Finite Group Databasebeta

← #10855686

Subgroups — #10855686

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3097C211yesmaximal, characteristic, complemented
2#83406C211yesmaximal, characteristic
3#82558C211yesmaximal, characteristic
4#79179C211yesmaximal, characteristic, complemented
5#71863C211yesmaximal, characteristic, complemented
6#71665C211yesmaximal, characteristic, complemented
7#70681C211yesmaximal, characteristic, complemented
8#70671C211yesmaximal, characteristic, complemented
9#70661C211yesmaximal, characteristic, complemented
10#70348C211yesmaximal, characteristic, complemented
11#70338C211yesmaximal, characteristic, complemented
12#70286C211yesmaximal, characteristic, complemented
13#70251C211yesmaximal, characteristic, complemented
14#70011C211yesmaximal, characteristic
15#69994C211yesmaximal, characteristic, complemented
16#26193—31nomaximal, Sylow
17C3×#536C2211yescharacteristic
18C3×#520C2211yescharacteristic
19C2×#950C2211yescharacteristic
20C2×#942C2211yescharacteristic
21C2×#901C2211yescharacteristic
22#5523C2211yescharacteristic
23#5523C2211yescharacteristic
24C3×#485—21no—
25C3×#485—21no—
26C3×#484—21no—
27C3×#483—21no—
28C3×#428C2211yescharacteristic
29C3×#427C2211yescharacteristic
30C3×#419C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C3×#408C2211yescharacteristic
33C2×#837C2211yescharacteristic
34#5232C2211yescharacteristic
35#5232C2211yescharacteristic
36#5229C2211yescharacteristic
37C2×#836C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#825C2211yescharacteristic
40C2×#818C2211yescharacteristic
41#5090—21no—
42#5083—21no—
43#5057—21no—
44#5054—21no—
45#5030—21no—
46C4×#215—21no—
47C4×#214—21no—
48#5014—21no—
49#4999C2211yescharacteristic
50#4997C2211yescharacteristic
51#4995C2211yescharacteristic
52#4995C2211yescharacteristic
53#4977C2211yescharacteristic
54#4966C2211yescharacteristic
55C2×#808C2211yescharacteristic
56#4880—21no—
57D3×#107—21no—
58#4861—21no—
59#4861—21no—
60#4558C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4552C2211yescharacteristic
64#4550C2211yescharacteristic
65#4532C2211yescharacteristic
66#4532C2211yescharacteristic
67#4530C2211yescharacteristic
68#3097D311yescharacteristic, complemented
69#3046—31no—
70#2952—31no—
71#2913—31no—
72#2097—31no—
73#2090—31no—
74#2078—31no—
75#2038—31no—
76#1930—31no—
77#1922—31no—
78#1908—31no—
79#1882—31no—
80#1876—31no—
81#1860—31no—
82#1860—31no—
83C2×C3×#41C2311yescharacteristic
84C22×C3×Q8C2311yescharacteristic
85C2×#235C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4⋊Q8—21no—
89C3×#124—21no—
90C3×#123—41no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×D4—21no—
95C3×C4×D4—41no—
96C3×#117—21no—
97C3×#117C2311yescharacteristic
98C3×#117C2311yescharacteristic
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2311yescharacteristic
101#950—22no—
102#950—22no—
103C2×#216—21no—
104C2×#216D411yescharacteristic
105#942—22no—
106#942—22no—
107C2×#215—21no—
108C2×#215D411yescharacteristic
109C2×#214D411yescharacteristic
110C2×#213D411yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C2×#208C2311yescharacteristic
115C2×#208C2311yescharacteristic
116C2×#208C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#901—22no—
119#901—22no—
120C2×ΓL(1,𝔽25)—21no—
121C2×D3×C8—21no—
122#892—21no—
123#892—21no—
124D3×C4⋊C4—21no—
125D3×C4⋊C4—21no—
126#67⋊C4—41no—
127#67⋊C4—41no—
128C4×D12—21no—
129C4×D12—21no—
130C4×Dic6—21no—
131C4×Dic6—21no—
132C3×#107—21no—
133C3×#107D411yescharacteristic
134C3×#107D411yescharacteristic
135C3×#106—21no—
136C3×#106D411yescharacteristic
137C3×#106D411yescharacteristic
138C3×#103—21no—
139C3×#103—21no—
140C3×#103—21no—
141C3×#103—21no—
142C3×C22⋊C8—21no—
143C3×C22⋊C8—21no—
144#837—22no—
145#836—22no—
146#835—21no—
147#835—21no—
148#835—21no—
149#835—21no—
150#832—21no—
151#832—21no—
152C3⋊#95—41no—
153C3⋊#95—41no—
154#825—22no—
155#818—21no—
156#818—21no—
157#810—21no—
158#809—21no—
159#808—21no—
160#808—21no—
161#808—21no—
162#808—21no—
163#807—21no—
164#807—21no—
165C4⋊(C3⋊C8)—21no—
166C4⋊(C3⋊C8)—21no—
167#803—21no—
168C4×C3⋊C8—21no—
169C2×#137—31no—
170C2×#135—31no—
171C2×#131—31no—
172#536C2×D311yescharacteristic
173#520—31no—
174#520—31no—
175#520C2×D311yescharacteristic
176#485—21no—
177#485—21no—
178#484—21no—
179#483—21no—
180#464—61no—
181#463—61no—
182#462—61no—
183#461—61no—
184#442—61no—
185#441—61no—
186C4×SD16—61no—
187C4×SD16—61no—
188#428—31no—
189#428—31no—
190#428—61no—
191#428—61no—
192#428C2×D311yescharacteristic
193#427—61no—
194#427C2×D311yescharacteristic
195C2×#107—31no—
196C2×#107—61no—
197C2×#106—31no—
198#424—31no—
199C2×#104—31no—
200C2×#104—31no—
201#419C2×D311yescharacteristic
202#417—31no—
203#417—31no—
204#417—31no—
205C2×#103—31no—
206C2×#103C2×D311yescharacteristic
207C2×#101—31no—
208#408C2×D311yescharacteristic
209#405—31no—
210#391—31no—
211#382—31no—
212#343—31no—
213#343—31no—
214#343—31no—
215#341—31no—
216#341—31no—
217#339—31no—
218#339—31no—
219#336—31no—
220C3×#41—21no—
221C3×#41—21no—
222C3×#41—21no—
223C3×#41—21no—
224C2×C3×Q8—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8C2×D411yescharacteristic
231C2×C3×D4—21no—
232C2×C3×D4C2×D411yescharacteristic
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—41no—
237C22×C3×C4C2411yescharacteristic
238C2×#67—21no—
239C22×Dic3—21no—
240C22×Dic3—21no—
241C22×Dic3—21no—
242#235—21no—
243#235—21no—
244#235#4112yes—
245#235#4112yes—
246C2×D12C2×D411yescharacteristic
247C2×C4×D3—21no—
248C2×C4×D3—41no—
249C2×C4×D3—41no—
250C2×Dic6C2×D411yescharacteristic
251C3×M16—21no—
252C3×M16—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8C2×D411yescharacteristic
257C2×C3×C8C2×D411yescharacteristic
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—41no—
261C3×C4⋊C4—41no—
262C3×C4⋊C4—41no—
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C4⋊C4C2×D411yescharacteristic
265C3×C4⋊C4C2×D411yescharacteristic
266C3×C4⋊C4C2×D411yescharacteristic
267C3×C22⋊C4—21no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—41no—
270C3×C22⋊C4—41no—
271C3×C22⋊C4—41no—
272C3×C42—21no—
273C3×C42—21no—
274C3×C42—41no—
275#216—22no—
276#216—42no—
277#215—22no—
278#215—42no—
279#214—22no—
280#213—22no—
281(C2×D3)⋊C4—41no—
282(C2×D3)⋊C4—41no—
283C4⋊Dic3—21no—
284C4⋊Dic3—21no—
285C4⋊Dic3—21no—
286C4⋊Dic3—21no—
287Dic3⋊C4—41no—
288Dic3⋊C4—41no—
289C4×Dic3—21no—
290C4×Dic3—21no—
291C4×Dic3—41no—
292C4×Dic3—41no—
293#208—21no—
294#208—21no—
295#208—21no—
296#208—21no—
297#208—21no—
298#208#4112yes—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8C2×D411yescharacteristic
305C2×C3⋊C8C2×D411yescharacteristic
306ΓL(1,𝔽25)—24no—
307D3×C8—24no—
308C2×#41—31no—
309C2×#41C22×D311yescharacteristic
310C22×Q8C22×D311yescharacteristic
311#137—62no—
312#137—62no—
313#135—62no—
314#135—62no—
315C2×Q16—31no—
316C2×Q16—61no—
317C2×SD16—31no—
318C2×SD16—31no—
319C2×SD16—61no—
320C2×D8—31no—
321#131—62no—
322#131—62no—
323C2×M16—31no—
324C2×M16—31no—
325C2×M16—31no—
326C2×M16—61no—
327C2×M16C22×D311yescharacteristic
328C22×C8—31no—
329C22×C8—61no—
330C22×C8C22×D311yescharacteristic
331C4⋊Q8—21no—
332#124—21no—
333#123—41no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C4×Q8—21no—
337C4×Q8—61no—
338C4×Q8—61no—
339C4×D4—21no—
340C4×D4—41no—
341C4×D4—61no—
342C4×D4—61no—
343C4×D4—121no—
344C4×D4—121no—
345#117—21no—
346#117—61no—
347#117—61no—
348#117C22×D311yescharacteristic
349#117C22×D311yescharacteristic
350C2×C4⋊C4—21no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—31no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4C22×D311yescharacteristic
356C2×C42—31no—
357#107—21no—
358#107—61no—
359#107—61no—
360#107—61no—
361#107—61no—
362#107#6711yescharacteristic
363#107#6711yescharacteristic
364#106—21no—
365#106—61no—
366#106—61no—
367#106—61no—
368#106—61no—
369#106#6711yescharacteristic
370#106#6711yescharacteristic
371C4⋊C8—61no—
372C4⋊C8—61no—
373#104—62no—
374#104—62no—
375#103—21no—
376#103—21no—
377#103—21no—
378#103—21no—
379#103—61no—
380#103—61no—
381#103—61no—
382#103—61no—
383#103—61no—
384#102—61no—
385#102—61no—
386#102—61no—
387#101—62no—
388C22⋊C8—21no—
389C22⋊C8—21no—
390#97—61no—
391C4×C8—61no—
392#95—121no—
393#95—121no—
394C23×C3—21no—
395C23×C3H(5,𝔽2)11yescharacteristic
396C22×D3—21no—
397C3×Q8—21no—
398C3×Q8—21no—
399C3×Q8—41no—
400C3×Q8—41no—
401C3×Q8—41no—
402C3×Q8—41no—
403C3×D4—21no—
404C3×D4—21no—
405C3×D4—41no—
406C3×D4—41no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—81no—
419C2×C3×C4—161no—
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C42−1+411yescharacteristic
426#67—22no—
427#67—22no—
428C2×Dic3—21no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—81no—
435C2×Dic3—81no—
436D12—21no—
437D12#12312yes—
438C4×D3—22no—
439C4×D3—22no—
440C4×D3—161no—
441C4×D3—161no—
442Dic6—21no—
443Dic6#12312yes—
444C3×C8—21no—
445C3×C8—21no—
446C3×C8—41no—
447C3⋊C8—21no—
448C3⋊C8—21no—
449C3⋊C8—22no—
450C3⋊C8—41no—
451C3⋊C8—41no—
452#41—21no—
453#41—21no—
454#41—21no—
455#41—21no—
456#41—32no—
457#41—32no—
458#41—61no—
459#41—61no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—31no—
467C2×Q8D3×D411yescharacteristic
468C2×D4—21no—
469C2×D4—31no—
470C2×D4—61no—
471C2×D4D3×D411yescharacteristic
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—41no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—121no—
481C22×C4—121no—
482C22×C4C23×D311yescharacteristic
483Q16—62no—
484Q16—122no—
485SD16—62no—
486SD16—62no—
487SD16—122no—
488D8—62no—
489M16—21no—
490M16—21no—
491M16—32no—
492M16—64no—
493M16—61no—
494M16—61no—
495M16—61no—
496M16—61no—
497M16—61no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—31no—
502C2×C8—31no—
503C2×C8—64no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8C2×#6711yescharacteristic
510C2×C8C2×#6711yescharacteristic
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—61no—
520C4⋊C4—121no—
521C4⋊C4—121no—
522C4⋊C4C2×#6711yescharacteristic
523C4⋊C4C2×#6711yescharacteristic
524C4⋊C4C2×#6711yescharacteristic
525C4⋊C4C2×#6711yescharacteristic
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—121no—
532C22⋊C4—121no—
533C42—21no—
534C42—21no—
535C42—41no—
536C42—61no—
537C42—61no—
538C42—121no—
539C42—121no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3#53911yescharacteristic
546C22×C3D4211yescharacteristic
547C22×C3#55311yescharacteristic
548C2×D3—22no—
549C2×D3—41no—
550C2×D3—41no—
551C3×C4—41no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—41no—
555C3×C4—41no—
556C3×C4—81no—
557C3×C4—81no—
558C3×C4C2×#12311yescharacteristic
559C3×C4#52511yescharacteristic
560C3×C4#54711yescharacteristic
561C3×C4#54811yescharacteristic
562Dic3—22no—
563Dic3—81no—
564Dic3—81no—
565Dic3—81no—
566C23—21no—
567C23—61no—
568C23#100411yescharacteristic
569Q8—21no—
570Q8—21no—
571Q8—32no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—41no—
576Q8—61no—
577D4—21no—
578D4—21no—
579D4—32no—
580D4—41no—
581D4—41no—
582D4—62no—
583D4—62no—
584D4—61no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—61no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—81no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—161no—
606C2×C4—241no—
607C2×C4—241no—
608C2×C4—481no—
609C2×C4—481no—
610C2×C4C2×D3×D411yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4#101011yescharacteristic
615C2×C4C22×#6711yescharacteristic
616C8—21no—
617C8—21no—
618C8—41no—
619C8—61no—
620C8—61no—
621C8—62no—
622C8—121no—
623C8—121no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—81no—
627C2×C3#244311yescharacteristic
628C2×C3#328911yescharacteristic
629C2×C3#329011yescharacteristic
630D3—42no—
631C22—21no—
632C22—21no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—62no—
637C22—121no—
638C22—121no—
639C22#566011yescharacteristic
640C22#566111yescharacteristic
641C22D4×#6711yescharacteristic
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—62no—
648C4—81no—
649C4—81no—
650C4—241no—
651C4—241no—
652C4—241no—
653C4D3×#12311yescharacteristic
654C4#565511yescharacteristic
655C4#582811yescharacteristic
656C4#583111yescharacteristic
657C3#2619311yescharacteristic, complemented, Sylow
658C2—21no—
659C2—21no—
660C2—81no—
661C2—122no—
662C2#8289711yescharacteristic
663C2#8401311yescharacteristic
664C2#8402011yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3097C2
socleC22×C3D42
prime radicalC2×#950C22
hypercenter#3097D3
hypocenterC3#26193
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3097

Complements

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

NH ≅ G/Ntype
1. C3×#3097C2semidirect
4. #79179C2semidirect
5. #71863C2semidirect
6. #71665C2semidirect
7. #70681C2semidirect
8. #70671C2semidirect
9. #70661C2semidirect
10. #70348C2semidirect
11. #70338C2semidirect
12. #70286C2semidirect
13. #70251C2semidirect
15. #69994C2semidirect
68. #3097D3semidirect
657. C3#26193semidirect

Maximal subgroups

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

subgroupindex
C3×#30972
#834062
#825582
#791792
#718632
#716652
#706812
#706712
#706612
#703482
#703382
#702862
#702512
#700112
#699942
#261933