Finite Group Databasebeta

← #365674

Subgroups — #365674

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×#1836C211yesmaximal, characteristic
5#14277C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11292C212yesmaximal
8#10885C212yesmaximal
9#10884C212yesmaximal
10#10842C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2331—24no—
16#2331—24no—
17C2×#391—21no—
18C2×#391—21no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#387—21no—
23C2×#387—21no—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1993C418yescomplemented
31#1990—24no—
32#1836—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—21no—
44#1444—21no—
45#1442—21no—
46#1442—21no—
47C4⋊#95C2212yes—
48C23×Q8—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8D414yes—
56C2×C4⋊Q8Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#97—22no—
75C2×#97—22no—
76C2×#97C2×C414yes—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80#391—28no—
81#391—28no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—24no—
87#389—24no—
88#387—28no—
89#387—28no—
90#384—22no—
91#384—22no—
92#382—22no—
93#382—22no—
94#382C2×C414yes—
95#382D414yes—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100#376—22no—
101#376—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340C2×C418yes—
124#340C2×C418yes—
125#336—24no—
126#336—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—82no—
131C22×Q8—82no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C22×C414yes—
158C22×C8#4114yes—
159C4⋊Q8—28no—
160C4⋊Q8—28no—
161C4⋊Q8—28no—
162C4⋊Q8C4⋊C4116yescomplemented
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
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—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
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⋊C4C22×C414yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×Q812yes—
208C2×C4⋊C4C2×Q812yes—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—82no—
219C2×C42—82no—
220C2×C42C22×C412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×Q812yes—
224C2×C42#4114yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227#97—28no—
228C4×C8—44no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—44no—
242#95—82no—
243#95—82no—
244C24C22×D411yescharacteristic
245C2×Q8—216no—
246C2×Q8—88no—
247C2×Q8—88no—
248C2×Q8—84no—
249C2×Q8—84no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4C4×D412yes—
307C22×C4C4×D412yes—
308C22×C4#12012yes—
309C22×C4#12012yes—
310C22×C4C22⋊Q812yes—
311C22×C4C22⋊Q812yes—
312C22×C4#12412yes—
313C22×C4#12412yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×Q811yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4112yes—
320C22×C4C2×#4112yes—
321C22×C4C2×#4111yescharacteristic
322C22×C42−1+412yes—
323C22×C42−1+412yes—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—216no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4—88no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4C2×C4⋊C414yes—
363C4⋊C4C2×C4⋊C414yes—
364C42—24no—
365C42—24no—
366C42—42no—
367C42—42no—
368C42—84no—
369C42—84no—
370C42C2×C4⋊C414yes—
371C42C4×D414yes—
372C42C22⋊Q814yes—
373C42#12314yes—
374C23C2×C4×D411yescharacteristic
375C23C2×#12011yescharacteristic
376C23C2×C22⋊Q811yescharacteristic
377C23C2×#12411yescharacteristic
378C23#53211yescharacteristic
379C23#53611yescharacteristic
380C23#53611yescharacteristic
381C23#57612yes—
382C23#57712yes—
383C23#57712yes—
384C23#57812yes—
385Q8—816no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#38514yes—
449C2×C4#38514yes—
450C2×C4C22×C4⋊C411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C4×#4112yes—
453C2×C4#51912yes—
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×#12311yescharacteristic
456C2×C4#52912yes—
457C2×C4#54112yes—
458C2×C4#54712yes—
459C2×C4#54812yes—
460C2×C4#54812yes—
461C2×C4#54812yes—
462C2×C4#55212yes—
463C2×C4#55612yes—
464C2×C4#55712yes—
465C2×C4#55712yes—
466C8—44no—
467C8—44no—
468C22C2×#38511yescharacteristic
469C22C4×#12411yescharacteristic
470C22#232711yescharacteristic
471C22#243211yescharacteristic
472C22#243611yescharacteristic
473C22#256511yescharacteristic
474C22#256811yescharacteristic
475C22#294612yes—
476C22#294712yes—
477C22#300811yescharacteristic
478C22#300911yescharacteristic
479C22#301011yescharacteristic
480C22#301111yescharacteristic
481C22#308112yes—
482C22#308212yes—
483C22#313612yes—
484C22#313712yes—
485C22#317312yes—
486C22#317512yes—
487C22#324112yes—
488C22#324212yes—
489C22#324312yes—
490C22#324412yes—
491C4—24no—
492C4—24no—
493C4—42no—
494C4—42no—
495C4—44no—
496C4—44no—
497C4—81no—
498C4—82no—
499C4—81no—
500C4—82no—
501C4—84no—
502C4—162no—
503C4#235112yes—
504C4#236212yes—
505C4#242112yes—
506C4#245312yes—
507C2#1397811yescharacteristic
508C2#2087611yescharacteristic
509C2#2087711yescharacteristic
510C2#2165412yes—
511C2#2225512yes—
512C2#2568011yescharacteristic
513C2#2568111yescharacteristic
514C2#2591411yescharacteristic
515C2#2591511yescharacteristic
516C2#2918712yes—
517C2#2922912yes—
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
30. #1993C4semidirect
162. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
C2×#19932
C2×#19902
C2×#18362
#142772
#112932
#112922
#108852
#108842
#108422
#106942