Finite Group Databasebeta

← #6467486

Subgroups — #6467486

Order 512.

2244 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3170C211yesmaximal, characteristic
2C2×#3113C212yesmaximal
3C22×C4×Q16C211yesmaximal, characteristic
4#35248C212yesmaximal
5#28532C212yesmaximal, complemented
6#28462C212yesmaximal, complemented
7#28417C214yesmaximal, complemented
8#28345C214yesmaximal
9#27931C212yesmaximal
10#25780C212yesmaximal
11#23814C214yesmaximal, complemented
12#21698C214yesmaximal, complemented
13#20476C211yesmaximal, characteristic
14C23×Q16C2211yescharacteristic
15C2×#533C2212yes—
16C22×C4×Q8C2212yes—
17#3170—28no—
18C2×#500C2211yescharacteristic
19C2×#494C2211yescharacteristic
20C2×#494C2214yes—
21C2×#487C2212yes—
22#3113—216no—
23C2×#483C2214yes—
24C2×#475C2214yes—
25C2×#467C2212yes—
26C2×#467C2212yes—
27C2×#462C2214yes—
28C2×#451C2214yes—
29C2×#441C2218yes—
30C2×C4×Q16—28no—
31C2×C4×Q16C2212yes—
32C2×C4×Q16C2212yes—
33C2×C4×Q16C2212yes—
34C2×C4×Q16C2212yes—
35C2×C4×Q16C2212yes—
36C2×C4×Q16C2212yes—
37C2×#432C2211yescharacteristic
38C22×#107C2211yescharacteristic
39C22×#103C2212yes—
40C22×C4×C8C2211yescharacteristic
41#2855—24no—
42#2634—24no—
43#2581—24no—
44#2568C2214yes—
45#2546C2214yes—
46#2503C2214yes—
47#2484—24no—
48#2434C2214yes—
49#2327C2218yes—
50#2283C2211yescharacteristic
51#2097C2218yes—
52#2097C2218yes—
53#2016C2218yes—
54#2016C2218yes—
55#1971C2218yes—
56#1969C2214yes—
57#1969C2214yes—
58#1960C2218yes—
59#1846C2214yes—
60#1770C2212yes—
61C8×C4⋊C4C2212yes—
62#1752C2212yes—
63C8×C22⋊C4C2212yes—
64#1746C2212yes—
65#1726C2212yes—
66C23×Q8C2312yes—
67C22×Q16—28no—
68C22×Q16—24no—
69C22×Q16C2312yes—
70C22×Q16C2312yes—
71C22×Q16C2318yes—
72C22×Q16C2312yes—
73C22×Q16C2312yes—
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76#533—28no—
77#533—28no—
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2318yes—
80C2×C4⋊Q8C2312yes—
81C2×#125—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8C2314yes—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×C4×Q8—22no—
87C2×C4×Q8—28no—
88C2×C4×Q8—28no—
89C2×C4×Q8C2314yes—
90C2×C4×Q8C2314yes—
91C2×C4×Q8C2314yes—
92C2×#117—22no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4C2312yes—
95C22×C42—22no—
96C22×C42C2311yescharacteristic
97#500—24no—
98#494—216no—
99#494—24no—
100#487—28no—
101#483—216no—
102#475—216no—
103#467—28no—
104#467—28no—
105#462—216no—
106#451—216no—
107#441—232no—
108C4×Q16—28no—
109C4×Q16—28no—
110C4×Q16—28no—
111C4×Q16—28no—
112C4×Q16—28no—
113C4×Q16—28no—
114#432D418yes—
115C2×#107—24no—
116C2×#107C2312yes—
117C2×#107C2312yes—
118C2×#107C2312yes—
119C2×#107C2312yes—
120C2×#106C2318yes—
121C2×C4⋊C8C2312yes—
122C2×#103—28no—
123C2×#103C2318yes—
124C2×#103C2318yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×C22⋊C8C2312yes—
130C2×#97C2314yes—
131C2×#97C2312yes—
132C2×C4×C8D418yes—
133C2×C4×C8C2314yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138#400—28no—
139#400—28no—
140#398—28no—
141#395—28no—
142#391—28no—
143#391C2318yes—
144#391C2318yes—
145#389—28no—
146#387—216no—
147#387—216no—
148#385—28no—
149#383D412yes—
150#382D412yes—
151#382C2318yes—
152#381—28no—
153#381D412yes—
154#380D412yes—
155C4×C4⋊C4C2312yes—
156C4×C22⋊C4C2312yes—
157#376C2312yes—
158#336C2314yes—
159#336C2314yes—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—28no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—44no—
166C22×Q8C2414yes—
167C22×Q8C2414yes—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C2×Q16—232no—
174C2×Q16—28no—
175C2×Q16—216no—
176C2×Q16—24no—
177C2×Q16—24no—
178C2×Q16—24no—
179C2×Q16—24no—
180C2×Q16—216no—
181C2×Q16—24no—
182C2×Q16—28no—
183C2×Q16—48no—
184C2×Q16—44no—
185C2×Q16#4118yes—
186C22×C8—24no—
187C22×C8—42no—
188C22×C8C2×D414yes—
189C22×C8C2×D414yes—
190C22×C8C2411yescharacteristic
191C22×C8C2411yescharacteristic
192C22×C8C2411yescharacteristic
193C22×C8C2414yes—
194C22×C8C2411yescharacteristic
195C4⋊Q8—216no—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—48no—
201#125—48no—
202C22⋊Q8—28no—
203C22⋊Q8—28no—
204C22⋊Q8—48no—
205C4×Q8—28no—
206C4×Q8—28no—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—28no—
211C4×Q8—48no—
212C4×Q8—48no—
213C4×Q8—48no—
214#117—48no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2414yes—
229C2×C4⋊C4C2414yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4C2×D414yes—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—48no—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D414yes—
239C2×C42C2×D414yes—
240C2×C42C2414yes—
241C2×C42C2411yescharacteristic
242C2×C42C2411yescharacteristic
243#107—24no—
244#107—24no—
245#107—24no—
246#107—24no—
247#106—216no—
248C4⋊C8C2×D418yes—
249#103—28no—
250#103—28no—
251#103—28no—
252#103—28no—
253#103—216no—
254#103—216no—
255C22⋊C8C2×D418yes—
256#97—44no—
257#97C2×D418yes—
258C4×C8—28no—
259C4×C8C2×D414yes—
260C4×C8C2×D414yes—
261C4×C8C2×D414yes—
262C4×C8C2×D414yes—
263#95—216no—
264#95—216no—
265#95—216no—
266#95C2×D414yes—
267#95C2×D414yes—
268#95C2×D414yes—
269#95C2×D414yes—
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—216no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—28no—
279C2×Q8—24no—
280C2×Q8—28no—
281C2×Q8—48no—
282C2×Q8—416no—
283C2×Q8—48no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—48no—
287C2×Q8C2×#4118yes—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—48no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—48no—
300C22×C4#12714yes—
301C22×C4#12714yes—
302C22×C4C22×D418yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C2511yescharacteristic
311Q16—216no—
312Q16—216no—
313Q16—432no—
314Q16—48no—
315Q16—48no—
316C2×C8—24no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—24no—
320C2×C8—22no—
321C2×C8—28no—
322C2×C8—44no—
323C2×C8C22×D412yes—
324C2×C8C22×D412yes—
325C2×C8C22×D412yes—
326C2×C8C22×D418yes—
327C2×C8C22×D412yes—
328C2×C8C2×#4114yes—
329C4⋊C4—28no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—432no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—48no—
341C4⋊C4—48no—
342C4⋊C4—44no—
343C22⋊C4—28no—
344C22⋊C4—48no—
345C42—24no—
346C42—24no—
347C42—48no—
348C42—48no—
349C42—48no—
350C42#12714yes—
351C42#12714yes—
352C42C22×D412yes—
353C42C22×D412yes—
354C42C22×D412yes—
355C42C22×D412yes—
356C23—22no—
357C23—22no—
358C23C2×#12712yes—
359C23#57814yes—
360C23C23×D411yescharacteristic
361Q8—28no—
362Q8—28no—
363Q8—416no—
364Q8—44no—
365Q8—44no—
366Q8—44no—
367Q8—44no—
368Q8—48no—
369Q8—88no—
370C2×C4—28no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—28no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—28no—
382C2×C4—28no—
383C2×C4—416no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—88no—
393C2×C4C2×#12712yes—
394C2×C4C2×#12712yes—
395C2×C4C2×#12712yes—
396C2×C4C2×#12712yes—
397C2×C4C2×#12712yes—
398C2×C4#57814yes—
399C2×C4C23×D411yescharacteristic
400C2×C4C23×D411yescharacteristic
401C2×C4C22×#4112yes—
402C2×C4#58512yes—
403C8—22no—
404C8—22no—
405C8—22no—
406C8—28no—
407C8#52514yes—
408C22—28no—
409C22—22no—
410C22—22no—
411C22#315614yes—
412C22C22×#12711yescharacteristic
413C22D4×#4112yes—
414C22C2×#57812yes—
415C22C2×#57812yes—
416C4—22no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—48no—
421C4—48no—
422C4—84no—
423C4—84no—
424C4—84no—
425C4#315614yes—
426C4C2×#52511yescharacteristic
427C4#344911yescharacteristic
428C4D4×#4112yes—
429C2—24no—
430C2#3493611yescharacteristic
431C2C2×#315611yescharacteristic
432C2C2×#315611yescharacteristic
433C2#6148414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #28532C2semidirect
6. #28462C2semidirect
7. #28417C2semidirect
11. #23814C2semidirect
12. #21698C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31702
C2×#31132
C22×C4×Q162
#352482
#285322
#284622
#284172
#283452
#279312
#257802
#238142
#216982
#204762