Finite Group Databasebeta

← #390223

Subgroups — #390223

Order 512.

799 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23601C211yesmaximal, characteristic
2#21178C211yesmaximal, characteristic, complemented
3#21072C211yesmaximal, characteristic
4#21071C211yesmaximal, characteristic
5#21042C211yesmaximal, characteristic, complemented
6#11289C211yesmaximal, characteristic, complemented
7#11250C211yesmaximal, characteristic, complemented
8#11165C211yesmaximal, characteristic
9#11137C211yesmaximal, characteristic
10#11065C211yesmaximal, characteristic
11#10930C211yesmaximal, characteristic
12#10902C211yesmaximal, characteristic, complemented
13#10900C211yesmaximal, characteristic, complemented
14#10883C211yesmaximal, characteristic, complemented
15#10865C211yesmaximal, characteristic, complemented
16C2×#430C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C22×#108C2211yescharacteristic
20#2911C2211yescharacteristic
21#2905C2211yescharacteristic
22C2×#424C2211yescharacteristic
23C2×#421C2211yescharacteristic
24C22×#104C2211yescharacteristic
25#2896C2211yescharacteristic
26C2×#416C2211yescharacteristic
27C2×#413C2211yescharacteristic
28#2879C2211yescharacteristic
29C2×#408C2211yescharacteristic
30#2873C2211yescharacteristic
31#2873C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2064C414yescomplemented
34#2063C414yescomplemented
35#1950—24no—
36#1929—24no—
37#1929—24no—
38#1922—24no—
39C2×#344C2211yescharacteristic
40C2×#344C2211yescharacteristic
41#1742C2211yescharacteristic
42#1738C2211yescharacteristic
43#1738C2211yescharacteristic
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48#1735C2211yescharacteristic
49#1733C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1728C2211yescharacteristic
56#1728C2211yescharacteristic
57C22×#41C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#430—22no—
73#430—41no—
74#430—41no—
75#430—41no—
76C2×#108—22no—
77C2×#108—22no—
78C2×#108—41no—
79C2×#108—41no—
80C2×#108—41no—
81C2×#108—41no—
82C2×#108C2×C414yes—
83C2×#108D414yes—
84#428—22no—
85#428C2×C414yes—
86#427—22no—
87#424—22no—
88#424—22no—
89#424—41no—
90#424C2×C414yes—
91#423—22no—
92C2×C4⋊C8—22no—
93#421—41no—
94#421—41no—
95C2×#104—22no—
96C2×#104—22no—
97C2×#104—22no—
98C2×#104—41no—
99#419C2×C412yes—
100#418C2×C412yes—
101#417—22no—
102#416—22no—
103#416—22no—
104#416—22no—
105C2×#103C2×C412yes—
106C2×#102C2×C412yes—
107#413—41no—
108#413—41no—
109C2×#101C2×C412yes—
110C2×#100C2×C412yes—
111#408—22no—
112#408—22no—
113#408—22no—
114#408—41no—
115#407—41no—
116C2×C22⋊C8—41no—
117#405—22no—
118#405—22no—
119#405—22no—
120#405—41no—
121C4×M16—22no—
122C4×M16—41no—
123C2×#97—22no—
124C2×#97—41no—
125C2×#97—41no—
126#344—22no—
127#344—41no—
128#343C2×C414yes—
129#341—22no—
130#341—22no—
131#341—22no—
132#341—22no—
133#341—22no—
134#341—41no—
135#340—22no—
136#339—22no—
137#339—22no—
138#339—41no—
139#339D414yes—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—41no—
144C2×#41—21no—
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—
156C22×Q8C22×C411yescharacteristic
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2411yescharacteristic
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
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—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16C22×C412yes—
204C2×M16C22×C412yes—
205C2×M16C22×C412yes—
206C2×M16C2×D412yes—
207C2×M16C2×D412yes—
208C2×M16C2×D412yes—
209C2×M16#4112yes—
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—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8C22×C412yes—
229C22×C8C2×D412yes—
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#117C22×C412yes—
243#117C2×D412yes—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4#4112yes—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—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×C42C2×D412yes—
259#108—44no—
260#108—44no—
261#108—44no—
262#108—44no—
263#107—24no—
264#106—24no—
265C4⋊C8—28no—
266C4⋊C8—44no—
267#104—44no—
268#104—44no—
269#103—24no—
270#103—24no—
271#102—24no—
272#102—24no—
273#101—42no—
274#100—42no—
275C22⋊C8—44no—
276C22⋊C8—44no—
277#97—42no—
278#97—42no—
279#97—44no—
280C4×C8—42no—
281C4×C8—42no—
282C24—21no—
283C242−1+411yescharacteristic
284#41—41no—
285#41—41no—
286#41—41no—
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—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—82no—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+411yescharacteristic
344C22×C42−1+411yescharacteristic
345C22×C42−1+411yescharacteristic
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—22no—
353M16—22no—
354M16—22no—
355M16—22no—
356M16—22no—
357M16—22no—
358M16—22no—
359M16—22no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
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—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—22no—
393C2×C8—22no—
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—41no—
415C2×C8C4×D414yes—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4C4×D414yes—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—82no—
443C23#52011yescharacteristic
444C23#54111yescharacteristic
445C23#54211yescharacteristic
446C23#54711yescharacteristic
447C23#54811yescharacteristic
448C23#56311yescharacteristic
449C23#56311yescharacteristic
450Q8—41no—
451Q8—41no—
452Q8—41no—
453Q8—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—81no—
459D4—81no—
460D4—81no—
461D4—81no—
462C2×C4—24no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C4×#4111yescharacteristic
482C2×C4#51811yescharacteristic
483C2×C4#51911yescharacteristic
484C2×C4#52011yescharacteristic
485C2×C4#53411yescharacteristic
486C2×C4#53611yescharacteristic
487C2×C4#53711yescharacteristic
488C2×C4#53711yescharacteristic
489C2×C4#53811yescharacteristic
490C2×C4#53811yescharacteristic
491C2×C4#53911yescharacteristic
492C2×C4#53911yescharacteristic
493C2×C4#54111yescharacteristic
494C2×C4#54211yescharacteristic
495C2×C4D4211yescharacteristic
496C2×C4#54611yescharacteristic
497C2×C4#54711yescharacteristic
498C2×C4#54811yescharacteristic
499C2×C4#55011yescharacteristic
500C2×C4#55211yescharacteristic
501C2×C4#55311yescharacteristic
502C2×C4#55311yescharacteristic
503C2×C4#55511yescharacteristic
504C2×C4#55611yescharacteristic
505C2×C4#56211yescharacteristic
506C2×C4#56211yescharacteristic
507C8—24no—
508C8—42no—
509C8—42no—
510C8—42no—
511C8—42no—
512C8—42no—
513C8—42no—
514C8—42no—
515C8—42no—
516C8—81no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—82no—
525C22—82no—
526C22—81no—
527C22—161no—
528C22#233411yescharacteristic
529C22#236111yescharacteristic
530C22#237811yescharacteristic
531C22#271511yescharacteristic
532C22#272111yescharacteristic
533C22#272511yescharacteristic
534C22#273111yescharacteristic
535C4—44no—
536C4—82no—
537C4—82no—
538C4—81no—
539C4—161no—
540C4#232311yescharacteristic
541C4#232511yescharacteristic
542C4#236211yescharacteristic
543C4#236311yescharacteristic
544C4#268111yescharacteristic
545C4#268211yescharacteristic
546C4#277911yescharacteristic
547C4#278311yescharacteristic
548C2—21no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—21no—
554C2—161no—
555C2#1456511yescharacteristic
556C2#2304312yes—
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2361
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21178C2semidirect
5. #21042C2semidirect
6. #11289C2semidirect
7. #11250C2semidirect
12. #10902C2semidirect
13. #10900C2semidirect
14. #10883C2semidirect
15. #10865C2semidirect
33. #2064C4semidirect
34. #2063C4semidirect

Maximal subgroups

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

subgroupindex
#236012
#211782
#210722
#210712
#210422
#112892
#112502
#111652
#111372
#110652
#109302
#109022
#109002
#108832
#108652