Finite Group Databasebeta

← #205384

Subgroups — #205384

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4#20536C211yesmaximal, characteristic
5C2×#1570C211yesmaximal, characteristic
6C2×#1549C211yesmaximal, characteristic
7C2×#1542C211yesmaximal, characteristic
8#9745C214yesmaximal, complemented
9#9742C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209C418yescomplemented
14C2×#494C2212yes—
15C2×#487C2212yes—
16C2×#462—22no—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1989—22no—
24#1988—22no—
25#1988C2212yes—
26#1988C2214yes—
27#1987C2212yes—
28#1770—22no—
29#1770C2214yes—
30#1749—22no—
31#1736—22no—
32C2×#340—22no—
33#1570—24no—
34C2×#334C2214yes—
35#1549—24no—
36#1542—24no—
37C2×#333C2212yes—
38C2×#332C2212yes—
39C2×#332C2214yes—
40C23×Q8—21no—
41C22×Q16—24no—
42C22×M16—21no—
43C23×C8—21no—
44#533—22no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8C2312yes—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×#125—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#494—24no—
70#494C2×C418yescomplemented
71#487—24no—
72#487C2×C418yescomplemented
73#462—216no—
74#451—216no—
75#431—22no—
76#431C2×C414yes—
77#431D414yes—
78#431D414yes—
79#423—22no—
80#423—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D412yes—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103—24no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8D412yes—
92C2×#97—24no—
93C2×C4×C8—21no—
94C2×C4×C8C2312yes—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4D412yes—
98C4×C4⋊C4C2312yes—
99C2×#95—21no—
100#340—24no—
101#340—28no—
102#340—24no—
103#336—24no—
104#334—28no—
105#333—24no—
106#332—24no—
107#332—28no—
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—22no—
117C23×C4—41no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C2×Q16—216no—
122C2×Q16—84no—
123C2×Q16—84no—
124C2×M16—42no—
125C2×M16—44no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—42no—
130C22×C8—44no—
131C22×C8C2×D414yes—
132C4⋊Q8—24no—
133C4⋊Q8—24no—
134C4⋊Q8—24no—
135C4⋊Q8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—22no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—22no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147C4⋊Q8C22×C414yes—
148C4⋊Q8C22×C412yes—
149C4⋊Q8C22×C412yes—
150#125—24no—
151#125—42no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—42no—
159C4×Q8—42no—
160#117—24no—
161#117—42no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
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—41no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—41no—
184C2×C42—82no—
185C2×C42C22×C412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42#4111yescharacteristic
191C2×C42#4112yes—
192C2×C42C2411yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—42no—
199C4⋊C8—42no—
200C4⋊C8C22⋊C418yes—
201C4⋊C8C2×D414yes—
202#103—216no—
203C22⋊C8—24no—
204C22⋊C8C22⋊C418yes—
205#97—84no—
206C4×C8—24no—
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—82no—
225C2×Q8—82no—
226C2×Q8—88no—
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—44no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4C2×C22⋊C412yes—
242C22×C4C4×D412yes—
243C22×C4C4×D412yes—
244C22×C4#12112yes—
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12412yes—
249C22×C4#12712yes—
250C22×C4#13714yes—
251C22×C4#13714yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C2×#4111yescharacteristic
255Q16—816no—
256M16—84no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—44no—
265C2×C8—44no—
266C2×C8—84no—
267C2×C8C2×C22⋊C418yes—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C22⋊C4—44no—
288C42—44no—
289C42—44no—
290C42—82no—
291C42C2×C22⋊C412yes—
292C42C4×D412yes—
293C42C4×D412yes—
294C42#12112yes—
295C42#12412yes—
296C42#12712yes—
297C42C23×C411yescharacteristic
298C42C22×D411yescharacteristic
299C42C22×D411yescharacteristic
300C42C2×#4111yescharacteristic
301C23—21no—
302C23—21no—
303C23—22no—
304C23#39012yes—
305C23#47014yes—
306C23C2×#12111yescharacteristic
307Q8—44no—
308Q8—48no—
309Q8—82no—
310Q8—82no—
311Q8—88no—
312Q8—82no—
313Q8—84no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—88no—
329C2×C4—88no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4#39012yes—
336C2×C4#39012yes—
337C2×C4#39012yes—
338C2×C4#44118yes—
339C2×C4#47014yes—
340C2×C4#49714yes—
341C2×C4#49714yes—
342C2×C4C22×C22⋊C411yescharacteristic
343C2×C4C2×C4×D411yescharacteristic
344C2×C4C2×C4×D411yescharacteristic
345C2×C4C2×#12111yescharacteristic
346C2×C4C2×#12411yescharacteristic
347C2×C4C2×#12711yescharacteristic
348C2×C4C2×#13712yes—
349C2×C4C2×#13712yes—
350C2×C4#57612yes—
351C2×C4#57712yes—
352C8—28no—
353C8—44no—
354C8—82no—
355C8—82no—
356C22—22no—
357C22—22no—
358C22—24no—
359C22—24no—
360C22#197114yes—
361C22C2×#39011yescharacteristic
362C22#304111yescharacteristic
363C22#304311yescharacteristic
364C22C2×#47012yes—
365C22#306912yes—
366C4—21no—
367C4—21no—
368C4—22no—
369C4—22no—
370C4—82no—
371C4—84no—
372C4—82no—
373C4—84no—
374C4#197114yes—
375C4#289712yes—
376C4C2×#44112yes—
377C4#304512yes—
378C4C2×#49711yescharacteristic
379C4C2×#49711yescharacteristic
380C2—24no—
381C2#1942312yes—
382C2#1943512yes—
383C2C2×#197111yescharacteristic
384C2#2125011yescharacteristic
385C2#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. #9745C2semidirect
9. #9742C2semidirect
13. #3209C4semidirect
70. #494C2×C4semidirect
72. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#213122
C2×#19882
#205362
C2×#15702
C2×#15492
C2×#15422
#97452
#97422