Finite Group Databasebeta

← #205350

Subgroups — #205350

Order 512.

1052 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21313C211yesmaximal, characteristic
3#21311C211yesmaximal, characteristic
4#20543C211yesmaximal, characteristic
5C2×#1571C211yesmaximal, characteristic
6C2×#1550C211yesmaximal, characteristic
7C22×#332C211yesmaximal, characteristic
8#9737C214yesmaximal, complemented
9#9733C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209C418yescomplemented
14C2×#494C2212yes—
15C2×#487C2212yes—
16C2×#462—22no—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1989—22no—
24#1988C2212yes—
25#1988C2214yes—
26#1987—22no—
27#1987C2212yes—
28C8×C4⋊C4C2214yes—
29C4×C4⋊C8—22no—
30C8×C22⋊C4—22no—
31#1736—22no—
32#1735—22no—
33#1571—24no—
34C2×#335C2214yes—
35#1550—24no—
36C2×#333C2212yes—
37C2×#332—24no—
38C2×#332C2212yes—
39C2×#332C2214yes—
40C23×Q8—21no—
41C22×Q16—24no—
42C22×M16—21no—
43C23×C8—21no—
44C23×C8—21no—
45#533—22no—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8C2312yes—
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×#117—21no—
65C2×#117D412yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C42C2311yescharacteristic
69#494—24no—
70#494C2×C418yes—
71#487—24no—
72#487C2×C418yes—
73#462—216no—
74#451—216no—
75#431C2×C414yes—
76#431D414yes—
77#423—22no—
78#423—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D414yes—
86C2×C4⋊C8D412yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×#103—24no—
91C2×C22⋊C8D412yes—
92C2×C4×C8—24no—
93C2×C4×C8C2312yes—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4C2312yes—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98#376—21no—
99C43—21no—
100#340—28no—
101#340—24no—
102#340—24no—
103#336—24no—
104#335—28no—
105#333—24no—
106#332—28no—
107#332—24no—
108C22×Q8—24no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—41no—
113C22×Q8—42no—
114C22×Q8—82no—
115C22×Q8—82no—
116C23×C4—41no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4#4111yescharacteristic
120C2×Q16—216no—
121C2×Q16—84no—
122C2×Q16—84no—
123C2×M16—42no—
124C2×M16—44no—
125C22×C8—22no—
126C22×C8—24no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—42no—
130C22×C8—44no—
131C22×C8—82no—
132C22×C8C2×D414yes—
133C4⋊Q8—24no—
134C4⋊Q8—24no—
135C4⋊Q8—24no—
136C4⋊Q8—22no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—22no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—22no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8C22×C414yes—
149C4⋊Q8C22×C412yes—
150C4⋊Q8C22×C412yes—
151#125—24no—
152#125—42no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—42no—
160C4×Q8—42no—
161#117—24no—
162#117—24no—
163#117—42no—
164#117Q1618yescomplemented
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4C2×D414yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—41no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—41no—
186C2×C42C22×C412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42#4111yescharacteristic
192C2×C42#4112yes—
193C2×C42C2411yescharacteristic
194C4⋊C8—24no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8C22⋊C418yes—
203C4⋊C8C2×D414yes—
204#103—216no—
205C22⋊C8C22⋊C418yes—
206C4×C8—84no—
207C4×C8C22×C414yes—
208C4×C8C2×D414yes—
209#95—22no—
210#95—22no—
211C24#12111yescharacteristic
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—44no—
219C2×Q8—42no—
220C2×Q8—44no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—88no—
224C2×Q8—84no—
225C2×Q8—84no—
226C2×Q8—84no—
227C2×Q8—82no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—84no—
237C22×C4C2×C22⋊C412yes—
238C22×C4C4×D412yes—
239C22×C4C4×D412yes—
240C22×C4#12112yes—
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12412yes—
245C22×C4#12712yes—
246C22×C4C2×Q1614yes—
247C22×C4#13514yes—
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C2×#4111yescharacteristic
251Q16—816no—
252M16—84no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—44no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—44no—
260C2×C8—44no—
261C2×C8—84no—
262C2×C8—84no—
263C2×C8C2×C22⋊C418yes—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4C2×Q1618yes—
282C22⋊C4—28no—
283C22⋊C4—44no—
284C42—28no—
285C42—42no—
286C42—42no—
287C42—44no—
288C42—82no—
289C42C2×C22⋊C412yes—
290C42C4×D412yes—
291C42C4×D412yes—
292C42#12112yes—
293C42#12412yes—
294C42#12712yes—
295C42C23×C411yescharacteristic
296C42C22×D411yescharacteristic
297C42C22×D411yescharacteristic
298C42C2×#4111yescharacteristic
299C23—22no—
300C23—22no—
301C23#39012yes—
302C23#46714yes—
303C23C2×#12111yescharacteristic
304Q8—44no—
305Q8—48no—
306Q8—82no—
307Q8—82no—
308Q8—88no—
309Q8—81no—
310Q8—82no—
311Q8—81no—
312Q8—82no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—28no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—88no—
328C2×C4—84no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4#39012yes—
332C2×C4#39012yes—
333C2×C4#39012yes—
334C2×C4C4×Q1618yes—
335C2×C4#46714yes—
336C2×C4#49414yes—
337C2×C4#49514yes—
338C2×C4C22×C22⋊C411yescharacteristic
339C2×C4C2×C4×D411yescharacteristic
340C2×C4C2×C4×D411yescharacteristic
341C2×C4C2×#12111yescharacteristic
342C2×C4C2×#12411yescharacteristic
343C2×C4C2×#12711yescharacteristic
344C2×C4C22×Q1612yes—
345C2×C4C2×#13512yes—
346C2×C4#57612yes—
347C2×C4#57712yes—
348C8—28no—
349C8—44no—
350C8—82no—
351C8—82no—
352C22—22no—
353C22—22no—
354C22—28no—
355C22#196914yes—
356C22C2×#39011yescharacteristic
357C22#304512yes—
358C22C2×#46712yes—
359C22#306912yes—
360C4—21no—
361C4—21no—
362C4—22no—
363C4—22no—
364C4—48no—
365C4—82no—
366C4—84no—
367C4—82no—
368C4#196914yes—
369C4#289712yes—
370C4C2×C4×Q1612yes—
371C4#304111yescharacteristic
372C4#304311yescharacteristic
373C4C2×#49411yescharacteristic
374C4C2×#49511yescharacteristic
375C2—24no—
376C2#1941114yes—
377C2C2×#196911yescharacteristic
378C2#2125011yescharacteristic
379C2#2126311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #9737C2semidirect
9. #9733C2semidirect
13. #3209C4semidirect
164. #117Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#213132
#213112
#205432
C2×#15712
C2×#15502
C22×#3322
#97372
#97332