Finite Group Databasebeta

← #416251

Subgroups — #416251

Order 512.

966 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
1C2×#1950C211yesmaximal, characteristic
2#21171C211yesmaximal, characteristic
3#20755C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1801C211yesmaximal, characteristic
6C4×#419C211yesmaximal, characteristic
7#11303C212yesmaximal
8#11249C212yesmaximal
9#11145C212yesmaximal
10#11128C212yesmaximal
11#10857C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×#419C2211yescharacteristic
19C2×#419C2211yescharacteristic
20C22×#103C2212yes—
21C22×#101C2212yes—
22C2×C4×M16C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1950—24no—
26#1948—21no—
27#1948—21no—
28#1947—21no—
29#1947—21no—
30#1832—22no—
31#1831—22no—
32#1803—22no—
33#1803—22no—
34#1801—24no—
35#1763—21no—
36#1763—21no—
37C4×#103—21no—
38C4×#103—21no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#341C2212yes—
43#1736C2211yescharacteristic
44#1736C2212yes—
45#1736C2211yescharacteristic
46C2×#340C2212yes—
47#1729C2212yes—
48#1729C2212yes—
49C23×Q8C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2312yes—
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C23×C8C2312yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8Q814yes—
63C2×#117C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C2×#108—24no—
73C2×#108—24no—
74C2×#108—42no—
75#428—42no—
76#428—42no—
77C2×#107—22no—
78C2×#106—22no—
79#423—22no—
80#423—22no—
81#423—22no—
82#423D414yes—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8D412yes—
90C2×C4⋊C8D412yes—
91#419—22no—
92#419—22no—
93#419—22no—
94#419—22no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#101—24no—
103C2×#101—24no—
104C2×#101—42no—
105C4×M16—22no—
106C4×M16—22no—
107C2×#97—21no—
108C2×C4×C8—21no—
109#384—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376—21no—
120C43—21no—
121#343—24no—
122#343—24no—
123#343—42no—
124#341—24no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336—24no—
133#336—24no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8C2×Q812yes—
146C22×Q8#4112yes—
147C22×Q8#4112yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4#4111yescharacteristic
152C23×C4#4111yescharacteristic
153C23×C4C2411yescharacteristic
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—22no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—21no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16C2×D414yes—
176C2×M16#4112yes—
177C2×M16#4112yes—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C2×D414yes—
189C4×Q8—24no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197#117—21no—
198#117—21no—
199#117—21no—
200#117—21no—
201#117—21no—
202#117—21no—
203#117—21no—
204#117—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×Q812yes—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—24no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—42no—
243C2×C42C2×D412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×Q812yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251#108—48no—
252#107—44no—
253#106—44no—
254C4⋊C8—24no—
255C4⋊C8—24no—
256C4⋊C8—24no—
257C4⋊C8—24no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260#103—24no—
261#103—24no—
262#103—24no—
263#103—24no—
264#103—48no—
265#101—48no—
266#97—42no—
267C4×C8—42no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272C24C2×#4111yescharacteristic
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8#12514yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4#12111yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4C22⋊Q812yes—
331C22×C4C22⋊Q812yes—
332C22×C4#12612yes—
333C22×C4#12612yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C42−1+411yescharacteristic
347C22×C42−1+411yescharacteristic
348M16—24no—
349M16—22no—
350M16—24no—
351M16—22no—
352M16—22no—
353M16—22no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—42no—
358M16—41no—
359M16—42no—
360M16—42no—
361M16—42no—
362M16—42no—
363M16—42no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—44no—
414C42—44no—
415C42—42no—
416C42—42no—
417C42—81no—
418C42—81no—
419C42#12112yes—
420C42#12112yes—
421C42#12112yes—
422C42#12112yes—
423C42C22⋊Q812yes—
424C42C22⋊Q812yes—
425C42C22⋊Q812yes—
426C42C22⋊Q812yes—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23C2×#12111yescharacteristic
432C23C2×C22⋊Q811yescharacteristic
433C23C2×#12611yescharacteristic
434C23#55211yescharacteristic
435C23#55311yescharacteristic
436C23#55511yescharacteristic
437C23#55611yescharacteristic
438Q8—24no—
439Q8—24no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443Q8—42no—
444Q8—42no—
445Q8—42no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—162no—
484C2×C4—162no—
485C2×C4#39612yes—
486C2×C4#39612yes—
487C2×C4#39612yes—
488C2×C4#39612yes—
489C2×C4#47112yes—
490C2×C4#47312yes—
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×C22⋊Q811yescharacteristic
494C2×C4C2×C22⋊Q811yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4#52511yescharacteristic
497C2×C4C2×#12511yescharacteristic
498C2×C4#52911yescharacteristic
499C2×C4#52911yescharacteristic
500C2×C4#53311yescharacteristic
501C2×C4#53511yescharacteristic
502C2×C4#53611yescharacteristic
503C2×C4#54211yescharacteristic
504C2×C4#54211yescharacteristic
505C2×C4#54411yescharacteristic
506C2×C4#54411yescharacteristic
507C2×C4#54611yescharacteristic
508C2×C4#54611yescharacteristic
509C2×C4#54611yescharacteristic
510C2×C4#54711yescharacteristic
511C2×C4#54811yescharacteristic
512C2×C4#54811yescharacteristic
513C2×C4#54811yescharacteristic
514C2×C4#55011yescharacteristic
515C2×C4#55211yescharacteristic
516C2×C4#55211yescharacteristic
517C2×C4#55311yescharacteristic
518C2×C4#55311yescharacteristic
519C8—44no—
520C8—44no—
521C8—42no—
522C8—42no—
523C8—82no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22C2×#39611yescharacteristic
537C22#241411yescharacteristic
538C22#241511yescharacteristic
539C22#243911yescharacteristic
540C22#244211yescharacteristic
541C22#249411yescharacteristic
542C22#249511yescharacteristic
543C22C2×#47111yescharacteristic
544C22C2×#47311yescharacteristic
545C22#306811yescharacteristic
546C22#306911yescharacteristic
547C4—22no—
548C4—22no—
549C4—22no—
550C4—44no—
551C4—82no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—162no—
557C4#206414yes—
558C4#242611yescharacteristic
559C4#242711yescharacteristic
560C4#244111yescharacteristic
561C4#244211yescharacteristic
562C4#253511yescharacteristic
563C4#253611yescharacteristic
564C4#262811yescharacteristic
565C4#285111yescharacteristic
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2#1522011yescharacteristic
571C2C2×#206411yescharacteristic
572C2#2360411yescharacteristic
573C2#2467211yescharacteristic
574C2#2467311yescharacteristic
575C2#2472211yescharacteristic
576C2#2472311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19502
#211712
#207552
C2×#18032
C2×#18012
C4×#4192
#113032
#112492
#111452
#111282
#108572