Finite Group Databasebeta

← #10855693

Subgroups — #10855693

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
1#82572C211yesmaximal, characteristic
2#82562C211yesmaximal, characteristic, complemented
3#79220C211yesmaximal, characteristic, complemented
4#78869C211yesmaximal, characteristic
5C3×#1860C211yesmaximal, characteristic, complemented
6#70671C211yesmaximal, characteristic, complemented
7#70377C211yesmaximal, characteristic
8#70365C211yesmaximal, characteristic, complemented
9#70304C211yesmaximal, characteristic, complemented
10#69562C211yesmaximal, characteristic, complemented
11#69561C211yesmaximal, characteristic, complemented
12#69010C211yesmaximal, characteristic, complemented
13#68982C211yesmaximal, characteristic, complemented
14#68738C211yesmaximal, characteristic, complemented
15#68734C211yesmaximal, characteristic, complemented
16#26193—31nomaximal, Sylow
17C3×#520C2211yescharacteristic
18#5827C2211yescharacteristic
19C2×#951C2211yescharacteristic
20C2×#950C2211yescharacteristic
21C2×#948C2211yescharacteristic
22#5526C2211yescharacteristic
23#5504C2211yescharacteristic
24C3×#428C2211yescharacteristic
25C2×C3×#104C2211yescharacteristic
26C3×#417C2211yescharacteristic
27C3×#382C2211yescharacteristic
28C2×#824C2211yescharacteristic
29#5133C2211yescharacteristic
30#5129C2211yescharacteristic
31#5122C2211yescharacteristic
32#5120C2211yescharacteristic
33#5031—21no—
34#5028—21no—
35#5019—21no—
36#5016—21no—
37#4999C2211yescharacteristic
38#4998—21no—
39#4998C2211yescharacteristic
40#4995—21no—
41#4995C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44#4966C2211yescharacteristic
45#4964—21no—
46#4964C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807—21no—
49C2×#807C2211yescharacteristic
50#4929C2211yescharacteristic
51C2×#805C2211yescharacteristic
52#4888—21no—
53#4884—21no—
54#4868—21no—
55#4862—21no—
56#4789—21no—
57#4788—21no—
58#4785—21no—
59#4758—21no—
60C3×#343C2211yescharacteristic
61C3×#339C2211yescharacteristic
62#4558C2211yescharacteristic
63#4558C2211yescharacteristic
64#4554C2211yescharacteristic
65#4530C2211yescharacteristic
66#4524C2211yescharacteristic
67#4523C2211yescharacteristic
68#3097—31no—
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#1860D311yescharacteristic, complemented
83C2×C3×#41C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C3×C4×Q8—21no—
89C3×C4×D4—21no—
90C3×C4×D4—41no—
91C3×#117—21no—
92C3×#117C2311yescharacteristic
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97#951—22no—
98#951—22no—
99#950—22no—
100#950—22no—
101#948—22no—
102#948—22no—
103Q8×Dic3—21no—
104C2×#216—21no—
105C2×#216D411yescharacteristic
106C2×#215D411yescharacteristic
107D4×Dic3—21no—
108D4×Dic3—41no—
109C2×#214—21no—
110C2×#214D411yescharacteristic
111C2×#213D411yescharacteristic
112C4×#67—41no—
113#926—21no—
114#926C2311yescharacteristic
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3C2311yescharacteristic
117C2×#208—21no—
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C22×C3⋊C8—21no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#897—21no—
124#897—21no—
125#893—21no—
126#892—21no—
127D3×C4⋊C4—21no—
128#887—21no—
129#886—41no—
130#875—21no—
131#869—21no—
132C3×#107—21no—
133C3×#106—21no—
134C3×#104—22no—
135C3×#103—21no—
136C3×#102—21no—
137C3×#95—41no—
138C3⋊#95—41no—
139#824—22no—
140#821—21no—
141(C2×D3)⋊C8—21no—
142(C2×D3)⋊C8—21no—
143#816—21no—
144#816—21no—
145#816—21no—
146#815—21no—
147C8×Dic3—21no—
148#810—21no—
149#810—21no—
150#810—21no—
151#810—21no—
152#810—21no—
153#809—21no—
154#808—21no—
155#808—21no—
156#808—21no—
157#808—21no—
158#808D411yescharacteristic
159#808D411yescharacteristic
160#807—21no—
161#807—21no—
162#807—21no—
163#807—21no—
164#807D411yescharacteristic
165#807D411yescharacteristic
166#805—22no—
167C4⋊(C3⋊C8)—21no—
168C4⋊(C3⋊C8)—21no—
169C2×#137—31no—
170C2×#135—31no—
171C2×#131—31no—
172#536—31no—
173#520—31no—
174#520—31no—
175#520C2×D311yescharacteristic
176#485—61no—
177#485—61no—
178#484—61no—
179#483—61no—
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—31no—
194#427—61no—
195C2×#107—31no—
196C2×#107—61no—
197C2×#106—31no—
198#424—31no—
199C2×#104—31no—
200C2×#104C2×D311yescharacteristic
201#419—31no—
202#417—31no—
203#417—31no—
204#417C2×D311yescharacteristic
205C2×#103—31no—
206C2×#103—31no—
207C2×#101—31no—
208#408—31no—
209#405—31no—
210#391—31no—
211#382C2×D311yescharacteristic
212#343—31no—
213#343—31no—
214#343C2×D311yescharacteristic
215#341—31no—
216#341—31no—
217#339—31no—
218#339C2×D311yescharacteristic
219#336—31no—
220C3×#41—21no—
221C3×#41—21no—
222C3×#41#4112yes—
223C3×#41#4112yes—
224C2×C3×Q8C2×D411yescharacteristic
225C2×C3×D4—21no—
226C2×C3×D4C2×D411yescharacteristic
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—41no—
232C22×C3×C4C2411yescharacteristic
233C2×#67—21no—
234C22×Dic3—21no—
235C22×Dic3—21no—
236C22×Dic3—41no—
237#235—21no—
238#235—21no—
239#235—21no—
240#235—21no—
241C2×D12C2×D411yescharacteristic
242C2×C4×D3—21no—
243C2×C4×D3—41no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6C2×D411yescharacteristic
251C3×M16—21no—
252C3×M16—21no—
253C3×M16—21no—
254C3×M16—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—41no—
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C22⋊C4—21no—
265C3×C22⋊C4—41no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—41no—
270#217—21no—
271#217—41no—
272#216—22no—
273#216—42no—
274#215—22no—
275#214—22no—
276#214—42no—
277#213—22no—
278(C2×D3)⋊C4—41no—
279(C2×D3)⋊C4—41no—
280(C2×D3)⋊C4—41no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—41no—
284C4⋊Dic3—41no—
285C4⋊Dic3—41no—
286C4⋊Dic3C2×D411yescharacteristic
287C4⋊Dic3C2×D411yescharacteristic
288Dic3⋊C4—41no—
289C4×Dic3—21no—
290C4×Dic3—41no—
291C4×Dic3—41no—
292#208—24no—
293#208—21no—
294#208—21no—
295#208—21no—
296#208#4112yes—
297C2×C3⋊C8—24no—
298C2×C3⋊C8—21no—
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
306C2×C3⋊C8C2×D411yescharacteristic
307C2×C3⋊C8C2×D411yescharacteristic
308C2×#41—31no—
309C2×#41C22×D311yescharacteristic
310C22×Q8—31no—
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—61no—
326C2×M16C22×D311yescharacteristic
327C2×M16C22×D311yescharacteristic
328C22×C8—31no—
329C22×C8—31no—
330C22×C8—61no—
331C4⋊Q8—61no—
332#124—61no—
333#123—121no—
334C22⋊Q8—61no—
335C22⋊Q8—61no—
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—31no—
347#117—61no—
348#117—61no—
349#117C22×D311yescharacteristic
350C2×C4⋊C4—21no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4C22×D311yescharacteristic
355C2×C4⋊C4C22×D311yescharacteristic
356C2×C42C22×D311yescharacteristic
357#107—21no—
358#107—31no—
359#107—31no—
360#107—61no—
361#107—61no—
362#107—61no—
363#107—61no—
364#106—21no—
365#106—31no—
366#106—31no—
367#106—61no—
368#106—61no—
369#106—61no—
370#106—61no—
371C4⋊C8—61no—
372C4⋊C8—61no—
373#104—22no—
374#104—62no—
375#103—21no—
376#103—61no—
377#103—61no—
378#103—61no—
379#103—61no—
380#103—61no—
381#103—61no—
382#103—61no—
383#103—61no—
384#102—21no—
385#102—61no—
386#102—61no—
387#101—62no—
388C22⋊C8—61no—
389C22⋊C8—61no—
390#97—61no—
391C4×C8—61no—
392#95—41no—
393#95—121no—
394C23×C3—21no—
395C23×C3H(5,𝔽2)11yescharacteristic
396C22×D3—21no—
397C3×Q8—21no—
398C3×Q8#12312yes—
399C3×D4—22no—
400C3×D4—22no—
401C3×D4—21no—
402C3×D4#12312yes—
403C2×C3×C4—21no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—81no—
414C2×C3×C4—81no—
415C2×C3×C4—161no—
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C4C2×#4111yescharacteristic
421C2×C3×C42−1+411yescharacteristic
422#67—41no—
423#67—41no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—81no—
432C2×Dic3—161no—
433D12—21no—
434D12—21no—
435C4×D3—41no—
436C4×D3—41no—
437C4×D3—161no—
438Dic6—21no—
439Dic6—21no—
440Dic6—41no—
441Dic6—41no—
442Dic6—41no—
443Dic6—41no—
444C3×C8—41no—
445C3×C8—41no—
446C3⋊C8—21no—
447C3⋊C8—21no—
448C3⋊C8—21no—
449C3⋊C8—22no—
450C3⋊C8—21no—
451C3⋊C8—41no—
452#41—21no—
453#41—21no—
454#41—61no—
455#41—61no—
456#41—61no—
457#41—61no—
458#41#23512yes—
459#41#23912yes—
460C2×Q8—31no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—61no—
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—21no—
476C22×C4—41no—
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—21no—
492M16—21no—
493M16—32no—
494M16—64no—
495M16—61no—
496M16—61no—
497M16—61no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—31no—
501C2×C8—31no—
502C2×C8—31no—
503C2×C8—31no—
504C2×C8—64no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—31no—
516C4⋊C4—31no—
517C4⋊C4—41no—
518C4⋊C4—61no—
519C4⋊C4—61no—
520C4⋊C4—121no—
521C4⋊C4—121no—
522C4⋊C4—121no—
523C4⋊C4—121no—
524C4⋊C4D3×D411yescharacteristic
525C4⋊C4D3×D411yescharacteristic
526C22⋊C4—21no—
527C22⋊C4—41no—
528C22⋊C4—61no—
529C22⋊C4—121no—
530C22⋊C4—121no—
531C22⋊C4—121no—
532C22⋊C4—121no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—41no—
537C42—61no—
538C42—121no—
539C42—121no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—22no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3#53911yescharacteristic
546C22×C3D4211yescharacteristic
547C22×C3#55311yescharacteristic
548C2×D3—41no—
549C2×D3—41no—
550C2×D3—41no—
551C3×C4—22no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—81no—
555C3×C4—81no—
556C3×C4—81no—
557C3×C4C2×#12311yescharacteristic
558C3×C4#52511yescharacteristic
559C3×C4#54711yescharacteristic
560C3×C4#54811yescharacteristic
561Dic3—41no—
562Dic3—41no—
563Dic3—41no—
564Dic3—81no—
565Dic3—81no—
566C23—21no—
567C23—61no—
568C23#100411yescharacteristic
569Q8—21no—
570Q8—61no—
571Q8—61no—
572Q8—121no—
573Q8—121no—
574Q8—121no—
575Q8—121no—
576Q8#89412yes—
577D4—22no—
578D4—22no—
579D4—21no—
580D4—61no—
581D4—61no—
582D4—121no—
583D4—121no—
584D4#89412yes—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—161no—
607C2×C4—241no—
608C2×C4—481no—
609C2×C4—481no—
610C2×C4C2×#23511yescharacteristic
611C2×C4C2×D3×D411yescharacteristic
612C2×C4C2×D3×D411yescharacteristic
613C2×C4C2×#23911yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C2×C4#101011yescharacteristic
616C8—41no—
617C8—41no—
618C8—61no—
619C8—61no—
620C8—61no—
621C8—61no—
622C8—62no—
623C8—121no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—42no—
627C2×C3#244311yescharacteristic
628C2×C3#328911yescharacteristic
629C2×C3#329011yescharacteristic
630D3—81no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—121no—
637C22—121no—
638C22—121no—
639C22#554011yescharacteristic
640C22#559011yescharacteristic
641C22#564511yescharacteristic
642C4—22no—
643C4—41no—
644C4—41no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—121no—
649C4—121no—
650C4—121no—
651C4—241no—
652C4—241no—
653C4C2×#89411yescharacteristic
654C4#555211yescharacteristic
655C4#562711yescharacteristic
656C4#565511yescharacteristic
657C3#2619311yescharacteristic, complemented, Sylow
658C2—21no—
659C2—21no—
660C2—42no—
661C2—241no—
662C2#7441811yescharacteristic
663C2#8028811yescharacteristic
664C2#8029611yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1860C2
socleC22×C3D42
prime radicalC2×#950C22
hypercenter#1860D3
hypocenterC3#26193
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1860

Complements

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

NH ≅ G/Ntype
2. #82562C2semidirect
3. #79220C2semidirect
5. C3×#1860C2semidirect
6. #70671C2semidirect
8. #70365C2semidirect
9. #70304C2semidirect
10. #69562C2semidirect
11. #69561C2semidirect
12. #69010C2semidirect
13. #68982C2semidirect
14. #68738C2semidirect
15. #68734C2semidirect
82. #1860D3semidirect
657. C3#26193semidirect

Maximal subgroups

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

subgroupindex
#825722
#825622
#792202
#788692
C3×#18602
#706712
#703772
#703652
#703042
#695622
#695612
#690102
#689822
#687382
#687342
#261933