Finite Group Databasebeta

← #195625

Subgroups — #195625

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#494C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#9066C218yesmaximal, complemented
7#7897C211yesmaximal, characteristic
8#7890C211yesmaximal, characteristic
9C23×Q16—22no—
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#494—22no—
13C2×#494—24no—
14C2×#494C418yescomplemented
15C2×#494C2214yes—
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2369C2214yes—
20#2365—22no—
21#2345—22no—
22Q8×C22⋊C4—22no—
23#2331—22no—
24#2316—21no—
25C4×C4⋊Q8—22no—
26#2285C2211yescharacteristic
27#2274—21no—
28C4×#117C2211yescharacteristic
29#1990—22no—
30#1988—22no—
31#1987C2214yes—
32#1987C2214yes—
33#1824—22no—
34#1736—21no—
35#1736—21no—
36#1548—24no—
37C2×#333C2214yes—
38#1279C2214yes—
39#1279C2214yes—
40C23×Q8—21no—
41C23×Q8D412yes—
42C22×Q16—216no—
43C22×Q16—28no—
44C22×Q16—416no—
45C22×M16—21no—
46C23×C8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yes—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8D412yes—
60C2×C4⋊Q8D414yes—
61C2×C4⋊Q8C2312yes—
62C2×C4⋊Q8C2312yes—
63C2×C4×Q8—24no—
64C2×#117—21no—
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C42C2311yescharacteristic
72#494—216no—
73#494—28no—
74#494C2×C4116yes—
75#431—22no—
76#431—22no—
77#423—22no—
78#423—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85C2×C4×C8—21no—
86C2×C4×C8C2×C414yes—
87C2×C4×C8C2312yes—
88#391—24no—
89#391—28no—
90#389—24no—
91#389—24no—
92#389—24no—
93#387—28no—
94#384D412yes—
95#382—21no—
96#380—24no—
97#380—21no—
98#380D412yes—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4C2312yes—
105C4×C4⋊C4C2312yes—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109#376—21no—
110C43—21no—
111#340—28no—
112#340—24no—
113#340—24no—
114#333—28no—
115C22×Q8—22no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—48no—
120C22×Q8—48no—
121C22×Q8—41no—
122C22×Q8—42no—
123C22×Q8C22⋊C418yes—
124C22×Q8C2×D414yes—
125C23×C4—22no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×Q16—232no—
131C2×Q16—432no—
132C2×Q16—464no—
133C2×M16—42no—
134C2×M16—42no—
135C2×M16—42no—
136C22×C8—28no—
137C22×C8—24no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8C22⋊C418yes—
158C4⋊Q8C22×C414yes—
159C4⋊Q8C22×C414yes—
160C4⋊Q8C2×D414yes—
161C4×Q8—88no—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—42no—
166#117—42no—
167#117Q1614yescomplemented
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—28no—
199C2×C42C22×C412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42C2411yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—42no—
212C22⋊C8—42no—
213C4×C8—24no—
214C4×C8—22no—
215C4×C8—42no—
216C4×C8C22×C414yes—
217#95—28no—
218#95—28no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223C24#12011yescharacteristic
224C2×Q8—24no—
225C2×Q8—28no—
226C2×Q8—24no—
227C2×Q8—48no—
228C2×Q8—416no—
229C2×Q8—48no—
230C2×Q8—416no—
231C2×Q8—42no—
232C2×Q8—44no—
233C2×Q8—48no—
234C2×Q8—88no—
235C2×Q8—88no—
236C2×Q8—84no—
237C2×Q8C2×C22⋊C418yes—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—28no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4C2×C22⋊C412yes—
250C22×C4C4×D412yes—
251C22×C4C4×D412yes—
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12112yes—
256C22×C4#12112yes—
257C22×C4#12312yes—
258C22×C4C2×Q1612yes—
259C22×C4#13512yes—
260C22×C4#13712yes—
261C22×C4#13712yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265Q16—464no—
266M16—84no—
267C2×C8—28no—
268C2×C8—216no—
269C2×C8—24no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—44no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—28no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—88no—
294C4⋊C4C2×Q1614yescomplemented
295C22⋊C4—24no—
296C22⋊C4—42no—
297C22⋊C4—44no—
298C42—24no—
299C42—42no—
300C42—42no—
301C42—44no—
302C42—42no—
303C42—44no—
304C42—88no—
305C42C2×C22⋊C412yes—
306C42C4×D412yes—
307C42C4×D412yes—
308C42#12112yes—
309C42#12112yes—
310C42#12312yes—
311C42C23×C411yescharacteristic
312C42C22×D411yescharacteristic
313C42C2×#4111yescharacteristic
314C42C2×#4111yescharacteristic
315C23—22no—
316C23—22no—
317C23#38612yes—
318C23#45112yes—
319C23#45212yes—
320C23C2×#12011yescharacteristic
321Q8—48no—
322Q8—48no—
323Q8—416no—
324Q8—84no—
325Q8—84no—
326Q8—88no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—48no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—88no—
346C2×C4—88no—
347C2×C4—84no—
348C2×C4#38612yes—
349C2×C4#38612yes—
350C2×C4#38612yes—
351C2×C4C4×Q1614yes—
352C2×C4#44114yes—
353C2×C4#45112yes—
354C2×C4#45212yes—
355C2×C4#45314yes—
356C2×C4#46214yes—
357C2×C4#46414yes—
358C2×C4C22×C22⋊C411yescharacteristic
359C2×C4C2×C4×D411yescharacteristic
360C2×C4C2×C4×D411yescharacteristic
361C2×C4C2×#12111yescharacteristic
362C2×C4C2×#12111yescharacteristic
363C2×C4C2×#12311yescharacteristic
364C2×C4C22×Q1611yescharacteristic
365C2×C4C2×#13511yescharacteristic
366C2×C4C2×#13711yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4#57814yes—
369C8—28no—
370C8—82no—
371C8—82no—
372C22—24no—
373C22—24no—
374C22—22no—
375C22—22no—
376C22#187814yes—
377C22C2×#38611yescharacteristic
378C22C2×#45111yescharacteristic
379C22C2×#45211yescharacteristic
380C22#301112yes—
381C22C2×#45312yes—
382C4—22no—
383C4—21no—
384C4—21no—
385C4—44no—
386C4—48no—
387C4—82no—
388C4—84no—
389C4—84no—
390C4#187814yes—
391C4#188714yes—
392C4C2×C4×Q1611yescharacteristic
393C4C2×#44111yescharacteristic
394C4#295012yes—
395C4C2×#46211yescharacteristic
396C4C2×#46411yescharacteristic
397C4#304412yes—
398C2—24no—
399C2#1862312yes—
400C2#1862712yes—
401C2C2×#187811yescharacteristic
402C2#2093511yescharacteristic
403C2C2×#188711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #9066C2semidirect
14. C2×#494C4semidirect
167. #117Q16semidirect
294. C4⋊C4C2×Q16semidirect

Maximal subgroups

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

subgroupindex
C22×#4942
#342162
#218342
#213132
C2×#15482
#90662
#78972
#78902