Finite Group Databasebeta

← #372079

Subgroups — #372079

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21083C211yesmaximal, characteristic
2#21021C211yesmaximal, characteristic, complemented
3#20384C211yesmaximal, characteristic
4#11317C211yesmaximal, characteristic, complemented
5#11291C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11068C211yesmaximal, characteristic
8#10924C211yesmaximal, characteristic
9#10879C211yesmaximal, characteristic, complemented
10#10822C211yesmaximal, characteristic, complemented
11#10809C211yesmaximal, characteristic, complemented
12#10809C211yesmaximal, characteristic, complemented
13#10791C211yesmaximal, characteristic
14#10791C211yesmaximal, characteristic
15#10694C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C22×#104C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C2×#416C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#408C2211yescharacteristic
23C2×#405C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2297C2211yescharacteristic
27#2296C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#1934C414yescomplemented
30#1933C414yescomplemented
31#1917—22no—
32#1917—22no—
33#1753—22no—
34#1752—22no—
35C2×#341C2211yescharacteristic
36C2×#341C2211yescharacteristic
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39#1735C2211yescharacteristic
40#1735C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47#1729C2211yescharacteristic
48#1729C2211yescharacteristic
49#1729C2211yescharacteristic
50#1729C2211yescharacteristic
51#1728C2211yescharacteristic
52#1728C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C22×#41C2311yescharacteristic
58C2×#131—22no—
59C2×#131—22no—
60C2×#131—22no—
61C2×#131—22no—
62C2×#131—41no—
63C2×#131—41no—
64C22×M16—21no—
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C23×C8—21no—
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#424—41no—
81#424—41no—
82C2×C4⋊C8—22no—
83C2×#104—22no—
84C2×#104—22no—
85C2×#104—22no—
86C2×#104—22no—
87C2×#104—41no—
88C2×#104—41no—
89#416—22no—
90#416—22no—
91#416—41no—
92#416—41no—
93C2×#103C2×C412yes—
94C2×#103C2×C412yes—
95C2×#102C2×C412yes—
96C2×#102C2×C412yes—
97#408—22no—
98#408—22no—
99C2×C22⋊C8—22no—
100#405—22no—
101#405—22no—
102C2×#97—21no—
103C2×#97—41no—
104C2×#97—41no—
105C2×#97D412yes—
106C2×#97D412yes—
107C2×C4×C8—21no—
108C2×C4×C8—41no—
109C2×C4×C8—41no—
110C2×C4×C8C2×C414yes—
111#391C2×C412yes—
112#390C2×C412yes—
113#389C2×C414yes—
114#386—22no—
115#385—22no—
116C4×C4⋊C4—22no—
117C4×C22⋊C4—22no—
118#376—41no—
119#376—41no—
120C2×#95—22no—
121#341—22no—
122#341—22no—
123#341—41no—
124#341—41no—
125#340C2×C414yes—
126#340C2×C414yes—
127#339—22no—
128#339—22no—
129#339—22no—
130#339—22no—
131#339—41no—
132#339—41no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41#4112yes—
156C2×#41#4112yes—
157C22×Q8C22×C411yescharacteristic
158C22×D4—21no—
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165#131—44no—
166#131—44no—
167#131—44no—
168#131—44no—
169C2×M16—24no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—42no—
193C2×M16—42no—
194C2×M16#4112yes—
195C2×M16#4112yes—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—24no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—42no—
224C22×C8—42no—
225C22×C8C22×C412yes—
226C22×C8C22×C412yes—
227C22×C8C2×D412yes—
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C22×C8#4112yes—
231C22×C8#4112yes—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—24no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42C22×C412yes—
270C2×C42C2×D412yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C4⋊C8—44no—
274#104—44no—
275#104—44no—
276#103—24no—
277#103—42no—
278#102—24no—
279#102—42no—
280C22⋊C8—28no—
281#97—42no—
282#97—42no—
283#97—42no—
284C4×C8—44no—
285C4×C8—42no—
286#95—24no—
287#95—24no—
288#95—44no—
289C24—21no—
290C24C2×#4111yescharacteristic
291#41—22no—
292#41—22no—
293#41—22no—
294#41—22no—
295#41—22no—
296#41—22no—
297#41—22no—
298#41—22no—
299#41—41no—
300#41—41no—
301#41—41no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8#11712yes—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4#11712yes—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—44no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4C4×D414yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C42−1+411yescharacteristic
361M16—22no—
362M16—22no—
363M16—22no—
364M16—22no—
365M16—48no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—48no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8#12412yes—
421C2×C8#12412yes—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—84no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—84no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C23—24no—
455C23—22no—
456C23—41no—
457C23—44no—
458C23—41no—
459C23C2×C4×D411yescharacteristic
460C23#52911yescharacteristic
461C23#52911yescharacteristic
462C23#52911yescharacteristic
463C23#52911yescharacteristic
464C23#53811yescharacteristic
465C23#53911yescharacteristic
466Q8—22no—
467Q8—22no—
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—22no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—41no—
477D4—41no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4C2×#11711yescharacteristic
505C2×C4C4×#4111yescharacteristic
506C2×C4C4×#4111yescharacteristic
507C2×C4C4×#4111yescharacteristic
508C2×C4#51811yescharacteristic
509C2×C4#52011yescharacteristic
510C2×C4#52511yescharacteristic
511C2×C4#52511yescharacteristic
512C2×C4C2×#12411yescharacteristic
513C2×C4#53211yescharacteristic
514C2×C4#53411yescharacteristic
515C2×C4#53711yescharacteristic
516C2×C4#55111yescharacteristic
517C2×C4#55111yescharacteristic
518C2×C4#55111yescharacteristic
519C2×C4#55111yescharacteristic
520C2×C4#55111yescharacteristic
521C2×C4#55111yescharacteristic
522C2×C4#55211yescharacteristic
523C2×C4#55211yescharacteristic
524C2×C4#55211yescharacteristic
525C2×C4#55311yescharacteristic
526C2×C4#55311yescharacteristic
527C2×C4#55311yescharacteristic
528C2×C4#55311yescharacteristic
529C2×C4#55311yescharacteristic
530C2×C4#55611yescharacteristic
531C2×C4#55911yescharacteristic
532C8—22no—
533C8—22no—
534C8—42no—
535C8—42no—
536C8—42no—
537C8—42no—
538C8—42no—
539C8—81no—
540C8—81no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—42no—
548C22—44no—
549C22—82no—
550C22—82no—
551C22#229011yescharacteristic
552C22C4×#12411yescharacteristic
553C22#232511yescharacteristic
554C22#257511yescharacteristic
555C22#257611yescharacteristic
556C22#257711yescharacteristic
557C22#257811yescharacteristic
558C22#295714yes—
559C4—42no—
560C4—44no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4#234311yescharacteristic
567C4#235311yescharacteristic
568C4#237311yescharacteristic
569C4#237411yescharacteristic
570C4#256111yescharacteristic
571C4#256211yescharacteristic
572C4#260111yescharacteristic
573C4#260211yescharacteristic
574C2—21no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—82no—
579C2#1408611yescharacteristic
580C2#2114012yes—
581C2#2165912yes—
582C2#2220612yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21021C2semidirect
4. #11317C2semidirect
5. #11291C2semidirect
9. #10879C2semidirect
10. #10822C2semidirect
11. #10809C2semidirect
12. #10809C2semidirect
15. #10694C2semidirect
29. #1934C4semidirect
30. #1933C4semidirect

Maximal subgroups

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

subgroupindex
#210832
#210212
#203842
#113172
#112912
#111912
#110682
#109242
#108792
#108222
#108092
#107912
#106942