Finite Group Databasebeta

← #6457913

Subgroups — #6457913

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35630C212yesmaximal
2#28023C214yesmaximal, complemented
3#28020C214yesmaximal, complemented
4#28015C214yesmaximal, complemented
5#28012C214yesmaximal, complemented
6#23629C212yesmaximal
7#23613C212yesmaximal
8#21232C211yesmaximal, characteristic
9#21232C211yesmaximal, characteristic
10C2×#1969C211yesmaximal, characteristic
11#21156C211yesmaximal, characteristic
12#21066C212yesmaximal
13#20797C211yesmaximal, characteristic
14#20455C211yesmaximal, characteristic
15#20435C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#427C2212yes—
20C22×#107C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C2×#419C2212yes—
23C2×#419C2212yes—
24C22×#103C2212yes—
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2514—24no—
29#2514C2218yes—
30#2511—24no—
31#2511C2218yes—
32#2493—28no—
33#2317C2212yes—
34#2310C2212yes—
35#2310C2212yes—
36#2296C2211yescharacteristic
37C2×#391C2212yes—
38#2285C2212yes—
39#2274C2212yes—
40#2074C2218yes—
41#2072C2218yes—
42#2068C2218yes—
43#2066C2218yes—
44#1971C2214yes—
45#1971C2214yes—
46#1969C2212yes—
47#1969C2214yes—
48#1969C2214yes—
49#1969C2212yes—
50#1969C2212yes—
51#1969C2212yes—
52#1944C2214yes—
53#1942C2214yes—
54#1928C2218yes—
55#1927C2218yes—
56#1846C2212yes—
57#1846C2212yes—
58#1845C2212yes—
59#1844C2212yes—
60#1776C2214yes—
61C4×#107C2214yes—
62#1771C2212yes—
63#1771C2212yes—
64C8×C4⋊C4C2212yes—
65C8×C4⋊C4C2212yes—
66#1736C2212yes—
67#1736C2212yes—
68#1729C2211yescharacteristic
69#1728C2211yescharacteristic
70C23×Q8C2312yes—
71C22×Q16—24no—
72C22×Q16—24no—
73C22×Q16—24no—
74C22×Q16C2312yes—
75C22×Q16C2312yes—
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C2×#117C2312yes—
81C2×#117C2312yes—
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C42C2311yescharacteristic
86#427—24no—
87#427—24no—
88#427—44no—
89C2×#107—24no—
90C2×#107C2312yes—
91C2×#107C2312yes—
92C2×#107C2312yes—
93C2×#107C2312yes—
94C2×#107C2314yes—
95C2×#106C2312yes—
96C2×#106C2312yes—
97C2×#106C2314yes—
98#424—24no—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101#419—44no—
102#419—44no—
103C2×#103—28no—
104C2×#103C2314yes—
105C2×#103C2314yes—
106C2×#103C2318yes—
107C2×#103C2318yes—
108#405—24no—
109#405—24no—
110C2×#97C2312yes—
111C2×#97C2312yes—
112C2×C4×C8—24no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2312yes—
118C2×C4×C8C2312yes—
119#401—28no—
120#400—28no—
121#400—28no—
122#400—28no—
123#391—22no—
124#391—24no—
125#391—22no—
126#391—24no—
127#391C2318yes—
128#391C2318yes—
129#391C2314yes—
130#391C2314yes—
131#389D412yes—
132#389D412yes—
133#389C2318yes—
134#387—28no—
135#387—28no—
136#387—28no—
137#385—24no—
138#385D414yes—
139#384—22no—
140#384C2314yes—
141#383—28no—
142#382—22no—
143#382C2314yes—
144#381—24no—
145#381—24no—
146#380—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4C2318yes—
150C4×C4⋊C4C2312yes—
151C4×C4⋊C4C2312yes—
152C4×C22⋊C4—24no—
153C2×#95—24no—
154#340C2318yes—
155#340C2318yes—
156#336C2312yes—
157#336C2312yes—
158#336C2312yes—
159#336C2312yes—
160C22×Q8—22no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8C2414yes—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C2×D412yes—
171C23×C4C2411yescharacteristic
172C2×Q16—24no—
173C2×Q16—24no—
174C2×Q16—24no—
175C2×Q16—24no—
176C2×Q16—24no—
177C2×Q16—24no—
178C2×Q16—416no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8#4114yes—
188C22×C8#4114yes—
189C22×C8#4114yes—
190C22×C8#4114yes—
191C22×C8C2412yes—
192C22×C8C2412yes—
193C22×C8C2411yescharacteristic
194C22×C8C2411yescharacteristic
195C22×C8C2411yescharacteristic
196C22×C8C2411yescharacteristic
197#117—22no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—24no—
202#117—24no—
203#117—24no—
204#117—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C4⋊C4C2414yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4C2×D412yes—
222C2×C22⋊C4C2×D412yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—28no—
228C2×C42C2×D412yes—
229C2×C42C2412yes—
230C2×C42C2412yes—
231C2×C42C2411yescharacteristic
232C2×C42C2411yescharacteristic
233#107—24no—
234#107—24no—
235#107—24no—
236#107—24no—
237#107—44no—
238#106—24no—
239#106—24no—
240#106—44no—
241C4⋊C8—24no—
242C4⋊C8—24no—
243#103—28no—
244#103—28no—
245#103—48no—
246#103—48no—
247#97—24no—
248#97—24no—
249C4×C8—22no—
250C4×C8—22no—
251C4×C8—22no—
252C4×C8—22no—
253C4×C8—24no—
254C4×C8—24no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95C2×D414yes—
266#95C2×D414yes—
267C24C22×D411yescharacteristic
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—48no—
280C2×Q8—48no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—84no—
293C22×C4C22×D412yes—
294C22×C4C22×D412yes—
295C22×C4C22×D414yes—
296C22×C4C22×D412yes—
297C22×C4C22×D412yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4112yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C4C2×#4112yes—
303C22×C4H(5,𝔽2)12yes—
304C22×C42−1+412yes—
305C22×C4C2511yescharacteristic
306Q16—48no—
307Q16—48no—
308M16—44no—
309M16—44no—
310C2×C8—28no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8#126116yes—
322C2×C8C2×#4112yes—
323C2×C8C2×#4112yes—
324C2×C8C2×#4112yes—
325C2×C8C2×#4112yes—
326C2×C8C2×#4112yes—
327C2×C8C2×#4112yes—
328C2×C8C2×#4112yes—
329C2×C8C2×#4112yes—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C4⋊C4—84no—
344C4⋊C4—88no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C42—24no—
350C42—21no—
351C42—21no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—21no—
356C42—21no—
357C42—22no—
358C42—44no—
359C42—44no—
360C42—88no—
361C23—22no—
362C23—22no—
363C23#53212yes—
364C23#54712yes—
365C23#55012yes—
366C23C23×D411yescharacteristic
367Q8—44no—
368Q8—44no—
369Q8—44no—
370Q8—44no—
371Q8—88no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—88no—
397C2×C4—164no—
398C2×C4C2×#12614yes—
399C2×C4#53212yes—
400C2×C4#54214yes—
401C2×C4#54712yes—
402C2×C4#55012yes—
403C2×C4#55314yes—
404C2×C4#55614yes—
405C2×C4#57814yes—
406C2×C4C22×#4111yescharacteristic
407C2×C4C22×#4111yescharacteristic
408C2×C4C22×#4111yescharacteristic
409C2×C4C22×#4111yescharacteristic
410C2×C4C2×H(5,𝔽2)11yescharacteristic
411C2×C4C2×2−1+411yescharacteristic
412C8—22no—
413C8—22no—
414C8—42no—
415C8—42no—
416C8C2×#12614yes—
417C8C2×#12614yes—
418C22—22no—
419C22—22no—
420C22—24no—
421C22—24no—
422C22#251214yes—
423C22C2×#53211yescharacteristic
424C22C2×#54711yescharacteristic
425C22C2×#55011yescharacteristic
426C22C2×#57812yes—
427C22#358412yes—
428C4—24no—
429C4—42no—
430C4—42no—
431C4—44no—
432C4—42no—
433C4—84no—
434C4—84no—
435C4—84no—
436C4—84no—
437C4#251214yes—
438C4C22×#12611yescharacteristic
439C4C2×#54211yescharacteristic
440C4C2×#55311yescharacteristic
441C4C2×#55611yescharacteristic
442C2—24no—
443C2C2×#251211yescharacteristic
444C2#6071311yescharacteristic
445C2#6071511yescharacteristic
446C2#6126412yes—
447C2#6140812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28023C2semidirect
3. #28020C2semidirect
4. #28015C2semidirect
5. #28012C2semidirect

Maximal subgroups

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

subgroupindex
#356302
#280232
#280202
#280152
#280122
#236292
#236132
#212322
C2×#19692
#211562
#210662
#207972
#204552
#204352