Finite Group Databasebeta

← #391001

Subgroups — #391001

Order 512.

807 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23474C211yesmaximal, characteristic
2#11395C211yesmaximal, characteristic, complemented
3#11391C211yesmaximal, characteristic, complemented
4#11319C211yesmaximal, characteristic, complemented
5#11319C211yesmaximal, characteristic, complemented
6#11088C211yesmaximal, characteristic
7#10961C211yesmaximal, characteristic
8#10941C211yesmaximal, characteristic
9#10920C211yesmaximal, characteristic, complemented
10#10914C211yesmaximal, characteristic, complemented
11#10831C211yesmaximal, characteristic, complemented
12#10803C211yesmaximal, characteristic, complemented
13#10771C211yesmaximal, characteristic
14#10757C211yesmaximal, characteristic
15#10748C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17#3458C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#413C2211yescharacteristic
20C2×#410C2211yescharacteristic
21C2×#410C2211yescharacteristic
22C2×#410C2211yescharacteristic
23#2879C2211yescharacteristic
24C2×#408C2211yescharacteristic
25#2873C2211yescharacteristic
26C2×#405C2211yescharacteristic
27#2032C418yes—
28C2×#344C2211yescharacteristic
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37C2×#344C2211yescharacteristic
38C2×#344C2211yescharacteristic
39C2×#344C2211yescharacteristic
40#1742C2211yescharacteristic
41#1742C2211yescharacteristic
42#1742C2211yescharacteristic
43#1742C2211yescharacteristic
44#1742C2211yescharacteristic
45#1740C2211yescharacteristic
46#1740C2211yescharacteristic
47#1740C2211yescharacteristic
48#1729C2211yescharacteristic
49#1729C2211yescharacteristic
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52C22×#41C2311yescharacteristic
53#568—22no—
54#568—22no—
55#568—41no—
56#568—41no—
57C2×#131—41no—
58C2×#131—41no—
59C22×M16—22no—
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C23×C8C2311yescharacteristic
68#544—41no—
69#544—41no—
70#544—41no—
71#543C2×C414yes—
72#533—41no—
73#533—41no—
74#533—41no—
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81#424—21no—
82#424—21no—
83#424—41no—
84C2×C4⋊C8—22no—
85#413—41no—
86#413—41no—
87C2×#101Q812yescomplemented
88C2×#100Q812yescomplemented
89#410—41no—
90#410—41no—
91#410—41no—
92#410—41no—
93#410—41no—
94#410—41no—
95C2×#99C2×C414yes—
96C2×#99C2×C414yes—
97C2×#99C2×C414yes—
98#408—21no—
99#408—21no—
100#408—22no—
101#408—41no—
102#408—41no—
103#407—22no—
104C2×C22⋊C8—22no—
105#405—22no—
106#405—41no—
107#405—41no—
108#405—41no—
109C4×M16—41no—
110C2×#97—21no—
111C2×C4×C8—21no—
112#344—22no—
113#344—22no—
114#344—41no—
115#344—41no—
116#344—41no—
117#344—41no—
118#344—41no—
119#344—41no—
120#344—41no—
121#344—41no—
122#344—41no—
123#344—41no—
124#343—22no—
125#343—22no—
126#343—41no—
127#343—41no—
128#343—41no—
129#342C2×C414yes—
130#342C2×C414yes—
131#342C2×C414yes—
132#336—41no—
133#336—41no—
134#336—41no—
135#336—41no—
136C2×#41—21no—
137C2×#41—21no—
138C2×#41—21no—
139C2×#41—21no—
140C2×#41—21no—
141C2×#41—21no—
142C2×#41—21no—
143C2×#41—21no—
144C2×#41—21no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C22×Q8C2×Q811yescharacteristic
149C22×D4C22×C412yes—
150C22×D4C2×Q811yescharacteristic
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153#131—44no—
154#131—44no—
155#131—44no—
156#131—44no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
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—44no—
196C2×M16—44no—
197C2×M16—42no—
198C2×M16C2×Q812yes—
199C2×M16C2×Q812yes—
200C2×M16#4112yes—
201C2×M16#4112yes—
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—42no—
215C22×C8#4111yescharacteristic
216C22×C8#4111yescharacteristic
217C22×C8#4112yes—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221#125—42no—
222#125—42no—
223#125—42no—
224C22⋊Q8—44no—
225C22⋊Q8—44no—
226C22⋊Q8—44no—
227C4×Q8—44no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
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—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C22⋊C4C22×C412yes—
259C2×C22⋊C4C22×C412yes—
260C2×C22⋊C4C22×C412yes—
261C2×C22⋊C4C22×C412yes—
262C2×C22⋊C4C22×C412yes—
263C2×C22⋊C4C22×C412yes—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C4⋊C8—42no—
271C4⋊C8—42no—
272#101—42no—
273#100—42no—
274#99—44no—
275#99—44no—
276#99—44no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C22⋊C8—44no—
280#97—42no—
281#97—42no—
282C4×C8—42no—
283C4×C8—42no—
284C24C4×Q812yes—
285C24C23×C411yescharacteristic
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—
300#41—41no—
301#41—41no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—44no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4C4×Q812yes—
334C22×C4C22×Q811yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C42−1+411yescharacteristic
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—22no—
353M16—22no—
354M16—22no—
355M16—22no—
356M16—48no—
357M16—41no—
358M16—41no—
359M16—41no—
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—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—22no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—48no—
391C2×C8—41no—
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—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—81no—
452C23—82no—
453C23C2×C4×Q811yescharacteristic
454C23C4×#4111yescharacteristic
455C23C4×#4111yescharacteristic
456C23#51811yescharacteristic
457C23#51811yescharacteristic
458C23#51811yescharacteristic
459C23#52011yescharacteristic
460Q8—41no—
461Q8—41no—
462Q8—41no—
463Q8—41no—
464Q8—164no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#53811yescharacteristic
493C2×C4#53811yescharacteristic
494C2×C4#53811yescharacteristic
495C2×C4#53811yescharacteristic
496C2×C4#53811yescharacteristic
497C2×C4#53911yescharacteristic
498C2×C4#54211yescharacteristic
499C2×C4#54211yescharacteristic
500C2×C4#54211yescharacteristic
501C2×C4#54211yescharacteristic
502C2×C4#54311yescharacteristic
503C2×C4#54411yescharacteristic
504C2×C4#55111yescharacteristic
505C2×C4#55111yescharacteristic
506C2×C4#55111yescharacteristic
507C2×C4#55111yescharacteristic
508C2×C4#55111yescharacteristic
509C2×C4#55111yescharacteristic
510C2×C4#55411yescharacteristic
511C2×C4#55411yescharacteristic
512C2×C4#55411yescharacteristic
513C2×C4#55411yescharacteristic
514C2×C4#55511yescharacteristic
515C2×C4#55511yescharacteristic
516C2×C4#56011yescharacteristic
517C2×C4#56011yescharacteristic
518C2×C4#56011yescharacteristic
519C2×C4H(3,𝔽4)11yescharacteristic
520C8—42no—
521C8—42no—
522C8—41no—
523C8—41no—
524C8—42no—
525C8—42no—
526C8—42no—
527C8—81no—
528C8—81no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22#232911yescharacteristic
541C22#233711yescharacteristic
542C22#233711yescharacteristic
543C22#236811yescharacteristic
544C22#236811yescharacteristic
545C22#237211yescharacteristic
546C22#237511yescharacteristic
547C4—42no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4#273611yescharacteristic
555C4#273711yescharacteristic
556C4#273711yescharacteristic
557C4#274211yescharacteristic
558C4#276011yescharacteristic
559C4#276311yescharacteristic
560C4#278411yescharacteristic
561C4#278411yescharacteristic
562C2—21no—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—82no—
570C2#1459611yescharacteristic
subgroupG/H
centerC4#2760
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC2#14596
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11395C2semidirect
3. #11391C2semidirect
4. #11319C2semidirect
5. #11319C2semidirect
9. #10920C2semidirect
10. #10914C2semidirect
11. #10831C2semidirect
12. #10803C2semidirect
87. C2×#101Q8semidirect
88. C2×#100Q8semidirect

Maximal subgroups

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

subgroupindex
#234742
#113952
#113912
#113192
#110882
#109612
#109412
#109202
#109142
#108312
#108032
#107712
#107572
#107482