Finite Group Databasebeta

← #359198

Subgroups — #359198

Order 512.

1294 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2C2×#2028C211yesmaximal, characteristic
3C4×#419C212yesmaximal
4#13786C211yesmaximal, characteristic
5#11392C211yesmaximal, characteristic
6#11296C211yesmaximal, characteristic
7#11276C212yesmaximal
8#11064C212yesmaximal
9#11055C211yesmaximal, characteristic
10C4×#343C211yesmaximal, characteristic
11#10701C212yesmaximal
12C2×#544C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#419C2212yes—
15C2×#419C2212yes—
16C2×C4×M16C2212yes—
17#2331—22no—
18#2330—22no—
19#2329—22no—
20#2317C2212yes—
21#2316C2211yescharacteristic
22C4×C4⋊Q8—22no—
23C4×#125—22no—
24C4×C22⋊Q8—22no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C4×#117C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#2028C418yescomplemented
31#1763—22no—
32#1763—22no—
33C4×#103—22no—
34C4×#103—22no—
35C2×#343C2212yes—
36C2×#343C2211yescharacteristic
37C2×#343C2211yescharacteristic
38#1736C2212yes—
39#1736C2212yes—
40C2×#339C2212yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44#1436—22no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47#1434—22no—
48C23×Q8C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2312yes—
51#544—24no—
52#544—22no—
53#544C2×C414yes—
54C2×C4⋊Q8—22no—
55C2×#125—22no—
56C2×C22⋊Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—24no—
62C2×C4×Q8Q814yescomplemented
63C2×#117C2312yes—
64C2×#117C2312yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42D412yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#419—24no—
75#419—24no—
76#419—24no—
77#419C2×C418yes—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C4×M16—24no—
83C4×M16—24no—
84C2×#97—22no—
85C2×C4×C8—22no—
86#391—24no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391—22no—
91#391C2×C414yes—
92#389—24no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#387—24no—
99#387—24no—
100#387—24no—
101#385—24no—
102#384—22no—
103#382—22no—
104#380—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4D412yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115#376—22no—
116#376—24no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D412yes—
124C43—22no—
125#343—24no—
126#343—22no—
127#343C2×C414yes—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#339—24no—
133#339—24no—
134#339—24no—
135#339C2×C418yes—
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8C22×C412yes—
145C22×Q8C2×Q812yes—
146C22×Q8#4112yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16C22×C414yes—
167C2×M16#4114yes—
168C22×C8—22no—
169C22×C8—22no—
170C4⋊Q8—28no—
171C4⋊Q8—28no—
172#125—28no—
173#125—28no—
174C22⋊Q8—28no—
175C22⋊Q8—28no—
176C4×Q8—28no—
177C4×Q8—28no—
178C4×Q8—44no—
179C4×Q8—44no—
180C4×Q8—84no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117C22×C414yes—
188#117#4114yes—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×Q812yes—
214C2×C4⋊C4#4112yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—82no—
235C2×C42C22×C412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×Q812yes—
239C2×C42#4112yes—
240C2×C42#4114yes—
241C2×C42#4112yes—
242#103—28no—
243#103—28no—
244#103—28no—
245#103—28no—
246#97—44no—
247C4×C8—44no—
248#95—28no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—28no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—42no—
259#95—42no—
260#95—82no—
261#95—82no—
262C24C22×D411yescharacteristic
263C2×Q8—28no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×Q8C4×Q814yescomplemented
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4C4×D414yes—
326C22×C4#12012yes—
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4C22⋊Q812yes—
330C22×C4C22⋊Q812yes—
331C22×C4#12412yes—
332C22×C4#12412yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C22×Q811yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4112yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343M16—24no—
344M16—24no—
345M16—24no—
346M16—24no—
347M16—42no—
348M16—42no—
349M16—42no—
350M16—42no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C4⋊C4—28no—
358C4⋊C4—216no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—84no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—84no—
377C4⋊C4C4×D414yes—
378C4⋊C4C4×D414yes—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—24no—
387C42—44no—
388C42—44no—
389C42—44no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—48no—
395C42—82no—
396C42—82no—
397C42—82no—
398C42C22⋊Q814yes—
399C42C22⋊Q814yes—
400C42#12414yes—
401C42#12414yes—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23C2×C4×D411yescharacteristic
407C23C2×#12011yescharacteristic
408C23C2×C22⋊Q811yescharacteristic
409C23#52511yescharacteristic
410C23#52511yescharacteristic
411C23C2×#12411yescharacteristic
412C23#53211yescharacteristic
413Q8—28no—
414Q8—44no—
415Q8—44no—
416Q8—44no—
417Q8—88no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
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—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4#39314yes—
477C2×C4#39312yes—
478C2×C4#39312yes—
479C2×C4C2×C4×D412yes—
480C2×C4C2×C4×Q811yescharacteristic
481C2×C4C4×#4112yes—
482C2×C4C4×#4111yescharacteristic
483C2×C4C2×C22⋊Q812yes—
484C2×C4#52512yes—
485C2×C4#52512yes—
486C2×C4C2×#12412yes—
487C2×C4#52912yes—
488C2×C4#52912yes—
489C2×C4#52911yescharacteristic
490C2×C4#52912yes—
491C2×C4#52912yes—
492C2×C4#53212yes—
493C2×C4#53311yescharacteristic
494C2×C4#53312yes—
495C8—44no—
496C8—44no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22C4×#12011yescharacteristic
510C22C4×C22⋊Q811yescharacteristic
511C22C4×#12411yescharacteristic
512C22C2×#39311yescharacteristic
513C22#238811yescharacteristic
514C22#239611yescharacteristic
515C22#239811yescharacteristic
516C22#301911yescharacteristic
517C22#302011yescharacteristic
518C22#302211yescharacteristic
519C22#302211yescharacteristic
520C4—22no—
521C4—22no—
522C4—24no—
523C4—44no—
524C4—44no—
525C4—44no—
526C4—44no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4C4×C22⋊Q812yes—
534C4C4×#12412yes—
535C4#239612yes—
536C4#239812yes—
537C2—21no—
538C2—21no—
539C2—21no—
540C2—21no—
541C2C4×#39311yescharacteristic
542C2#2147411yescharacteristic
543C2#2147611yescharacteristic
544C2#2346211yescharacteristic
545C2#2346411yescharacteristic
546C2#2359311yescharacteristic
547C2#2359411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #2028C4semidirect
62. C2×C4×Q8Q8semidirect
281. C2×Q8C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5442
C2×#20282
C4×#4192
#137862
#113922
#112962
#112762
#110642
#110552
C4×#3432
#107012