Finite Group Databasebeta

← #415591

Subgroups — #415591

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2#21172C212yesmaximal
3#13909C211yesmaximal, characteristic
4#11392C211yesmaximal, characteristic
5#11392C211yesmaximal, characteristic
6#11342C211yesmaximal, characteristic
7#11342C211yesmaximal, characteristic
8#11340C212yesmaximal
9#11339C212yesmaximal
10#10852C212yesmaximal
11#10847C211yesmaximal, characteristic
12C2×#544C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#423C2212yes—
16#2331—22no—
17#2330—22no—
18#2329—22no—
19#2317C2212yes—
20#2316C2211yescharacteristic
21C4×C4⋊Q8—22no—
22C4×#125—22no—
23C4×C22⋊Q8—22no—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C4×#117C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1949—22no—
31#1949—22no—
32#1949—22no—
33#1949—22no—
34C2×#343C2211yescharacteristic
35C2×#343C2211yescharacteristic
36C2×#343C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39C2×#339C2212yes—
40C2×#339C2212yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43#1441—22no—
44#1436—22no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C22×M16C2312yes—
48C22×M16C2312yes—
49#544—24no—
50#544—22no—
51#544Q814yes—
52C2×C4⋊Q8—22no—
53C2×#125—22no—
54C2×C22⋊Q8—22no—
55C2×C4×Q8—22no—
56C2×#117C2312yes—
57C2×#117C2312yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#428—42no—
69#428—42no—
70#428—42no—
71#428—42no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#106—22no—
75C2×#106—22no—
76#423—24no—
77#423Q818yescomplemented
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80#391—24no—
81#391—22no—
82#389—24no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#387—24no—
89#387—24no—
90#387—24no—
91#385—24no—
92#384—22no—
93#384—42no—
94#382—22no—
95#382—22no—
96#382—22no—
97#382—42no—
98#380—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4Q814yescomplemented
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111#376—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95D412yes—
119C43—22no—
120#343—22no—
121#343—22no—
122#343—42no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#339—24no—
128#339—24no—
129#339—42no—
130#339—42no—
131C22×Q8—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16C2×Q814yes—
152C2×M16C2×Q814yes—
153C22×C8—22no—
154C22×C8—22no—
155C4⋊Q8—28no—
156C4⋊Q8—28no—
157#125—28no—
158#125—28no—
159C22⋊Q8—28no—
160C22⋊Q8—28no—
161C4×Q8—84no—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117C2×Q814yes—
169#117#4114yes—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×Q812yes—
201C2×C4⋊C4#4112yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—82no—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×Q812yes—
222C2×C42C2×Q812yes—
223C2×C42C2×Q814yes—
224C2×C42C2×Q812yes—
225C2×C42#4112yes—
226#107—44no—
227#107—44no—
228#106—44no—
229#106—44no—
230C4⋊C8—28no—
231C4⋊C8—28no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—24no—
241#95—44no—
242#95—44no—
243#95—41no—
244#95—42no—
245#95—41no—
246#95—82no—
247C24C22×D411yescharacteristic
248C2×Q8—28no—
249C2×Q8—82no—
250C2×Q8—82no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4#12012yes—
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4C22⋊Q814yes—
296C22×C4C22⋊Q812yes—
297C22×C4C22⋊Q812yes—
298C22×C4C4⋊Q812yes—
299C22×C4C4⋊Q812yes—
300C22×C4C22×D412yes—
301C22×C4C22×Q811yescharacteristic
302C22×C4C22×Q812yes—
303C22×C4C22×Q811yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4112yes—
306C22×C4C2×#4111yescharacteristic
307C22×C4H(5,𝔽2)11yescharacteristic
308C22×C4H(5,𝔽2)12yes—
309C22×C42−1+411yescharacteristic
310M16—24no—
311M16—24no—
312M16—24no—
313M16—24no—
314M16—42no—
315M16—42no—
316M16—42no—
317M16—42no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—28no—
325C4⋊C4—216no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—48no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—84no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—84no—
345C4⋊C4—88no—
346C4⋊C4C22⋊Q814yes—
347C4⋊C4C22⋊Q814yes—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C42—24no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—44no—
357C42—44no—
358C42—44no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—82no—
364C42—82no—
365C42—82no—
366C42C22⋊Q814yes—
367C42C22⋊Q814yes—
368C42C4⋊Q814yes—
369C42C4⋊Q814yes—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23C2×#12011yescharacteristic
375C23C2×C22⋊Q811yescharacteristic
376C23C2×C22⋊Q811yescharacteristic
377C23#52511yescharacteristic
378C23C2×C4⋊Q811yescharacteristic
379C23#53511yescharacteristic
380C23#53711yescharacteristic
381Q8—88no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4#39514yes—
439C2×C4#39512yes—
440C2×C4#39512yes—
441C2×C4C2×C22⋊Q812yes—
442C2×C4C2×C22⋊Q812yes—
443C2×C4#52512yes—
444C2×C4C2×C4⋊Q812yes—
445C2×C4#53512yes—
446C2×C4#53712yes—
447C2×C4#55412yes—
448C2×C4#55412yes—
449C2×C4#55412yes—
450C2×C4#55711yescharacteristic
451C2×C4#55712yes—
452C2×C4#55711yescharacteristic
453C2×C4Q8211yescharacteristic
454C2×C4Q8212yes—
455C2×C4#55911yescharacteristic
456C2×C4#55912yes—
457C8—44no—
458C8—44no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22C2×#39511yescharacteristic
472C22#256711yescharacteristic
473C22#258011yescharacteristic
474C22#258211yescharacteristic
475C22#258411yescharacteristic
476C22#261511yescharacteristic
477C22#284811yescharacteristic
478C22#301811yescharacteristic
479C22#301811yescharacteristic
480C22#301911yescharacteristic
481C22#302011yescharacteristic
482C4—24no—
483C4—22no—
484C4—22no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4—44no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—82no—
493C4—161no—
494C4—161no—
495C4#256712yes—
496C4#258012yes—
497C4#258212yes—
498C4#258412yes—
499C2—21no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2#1520211yescharacteristic
504C2#2356011yescharacteristic
505C2#2356111yescharacteristic
506C2#2968211yescharacteristic
507C2#2968311yescharacteristic
508C2#3297311yescharacteristic
509C2#3297411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×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
77. #423Q8semidirect
108. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5442
#211722
#139092
#113922
#113422
#113402
#113392
#108522
#108472