Finite Group Databasebeta

← #356197

Subgroups — #356197

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic
2C2×#1805C211yesmaximal, characteristic
3C2×#1770C211yesmaximal, characteristic
4C2×#1763C212yesmaximal
5C2×C4×#103C211yesmaximal, characteristic
6#13804C211yesmaximal, characteristic
7#10885C212yesmaximal
8#10881C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10842C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4×Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#97C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2351—24no—
21#2351C418yescomplemented
22C2×#389—22no—
23C2×#389C2212yes—
24C2×#382—21no—
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1805C418yescomplemented
32#1770—24no—
33#1763—28no—
34C4×#103C418yescomplemented
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#336C2212yes—
43#1436—22no—
44C4⋊#95C2212yes—
45#1434—21no—
46C4×#95—21no—
47C23×Q8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8—42no—
59C2×C4×Q8C2×C418yes—
60C2×C4×Q8C2×C414yes—
61C2×C4×Q8C2×C414yes—
62C2×C4×Q8C2×C414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—21no—
73C22×C42—22no—
74C22×C42—21no—
75C22×C42—22no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8C2×C414yes—
81C2×#103—24no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—24no—
85C2×#103C2×C418yes—
86C2×#103C2×C414yes—
87C2×#103C2×C414yes—
88C2×#103D414yes—
89C2×#103Q814yes—
90C2×#97—24no—
91C2×#97—24no—
92C2×#97—42no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8C2×C414yes—
96#389—216no—
97#389—24no—
98#389C2×C418yes—
99#382—22no—
100#382—22no—
101#382—28no—
102#382C2×C414yes—
103#382C2×C414yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—28no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—42no—
108C4×C4⋊C4C2×C414yes—
109C4×C4⋊C4C2×C414yes—
110C4×C4⋊C4D414yes—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#340C2×C418yes—
127#340C2×C414yes—
128#340C2×C414yes—
129#336—24no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8C22×C412yes—
138C22×Q8C22×C412yes—
139C22×Q8C22×C412yes—
140C22×Q8C2×D412yes—
141C22×Q8C2×Q812yes—
142C22×Q8#4112yes—
143C22×Q8#4112yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8C22×C412yes—
172C22×C8C22×C414yes—
173C22×C8C22×C412yes—
174C22×C8C2×D412yes—
175C22×C8C2×Q812yes—
176C4×Q8—28no—
177C4×Q8—28no—
178C4×Q8—28no—
179C4×Q8—216no—
180C4×Q8—48no—
181C4×Q8—48no—
182C4×Q8—48no—
183C4×Q8C42116yescomplemented
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C22⋊C418yes—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C414yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×Q812yes—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—28no—
234C2×C42—24no—
235C2×C42—44no—
236C2×C42—44no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—82no—
240C2×C42—82no—
241C2×C42C22×C414yes—
242C2×C42C22×C412yes—
243C2×C42C22×C412yes—
244C2×C42C22×C412yes—
245C2×C42C2×D414yes—
246C2×C42C2×D412yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C4⋊C8—28no—
250#103—28no—
251#103—216no—
252#103C4⋊C4116yescomplemented
253#97—48no—
254C4×C8—28no—
255#95—24no—
256#95—28no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—44no—
262#95—44no—
263#95—82no—
264#95C22⋊C414yes—
265#95C22⋊C414yes—
266C24C22×D411yescharacteristic
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
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×Q8C2×C4214yes—
289C2×Q8C2×C4⋊C414yes—
290C2×Q8#11714yes—
291C2×Q8C4×D414yes—
292C2×Q8C4×D414yes—
293C2×Q8C4×Q814yes—
294C2×Q8C4×Q814yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4C2×C22⋊C414yes—
348C22×C4C2×C22⋊C414yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4C4×D412yes—
354C22×C4C4×D412yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C23×C411yescharacteristic
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×Q811yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C42−1+411yescharacteristic
368C22×C42−1+411yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8C2×C4⋊C414yes—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—216no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—48no—
404C4⋊C4—416no—
405C4⋊C4—416no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4C2×C4214yes—
413C4⋊C4C2×C4⋊C414yes—
414C42—24no—
415C42—24no—
416C42—24no—
417C42—24no—
418C42—216no—
419C42—416no—
420C42—84no—
421C42—88no—
422C42C2×C4214yes—
423C42#11714yes—
424C42C4×D414yes—
425C42C4×D414yes—
426C23C22×C22⋊C411yescharacteristic
427C23C2×C4×D411yescharacteristic
428C23C2×C4×D411yescharacteristic
429C23D4211yescharacteristic
430C23#54611yescharacteristic
431C23#54811yescharacteristic
432C23D4×Q811yescharacteristic
433C23#57512yes—
434C23#57512yes—
435C23#57712yes—
436C23#57812yes—
437Q8—24no—
438Q8—24no—
439Q8—24no—
440Q8—24no—
441Q8—24no—
442Q8—24no—
443Q8—24no—
444Q8C4×C4⋊C418yes—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4C4×C22⋊C414yes—
504C2×C4C4×C22⋊C414yes—
505C2×C4C22×C4211yescharacteristic
506C2×C4C22×C4⋊C411yescharacteristic
507C2×C4C2×#11711yescharacteristic
508C2×C4C2×#11711yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×C4×D411yescharacteristic
513C2×C4C2×C4×Q811yescharacteristic
514C2×C4C2×C4×Q811yescharacteristic
515C2×C4#51811yescharacteristic
516C2×C4#51911yescharacteristic
517C2×C4#52011yescharacteristic
518C2×C4#52011yescharacteristic
519C2×C4#52011yescharacteristic
520C2×C4#52011yescharacteristic
521C2×C4D4211yescharacteristic
522C2×C4#54611yescharacteristic
523C2×C4#54711yescharacteristic
524C2×C4#54711yescharacteristic
525C2×C4#54811yescharacteristic
526C2×C4D4×Q811yescharacteristic
527C2×C4#55211yescharacteristic
528C2×C4#55211yescharacteristic
529C2×C4#55311yescharacteristic
530C2×C4#55311yescharacteristic
531C2×C4#55411yescharacteristic
532C2×C4#55411yescharacteristic
533C8—24no—
534C8—42no—
535C8—44no—
536C22C2×C4×C22⋊C411yescharacteristic
537C22D4×C22⋊C411yescharacteristic
538C22Q8×C22⋊C411yescharacteristic
539C22D4×C4⋊C411yescharacteristic
540C22#235211yescharacteristic
541C22#236011yescharacteristic
542C22#236111yescharacteristic
543C22#289612yes—
544C22#289812yes—
545C22#294311yescharacteristic
546C22#294311yescharacteristic
547C22#294312yes—
548C22#294712yes—
549C22#294911yescharacteristic
550C22#295011yescharacteristic
551C22#327512yes—
552C22#327712yes—
553C22#332612yes—
554C22#332712yes—
555C22#337112yes—
556C22#337712yes—
557C22#338812yes—
558C22#339012yes—
559C4—24no—
560C4—24no—
561C4—28no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—44no—
566C4—84no—
567C4—84no—
568C4—82no—
569C4C2×C4×C4⋊C411yescharacteristic
570C4#227411yescharacteristic
571C4#234411yescharacteristic
572C4#234511yescharacteristic
573C4D4×C4⋊C411yescharacteristic
574C4#235211yescharacteristic
575C4#236011yescharacteristic
576C4#236111yescharacteristic
577C2C22⋊C4×C4⋊C411yescharacteristic
578C2#2041111yescharacteristic
579C2#2041711yescharacteristic
580C2#2242912yes—
581C2#2252712yes—
582C2#2267311yescharacteristic
583C2#2267511yescharacteristic
584C2#2276111yescharacteristic
585C2#2276511yescharacteristic
586C2#2297312yes—
587C2#2300612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #2351C4semidirect
31. #1805C4semidirect
34. C4×#103C4semidirect
183. C4×Q8C42semidirect
252. #103C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
C2×#18052
C2×#17702
C2×#17632
C2×C4×#1032
#138042
#108852
#108812
#108502
#108422
#107892