Finite Group Databasebeta

← #368594

Subgroups — #368594

Order 512.

875 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21337C211yesmaximal, characteristic
2#21306C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4C4×#430C211yesmaximal, characteristic
5C4⋊C4×M16C211yesmaximal, characteristic
6#11304C212yesmaximal
7#11289C212yesmaximal
8#10897C212yesmaximal
9#10851C211yesmaximal, characteristic
10#10848C211yesmaximal, characteristic
11#10826C212yesmaximal
12C2×#533C2211yescharacteristic
13C2×#430C2211yescharacteristic
14C2×#430C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C2×C4×M16C2212yes—
18C2×C4×M16C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1998C414yescomplemented
23#1997C412yescomplemented
24#1997C412yescomplemented
25#1986—22no—
26#1986—22no—
27#1986—21no—
28#1986—21no—
29#1986—22no—
30#1778—22no—
31C4×#108—22no—
32#1771—22no—
33#1770—21no—
34C8×C4⋊C4—21no—
35C2×#344C2212yes—
36C2×#344C2212yes—
37#1742C2212yes—
38#1742C2212yes—
39#1738C2211yescharacteristic
40#1738C2211yescharacteristic
41#1733C2212yes—
42#1733C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47#1728C2212yes—
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#125C2×C412yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×#117C2312yes—
71C2×#117C2312yes—
72C22×C4⋊C4—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#430—22no—
76#430—22no—
77#430—22no—
78#430Q814yescomplemented
79C2×#108—22no—
80C2×#108—22no—
81#423—21no—
82#423—21no—
83#423C2×C412yes—
84#423C2×C412yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16—21no—
91C4×M16—24no—
92C4×M16—21no—
93C4×M16—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—42no—
97C4×M16C2×C412yes—
98C4×M16C2×C412yes—
99C4×M16D412yes—
100C4×M16D412yes—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—22no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—22no—
108C2×C4×C8—21no—
109#384—21no—
110#382—21no—
111#380—21no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D412yes—
120C2×#95—21no—
121#344—24no—
122#344—42no—
123#343C2×C414yes—
124#343C2×C414yes—
125#343C2×C414yes—
126#343C2×C414yes—
127#341—22no—
128#341—22no—
129#339—24no—
130#339—24no—
131#339—24no—
132#339—22no—
133#339—22no—
134#339—22no—
135#339—22no—
136#339—42no—
137#336—22no—
138#336—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C23×C4—22no—
142C23×C4#4111yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4C2411yescharacteristic
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—22no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—22no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—22no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16C22×C412yes—
177C2×M16C22×C414yes—
178C2×M16C22×C412yes—
179C2×M16C2×D412yes—
180C2×M16C2×D412yes—
181C2×M16C2×Q812yes—
182C2×M16C2×Q812yes—
183C2×M16C2×Q812yes—
184C2×M16#4112yes—
185C2×M16#4112yes—
186C2×M16#4112yes—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202#125—22no—
203#125—22no—
204#125—22no—
205#125—22no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—41no—
220C4×Q8—41no—
221C4×Q8—41no—
222#117—22no—
223#117—22no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C412yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266#108—28no—
267#108—28no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272#97—22no—
273#97—22no—
274#97—22no—
275#97—22no—
276#97—44no—
277#97—41no—
278#97—41no—
279C4×C8—22no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—22no—
283C4×C8—44no—
284C4×C8—41no—
285C4×C8—41no—
286#95—21no—
287#95—22no—
288#95—21no—
289#95—41no—
290#95—41no—
291C24C2×#4111yescharacteristic
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—82no—
297C2×Q8—82no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—81no—
308C22×C4—82no—
309C22×C4#11711yescharacteristic
310C22×C4#11711yescharacteristic
311C22×C4#11711yescharacteristic
312C22×C4#11711yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4#12311yescharacteristic
316C22×C4#12311yescharacteristic
317C22×C4#12411yescharacteristic
318C22×C4#12411yescharacteristic
319C22×C4#12411yescharacteristic
320C22×C4#12411yescharacteristic
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×Q811yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)11yescharacteristic
332C22×C42−1+411yescharacteristic
333C22×C42−1+411yescharacteristic
334C22×C42−1+411yescharacteristic
335M16—22no—
336M16—22no—
337M16—22no—
338M16—22no—
339M16—22no—
340M16—22no—
341M16—24no—
342M16—22no—
343M16—22no—
344M16—24no—
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—41no—
352M16—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—42no—
359M16—41no—
360M16—42no—
361M16C4×Q814yes—
362M16C22⋊Q814yes—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—42no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42C4×D411yescharacteristic
418C42C4×D411yescharacteristic
419C42C4×D411yescharacteristic
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42C4×D411yescharacteristic
423C42C4×D411yescharacteristic
424C42C4×D411yescharacteristic
425C42#12111yescharacteristic
426C42#12111yescharacteristic
427C42#12111yescharacteristic
428C42#12111yescharacteristic
429C42#12611yescharacteristic
430C42#12611yescharacteristic
431C42#12611yescharacteristic
432C42#12611yescharacteristic
433C23—41no—
434C23—41no—
435C23C2×#11711yescharacteristic
436C23C2×#12311yescharacteristic
437C23C2×#12411yescharacteristic
438C23#55611yescharacteristic
439C23#55611yescharacteristic
440C23#55711yescharacteristic
441C23#55911yescharacteristic
442Q8—82no—
443Q8—82no—
444Q8—82no—
445Q8—82no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#38811yescharacteristic
474C2×C4#38811yescharacteristic
475C2×C4#38811yescharacteristic
476C2×C4#38811yescharacteristic
477C2×C4#38811yescharacteristic
478C2×C4#38811yescharacteristic
479C2×C4#38811yescharacteristic
480C2×C4#38811yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×C4×Q811yescharacteristic
484C2×C4C4×#4111yescharacteristic
485C2×C4#51911yescharacteristic
486C2×C4#52011yescharacteristic
487C2×C4C2×#12111yescharacteristic
488C2×C4C2×C22⋊Q811yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4C2×#12611yescharacteristic
491C2×C4#52911yescharacteristic
492C2×C4#53311yescharacteristic
493C2×C4#53611yescharacteristic
494C2×C4#53711yescharacteristic
495C2×C4#54111yescharacteristic
496C2×C4#54211yescharacteristic
497C2×C4#54611yescharacteristic
498C2×C4#54711yescharacteristic
499C2×C4#54811yescharacteristic
500C2×C4D4×Q811yescharacteristic
501C2×C4#55211yescharacteristic
502C2×C4#55211yescharacteristic
503C2×C4#55411yescharacteristic
504C2×C4#55611yescharacteristic
505C2×C4#55611yescharacteristic
506C2×C4#55611yescharacteristic
507C2×C4#55711yescharacteristic
508C2×C4#55911yescharacteristic
509C8—24no—
510C8—24no—
511C8—42no—
512C8—44no—
513C8—42no—
514C8—42no—
515C8—42no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22C2×#38811yescharacteristic
527C22#235511yescharacteristic
528C22#235611yescharacteristic
529C22#244611yescharacteristic
530C22#244711yescharacteristic
531C22#256811yescharacteristic
532C22#259211yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—44no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—162no—
545C4#193212yes—
546C4C4×C22⋊Q811yescharacteristic
547C4#232311yescharacteristic
548C4#236511yescharacteristic
549C4#236711yescharacteristic
550C4#241511yescharacteristic
551C4#241711yescharacteristic
552C4#251111yescharacteristic
553C4#251411yescharacteristic
554C2—21no—
555C2—21no—
556C2—21no—
557C2—21no—
558C2—21no—
559C2—21no—
560C2#1402111yescharacteristic
561C2C2×#193211yescharacteristic
562C2#2108011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#388
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1998C4semidirect
23. #1997C4semidirect
24. #1997C4semidirect
78. #430Q8semidirect

Maximal subgroups

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

subgroupindex
#213372
#213062
C2×#19862
C4×#4302
C4⋊C4×M162
#113042
#112892
#108972
#108512
#108482
#108262