Finite Group Databasebeta

← #200621

Subgroups — #200621

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34955C211yesmaximal, characteristic
2C4×#519C211yesmaximal, characteristic, complemented
3#17544C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9410C218yesmaximal, complemented
6#8048C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10#2920C2212yes—
11#2592—22no—
12#2591C2214yes—
13#2568C412yescomplemented
14#2548C412yescomplemented
15#2504—21no—
16#2436—21no—
17#2398C414yescomplemented
18C4×#124—22no—
19#2297—22no—
20#2276—24no—
21#2274—21no—
22#2274C2214yes—
23C42×Q8—24no—
24C4×#117—22no—
25C4×#117C2211yescharacteristic
26C4×#117C2212yes—
27#1507C2218yes—
28#1505—24no—
29#1502—24no—
30#1467—22no—
31#1463C2214yes—
32#1461—22no—
33#1297C2218yes—
34C22×#41—21no—
35C23×Q8D411yescharacteristic
36#532—22no—
37#532—22no—
38#532—22no—
39#532—22no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#127C2×C411yescharacteristic
42C2×#124—22no—
43C2×#124—21no—
44C2×#121—21no—
45C2×#121C2312yes—
46#519—216no—
47C2×C4×Q8—28no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8D412yes—
56C2×C4×D4—21no—
57C2×C4×D4C2×C412yes—
58C2×#117—24no—
59C2×#117—22no—
60C2×#117—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65#432—22no—
66#432—22no—
67#431—22no—
68#431—22no—
69#423—22no—
70#407C2314yes—
71C4×M16—22no—
72#400C2×C418yes—
73#397—24no—
74#394C2×C418yes—
75#393—24no—
76#391—22no—
77#391—24no—
78#390—22no—
79#390—24no—
80#388—28no—
81#386—24no—
82#386—24no—
83#382—22no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—28no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—216no—
90C4×C4⋊C4C2×C412yes—
91C4×C4⋊C4D414yes—
92C4×C22⋊C4—21no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—28no—
96C4×C22⋊C4C2312yes—
97C4×C22⋊C4C2314yes—
98#376—24no—
99#376C2×C412yes—
100#376D414yes—
101C43—24no—
102C43—22no—
103C43—21no—
104#327—216no—
105#324—28no—
106C2×#41—48no—
107C2×#41—41no—
108C2×#41—41no—
109C2×#41—41no—
110C2×#41—41no—
111C22×Q8—28no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8C22⋊C412yes—
116C22×Q8C22⋊C412yes—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4C22×C412yes—
121C23×C4—22no—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2411yescharacteristic
127C2×M16—24no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130#127—22no—
131#127—22no—
132#124—48no—
133#124—42no—
134#124—42no—
135#121—24no—
136#121—24no—
137#121—42no—
138#121—42no—
139C4×Q8—264no—
140C4×Q8—24no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—42no—
151C4×D4—42no—
152#117—232no—
153#117—216no—
154#117—44no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C414yes—
167C2×C4⋊C4C2×D414yes—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—28no—
174C2×C42—28no—
175C2×C42—24no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—28no—
184C2×C42—24no—
185C2×C42—22no—
186C2×C42C22⋊C414yes—
187C2×C42C22⋊C414yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4112yes—
195C2×C42#4112yes—
196C4⋊C8—42no—
197C4⋊C8—42no—
198C22⋊C8—24no—
199C22⋊C8—24no—
200#97—42no—
201C4×C8—42no—
202#95—28no—
203#95—22no—
204#95—28no—
205#95—28no—
206#95—24no—
207#95C22×C414yes—
208#95C2×D418yes—
209C24—22no—
210C24C22×D411yescharacteristic
211#41—84no—
212#41—84no—
213#41—84no—
214#41—84no—
215C2×Q8—216no—
216C2×Q8—216no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—28no—
220C2×Q8—24no—
221C2×Q8—44no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8#10418yes—
225C2×D4—24no—
226C2×D4—44no—
227C2×D4—48no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—44no—
234C2×D4—82no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—28no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—28no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—48no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—81no—
249C22×C4—82no—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C414yes—
254C22×C4C2×C22⋊C414yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12012yes—
258C22×C4#12011yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12312yes—
261C22×C4#12311yescharacteristic
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D412yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4112yes—
266M16—84no—
267C2×C8—44no—
268C4⋊C4—232no—
269C4⋊C4—216no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C22⋊C4—216no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—416no—
282C22⋊C4—88no—
283C42—232no—
284C42—216no—
285C42—24no—
286C42—24no—
287C42—24no—
288C42—24no—
289C42—44no—
290C42—44no—
291C42—48no—
292C42—84no—
293C42—82no—
294C42—88no—
295C42C4×D412yes—
296C42C4×D412yes—
297C42C4×D412yes—
298C42C4×D412yes—
299C42#12112yes—
300C42#12112yes—
301C42#12112yes—
302C42#12112yes—
303C23—22no—
304C23—22no—
305C23—41no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—42no—
310C23C22×C22⋊C411yescharacteristic
311C23C2×#12011yescharacteristic
312C23C2×#12311yescharacteristic
313Q8—216no—
314Q8—216no—
315Q8—82no—
316Q8—82no—
317D4—84no—
318D4—82no—
319D4—84no—
320D4—84no—
321D4—82no—
322D4—84no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—216no—
336C2×C4—24no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—48no—
342C2×C4—44no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—84no—
351C2×C4—82no—
352C2×C4—84no—
353C2×C4#38611yescharacteristic
354C2×C4#38611yescharacteristic
355C2×C4#38612yes—
356C2×C4#38611yescharacteristic
357C2×C4#38612yes—
358C2×C4#38611yescharacteristic
359C2×C4C2×#10418yes—
360C2×C4C2×#10414yes—
361C2×C4#45414yes—
362C2×C4C2×C4×D412yes—
363C2×C4C2×#12112yes—
364C8—84no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—44no—
373C22—88no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22C2×#38611yescharacteristic
378C22C22×#10412yes—
379C22C2×#45412yes—
380C4—22no—
381C4—22no—
382C4—21no—
383C4—21no—
384C4—22no—
385C4—28no—
386C4—416no—
387C4—44no—
388C4—84no—
389C4—82no—
390C4#188812yes—
391C4#188812yes—
392C4#189614yes—
393C2—22no—
394C2—22no—
395C2—41no—
396C2—162no—
397C2C2×#188811yescharacteristic
398C2C2×#189611yescharacteristic
399C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#519C2semidirect
4. #17365C2semidirect
5. #9410C2semidirect
6. #8048C2semidirect
13. #2568C4semidirect
14. #2548C4semidirect
17. #2398C4semidirect

Maximal subgroups

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

subgroupindex
#349552
C4×#5192
#175442
#173652
#94102
#80482