Finite Group Databasebeta

← #213772

Subgroups — #213772

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#17353C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#9102C212yesmaximal, complemented
5#9099C212yesmaximal, complemented
6#8959C212yesmaximal, complemented
7#8958C212yesmaximal, complemented
8#8730C212yesmaximal, complemented
9#8324C212yesmaximal, complemented
10C2×#532C411yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#520C2211yescharacteristic
13#2982C2212yes—
14#2972C2212yes—
15#2361—22no—
16#2361—22no—
17#2354—22no—
18D4×C4⋊C4—22no—
19#2326C411yescharacteristic
20#2326C411yescharacteristic
21#2326C2211yescharacteristic
22#2326C2211yescharacteristic
23#2311C411yescharacteristic
24#2311C411yescharacteristic
25C4×C4⋊Q8C411yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×#127C411yescharacteristic
28C4×#127C2211yescharacteristic
29C2×#384C411yescharacteristic
30C4×#117C2211yescharacteristic
31#1479C2214yes—
32#1478C2214yes—
33#1461C2212yes—
34#1460—22no—
35#1460C2212yes—
36#1459—22no—
37#1458—22no—
38#1458C2212yes—
39#1458C2212yes—
40C2×#324—22no—
41#1276C2214yes—
42#1274C2214yes—
43C22×#41C2×C411yescharacteristic
44C22×#41C2×C411yescharacteristic
45#532—24no—
46#532C818yescomplemented
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#127C2×C411yescharacteristic
49C2×#124C2×C411yescharacteristic
50C2×#124C2×C411yescharacteristic
51C2×#121—22no—
52#520—44no—
53#520—44no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D412yes—
66C2×C4×D4D412yes—
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117C2×C411yescharacteristic
71C2×#117D412yes—
72C2×#117C2311yescharacteristic
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4D412yes—
78C22×C42C2×C411yescharacteristic
79C8×Q8—44no—
80C8×D4—44no—
81#431—24no—
82#431C2314yes—
83#431C2314yes—
84C2×C22⋊C8—24no—
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#390—22no—
90#390—22no—
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#390C2×C411yescharacteristic
94#390C2×C411yescharacteristic
95#389—22no—
96#389—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#386—22no—
102#386—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#384—22no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#380—22no—
113#380—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376C2×C411yescharacteristic
121#376C2311yescharacteristic
122C43C2×C411yescharacteristic
123C43C2311yescharacteristic
124Q8⋊C8—44no—
125D4⋊C8—44no—
126#324—22no—
127#324—24no—
128#324—22no—
129#323—24no—
130#323—22no—
131#323—22no—
132C2×#41—22no—
133C2×#41—42no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41—42no—
137C2×#41C2×C814yes—
138C2×#41C2×C814yescomplemented
139C2×#41M1614yes—
140C22×Q8C22×C411yescharacteristic
141C22×Q8C22×C411yescharacteristic
142C22×D4—22no—
143C22×D4—21no—
144C22×D4C22⋊C412yes—
145C22×D4C22×C411yescharacteristic
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22⋊C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D412yes—
157C22×C8—44no—
158C4⋊Q8—22no—
159C4⋊Q8C2×C814yescomplemented
160#127—22no—
161#127C2×C814yes—
162#124—22no—
163#124—22no—
164#124C2×C814yes—
165#124C2×C814yescomplemented
166#121—216no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×D4—216no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—44no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—44no—
178C4×D4—44no—
179C4×D4—44no—
180C4×D4—84no—
181C4×D4—84no—
182#117—24no—
183#117—42no—
184#117—44no—
185#117—44no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—84no—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C2×D412yes—
220C2×C22⋊C4C2×D411yescharacteristic
221C2×C22⋊C4C2×D411yescharacteristic
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42C22⋊C412yes—
231C2×C42C2×C814yescomplemented
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2411yescharacteristic
239C4⋊C8—24no—
240C4⋊C8—24no—
241C4⋊C8—44no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—44no—
245C4×C8—24no—
246C4×C8—24no—
247C4×C8—44no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—21no—
255#95—21no—
256C24—21no—
257C24C2×C22⋊C412yes—
258C24C2×C22⋊C411yescharacteristic
259#41—48no—
260#41—48no—
261#41—48no—
262#41—48no—
263C2×Q8—22no—
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8C22×C812yes—
269C2×Q8C22×C812yes—
270C2×Q8C2×M1612yes—
271C2×D4—28no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—21no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4C22⋊C818yes—
285C2×D4C22×C812yes—
286C2×D4C22×C812yes—
287C2×D4C2×M1612yes—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4C22⋊C818yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C22×C812yes—
325C22×C4C2×M1612yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—164no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—28no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C22⋊C4—28no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—24no—
354C22⋊C4—28no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C42—22no—
364C42—24no—
365C42—21no—
366C42—21no—
367C42—24no—
368C42—22no—
369C42—22no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42C22×C812yes—
378C42C22×C812yes—
379C42H(5,𝔽2)14yes—
380C23—21no—
381C23—22no—
382C23—21no—
383C23—22no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23C2×C22⋊C814yes—
394C23C2×C22⋊C812yes—
395C23C22×C22⋊C411yescharacteristic
396Q8—24no—
397Q8—24no—
398Q8—24no—
399Q8—42no—
400D4—24no—
401D4—24no—
402D4—24no—
403D4—48no—
404D4—416no—
405D4—48no—
406D4—42no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—84no—
446C2×C4—162no—
447C2×C4—162no—
448C2×C4—162no—
449C2×C4—162no—
450C2×C4—162no—
451C2×C4C2×C22⋊C812yes—
452C2×C4C2×C22⋊C814yes—
453C2×C4#51812yes—
454C2×C4#53412yes—
455C2×C4#53412yes—
456C2×C4C23×C811yescharacteristic
457C2×C4C22×M1611yescharacteristic
458C2×C4#56812yes—
459C8—84no—
460C8—84no—
461C22—21no—
462C22—21no—
463C22—22no—
464C22—24no—
465C22—42no—
466C22—44no—
467C22—42no—
468C22—44no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—42no—
473C22—42no—
474C22—82no—
475C22—82no—
476C22—81no—
477C22—81no—
478C22#230811yescharacteristic
479C22C22×C22⋊C811yescharacteristic
480C22#287911yescharacteristic
481C4—24no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—24no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—42no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4#297212yes—
497C4#297312yes—
498C2—21no—
499C2—21no—
500C2—42no—
501C2—44no—
502C2—82no—
503C2#2173811yescharacteristic
504C2#2178211yescharacteristic
505C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC2×#532C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17353C2semidirect
3. #17351C2semidirect
4. #9102C2semidirect
5. #9099C2semidirect
6. #8959C2semidirect
7. #8958C2semidirect
8. #8730C2semidirect
9. #8324C2semidirect
46. #532C8semidirect
138. C2×#41C2×C8semidirect
159. C4⋊Q8C2×C8semidirect
165. #124C2×C8semidirect
231. C2×C42C2×C8semidirect

Maximal subgroups

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

subgroupindex
#342072
#173532
#173512
#91022
#90992
#89592
#89582
#87302
#83242