Finite Group Databasebeta

← #201569

Subgroups — #201569

Order 512.

1161 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2982C211yesmaximal, characteristic, complemented
2#35245C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic, complemented
4#17487C212yesmaximal
5#9486C214yesmaximal, complemented
6#9484C214yesmaximal, complemented
7#7958C212yesmaximal, complemented
8C2×#529C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2982—24no—
11#2982—22no—
12#2982C2214yes—
13C2×C8×Q8—24no—
14#2920C2212yes—
15C2×#431—22no—
16C2×#431C2212yes—
17#2853—22no—
18#2634C412yes—
19#2632C412yes—
20#2604—22no—
21#2591C412yes—
22#2583C412yes—
23#2581C2212yes—
24#2536C2212yes—
25#2487—21no—
26#2409—21no—
27#2323—22no—
28#2283C2211yescharacteristic
29#1821—21no—
30#1801C2214yes—
31#1784—21no—
32C4×#101—22no—
33#1494—24no—
34#1487C2214yes—
35#1486C2214yes—
36#1482—24no—
37#1286C2218yes—
38C23×Q8D411yescharacteristic
39#529—22no—
40#529—22no—
41#529—22no—
42#529—22no—
43C2×#126—21no—
44C2×#125C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#123—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8C2×C411yescharacteristic
49C2×#121C2×C411yescharacteristic
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8D412yes—
56C2×C4×Q8D412yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×Q8C2312yes—
59C2×C4×D4—21no—
60C2×C4×D4C2311yescharacteristic
61C2×#117—21no—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4D411yescharacteristic
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66C8×Q8—216no—
67C8×Q8—216no—
68#432—22no—
69#432—22no—
70#431—28no—
71#431—28no—
72#431—22no—
73#431—22no—
74#431—22no—
75#431—22no—
76#431D414yes—
77#431C2314yes—
78#423C2314yes—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—22no—
81C2×#101—24no—
82#407—22no—
83C2×C22⋊C8—22no—
84C4×M16—22no—
85C2×C4×C8—24no—
86C2×C4×C8D414yes—
87#400—22no—
88#400C2×C414yes—
89#399—22no—
90#398C2×C414yes—
91#395—22no—
92#395C2×C414yes—
93#394C2×C414yes—
94#393—22no—
95#391—22no—
96#389—24no—
97#389—22no—
98#388—24no—
99#388—24no—
100#388—24no—
101#387—24no—
102#387—24no—
103#386—22no—
104#386—22no—
105#385—22no—
106#385—22no—
107#383C2×C412yes—
108#382C2312yes—
109#381—21no—
110#380—21no—
111#380—22no—
112C4×C4⋊C4—21no—
113C4×C22⋊C4C2×C412yes—
114#376—21no—
115#343—24no—
116Q8⋊C8—28no—
117D4⋊C8—28no—
118C22×Q8—28no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8C22⋊C412yes—
122C22×Q8C22×C411yescharacteristic
123C22×Q8C2×D412yes—
124C22×D4—21no—
125C22×D4C22×C411yescharacteristic
126C23×C4—21no—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—24no—
133C22×C8—24no—
134#126—42no—
135#126—42no—
136#125—22no—
137#125—22no—
138#124—22no—
139#124—22no—
140#123—41no—
141#123—41no—
142#123—41no—
143#123—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—48no—
147#121—22no—
148#121—22no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—42no—
162C4×D4—42no—
163#117—42no—
164#117—42no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4C22⋊C414yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C414yes—
181C2×C4⋊C4C2×D412yes—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—24no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42C22⋊C414yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D414yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42C2411yescharacteristic
206C4⋊C8—216no—
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8—24no—
211#101—48no—
212C22⋊C8—24no—
213C22⋊C8—24no—
214C22⋊C8—42no—
215C22⋊C8—42no—
216#97—42no—
217C4×C8—216no—
218C4×C8—24no—
219C4×C8—42no—
220C4×C8C2×D418yes—
221#95—22no—
222#95—22no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—22no—
228#95C22×C414yes—
229C24—21no—
230C24C2×C22⋊C411yescharacteristic
231C2×Q8—216no—
232C2×Q8—216no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—28no—
237C2×Q8—22no—
238C2×Q8—81no—
239C2×Q8#13518yes—
240C2×D4—22no—
241C2×D4—44no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—81no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—81no—
259C22×C4—82no—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C414yes—
262C22×C4C2×C22⋊C414yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C4×D412yes—
266C22×C4C4×D412yes—
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12112yes—
270C22×C4#12112yes—
271C22×C4#12311yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276M16—48no—
277C2×C8—28no—
278C2×C8—48no—
279C2×C8—44no—
280C4⋊C4—28no—
281C4⋊C4—24no—
282C4⋊C4—28no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—48no—
294C4⋊C4—84no—
295C22⋊C4—44no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—42no—
304C22⋊C4—48no—
305C22⋊C4—84no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—24no—
310C42—42no—
311C42—82no—
312C42C4×D412yes—
313C42C4×D412yes—
314C42#12012yes—
315C42#12112yes—
316C42#12112yes—
317C42#12312yes—
318C42C22×D412yes—
319C42C2×#4112yes—
320C23—22no—
321C23—22no—
322C23—41no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23#38611yescharacteristic
328C23#38611yescharacteristic
329C23C22×C22⋊C411yescharacteristic
330Q8—216no—
331Q8—216no—
332Q8—44no—
333D4—44no—
334D4—82no—
335D4—82no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4#38611yescharacteristic
368C2×C4#38611yescharacteristic
369C2×C4#38612yes—
370C2×C4#38612yes—
371C2×C4#41618yes—
372C2×C4C2×C4×D412yes—
373C2×C4C2×#12011yescharacteristic
374C2×C4C2×#12112yes—
375C2×C4C2×#12311yescharacteristic
376C2×C4C2×#13514yes—
377C8—48no—
378C8—84no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—44no—
387C22—84no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#191212yes—
392C22C2×#38611yescharacteristic
393C22C2×#41612yes—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—28no—
402C4—44no—
403C4—84no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4#305014yes—
408C2—22no—
409C2—22no—
410C2—41no—
411C2—161no—
412C2#2096411yescharacteristic
413C2#2096711yescharacteristic
414C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2982C2semidirect
3. #20727C2semidirect
5. #9486C2semidirect
6. #9484C2semidirect
7. #7958C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29822
#352452
#207272
#174872
#94862
#94842
#79582