Finite Group Databasebeta

← #201316

Subgroups — #201316

Order 512.

1241 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#34371C211yesmaximal, characteristic, complemented
3#21829C211yesmaximal, characteristic, complemented
4#17589C211yesmaximal, characteristic
5#17561C211yesmaximal, characteristic
6#9465C218yesmaximal, 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#2563C414yescomplemented
17#2547—22no—
18#2504C2214yes—
19#2436—22no—
20#2381—22no—
21#2365—22no—
22#2345—22no—
23Q8×C22⋊C4—22no—
24#2331—22no—
25#2327—22no—
26#2313C2214yes—
27#2311—22no—
28#2283C2211yescharacteristic
29#2283C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1997—22no—
32#1997—22no—
33#1821C2214yes—
34#1517—22no—
35#1514—22no—
36#1510C2214yes—
37#1510C2214yes—
38#1505—22no—
39#1502—22no—
40#1284C2214yes—
41#1284C2214yes—
42C22×#41—21no—
43C23×Q8D412yes—
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—24no—
67C2×#124—21no—
68C2×#121—21no—
69C2×#121C2312yes—
70C2×C4×Q8—24no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C412yes—
73C2×#117—21no—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4C2312yes—
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—
87#407C2312yes—
88#407C2312yes—
89C4×M16—21no—
90C4×M16—21no—
91#397—28no—
92#393—28no—
93#391—24no—
94#391—28no—
95#391—24no—
96#391—24no—
97#390—24no—
98#390C2×C418yes—
99#389—24no—
100#387—28no—
101#387—28no—
102#387—28no—
103#386—28no—
104#383—21no—
105#383D412yes—
106#382—21no—
107#382C2×C418yes—
108#382D412yes—
109#381C2312yes—
110#381C2312yes—
111#380—24no—
112#380—21no—
113#380—21no—
114#380—28no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D412yes—
120C4×C22⋊C4—24no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4C2312yes—
125#376—21no—
126#376—21no—
127#376D412yes—
128C43C2×C412yes—
129#343—28no—
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×Q8C22⋊C414yes—
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×Q8—88no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202#117—44no—
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×C412yes—
219C2×C4⋊C4C22×C414yes—
220C2×C4⋊C4C2×D414yes—
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—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253#97—41no—
254#97—41no—
255C4×C8—41no—
256C4×C8—41no—
257#95—28no—
258#95—28no—
259#95—22no—
260#95—216no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95C2×D414yes—
269#95C2×D414yes—
270C24—22no—
271C24C22×D411yescharacteristic
272#41—84no—
273#41—84no—
274#41—84no—
275#41—84no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—44no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—84no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—48no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—48no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—84no—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C418yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4#12311yescharacteristic
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333M16—44no—
334M16—44no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—44no—
352C4⋊C4—88no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—816no—
361C42—44no—
362C42—44no—
363C42—44no—
364C42—44no—
365C42—44no—
366C42—88no—
367C42C4×D411yescharacteristic
368C42C4×D411yescharacteristic
369C42C4×D411yescharacteristic
370C42C4×D411yescharacteristic
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42C4×D411yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42#12111yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C42#12111yescharacteristic
383C23—22no—
384C23—22no—
385C23—41no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—42no—
390C23C22×C22⋊C411yescharacteristic
391C23C2×#12011yescharacteristic
392C23C2×#12311yescharacteristic
393Q8—48no—
394Q8—48no—
395Q8—84no—
396Q8—84no—
397Q8—82no—
398Q8—82no—
399D4—82no—
400D4—82no—
401D4—82no—
402D4—82no—
403D4—84no—
404D4—84no—
405D4—82no—
406D4—82no—
407C2×C4—22no—
408C2×C4—22no—
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—28no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4#38611yescharacteristic
442C2×C4#38611yescharacteristic
443C2×C4#38611yescharacteristic
444C2×C4#38611yescharacteristic
445C2×C4#38611yescharacteristic
446C2×C4#38611yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#45414yes—
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×#12111yescharacteristic
454C8—82no—
455C8—82no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—88no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22C2×#38611yescharacteristic
469C22#288412yes—
470C22C2×#45412yes—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—48no—
481C4—84no—
482C4—84no—
483C4—84no—
484C4—82no—
485C4#188812yes—
486C4#188812yes—
487C4#191112yes—
488C2—22no—
489C2—22no—
490C2—41no—
491C2—162no—
492C2C2×#188811yescharacteristic
493C2#2098611yescharacteristic
494C2C2×#191111yescharacteristic
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. #34371C2semidirect
3. #21829C2semidirect
6. #9465C2semidirect
7. #7941C2semidirect
8. #7940C2semidirect
13. #2862C4semidirect
14. #2858C4semidirect
16. #2563C4semidirect

Maximal subgroups

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

subgroupindex
#352892
#343712
#218292
#175892
#175612
#94652
#79412
#79402