Finite Group Databasebeta

← #375390

Subgroups — #375390

Order 512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic
4C2×#1991C211yesmaximal, characteristic
5C2×#1990C211yesmaximal, characteristic
6C2×#1768C211yesmaximal, characteristic
7#13779C211yesmaximal, characteristic
8#11358C212yesmaximal
9#11357C212yesmaximal
10#10850C212yesmaximal
11#10849C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#97C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2660—24no—
18#2660C418yescomplemented
19C2×#398—22no—
20C2×#395—22no—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×#376C2211yescharacteristic
29#1993—24no—
30#1992—24no—
31#1991—24no—
32#1990—24no—
33#1768—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1436—22no—
44#1436—22no—
45C4⋊#95C2212yes—
46C23×Q8—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yes—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125C2×C414yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C42—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8Q818yescomplemented
81C2×#97—22no—
82C2×#97—22no—
83C2×#97—22no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87#398—216no—
88#395—216no—
89#384—22no—
90#384—22no—
91#384—22no—
92#384C2×C414yes—
93#384C2×C414yes—
94#384D414yes—
95#382—22no—
96#382—22no—
97#382—22no—
98#382C2×C414yes—
99#382C2×C414yes—
100#382D414yes—
101C4×C4⋊C4—28no—
102#376C2×C414yes—
103#376D414yes—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D412yes—
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#336—24no—
122C22×Q8—24no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—82no—
126C22×Q8—82no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C2×Q814yes—
151C22×C8C2×Q814yes—
152C4⋊Q8—28no—
153C4⋊Q8—28no—
154C4⋊Q8—44no—
155C4⋊Q8—44no—
156#125—28no—
157#125—28no—
158#125—44no—
159#125—44no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—44no—
214C2×C42—82no—
215C2×C42—82no—
216C2×C42C22×C412yes—
217C2×C42C22×C412yes—
218C2×C42C22×C412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×Q814yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42#4114yes—
228C4⋊C8—216no—
229#97—44no—
230C4×C8—44no—
231#95—28no—
232#95—28no—
233#95—24no—
234#95—24no—
235#95—44no—
236#95—82no—
237#95—82no—
238#95C22⋊C418yes—
239#95C22⋊C418yes—
240C24C22×D411yescharacteristic
241C2×Q8—48no—
242C2×Q8—88no—
243C2×Q8—84no—
244C2×Q8—84no—
245C2×Q8—88no—
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—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—24no—
265C22×C4—24no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
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—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4C2×C22⋊C414yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C414yes—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#12414yes—
303C22×C4C4⋊Q814yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×Q812yes—
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4112yes—
312C22×C42−1+412yes—
313C22×C42−1+412yes—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—88no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C4⋊C4—88no—
355C42—24no—
356C42—24no—
357C42—88no—
358C42C4×D414yes—
359C42C4×D414yes—
360C42C4×Q818yes—
361C42C22⋊Q818yes—
362C42#12414yes—
363C42#12714yes—
364C23C22×C22⋊C411yescharacteristic
365C23C2×#12111yescharacteristic
366C23C2×#12111yescharacteristic
367C23C2×#12411yescharacteristic
368C23C2×C4⋊Q811yescharacteristic
369C23#53611yescharacteristic
370C23#53611yescharacteristic
371C23#57612yes—
372C23#57712yes—
373C23#57712yes—
374C23#57812yes—
375Q8—816no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
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#39014yes—
432C2×C4#39014yes—
433C2×C4#39118yes—
434C2×C4C2×C4×D411yescharacteristic
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×C4×Q812yes—
437C2×C4#51912yes—
438C2×C4C2×C22⋊Q812yes—
439C2×C4C2×#12411yescharacteristic
440C2×C4C2×#12711yescharacteristic
441C2×C4#53612yes—
442C2×C4#54712yes—
443C2×C4#54712yes—
444C2×C4D4×Q812yes—
445C2×C4D4×Q812yes—
446C2×C4#55612yes—
447C2×C4#55612yes—
448C2×C4#55712yes—
449C2×C4#55712yes—
450C8—44no—
451C8—44no—
452C22C2×#39011yescharacteristic
453C22C2×#39111yescharacteristic
454C22#231011yescharacteristic
455C22#254611yescharacteristic
456C22#254611yescharacteristic
457C22#254711yescharacteristic
458C22#254711yescharacteristic
459C22#289712yes—
460C22#289812yes—
461C22#304111yescharacteristic
462C22#304211yescharacteristic
463C22#304311yescharacteristic
464C22#304411yescharacteristic
465C22#304512yes—
466C22#304612yes—
467C22#313612yes—
468C22#313712yes—
469C22#316312yes—
470C22#316412yes—
471C22#324112yes—
472C22#324212yes—
473C22#324312yes—
474C22#324412yes—
475C4—24no—
476C4—24no—
477C4—28no—
478C4—44no—
479C4—82no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—84no—
484C4—84no—
485C4#236512yes—
486C4#236512yes—
487C4#254212yes—
488C4#256812yes—
489C2#1415511yescharacteristic
490C2#2126211yescharacteristic
491C2#2126311yescharacteristic
492C2#2132612yes—
493C2#2184312yes—
494C2#2869611yescharacteristic
495C2#2869711yescharacteristic
496C2#2869812yes—
497C2#2872811yescharacteristic
498C2#2872911yescharacteristic
499C2#2873012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #2660C4semidirect
80. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#19932
C2×#19922
C2×#19912
C2×#19902
C2×#17682
#137792
#113582
#113572
#108502
#108492