Finite Group Databasebeta

← #468181

Subgroups — #468181

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3C2×#1921C211yesmaximal, characteristic
4#15202C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11380C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11328C212yesmaximal
9#11293C212yesmaximal
10#11219C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16#2567—24no—
17#2567—24no—
18C2×#395—22no—
19C2×#391—21no—
20C2×#389C2212yes—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#384—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#2057—24no—
27#1921—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1446—21no—
41#1444—22no—
42#1444—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8D411yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8D414yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—42no—
57C2×C4×Q8—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—24no—
77C2×#107—42no—
78C2×#107—42no—
79C2×#107Q814yescomplemented
80#395—216no—
81#391—28no—
82#389—24no—
83#389—24no—
84#389—24no—
85#389—24no—
86#389—22no—
87#389—22no—
88#389—42no—
89#389—42no—
90#384—28no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4D414yes—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102#340—24no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—24no—
107#340—42no—
108#340—42no—
109#340Q818yescomplemented
110#336—22no—
111#336—22no—
112#336—42no—
113#336—42no—
114C22×Q8—22no—
115C22×Q8—24no—
116C22×Q8—22no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—44no—
120C22×Q8—81no—
121C22×Q8—82no—
122C22×Q8—81no—
123C22×Q8Q1614yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8C2×Q814yes—
151C22×C8C2×Q812yes—
152C4⋊Q8—48no—
153C4⋊Q8—44no—
154C4⋊Q8—44no—
155C4×Q8—48no—
156C4×Q8—44no—
157C4×Q8—44no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×Q814yes—
204C2×C4⋊C4C2×Q812yes—
205C2×C4⋊C4C2×Q812yes—
206C2×C4⋊C4C2×Q814yes—
207C2×C4⋊C4#4112yes—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—44no—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220#107—28no—
221#107—48no—
222#95—28no—
223#95—28no—
224#95—24no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—42no—
229#95—81no—
230#95—82no—
231C24C22×D411yescharacteristic
232C2×Q8—28no—
233C2×Q8—48no—
234C2×Q8—44no—
235C2×Q8—48no—
236C2×Q8—44no—
237C2×Q8—48no—
238C2×Q8—84no—
239C2×Q8—84no—
240C2×Q8—88no—
241C2×Q8—88no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4#12012yes—
296C22×C4#12012yes—
297C22×C4C2×Q1614yes—
298C22×C4C2×Q1612yes—
299C22×C4#13712yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×Q811yescharacteristic
303C22×C4C22×Q812yes—
304C22×C4C2×#4111yescharacteristic
305C22×C4H(5,𝔽2)12yes—
306C22×C4H(5,𝔽2)12yes—
307C22×C42−1+412yes—
308C22×C42−1+411yescharacteristic
309C22×C42−1+411yescharacteristic
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—84no—
351C4⋊C4—88no—
352C4⋊C4—88no—
353C4⋊C4—88no—
354C4⋊C4C22⋊Q814yes—
355C4⋊C4C4⋊Q814yes—
356C42—24no—
357C42—22no—
358C42—22no—
359C42—48no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—42no—
364C23C2×#12011yescharacteristic
365C23#53412yes—
366C23#54811yescharacteristic
367C23D4×Q812yes—
368C23D4×Q811yescharacteristic
369C23C22×Q1611yescharacteristic
370C23C2×#13711yescharacteristic
371C23#57512yes—
372C23#57612yes—
373C23#57611yescharacteristic
374C23#57711yescharacteristic
375Q8—416no—
376Q8—816no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
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—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4#45114yes—
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×C4⋊Q811yescharacteristic
437C2×C4#53412yes—
438C2×C4#53511yescharacteristic
439C2×C4#53511yescharacteristic
440C2×C4#53611yescharacteristic
441C2×C4#53611yescharacteristic
442C2×C4#54212yes—
443C2×C4#54412yes—
444C2×C4#54412yes—
445C2×C4#54412yes—
446C2×C4#54811yescharacteristic
447C2×C4D4×Q812yes—
448C2×C4D4×Q811yescharacteristic
449C2×C4#55412yes—
450C2×C4Q8212yes—
451C2×C4#56312yes—
452C2×C4#56312yes—
453C8—42no—
454C8—44no—
455C8—81no—
456C22#240911yescharacteristic
457C22#243411yescharacteristic
458C22#261311yescharacteristic
459C22#272912yes—
460C22#273412yes—
461C22C2×#45111yescharacteristic
462C22#300511yescharacteristic
463C22#300811yescharacteristic
464C22#301211yescharacteristic
465C22#318912yes—
466C22#319612yes—
467C22#319812yes—
468C22#319912yes—
469C22#336311yescharacteristic
470C22#337012yes—
471C22#337511yescharacteristic
472C22Q8×Q1611yescharacteristic
473C22Q8×Q1612yes—
474C22#338512yes—
475C22#338711yescharacteristic
476C22#338812yes—
477C22#338912yes—
478C22#338912yes—
479C4—24no—
480C4—44no—
481C4—44no—
482C4—42no—
483C4—44no—
484C4—82no—
485C4—84no—
486C4—82no—
487C4—84no—
488C4—161no—
489C4#256711yescharacteristic
490C4#262811yescharacteristic
491C4#266011yescharacteristic
492C4#272912yes—
493C4#273412yes—
494C4#282111yescharacteristic
495C2#1660311yescharacteristic
496C2#2433311yescharacteristic
497C2#2433911yescharacteristic
498C2#2584111yescharacteristic
499C2#2584511yescharacteristic
500C2#2959711yescharacteristic
501C2#2960211yescharacteristic
502C2#3257012yes—
503C2#3258012yes—
504C2#3268412yes—
505C2#3268912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
79. C2×#107Q8semidirect
109. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#20572
C2×#19212
#152022
#113842
#113802
#113282
#112932
#112192
#108072