Finite Group Databasebeta

← #468179

Subgroups — #468179

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#1921C211yesmaximal, characteristic
4#15202C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11380C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11327C212yesmaximal
9#11292C212yesmaximal
10#11219C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2212yes—
17#2546—24no—
18#2546—24no—
19C2×#398—22no—
20C2×#391—21no—
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26#2058—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—28no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8—48no—
55C2×C4⋊Q8D414yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—42no—
59C2×C4×Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#107—22no—
77C2×#107—22no—
78C2×#107Q814yescomplemented
79C2×#106—24no—
80C2×#106—42no—
81C2×#106—42no—
82#398—216no—
83#391—28no—
84#389—24no—
85#389—24no—
86#389—24no—
87#389—24no—
88#389—22no—
89#389—22no—
90#389—42no—
91#389—42no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4D414yes—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103#340—24no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—24no—
108#340—42no—
109#340—42no—
110#340Q818yescomplemented
111#336—22no—
112#336—22no—
113#336—42no—
114#336—42no—
115C22×Q8—22no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×Q8—44no—
122C22×Q8—44no—
123C22×Q8—42no—
124C22×Q8—42no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8C2×Q814yes—
152C22×C8C2×Q812yes—
153C4⋊Q8—48no—
154C4⋊Q8—44no—
155C4⋊Q8—44no—
156C4⋊Q8—432no—
157C4×Q8—48no—
158C4×Q8—44no—
159C4×Q8—44no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×Q814yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C4⋊C4C2×Q812yes—
208C2×C4⋊C4C2×Q814yes—
209C2×C4⋊C4#4112yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—44no—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222#107—28no—
223#106—48no—
224#95—28no—
225#95—28no—
226#95—24no—
227#95—44no—
228#95—42no—
229#95—42no—
230#95—44no—
231#95—81no—
232#95—82no—
233C24C22×D411yescharacteristic
234C2×Q8—28no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—44no—
238C2×Q8—48no—
239C2×Q8—48no—
240C2×Q8—44no—
241C2×Q8—48no—
242C2×Q8—88no—
243C2×Q8—88no—
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—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
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—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4#12012yes—
298C22×C4#12012yes—
299C22×C4#13512yes—
300C22×C4AGL(1,ℤ/8)14yes—
301C22×C4#13712yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×Q811yescharacteristic
305C22×C4C22×Q812yes—
306C22×C4C2×#4111yescharacteristic
307C22×C4H(5,𝔽2)12yes—
308C22×C4H(5,𝔽2)12yes—
309C22×C42−1+412yes—
310C22×C42−1+411yescharacteristic
311C22×C42−1+411yescharacteristic
312C2×C8—22no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—416no—
344C4⋊C4—416no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—84no—
355C4⋊C4—88no—
356C4⋊C4C22⋊Q814yes—
357C4⋊C4C4⋊Q814yes—
358C42—24no—
359C42—22no—
360C42—22no—
361C42—48no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C23C2×#12011yescharacteristic
367C23#53412yes—
368C23#54811yescharacteristic
369C23D4×Q812yes—
370C23D4×Q811yescharacteristic
371C23C2×#13511yescharacteristic
372C23C2×AGL(1,ℤ/8)12yes—
373C23C2×#13711yescharacteristic
374C23#57711yescharacteristic
375C23#57712yes—
376C23#57811yescharacteristic
377Q8—416no—
378Q8—816no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
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—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4#45214yes—
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4C2×C4⋊Q811yescharacteristic
439C2×C4#53412yes—
440C2×C4#53511yescharacteristic
441C2×C4#53511yescharacteristic
442C2×C4#53611yescharacteristic
443C2×C4#53611yescharacteristic
444C2×C4#54212yes—
445C2×C4#54412yes—
446C2×C4#54412yes—
447C2×C4#54412yes—
448C2×C4#54811yescharacteristic
449C2×C4D4×Q812yes—
450C2×C4D4×Q811yescharacteristic
451C2×C4#55412yes—
452C2×C4Q8212yes—
453C2×C4#56312yes—
454C2×C4#56312yes—
455C8—42no—
456C8—44no—
457C8—81no—
458C22#240911yescharacteristic
459C22#243411yescharacteristic
460C22#261311yescharacteristic
461C22#272912yes—
462C22#273412yes—
463C22C2×#45211yescharacteristic
464C22#300311yescharacteristic
465C22#301011yescharacteristic
466C22#301311yescharacteristic
467C22#318812yes—
468C22#319312yes—
469C22#320012yes—
470C22#320112yes—
471C22#336511yescharacteristic
472C22#337312yes—
473C22#337411yescharacteristic
474C22#338112yes—
475C22#338311yescharacteristic
476C22#338412yes—
477C22#338412yes—
478C22#338612yes—
479C22#338711yescharacteristic
480C22#338712yes—
481C4—24no—
482C4—44no—
483C4—44no—
484C4—42no—
485C4—44no—
486C4—82no—
487C4—84no—
488C4—84no—
489C4—82no—
490C4—161no—
491C4#256711yescharacteristic
492C4#262811yescharacteristic
493C4#266011yescharacteristic
494C4#272912yes—
495C4#273412yes—
496C4#282111yescharacteristic
497C2#1660311yescharacteristic
498C2#2433211yescharacteristic
499C2#2433811yescharacteristic
500C2#2583611yescharacteristic
501C2#2584411yescharacteristic
502C2#2959811yescharacteristic
503C2#2960311yescharacteristic
504C2#3257712yes—
505C2#3259112yes—
506C2#3268812yes—
507C2#3269412yes—
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
78. C2×#107Q8semidirect
110. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#20582
C2×#19212
#152022
#113842
#113802
#113272
#112922
#112192
#108072