmaximal, characteristic, complemented
4C2×#5005C211yesmaximal, characteristic
5C2×C3×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
6C2×#4554C211yesmaximal, characteristic, complemented
7C2×#4531C211yesmaximal, characteristic, complemented
8#11072—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5139—22no—
17#5139—22no—
18C2×#825—22no—
19#5005—22no—
20#5005—22no—
21#4987—22no—
22#4972—22no—
23#4958—22no—
24#4942—22no—
25#4939C412yescomplemented
26#4929C412yescomplemented
27#4928—22no—
28C2×#805—22no—
29C3×H(3,ℤ/4)—22no—
30#4554—22no—
31#4531—22no—
32C2×#421—31no—
33C2×#421—31no—
34#2316—31no—
35#2311—31no—
36C2×#343—31no—
37C2×H(3,ℤ/4)—31no—
38C2×H(3,ℤ/4)D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C23×Dic3D412yes—
42C2×#235—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×#235D412yes—
53C2×#235Q812yes—
54C22×D12C2×C411yescharacteristic
55C22×C4×D3—21no—
56C22×Dic6C2×C411yescharacteristic
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16D412yes—
63C22×C3×C8—21no—
64C3×#117—21no—
65C3×#117—21no—
66C3×#117—21no—
67C3×#117—21no—
68C3×#117—21no—
69C3×#117—21no—
70C3×#117—21no—
71C3×#117—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42C2×C412yes—
77C2×C3×C42D412yes—
78C2×#217—21no—
79C2×(C2×D3)⋊C4—24no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2×C412yes—
90C2×C4⋊Dic3D412yes—
91C2×Dic3⋊C4—24no—
92C2×C4×Dic3D412yes—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208—21no—
98C2×#208D412yes—
99C22×C3⋊C8—21no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102#825—28no—
103#805—28no—
104C22×#41—31no—
105C22×M16—31no—
106C22×M16C2×D311yescharacteristic
107C2×#117—31no—
108C2×#117C2×D311yescharacteristic
109C2×#117C2×D311yescharacteristic
110C22×C4⋊C4—31no—
111#421—62no—
112#421—62no—
113#421—62no—
114#421—62no—
115C2×#104—62no—
116C2×#104—62no—
117#391—32no—
118#390—32no—
119#389—62no—
120#389—62no—
121#386—62no—
122#380—62no—
123#380—62no—
124#343—62no—
125H(3,ℤ/4)—22no—
126H(3,ℤ/4)—62no—
127C24×C3C2×D411yescharacteristic
128C23×D3—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×Q811yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C2×#67—22no—
146C2×#67—22no—
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—41no—
150C2×#67—41no—
151C22×Dic3—21no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—82no—
156C22×Dic3C22⋊C414yes—
157#235—22no—
158#235—22no—
159#235—22no—
160#235—22no—
161#235—22no—
162#235—22no—
163#235—22no—
164#235—22no—
165#235—41no—
166#235—41no—
167#235—41no—
168#235—41no—
169#235—41no—
170#235—41no—
171#235—41no—
172#235—41no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12C4⋊C412yescomplemented
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—41no—
182C2×C4×D3—41no—
183C2×C4×D3—41no—
184C2×C4×D3—41no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6C4⋊C412yescomplemented
191C3×M16—22no—
192C3×M16—22no—
193C3×M16—41no—
194C3×M16—41no—
195C2×C3×C8—22no—
196C2×C3×C8—41no—
197C2×C3×C8—41no—
198C3×C4⋊C4—41no—
199C3×C4⋊C4—41no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—41no—
202C3×C22⋊C4—41no—
203C3×C22⋊C4—41no—
204C3×C22⋊C4—41no—
205C3×C22⋊C4—41no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42—22no—
210#217—41no—
211#217—41no—
212(C2×D3)⋊C4—84no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—41no—
218C4⋊Dic3—41no—
219Dic3⋊C4—84no—
220C4×Dic3—22no—
221C4×Dic3—22no—
222#208—22no—
223#208—22no—
224#208—41no—
225#208—41no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—41no—
228C2×C3⋊C8—41no—
229C2×#41—32no—
230C2×#41—32no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—61no—
234C2×#41—61no—
235C2×#41—61no—
236C2×#41—61no—
237C2×#41—61no—
238C2×#41—61no—
239C2×#41—61no—
240C2×#41—61no—
241C22×Q8—31no—
242C22×D4—31no—
243C22×D4—61no—
244C23×C4—32no—
245C23×C4—61no—
246C23×C4C22×D311yescharacteristic
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—32no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16#6712yes—
259C22×C8—21no—
260C22×C8—61no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—61no—
270#117—61no—
271#117—61no—
272#117—61no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—32no—
276C2×C4⋊C4—32no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—64no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—64no—
287C2×C42—32no—
288C2×C42C4×D312yes—
289C2×C42D1212yes—
290#104—68no—
291#104—68no—
292#95—64no—
293#95—64no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3C2×C22⋊C411yescharacteristic
297C23×C3#12011yescharacteristic
298C23×C3#12111yescharacteristic
299C23×C3C22⋊Q811yescharacteristic
300C23×C3#12311yescharacteristic
301C23×C3#12411yescharacteristic
302C23×C3#12711yescharacteristic
303C22×D3—22no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—41no—
307C22×D3—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4C2×C4⋊C411yescharacteristic
315C2×C3×C4#11711yescharacteristic
316C2×C3×C4C4×D411yescharacteristic
317C2×C3×C4C4×D411yescharacteristic
318C2×C3×C4C4×D411yescharacteristic
319C2×C3×C4C4×D411yescharacteristic
320C2×C3×C4#12011yescharacteristic
321C2×C3×C4#12011yescharacteristic
322C2×C3×C4#12111yescharacteristic
323C2×C3×C4#12111yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4#12111yescharacteristic
327C2×C3×C4#12111yescharacteristic
328C2×C3×C4#12111yescharacteristic
329C2×C3×C4C22⋊Q811yescharacteristic
330C2×C3×C4C22⋊Q811yescharacteristic
331C2×C3×C4C22⋊Q811yescharacteristic
332C2×C3×C4C22⋊Q811yescharacteristic
333C2×C3×C4C22⋊Q811yescharacteristic
334C2×C3×C4#12311yescharacteristic
335C2×C3×C4#12311yescharacteristic
336C2×C3×C4#12311yescharacteristic
337C2×C3×C4#12311yescharacteristic
338C2×C3×C4#12411yescharacteristic
339C2×C3×C4#12411yescharacteristic
340C2×C3×C4#12611yescharacteristic
341C2×C3×C4#12611yescharacteristic
342#67—42no—
343#67—42no—
344#67—42no—
345#67—42no—
346C2×Dic3—22no—
347C2×Dic3—24no—
348C2×Dic3—44no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—44no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—82no—
356D12—22no—
357D12—22no—
358D12—41no—
359D12—41no—
360C4×D3—42no—
361C4×D3—42no—
362C4×D3—42no—
363C4×D3—42no—
364Dic6—22no—
365Dic6—22no—
366Dic6—41no—
367Dic6—41no—
368C3×C8—42no—
369C3⋊C8—42no—
370C24—61no—
371C24D3×D411yescharacteristic
372#41—62no—
373#41—62no—
374#41—62no—
375#41—62no—
376#41—62no—
377#41—62no—
378#41—62no—
379#41—62no—
380#41—121no—
381#41—121no—
382#41—121no—
383#41—121no—
384#41—121no—
385#41—121no—
386#41—121no—
387#41—121no—
388C2×Q8—32no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×D4—32no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—121no—
403C2×D4—121no—
404C2×D4—121no—
405C2×D4—121no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—34no—
409C22×C4—61no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—121no—
416C22×C4—121no—
417C22×C4—121no—
418C22×C4—121no—
419C22×C4—242no—
420C22×C4C2×C4×D311yescharacteristic
421C22×C4C2×D1211yescharacteristic
422C22×C4#23511yescharacteristic
423C22×C4#23511yescharacteristic
424C22×C4#23511yescharacteristic
425C22×C4#23511yescharacteristic
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4#23711yescharacteristic
431C22×C4D3×Q811yescharacteristic
432C22×C4#23911yescharacteristic
433C22×C4C2×#6711yescharacteristic
434M16—22no—
435M16—22no—
436M16—41no—
437M16—41no—
438M16—62no—
439M16—62no—
440M16—121no—
441M16—121no—
442C2×C8—22no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—62no—
446C2×C8—121no—
447C2×C8—121no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4—62no—
456C4⋊C4—121no—
457C4⋊C4—121no—
458C4⋊C4—244no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—121no—
464C22⋊C4—121no—
465C22⋊C4—244no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—62no—
471C42—62no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3#38011yescharacteristic
483C22×C3#38611yescharacteristic
484C22×C3#39011yescharacteristic
485C22×C3#39211yescharacteristic
486C22×C3#39311yescharacteristic
487C22×C3#39711yescharacteristic
488C22×C3#39911yescharacteristic
489C2×D3—44no—
490C2×D3—42no—
491C2×D3—82no—
492C2×D3—81no—
493C2×D3—81no—
494C3×C4—82no—
495C3×C4—82no—
496C3×C4#38511yescharacteristic
497C3×C4#38511yescharacteristic
498C3×C4#38811yescharacteristic
499C3×C4#38811yescharacteristic
500C3×C4#39411yescharacteristic
501C3×C4#39411yescharacteristic
502C3×C4#39611yescharacteristic
503C3×C4#39611yescharacteristic
504Dic3—44no—
505Dic3—42no—
506Dic3—82no—
507C23—41no—
508C23—41no—
509C23—62no—
510C23—122no—
511C23—122no—
512C23—121no—
513C23—121no—
514C23D3×C22⋊C411yescharacteristic
515C23#88211yescharacteristic
516C23#88411yescharacteristic
517C23#88511yescharacteristic
518C23#89411yescharacteristic
519C23#89611yescharacteristic
520C23#94011yescharacteristic
521Q8—62no—
522Q8—62no—
523Q8—121no—
524Q8—121no—
525D4—62no—
526D4—62no—
527D4—122no—
528D4—122no—
529D4—122no—
530D4—122no—
531D4—121no—
532D4—121no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—64no—
540C2×C4—62no—
541C2×C4—124no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—124no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—242no—
553C2×C4#87211yescharacteristic
554C2×C4C4×D1211yescharacteristic
555C2×C4#87511yescharacteristic
556C2×C4#87611yescharacteristic
557C2×C4#67⋊C411yescharacteristic
558C2×C4#88211yescharacteristic
559C2×C4#88311yescharacteristic
560C2×C4#88411yescharacteristic
561C2×C4#88411yescharacteristic
562C2×C4#88411yescharacteristic
563C2×C4#88411yescharacteristic
564C2×C4#88511yescharacteristic
565C2×C4D3×C4⋊C411yescharacteristic
566C2×C4#89311yescharacteristic
567C2×C4#89411yescharacteristic
568C2×C4#89411yescharacteristic
569C2×C4#89511yescharacteristic
570C2×C4#89611yescharacteristic
571C2×C4#89611yescharacteristic
572C2×C4#89611yescharacteristic
573C2×C4#89711yescharacteristic
574C2×C4#89811yescharacteristic
575C2×C4C4×#6711yescharacteristic
576C2×C4#92911yescharacteristic
577C2×C4#93011yescharacteristic
578C2×C4#93711yescharacteristic
579C2×C4#93911yescharacteristic
580C2×C4#94611yescharacteristic
581C8—42no—
582C8—122no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3#143911yescharacteristic
590C2×C3#191111yescharacteristic
591C2×C3#191211yescharacteristic
592D3—82no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—124no—
604C22—122no—
605C22—242no—
606C22—241no—
607C22—241no—
608C22#466511yescharacteristic
609C22#466711yescharacteristic
610C22#467311yescharacteristic
611C22#467511yescharacteristic
612C22#495711yescharacteristic
613C22#496011yescharacteristic
614C22#499311yescharacteristic
615C4—82no—
616C4—82no—
617C4—122no—
618C4—124no—
619C4—242no—
620C4#466811yescharacteristic
621C4#466911yescharacteristic
622C4#467211yescharacteristic
623C4#467611yescharacteristic
624C4#494011yescharacteristic
625C4#495911yescharacteristic
626C4#498911yescharacteristic
627C4#499411yescharacteristic
628C3#1107211yescharacteristic, complemented, Sylow
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—242no—
636C2#6870511yescharacteristic
637C2#6870811yescharacteristic
638C2#6942811yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×H(3,ℤ/4)C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×H(3,ℤ/4)D3
hypocenterC3#11072
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
3. #82232C2semidirect
5. C2×C3×H(3,ℤ/4)C2semidirect
6. C2×#4554C2semidirect
7. C2×#4531C2semidirect
25. #4939C4semidirect
26. #4929C4semidirect
38. C2×H(3,ℤ/4)D3semidirect
178. C2×D12C4⋊C4semidirect
190. C2×Dic6C4⋊C4semidirect
628. C3#11072semidirect

Maximal subgroups

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

subgroupindex
C2×#51392
#822702
#822322
C2×#50052
C2×C3×H(3,ℤ/4)2
C2×#45542
C2×#45312
#110723