Finite Group Databasebeta

← #393836

Subgroups — #393836

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 467 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×#1726C211yesmaximal, characteristic
7#14354C211yesmaximal, characteristic
8#10843C212yesmaximal
9#10843C212yesmaximal
10#10842C212yesmaximal
11#10841C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C22×#97C2212yes—
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2212yes—
18#2660—24no—
19#2660C418yescomplemented
20C2×#398—22no—
21C2×#395—22no—
22C2×#384C2211yescharacteristic
23C2×#384C2212yes—
24C2×#384C2212yes—
25C2×#382C2211yescharacteristic
26C2×#382C2212yes—
27C2×#382C2212yes—
28C2×#376C2211yescharacteristic
29#1993—24no—
30#1992—24no—
31#1991—24no—
32#1990—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1726—24no—
44#1444—22no—
45#1444—22no—
46C23×Q8—21no—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125C2×C414yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C42C2311yescharacteristic
76C22×C42C2312yes—
77C2×#97—22no—
78C2×#97—24no—
79C2×#97—22no—
80C2×#97—24no—
81C2×#97D414yes—
82C2×#97Q818yescomplemented
83C2×C4×C8—22no—
84C2×C4×C8—24no—
85C2×C4×C8—22no—
86C2×C4×C8—24no—
87C2×C4×C8D414yes—
88C2×C4×C8Q818yescomplemented
89#398—216no—
90#395—216no—
91#384—24no—
92#384—22no—
93#384—22no—
94#384—24no—
95#384—24no—
96#384C2×C418yes—
97#382—24no—
98#382—22no—
99#382—22no—
100#382—24no—
101#382—24no—
102#382C2×C418yes—
103#376C2×C414yes—
104#376D414yes—
105C2×#95—24no—
106C2×#95—24no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95D412yes—
110C2×#95D412yes—
111#340—24no—
112#340—24no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121C22×Q8—24no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—82no—
125C22×Q8—82no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D414yes—
135C23×C4C2411yescharacteristic
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C22×C8C2×D414yes—
149C22×C8C2×Q814yes—
150C22×C8C2×Q814yes—
151C22×C8C2×Q814yes—
152C22×C8C2×Q814yes—
153C4⋊Q8—28no—
154C4⋊Q8—28no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157#125—28no—
158#125—28no—
159#125—44no—
160#125—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
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—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—216no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—84no—
204C2×C4⋊C4—84no—
205C2×C4⋊C4—84no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—84no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C414yes—
212C2×C42C22×C412yes—
213C2×C42C22×C414yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D414yes—
218C2×C42C2×Q814yes—
219C2×C42C2×Q814yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4114yes—
223C2×C42#4114yes—
224#97—28no—
225#97—216no—
226C4×C8—28no—
227C4×C8—216no—
228#95—216no—
229#95—48no—
230#95—42no—
231#95—42no—
232#95C22⋊C418yes—
233#95C22⋊C418yes—
234C24C22×D411yescharacteristic
235C2×Q8—48no—
236C2×Q8—88no—
237C2×Q8—88no—
238C2×Q8—84no—
239C2×Q8—84no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C418yes—
279C22×C4#12412yes—
280C22×C4#12412yes—
281C22×C4#12418yes—
282C22×C4#12712yes—
283C22×C4#12712yes—
284C22×C4C4⋊Q818yes—
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×Q812yes—
290C22×C4C22×Q812yes—
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4C2×#4112yes—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—24no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8C4⋊Q814yes—
310C2×C8C4⋊Q814yes—
311C2×C8C4⋊Q818yes—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—816no—
338C4⋊C4—816no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C42C4×D414yes—
342C42C4×D414yes—
343C42C4×Q818yes—
344C42C4×Q818yes—
345C42#12114yes—
346C42#12114yes—
347C42C22⋊Q818yes—
348C42C22⋊Q818yes—
349C23C22×C22⋊C411yescharacteristic
350C23C2×#12412yes—
351C23C2×#12411yescharacteristic
352C23C2×#12711yescharacteristic
353C23C2×C4⋊Q812yes—
354C23#57612yes—
355C23#57712yes—
356C23#57712yes—
357C23#57812yes—
358Q8—816no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—28no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—82no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4#39014yes—
402C2×C4#39014yes—
403C2×C4#391116yes—
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×C4×D411yescharacteristic
406C2×C4C2×C4×Q812yes—
407C2×C4C2×C4×Q812yes—
408C2×C4C2×#12111yescharacteristic
409C2×C4C2×#12111yescharacteristic
410C2×C4C2×C22⋊Q812yes—
411C2×C4C2×C22⋊Q812yes—
412C2×C4C2×C4⋊Q811yescharacteristic
413C2×C4C2×C4⋊Q811yescharacteristic
414C2×C4C2×C4⋊Q812yes—
415C2×C4#53211yescharacteristic
416C2×C4#53212yes—
417C2×C4#53211yescharacteristic
418C2×C4#53312yes—
419C2×C4#53312yes—
420C2×C4#53312yes—
421C2×C4#53312yes—
422C8—24no—
423C8—24no—
424C8—28no—
425C22C2×#39011yescharacteristic
426C22C2×#39112yes—
427C22#285811yescharacteristic
428C22#286212yes—
429C22#286811yescharacteristic
430C22#289712yes—
431C22#289812yes—
432C22#313612yes—
433C22#313614yes—
434C22#313712yes—
435C22#313714yes—
436C22#315411yescharacteristic
437C22#315411yescharacteristic
438C22#315511yescharacteristic
439C22#315611yescharacteristic
440C22#316314yes—
441C22#316414yes—
442C4—216no—
443C4—24no—
444C4—24no—
445C4—82no—
446C4—82no—
447C4—84no—
448C4—161no—
449C4—162no—
450C4—161no—
451C4C4×C4⋊Q811yescharacteristic
452C4C4×C4⋊Q811yescharacteristic
453C4C4×C4⋊Q812yes—
454C4#285711yescharacteristic
455C4#285711yescharacteristic
456C4#285712yes—
457C2#1471411yescharacteristic
458C2#2120511yescharacteristic
459C2#2120611yescharacteristic
460C2#2132614yes—
461C2#3322811yescharacteristic
462C2#3322911yescharacteristic
463C2#3329214yes—
464C2#3331811yescharacteristic
465C2#3331911yescharacteristic
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
19. #2660C4semidirect
82. C2×#97Q8semidirect
88. 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×#17262
#143542
#108432
#108422
#108412