Finite Group Databasebeta

← #372051

Subgroups — #372051

Order 512.

861 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21083C211yesmaximal, characteristic
2#21047C211yesmaximal, characteristic
3#20436C211yesmaximal, characteristic, complemented
4#11317C211yesmaximal, characteristic, complemented
5#11291C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11068C211yesmaximal, characteristic
8#10934C211yesmaximal, characteristic
9#10934C211yesmaximal, characteristic
10#10924C211yesmaximal, characteristic
11#10887C211yesmaximal, characteristic, complemented
12#10887C211yesmaximal, characteristic, complemented
13#10879C211yesmaximal, characteristic, complemented
14#10822C211yesmaximal, characteristic, complemented
15#10694C211yesmaximal, characteristic, complemented
16C2×#424C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C22×#104C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C2×#416C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#408C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2297C2211yescharacteristic
29#2296C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1934C414yescomplemented
32#1933C414yescomplemented
33#1924—22no—
34#1924—22no—
35#1771—22no—
36#1770—22no—
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41#1735C2211yescharacteristic
42#1735C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49#1729C2211yescharacteristic
50#1729C2211yescharacteristic
51#1728C2211yescharacteristic
52#1728C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C22×#41C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#424—22no—
73#424—22no—
74#424—22no—
75#424—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×#104—22no—
80C2×#104—22no—
81C2×#104—22no—
82C2×#104—22no—
83C2×#104—41no—
84C2×#104—41no—
85#416—22no—
86#416—22no—
87#416—41no—
88#416—41no—
89C2×#103C2×C412yes—
90C2×#103C2×C412yes—
91C2×#102C2×C412yes—
92C2×#102C2×C412yes—
93#408—22no—
94#408—22no—
95#408—41no—
96#408—41no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100#405—22no—
101#405—22no—
102#405—41no—
103#405—41no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—41no—
107C2×#97—41no—
108C2×#97D412yes—
109C2×#97D412yes—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—41no—
113C2×C4×C8—41no—
114C2×C4×C8C2×C414yes—
115#391C2×C412yes—
116#390C2×C412yes—
117#389C2×C414yes—
118#386—22no—
119#385—22no—
120C4×C4⋊C4—22no—
121C4×C22⋊C4—22no—
122#376—41no—
123#376—41no—
124C2×#95—22no—
125#341—22no—
126#341—22no—
127#341—41no—
128#341—41no—
129#340C2×C414yes—
130#340C2×C414yes—
131#339—22no—
132#339—22no—
133#339—22no—
134#339—22no—
135#339—41no—
136#339—41no—
137#336—22no—
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—21no—
156C2×#41—21no—
157C22×Q8C22×C411yescharacteristic
158C22×D4—21no—
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
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#4112yes—
188C2×M16#4112yes—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
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×C8C22×C412yes—
216C22×C8C22×C412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8#4112yes—
221C22×C8#4112yes—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117#4112yes—
233#117#4112yes—
234C2×C4⋊C4—24no—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—24no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42C22×C412yes—
260C2×C42C2×D412yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C4⋊C8—28no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266#104—44no—
267#104—44no—
268#103—24no—
269#103—42no—
270#102—24no—
271#102—42no—
272C22⋊C8—44no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275#97—24no—
276#97—42no—
277#97—42no—
278#97—42no—
279C4×C8—24no—
280C4×C8—44no—
281C4×C8—42no—
282#95—24no—
283#95—24no—
284#95—44no—
285C24—21no—
286C24C2×#4111yescharacteristic
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41—41no—
302#41—41no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4C4×D414yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C42−1+411yescharacteristic
357M16—22no—
358M16—22no—
359M16—22no—
360M16—22no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
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—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
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#12412yes—
415C2×C8#12412yes—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—84no—
422C4⋊C4#11714yes—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—84no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C23—24no—
449C23—41no—
450C23—44no—
451C23—41no—
452C23—41no—
453C23C2×C4×D411yescharacteristic
454C23#52911yescharacteristic
455C23#52911yescharacteristic
456C23#52911yescharacteristic
457C23#52911yescharacteristic
458C23#53811yescharacteristic
459C23#53911yescharacteristic
460Q8—41no—
461Q8—41no—
462Q8—41no—
463Q8—41no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—41no—
468D4—81no—
469D4—81no—
470D4—81no—
471D4—81no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4C2×#11711yescharacteristic
499C2×C4C4×#4111yescharacteristic
500C2×C4C4×#4111yescharacteristic
501C2×C4C4×#4111yescharacteristic
502C2×C4#51811yescharacteristic
503C2×C4#52011yescharacteristic
504C2×C4#52511yescharacteristic
505C2×C4#52511yescharacteristic
506C2×C4C2×#12411yescharacteristic
507C2×C4#53211yescharacteristic
508C2×C4#53411yescharacteristic
509C2×C4#53711yescharacteristic
510C2×C4#55111yescharacteristic
511C2×C4#55111yescharacteristic
512C2×C4#55211yescharacteristic
513C2×C4#55311yescharacteristic
514C2×C4#55311yescharacteristic
515C2×C4#55311yescharacteristic
516C2×C4#55311yescharacteristic
517C2×C4#55311yescharacteristic
518C2×C4#55311yescharacteristic
519C2×C4#55311yescharacteristic
520C2×C4#55611yescharacteristic
521C2×C4#55611yescharacteristic
522C2×C4#55611yescharacteristic
523C2×C4#55911yescharacteristic
524C2×C4#55911yescharacteristic
525C2×C4#55911yescharacteristic
526C8—22no—
527C8—22no—
528C8—42no—
529C8—42no—
530C8—42no—
531C8—42no—
532C8—42no—
533C8—81no—
534C8—81no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—81no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22#229011yescharacteristic
546C22C4×#12411yescharacteristic
547C22#232511yescharacteristic
548C22#259411yescharacteristic
549C22#259511yescharacteristic
550C22#259611yescharacteristic
551C22#259711yescharacteristic
552C22#295714yes—
553C4—44no—
554C4—44no—
555C4—82no—
556C4—82no—
557C4—81no—
558C4—82no—
559C4—82no—
560C4#234711yescharacteristic
561C4#235711yescharacteristic
562C4#237311yescharacteristic
563C4#237411yescharacteristic
564C4#259111yescharacteristic
565C4#259211yescharacteristic
566C4#260111yescharacteristic
567C4#260211yescharacteristic
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—161no—
573C2#1408511yescharacteristic
574C2#2114012yes—
575C2#2165912yes—
576C2#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
3. #20436C2semidirect
4. #11317C2semidirect
5. #11291C2semidirect
11. #10887C2semidirect
12. #10887C2semidirect
13. #10879C2semidirect
14. #10822C2semidirect
15. #10694C2semidirect
31. #1934C4semidirect
32. #1933C4semidirect

Maximal subgroups

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

subgroupindex
#210832
#210472
#204362
#113172
#112912
#111912
#110682
#109342
#109242
#108872
#108792
#108222
#106942