Finite Group Databasebeta

← #10702178

Subgroups — #10702178

Order 768.

910 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2316C211yesmaximal, characteristic, complemented
2C2×#5006C211yesmaximal, characteristic
3C2×#5005C211yesmaximal, characteristic
4C2×#4966C211yesmaximal, characteristic
5C2×#4533C211yesmaximal, characteristic, complemented
6C2×#4531C211yesmaximal, characteristic, complemented
7#71558C211yesmaximal, characteristic, complemented
8#11057—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C3×#389—22no—
17C3×#389—22no—
18C3×#380—22no—
19#5006—22no—
20#5006—22no—
21#5005—22no—
22#5005—22no—
23C2×#810C412yescomplemented
24#4966—22no—
25#4966—22no—
26C2×#809C412yescomplemented
27C2×#806—22no—
28C2×#805—22no—
29#4533—22no—
30#4531—22no—
31#4529—22no—
32C2×#421—31no—
33C2×#417—31no—
34C2×#410—31no—
35#2316D311yescharacteristic, complemented
36C2×#344—31no—
37#1736—31no—
38C2×H(3,ℤ/4)—31no—
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
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
57C3×#117—21no—
58C3×#117—21no—
59C3×#117—21no—
60C3×#117—21no—
61C3×#117—21no—
62C3×#117—21no—
63C3×#117—21no—
64C3×#117—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4C2×C412yes—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4D412yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4D412yes—
76C2×C3×C42—21no—
77C2×C3×C42D412yes—
78C2×#217—21no—
79#926—21no—
80#926—21no—
81#926—21no—
82#926D412yes—
83C2×C4⋊Dic3—21no—
84C2×C4×Dic3—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208D412yes—
97C22×C3⋊C8—21no—
98C22×C3⋊C8C2×C412yes—
99C3×#95—24no—
100#810—24no—
101#809—24no—
102#806—28no—
103#805—28no—
104C22×#41—31no—
105C22×M16—31no—
106C22×M16—31no—
107C2×#117—31no—
108C2×#117C2×D311yescharacteristic
109C2×#117C2×D311yescharacteristic
110C22×C4⋊C4C2×D311yescharacteristic
111#421—62no—
112#421—62no—
113C2×#104—62no—
114#417—62no—
115#417—62no—
116C2×#103—32no—
117C2×#102—32no—
118#410—62no—
119#410—62no—
120C2×#99—62no—
121#389—22no—
122#389—22no—
123#380—22no—
124#344—62no—
125#340—62no—
126H(3,ℤ/4)—62no—
127C24×C3C2×D411yescharacteristic
128C23×D3—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—82no—
135C22×C3×C4C22×C411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×Q811yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C2×#67—22no—
150C2×#67—22no—
151C2×#67—41no—
152C2×#67—41no—
153C2×#67—41no—
154C2×#67—41no—
155C22×Dic3—21no—
156C22×Dic3—41no—
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×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—41no—
195C3×C4⋊C4—41no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—41no—
198C3×C4⋊C4—84no—
199C3×C4⋊C4C22⋊C414yes—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—41no—
203C3×C22⋊C4—41no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—41no—
207C3×C42—41no—
208#217—22no—
209#217—41no—
210C4⋊Dic3—22no—
211C4⋊Dic3—41no—
212C4×Dic3—22no—
213C4×Dic3—41no—
214#208—22no—
215#208—22no—
216#208—41no—
217#208—41no—
218#208—41no—
219#208—41no—
220#208—41no—
221#208—41no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—41no—
227C2×C3⋊C8—41no—
228C2×#41—32no—
229C2×#41—32no—
230C2×#41—61no—
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—
240C22×Q8—31no—
241C22×D4—31no—
242C22×D4—61no—
243C23×C4—21no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×M16—32no—
247C2×M16—61no—
248C2×M16—61no—
249C2×M16—61no—
250C2×M16—61no—
251C2×M16—61no—
252C2×M16—61no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C22×C8—32no—
259C22×C8—61no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—32no—
269#117—61no—
270#117—61no—
271#117—61no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4C4×D312yes—
280C2×C4⋊C4D1212yes—
281C2×C4⋊C4#6712yes—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4#6712yes—
285C2×C42—21no—
286C2×C42—61no—
287C2×C42#6712yes—
288#104—68no—
289#103—64no—
290#102—64no—
291#99—68no—
292#95—24no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3C4×D411yescharacteristic
296C23×C3#12011yescharacteristic
297C23×C3#12011yescharacteristic
298C23×C3#12111yescharacteristic
299C23×C3#12111yescharacteristic
300C23×C3C22⋊Q811yescharacteristic
301C23×C3#12411yescharacteristic
302C22×D3—22no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—81no—
306C22×D3—81no—
307C2×C3×C4—24no—
308C2×C3×C4—44no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—82no—
316C2×C3×C4—82no—
317C2×C3×C4C2×C22⋊C411yescharacteristic
318C2×C3×C4C2×C4⋊C411yescharacteristic
319C2×C3×C4#11711yescharacteristic
320C2×C3×C4C4×D411yescharacteristic
321C2×C3×C4C4×D411yescharacteristic
322C2×C3×C4C4×D411yescharacteristic
323C2×C3×C4#12011yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4#12111yescharacteristic
327C2×C3×C4#12111yescharacteristic
328C2×C3×C4#12111yescharacteristic
329C2×C3×C4#12111yescharacteristic
330C2×C3×C4C22⋊Q811yescharacteristic
331C2×C3×C4C22⋊Q811yescharacteristic
332C2×C3×C4C22⋊Q811yescharacteristic
333C2×C3×C4C22⋊Q811yescharacteristic
334C2×C3×C4C22⋊Q811yescharacteristic
335C2×C3×C4#12311yescharacteristic
336C2×C3×C4#12311yescharacteristic
337C2×C3×C4#12311yescharacteristic
338C2×C3×C4#12311yescharacteristic
339C2×C3×C4#12311yescharacteristic
340C2×C3×C4#12411yescharacteristic
341C2×C3×C4#12411yescharacteristic
342C2×C3×C4#12611yescharacteristic
343C2×C3×C4#12611yescharacteristic
344C2×C3×C4#12711yescharacteristic
345#67—42no—
346#67—42no—
347#67—42no—
348#67—42no—
349C2×Dic3—22no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—42no—
353C2×Dic3—81no—
354C2×Dic3—81no—
355D12—22no—
356D12—22no—
357D12—41no—
358D12—41no—
359C4×D3—42no—
360C4×D3—42no—
361C4×D3—42no—
362C4×D3—42no—
363Dic6—22no—
364Dic6—22no—
365Dic6—41no—
366Dic6—41no—
367C3⋊C8—42no—
368C3⋊C8—42no—
369C24—61no—
370C24C2×#6711yescharacteristic
371#41—62no—
372#41—62no—
373#41—62no—
374#41—62no—
375#41—62no—
376#41—62no—
377#41—62no—
378#41—62no—
379#41—121no—
380#41—121no—
381#41—121no—
382#41—121no—
383#41—121no—
384#41—121no—
385#41—121no—
386#41—121no—
387C2×Q8—32no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×D4—32no—
394C2×D4—62no—
395C2×D4—62no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—121no—
402C2×D4—121no—
403C2×D4—121no—
404C2×D4—121no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—82no—
414C22×C4—121no—
415C22×C4—121no—
416C22×C4—121no—
417C22×C4—121no—
418C22×C4—121no—
419C22×C4C2×C4×D311yescharacteristic
420C22×C4C2×D1211yescharacteristic
421C22×C4#23511yescharacteristic
422C22×C4#23511yescharacteristic
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4D3×D411yescharacteristic
426C22×C4#23711yescharacteristic
427C22×C4D3×Q811yescharacteristic
428C22×C4#23911yescharacteristic
429C22×C4#23911yescharacteristic
430C22×C4#23911yescharacteristic
431C22×C4C2×#6711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433M16—62no—
434M16—62no—
435M16—121no—
436M16—121no—
437M16—121no—
438M16—121no—
439M16—121no—
440M16—121no—
441C2×C8—62no—
442C2×C8—62no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—121no—
446C2×C8—121no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—62no—
455C4⋊C4—84no—
456C4⋊C4—121no—
457C4⋊C4(C2×D3)⋊C414yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—62no—
463C22⋊C4—121no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—62no—
469C42—121no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3#38511yescharacteristic
481C22×C3#38611yescharacteristic
482C22×C3#38811yescharacteristic
483C22×C3#39211yescharacteristic
484C22×C3#39311yescharacteristic
485C22×C3#39411yescharacteristic
486C22×C3#39611yescharacteristic
487C2×D3—42no—
488C2×D3—82no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—82no—
492C3×C4—44no—
493C3×C4—82no—
494C3×C4—82no—
495C3×C4#38011yescharacteristic
496C3×C4#38511yescharacteristic
497C3×C4#38811yescharacteristic
498C3×C4#39011yescharacteristic
499C3×C4#39411yescharacteristic
500C3×C4#39611yescharacteristic
501C3×C4#39711yescharacteristic
502C3×C4#39911yescharacteristic
503Dic3—42no—
504Dic3—82no—
505C23—41no—
506C23—41no—
507C23—62no—
508C23—121no—
509C23—121no—
510C23—241no—
511C23—241no—
512C23C4×#6711yescharacteristic
513C23#93011yescharacteristic
514C23#93711yescharacteristic
515C23#93911yescharacteristic
516C23#94611yescharacteristic
517C23#94711yescharacteristic
518C23#95311yescharacteristic
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—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—62no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—241no—
550C2×C4—241no—
551C2×C4D3×C4⋊C411yescharacteristic
552C2×C4#89211yescharacteristic
553C2×C4#89311yescharacteristic
554C2×C4#89311yescharacteristic
555C2×C4#89411yescharacteristic
556C2×C4#89411yescharacteristic
557C2×C4#89411yescharacteristic
558C2×C4#89411yescharacteristic
559C2×C4#89511yescharacteristic
560C2×C4#89511yescharacteristic
561C2×C4#89511yescharacteristic
562C2×C4#89611yescharacteristic
563C2×C4#89611yescharacteristic
564C2×C4#89711yescharacteristic
565C2×C4#89811yescharacteristic
566C2×C4#89811yescharacteristic
567C2×C4C2×(C2×D3)⋊C411yescharacteristic
568C2×C4C4×#6711yescharacteristic
569C2×C4#92911yescharacteristic
570C2×C4#93011yescharacteristic
571C2×C4#93711yescharacteristic
572C2×C4#93811yescharacteristic
573C2×C4#93911yescharacteristic
574C2×C4#94011yescharacteristic
575C2×C4#94611yescharacteristic
576C2×C4#94611yescharacteristic
577C2×C4#94711yescharacteristic
578C2×C4#94711yescharacteristic
579C8—122no—
580C8—122no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3#143911yescharacteristic
588C2×C3#190411yescharacteristic
589C2×C3#190511yescharacteristic
590D3—82no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—122no—
602C22—242no—
603C22—242no—
604C22—241no—
605C22—241no—
606C22#498911yescharacteristic
607C22#499011yescharacteristic
608C22#499111yescharacteristic
609C22#499411yescharacteristic
610C22#521311yescharacteristic
611C22#522311yescharacteristic
612C22#523111yescharacteristic
613C4—44no—
614C4—82no—
615C4—82no—
616C4—122no—
617C4—242no—
618C4#498711yescharacteristic
619C4#498811yescharacteristic
620C4#498911yescharacteristic
621C4#499011yescharacteristic
622C4#499111yescharacteristic
623C4#499211yescharacteristic
624C4#499311yescharacteristic
625C4#499411yescharacteristic
626C3#1105711yescharacteristic, complemented, Sylow
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—242no—
634C2#7157011yescharacteristic
635C2#7165111yescharacteristic
636C2#7166211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2316C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#2316D3
hypocenterC3#11057
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
1. C3×#2316C2semidirect
5. C2×#4533C2semidirect
6. C2×#4531C2semidirect
7. #71558C2semidirect
23. C2×#810C4semidirect
26. C2×#809C4semidirect
35. #2316D3semidirect
178. C2×D12C4⋊C4semidirect
190. C2×Dic6C4⋊C4semidirect
626. C3#11057semidirect

Maximal subgroups

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

subgroupindex
C3×#23162
C2×#50062
C2×#50052
C2×#49662
C2×#45332
C2×#45312
#715582
#110573