Finite Group Databasebeta

← #368960

Subgroups — #368960

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11219C212yesmaximal
8#10908C212yesmaximal
9#10907C212yesmaximal
10#10843C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#501—24no—
15C2×#501—24no—
16C2×#501C418yescomplemented
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2331—24no—
22#2331—24no—
23C2×#391—22no—
24C2×#389C2212yes—
25C2×#387—22no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×#376C2211yescharacteristic
30#1991—24no—
31#1990—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1441—22no—
41#1441—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C23×Q8—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8C2×C414yes—
53C2×C4⋊Q8D414yes—
54C2×C4⋊Q8D414yes—
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—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C42—22no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#501—232no—
78C2×#107—24no—
79C2×#107—24no—
80C2×#107C2×C418yes—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106C2×C418yes—
84C2×#97—22no—
85C2×#97—22no—
86C2×#97C2×C414yes—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90#391—216no—
91#389—24no—
92#389—24no—
93#387—216no—
94#384—22no—
95#384—22no—
96#382—22no—
97#382D414yes—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100#376—22no—
101#376—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D412yes—
111C2×#95D412yes—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#336—24no—
119#336Q818yescomplemented
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8—82no—
124C22×Q8—82no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
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×C8C22×C414yes—
147C22×C8C2×Q814yes—
148C22×C8C2×Q814yes—
149C22×C8#4114yes—
150C4⋊Q8—28no—
151C4⋊Q8—28no—
152C4⋊Q8—28no—
153C4⋊Q8C22⋊C4116yes—
154#125—28no—
155#125—28no—
156#125—44no—
157#125—44no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
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—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C414yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D414yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—28no—
214C2×C42—82no—
215C2×C42—82no—
216C2×C42C22×C412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×Q814yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223#107—216no—
224#106—216no—
225#97—28no—
226C4×C8—44no—
227#95—28no—
228#95—28no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—82no—
234#95—82no—
235#95—82no—
236#95—82no—
237C24C22×D411yescharacteristic
238C2×Q8—216no—
239C2×Q8—88no—
240C2×Q8—88no—
241C2×Q8—84no—
242C2×Q8—84no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
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—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
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—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4C4×D412yes—
295C22×C4C4×D412yes—
296C22×C4#12012yes—
297C22×C4#12012yes—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4C4⋊Q814yes—
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×Q812yes—
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4C2×#4111yescharacteristic
309C22×C42−1+412yes—
310C22×C42−1+412yes—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
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×C8C4×Q818yes—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—216no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4—88no—
350C4⋊C4C2×C22⋊C418yes—
351C42—24no—
352C42—24no—
353C42—42no—
354C42—42no—
355C42—88no—
356C42C2×C22⋊C414yes—
357C42C4×D414yes—
358C42#12114yes—
359C42#12314yes—
360C23C2×C4×D411yescharacteristic
361C23C2×#12011yescharacteristic
362C23C2×#12111yescharacteristic
363C23C2×C4⋊Q811yescharacteristic
364C23#53211yescharacteristic
365C23#53611yescharacteristic
366C23#53611yescharacteristic
367C23#57612yes—
368C23#57712yes—
369C23#57712yes—
370C23#57812yes—
371Q8—816no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—28no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4#38614yes—
429C2×C4#38614yes—
430C2×C4C22×C22⋊C411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×Q812yes—
433C2×C4#51912yes—
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12311yescharacteristic
436C2×C4#53312yes—
437C2×C4#54112yes—
438C2×C4#54712yes—
439C2×C4#54812yes—
440C2×C4#54812yes—
441C2×C4D4×Q812yes—
442C2×C4D4×Q812yes—
443C2×C4D4×Q812yes—
444C2×C4#55612yes—
445C2×C4#55712yes—
446C8—28no—
447C8—44no—
448C22C2×#38611yescharacteristic
449C22C4×C4⋊Q811yescharacteristic
450C22#232711yescharacteristic
451C22#243211yescharacteristic
452C22#243411yescharacteristic
453C22#254511yescharacteristic
454C22#254611yescharacteristic
455C22#294612yes—
456C22#294712yes—
457C22#301212yes—
458C22#301312yes—
459C22#304111yescharacteristic
460C22#304211yescharacteristic
461C22#304311yescharacteristic
462C22#304411yescharacteristic
463C22#316312yes—
464C22#316412yes—
465C22#317812yes—
466C22#317912yes—
467C22#324112yes—
468C22#324212yes—
469C22#324312yes—
470C22#324412yes—
471C4—24no—
472C4—24no—
473C4—48no—
474C4—44no—
475C4—82no—
476C4—84no—
477C4—82no—
478C4—82no—
479C4—161no—
480C4—161no—
481C4Q8×C22⋊C412yes—
482C4#236512yes—
483C4#242912yes—
484C4#244612yes—
485C2#1402411yescharacteristic
486C2#2092911yescharacteristic
487C2#2093011yescharacteristic
488C2#2173012yes—
489C2#2225512yes—
490C2#2568412yes—
491C2#2582212yes—
492C2#2866611yescharacteristic
493C2#2866711yescharacteristic
494C2#2869611yescharacteristic
495C2#2869711yescharacteristic
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
16. C2×#501C4semidirect
119. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#23312
C2×#19912
C2×#19902
#142832
#112202
#112192
#109082
#109072
#108432
#106942