Finite Group Databasebeta

← #364172

Subgroups — #364172

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2930C211yesmaximal, characteristic
2#20971C212yesmaximal
3C2×#1809C211yesmaximal, characteristic, complemented
4C2×#1738C211yesmaximal, characteristic, complemented
5#11239C211yesmaximal, characteristic, complemented
6#11238C211yesmaximal, characteristic, complemented
7#11107C211yesmaximal, characteristic, complemented
8#11106C211yesmaximal, characteristic, complemented
9#10827C212yesmaximal, complemented
10#10774C212yesmaximal, complemented
11#10739C212yesmaximal, complemented
12C22×#131C2211yescharacteristic
13C23×M16C2212yes—
14C23×M16C2211yescharacteristic
15C2×#525C2211yescharacteristic
16#2930—24no—
17#2930C418yescomplemented
18C2×#436—22no—
19C2×#435—22no—
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C2×#424C2211yescharacteristic
23C2×#417C2212yes—
24C2×#417C2212yes—
25C2×#408C2211yescharacteristic
26C2×#407C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#380C2212yes—
29#1895—22no—
30#1894—22no—
31#1893—22no—
32#1892—22no—
33#1809—22no—
34#1809—22no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39#1738—24no—
40C2×#341C2212yes—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43#1736C2212yes—
44#1736C2212yes—
45C2×#340C2212yes—
46C22×#41C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#131—22no—
49C2×#131—42no—
50C2×#131—42no—
51C2×#131C2×C414yes—
52C22×M16—22no—
53C22×M16—21no—
54C22×M16—24no—
55C22×M16—24no—
56C22×M16—24no—
57C22×M16—24no—
58C22×M16—42no—
59C22×M16—42no—
60C22×M16—42no—
61C22×M16—42no—
62C22×M16C2×C414yes—
63C22×M16D412yes—
64C22×M16D412yes—
65C22×M16D414yes—
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2312yes—
69C22×M16C2312yes—
70C23×C8—21no—
71C23×C8—22no—
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74#525—22no—
75#525—22no—
76#525C2×C414yes—
77#525Q814yes—
78C2×#123—21no—
79C2×C22⋊Q8D412yes—
80C2×#121D412yes—
81C2×#120—21no—
82C2×C4×D4—22no—
83C2×#117C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—22no—
87#436—216no—
88#435—216no—
89C2×#107—22no—
90C2×#107—42no—
91C2×#106—22no—
92C2×#106—42no—
93#424—22no—
94#424C2×C414yes—
95C2×C4⋊C8—22no—
96#417—24no—
97#417—24no—
98#417—42no—
99#417—42no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#102—22no—
103C2×#102—22no—
104#408—22no—
105#408C2×C414yes—
106#407—22no—
107#407C2×C414yes—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110#405—22no—
111#405C2×C414yes—
112C2×#97—21no—
113C2×C4×C8—21no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118C2×#95—22no—
119#343—24no—
120#343—24no—
121#343—42no—
122#343—42no—
123#341—24no—
124#341—22no—
125#341—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132C25#4111yescharacteristic
133C2×#41—21no—
134C2×#41—21no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—22no—
139C2×#41C22×C412yes—
140C2×#41C2×Q812yes—
141C22×Q8C2×D411yescharacteristic
142C22×D4—21no—
143C22×D4—22no—
144C22×D4C2×D411yescharacteristic
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4C22×C412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×Q812yes—
157C23×C4#4111yescharacteristic
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160#131—48no—
161#131—48no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—24no—
167C2×M16—216no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—48no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—42no—
184C2×M16—42no—
185C2×M16—48no—
186C2×M16—44no—
187C2×M16—44no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—48no—
191C2×M16—48no—
192C2×M16—48no—
193C2×M16—44no—
194C2×M16C22⋊C418yes—
195C2×M16C22⋊C418yes—
196C2×M16C22×C412yes—
197C2×M16C22×C412yes—
198C2×M16#4114yes—
199C2×M16#4114yes—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—24no—
209C22×C8—44no—
210C22×C8—44no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—44no—
214C22×C8C22×C412yes—
215C22×C8C22×C412yes—
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D412yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221#123—24no—
222#123—24no—
223C22⋊Q8—24no—
224C22⋊Q8C4⋊C418yes—
225#121—24no—
226#121C4⋊C418yes—
227#120—24no—
228#120—24no—
229C4×D4—28no—
230C4×D4—28no—
231#117—21no—
232#117—21no—
233#117C22×C412yes—
234#117C2×Q812yes—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C2×D412yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—84no—
255C2×C42—21no—
256#107—44no—
257#106—44no—
258C4⋊C8—28no—
259#103—44no—
260#103—44no—
261#102—44no—
262#102—44no—
263C22⋊C8—28no—
264C22⋊C8—28no—
265#97—24no—
266C4×C8—24no—
267#95—42no—
268#95—42no—
269C24—22no—
270C24—22no—
271C24—21no—
272C24—42no—
273C24—42no—
274C24#11712yes—
275C24#12112yes—
276C24C22⋊Q812yes—
277C24C2×#4111yescharacteristic
278#41—24no—
279#41—24no—
280#41—24no—
281#41—24no—
282#41—42no—
283#41—42no—
284#41—42no—
285#41—42no—
286C2×Q8—21no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8C2×C4⋊C412yes—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—28no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4C2×C4⋊C412yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4C2×C22⋊C414yes—
338C22×C4C2×C4⋊C414yes—
339C22×C4#11712yes—
340C22×C4#12114yes—
341C22×C4#12112yes—
342C22×C4C22⋊Q814yes—
343C22×C4C22⋊Q812yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×Q811yescharacteristic
349C22×C4C2×#4112yes—
350C22×C4C2×#4112yes—
351C22×C4C2×#4111yescharacteristic
352C22×C4H(5,𝔽2)12yes—
353C22×C4H(5,𝔽2)12yes—
354M16—24no—
355M16—24no—
356M16—24no—
357M16—24no—
358M16—24no—
359M16—24no—
360M16—48no—
361M16—48no—
362M16—48no—
363M16—416no—
364M16—42no—
365M16—42no—
366M16—42no—
367M16—42no—
368M16—42no—
369M16—42no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—24no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—24no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—48no—
385C2×C8—416no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8C2×C22⋊C412yes—
395C2×C8C2×C22⋊C412yes—
396C2×C8C2×C22⋊C414yes—
397C4⋊C4—22no—
398C4⋊C4—28no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4C2×C4⋊C414yes—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C42—22no—
415C42—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—44no—
428C23—44no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23#38014yes—
438C23C2×#11711yescharacteristic
439C23C2×#12111yescharacteristic
440C23C2×C22⋊Q811yescharacteristic
441C23#55312yes—
442C23#55312yes—
443Q8—24no—
444Q8—42no—
445D4—24no—
446D4—44no—
447D4—44no—
448D4—42no—
449D4—88no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—162no—
479C2×C4—162no—
480C2×C4—162no—
481C2×C4—162no—
482C2×C4#38014yes—
483C2×C4C22×C22⋊C411yescharacteristic
484C2×C4C22×C4⋊C411yescharacteristic
485C2×C4C4×#4112yes—
486C2×C4#51812yes—
487C2×C4C2×#12111yescharacteristic
488C2×C4C2×C22⋊Q811yescharacteristic
489C2×C4#52512yes—
490C2×C4#53211yescharacteristic
491C2×C4#53211yescharacteristic
492C2×C4#53411yescharacteristic
493C2×C4#53411yescharacteristic
494C2×C4#53412yes—
495C2×C4#54612yes—
496C2×C4#54612yes—
497C2×C4#55112yes—
498C2×C4#55112yes—
499C2×C4#55412yes—
500C2×C4#55412yes—
501C8—22no—
502C8—24no—
503C8—22no—
504C8—42no—
505C8—44no—
506C8—44no—
507C8—42no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—44no—
523C22—84no—
524C22—84no—
525C22—81no—
526C22—81no—
527C22—82no—
528C22C2×#38011yescharacteristic
529C22#235312yes—
530C22#254311yescharacteristic
531C22#254311yescharacteristic
532C22#257912yes—
533C22#306812yes—
534C22#306912yes—
535C4—24no—
536C4—42no—
537C4—44no—
538C4—162no—
539C4—162no—
540C4#229711yescharacteristic
541C4#230811yescharacteristic
542C4#234912yes—
543C4#253412yes—
544C4#256411yescharacteristic
545C4#256411yescharacteristic
546C2—22no—
547C2—22no—
548C2—44no—
549C2—82no—
550C2#1396011yescharacteristic
551C2#2067912yes—
552C2#2859411yescharacteristic
553C2#2859511yescharacteristic
554C2#2859612yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1809C2semidirect
4. C2×#1738C2semidirect
5. #11239C2semidirect
6. #11238C2semidirect
7. #11107C2semidirect
8. #11106C2semidirect
9. #10827C2semidirect
10. #10774C2semidirect
11. #10739C2semidirect
17. #2930C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29302
#209712
C2×#18092
C2×#17382
#112392
#112382
#111072
#111062
#108272
#107742
#107392