Finite Group Databasebeta

← #213775

Subgroups — #213775

Order 512.

1069 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#17353C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#9100C212yesmaximal, complemented
5#9099C212yesmaximal, complemented
6#8958C212yesmaximal, complemented
7#8957C212yesmaximal, complemented
8#8731C212yesmaximal, complemented
9#8323C212yesmaximal, complemented
10C2×#536C411yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#519C2211yescharacteristic
13#2982C2212yes—
14#2972C2212yes—
15#2362—22no—
16#2360—22no—
17#2359—22no—
18#2359—22no—
19#2327C411yescharacteristic
20#2327C2211yescharacteristic
21#2326C411yescharacteristic
22#2326C2211yescharacteristic
23#2317C411yescharacteristic
24C4×C4⋊Q8C411yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#124C411yescharacteristic
27C4×#124C2211yescharacteristic
28#2297C411yescharacteristic
29C2×#391C411yescharacteristic
30C4×#117C2211yescharacteristic
31#1479C2214yes—
32#1479C2214yes—
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
44C23×Q8C2×C411yescharacteristic
45#536—24no—
46#536C818yescomplemented
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#124C2×C411yescharacteristic
50C2×#124C2×C411yescharacteristic
51C2×#123—22no—
52C2×C22⋊Q8—22no—
53#520—44no—
54#519—44no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C2×C4×D4D412yes—
66C2×C4×D4C2311yescharacteristic
67C2×#117—21no—
68C2×#117—21no—
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117D412yes—
72C2×#117C2311yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4D412yes—
75C22×C42C2×C411yescharacteristic
76C8×Q8—44no—
77C8×D4—44no—
78#431—24no—
79#431C2314yes—
80#431C2314yes—
81C2×C22⋊C8—24no—
82#391—22no—
83#391C2×C411yescharacteristic
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#390C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#389—22no—
92#389—22no—
93#389—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#386—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#384C2×C411yescharacteristic
107#384C2×C411yescharacteristic
108#382C2×C411yescharacteristic
109#382C2×C411yescharacteristic
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376C2×C411yescharacteristic
120#376C2311yescharacteristic
121C2×#95—22no—
122C43C2×C411yescharacteristic
123C43C2311yescharacteristic
124Q8⋊C8—44no—
125Q8⋊C8—44no—
126#324—22no—
127#324—24no—
128#324—22no—
129#323—24no—
130#323—22no—
131#323—22no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41C2×C814yes—
135C2×#41M1614yes—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8C2×C814yescomplemented
141C22×Q8C22×C411yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×D4C22⋊C412yes—
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C22×C8—44no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8C2×C814yescomplemented
157C4⋊Q8C2×C814yescomplemented
158#124—22no—
159#124—22no—
160#124C2×C814yes—
161#124C2×C814yes—
162#123—216no—
163C22⋊Q8—216no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—44no—
171C4×Q8—44no—
172C4×D4—24no—
173C4×D4—44no—
174C4×D4—42no—
175C4×D4—44no—
176#117—22no—
177#117—24no—
178#117—44no—
179#117—44no—
180#117—44no—
181#117—42no—
182#117C2×C814yescomplemented
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C2×D411yescharacteristic
211C2×C22⋊C4C2×D411yescharacteristic
212C2×C22⋊C4C2×D412yes—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—42no—
224C2×C42C22⋊C412yes—
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2411yescharacteristic
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—44no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—44no—
238C4×C8—24no—
239C4×C8—24no—
240C4×C8—44no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253C24C2×C22⋊C412yes—
254C24C2×C22⋊C411yescharacteristic
255#41—48no—
256#41—48no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—48no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—48no—
270C2×Q8C22×C812yes—
271C2×Q8C22×C812yes—
272C2×Q8C22×C812yes—
273C2×Q8C2×M1612yes—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4C22⋊C818yes—
279C2×D4C22×C812yes—
280C2×D4C2×M1612yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
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—81no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4C22⋊C818yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C22×C812yes—
313C22×C4C2×M1612yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—164no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—24no—
339C22⋊C4—28no—
340C22⋊C4—28no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—24no—
344C22⋊C4—42no—
345C22⋊C4—44no—
346C22⋊C4—42no—
347C22⋊C4—82no—
348C42—22no—
349C42—24no—
350C42—21no—
351C42—21no—
352C42—22no—
353C42—22no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—81no—
364C42—81no—
365C42C22×C812yes—
366C42C22×C812yes—
367C42H(5,𝔽2)14yes—
368C23—21no—
369C23—22no—
370C23—21no—
371C23—22no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—42no—
376C23C2×C22⋊C814yes—
377C23C2×C22⋊C812yes—
378C23C22×C22⋊C411yescharacteristic
379Q8—24no—
380Q8—24no—
381Q8—24no—
382Q8—24no—
383Q8—48no—
384Q8—42no—
385Q8—42no—
386D4—24no—
387D4—24no—
388D4—48no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—48no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—48no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—162no—
432C2×C4—162no—
433C2×C4C2×C22⋊C814yes—
434C2×C4C2×C22⋊C812yes—
435C2×C4#51812yes—
436C2×C4#53412yes—
437C2×C4#53412yes—
438C2×C4C23×C811yescharacteristic
439C2×C4C22×M1611yescharacteristic
440C2×C4#56812yes—
441C8—84no—
442C8—84no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—24no—
447C22—42no—
448C22—44no—
449C22—42no—
450C22—41no—
451C22—42no—
452C22—42no—
453C22—41no—
454C22—82no—
455C22#230811yescharacteristic
456C22C22×C22⋊C811yescharacteristic
457C22#287911yescharacteristic
458C4—22no—
459C4—22no—
460C4—22no—
461C4—24no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—41no—
466C4—42no—
467C4—41no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4#297212yes—
475C4#297312yes—
476C2—21no—
477C2—21no—
478C2—42no—
479C2—44no—
480C2#2173811yescharacteristic
481C2#2178311yescharacteristic
482C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
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
4. #9100C2semidirect
5. #9099C2semidirect
6. #8958C2semidirect
7. #8957C2semidirect
8. #8731C2semidirect
9. #8323C2semidirect
46. #536C8semidirect
140. C22×Q8C2×C8semidirect
156. C4⋊Q8C2×C8semidirect
157. C4⋊Q8C2×C8semidirect
182. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
#342762
#173532
#173512
#91002
#90992
#89582
#89572
#87312
#83232