Finite Group Databasebeta

← #201317

Subgroups — #201317

Order 512.

1257 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#34365C211yesmaximal, characteristic, complemented
3#21799C211yesmaximal, characteristic, complemented
4#17589C211yesmaximal, characteristic
5#17561C211yesmaximal, characteristic
6#9466C218yesmaximal, complemented
7#7941C211yesmaximal, characteristic, complemented
8#7940C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2862C412yescomplemented
14#2858C412yescomplemented
15#2640—22no—
16#2563—22no—
17#2547C414yescomplemented
18#2504—22no—
19#2436C2214yes—
20#2381—22no—
21#2369—22no—
22#2344—22no—
23#2344—22no—
24#2331—22no—
25#2331—22no—
26#2311—22no—
27#2310C2214yes—
28#2283C2211yescharacteristic
29#2283C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1966—22no—
32#1966—22no—
33#1784C2214yes—
34#1517—22no—
35#1514—22no—
36#1510C2214yes—
37#1510C2214yes—
38#1505—22no—
39#1502—22no—
40#1284C2214yes—
41#1284C2214yes—
42C22×#41—21no—
43C23×Q8C2312yes—
44#532—21no—
45#532—21no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8D412yes—
65C2×#127C2×C411yescharacteristic
66C2×#124—21no—
67C2×#124C2×C418yes—
68C2×#121—21no—
69C2×#121C2312yes—
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C2×#117—21no—
73C2×#117—24no—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4D412yes—
76C22×C42C2311yescharacteristic
77#432—21no—
78#432—21no—
79#432—21no—
80#432—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#423—21no—
86#423—21no—
87C2×#101—28no—
88#407C2312yes—
89#407C2312yes—
90C4×M16—21no—
91C4×M16—21no—
92#397—28no—
93#393—28no—
94#391—24no—
95#391—28no—
96#391—28no—
97#391C2×C418yes—
98#390—24no—
99#390—24no—
100#389—24no—
101#389—24no—
102#387—28no—
103#387—28no—
104#386—28no—
105#383—21no—
106#383D412yes—
107#382—24no—
108#382—21no—
109#382D412yes—
110#381—24no—
111#381C2312yes—
112#381C2312yes—
113#380—21no—
114#380—21no—
115#380—28no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D412yes—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4C2312yes—
126#376—21no—
127#376—21no—
128#376D412yes—
129C43C2×C412yes—
130#327—28no—
131#327—28no—
132C2×#41—48no—
133C2×#41—41no—
134C2×#41—41no—
135C2×#41—41no—
136C2×#41—41no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—84no—
143C22×Q8C22×C414yes—
144C22×Q8C2×D414yes—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D414yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C2×M16—22no—
155C2×M16—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176#127—21no—
177#127—21no—
178#127—21no—
179#127—21no—
180#124—41no—
181#124—41no—
182#124—41no—
183#124—41no—
184#124—88no—
185#121—22no—
186#121—22no—
187#121—22no—
188#121—22no—
189#121—41no—
190#121—41no—
191#121—41no—
192#121—41no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201#117—44no—
202#117—88no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4C22⋊C414yes—
217C2×C4⋊C4C22⋊C414yes—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4C22⋊C414yes—
220C2×C4⋊C4C22×C412yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—216no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—28no—
232C2×C42C22×C414yes—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249#101—88no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254#97—41no—
255#97—41no—
256C4×C8—41no—
257C4×C8—41no—
258#95—28no—
259#95—28no—
260#95—22no—
261#95—216no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95C2×D414yes—
270#95C2×D414yes—
271C24—22no—
272C24C22×D411yescharacteristic
273#41—84no—
274#41—84no—
275#41—84no—
276#41—84no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—48no—
282C2×Q8—48no—
283C2×Q8—44no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—82no—
287C2×Q8—84no—
288C2×Q8—82no—
289C2×Q8—84no—
290C2×Q8—84no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—48no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—42no—
311C22×C4—48no—
312C22×C4—42no—
313C22×C4—48no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4C2×C22⋊C418yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4#12311yescharacteristic
327C22×C4#12311yescharacteristic
328C22×C4#12311yescharacteristic
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335M16—44no—
336M16—44no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—44no—
354C4⋊C4—88no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—816no—
363C42—44no—
364C42—44no—
365C42—44no—
366C42—44no—
367C42—44no—
368C42—88no—
369C42C4×D411yescharacteristic
370C42C4×D411yescharacteristic
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42C4×D411yescharacteristic
375C42C4×D411yescharacteristic
376C42C4×D411yescharacteristic
377C42#12111yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C42#12111yescharacteristic
383C42#12111yescharacteristic
384C42#12111yescharacteristic
385C23—22no—
386C23—22no—
387C23—41no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—42no—
392C23C22×C22⋊C411yescharacteristic
393C23C2×#12011yescharacteristic
394C23C2×#12311yescharacteristic
395Q8—84no—
396Q8—84no—
397Q8—84no—
398Q8—84no—
399Q8—82no—
400Q8—82no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405D4—84no—
406D4—84no—
407D4—82no—
408D4—82no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—216no—
420C2×C4—28no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4#38611yescharacteristic
445C2×C4#38611yescharacteristic
446C2×C4#38611yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4#38611yescharacteristic
451C2×C4#38611yescharacteristic
452C2×C4#45414yes—
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×#12111yescharacteristic
456C2×C4C2×#12111yescharacteristic
457C8—82no—
458C8—82no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—44no—
467C22—88no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22C2×#38611yescharacteristic
472C22#288312yes—
473C22C2×#45412yes—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—48no—
484C4—84no—
485C4—84no—
486C4—84no—
487C4—82no—
488C4#188812yes—
489C4#188812yes—
490C4#191212yes—
491C2—22no—
492C2—22no—
493C2—41no—
494C2—162no—
495C2C2×#188811yescharacteristic
496C2#2098511yescharacteristic
497C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34365C2semidirect
3. #21799C2semidirect
6. #9466C2semidirect
7. #7941C2semidirect
8. #7940C2semidirect
13. #2862C4semidirect
14. #2858C4semidirect
17. #2547C4semidirect

Maximal subgroups

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

subgroupindex
#352892
#343652
#217992
#175892
#175612
#94662
#79412
#79402