Finite Group Databasebeta

← #6392790

Subgroups — #6392790

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2C2×#2983C211yesmaximal, characteristic
3C2×Q8×M16C211yesmaximal, characteristic
4#22747C218yesmaximal, complemented
5#22733C218yesmaximal, complemented
6#21994C214yesmaximal
7#21993C212yesmaximal
8#21371C214yesmaximal
9#21370C212yesmaximal
10C22×C4×Q8C2211yescharacteristic
11#2984—24no—
12#2983—21no—
13#2983—22no—
14#2983C412yes—
15#2983C412yes—
16#2983C414yes—
17#2983C418yes—
18#2982—21no—
19Q8×M16—24no—
20Q8×M16C2218yes—
21Q8×M16C2214yes—
22Q8×M16C2214yes—
23C2×#446—21no—
24C2×#446—22no—
25C2×#446C2218yes—
26C2×C8×Q8—21no—
27#2920C2212yes—
28#2920C2214yes—
29C2×#432C2214yes—
30C2×#431C2212yes—
31C2×#423C2211yescharacteristic
32C2×#423C2212yes—
33C2×C4×M16C2211yescharacteristic
34C2×C4×M16C2212yes—
35#1950C2214yes—
36#1950C2214yes—
37#1950C2218yes—
38#1950C2218yes—
39#1950C2218yes—
40#1950C22116yes—
41#1855C2214yes—
42#1855C2214yes—
43#1855C2218yes—
44#1849C2218yes—
45#1848C2214yes—
46#1838C2218yes—
47#1837C2214yes—
48#1778C22116yes—
49C4×#108C2218yes—
50C23×Q8Q811yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2312yes—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2×C414yes—
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D412yes—
61C2×C4×Q8D414yes—
62C2×C4×Q8Q814yes—
63C2×C4×Q8C2314yes—
64C22×C4⋊C4Q811yescharacteristic
65C22×C4⋊C4Q812yes—
66C22×C42C2311yescharacteristic
67C22×C42C2312yes—
68#446—28no—
69#446—216no—
70#446—24no—
71#446—24no—
72#446—28no—
73#446—28no—
74#446C2×C4116yes—
75#446C2×C4116yes—
76#446C2×C4132yes—
77C8×Q8—28no—
78C8×Q8—24no—
79C8×Q8—24no—
80#432—24no—
81#432—22no—
82#432—22no—
83#432—24no—
84#432C2×C418yes—
85#432C2×C418yes—
86#432C2×C418yes—
87#432C2×C418yes—
88#431—22no—
89#431—24no—
90#431—22no—
91#431—24no—
92#431C2×C414yes—
93#431C2×C414yes—
94#431C2×C414yes—
95#431C2×C414yes—
96C2×#108C2318yes—
97C2×#108C23116yes—
98C2×#108C2318yes—
99#423—22no—
100#423—24no—
101#423C2312yes—
102#423C2314yes—
103#423C2314yes—
104#423C2318yes—
105#423C2312yes—
106#423C2314yes—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8C2314yes—
112C2×C4⋊C8C2318yes—
113#407C2314yes—
114C2×C22⋊C8C2314yes—
115C4×M16—22no—
116C4×M16—24no—
117C4×M16C2314yes—
118C4×M16C2318yes—
119C4×M16C2314yes—
120C4×M16C2318yes—
121C2×#97—22no—
122C2×#97—21no—
123C2×#97C2318yes—
124C2×C4×C8—21no—
125C2×C4×C8—22no—
126C2×C4×C8C2314yes—
127#341C23116yes—
128#341C2318yes—
129#341C2318yes—
130C22×Q8—44no—
131C22×Q8C4⋊C411yescharacteristic
132C22×Q8C4⋊C411yescharacteristic
133C22×Q8C4⋊C412yes—
134C22×Q8C4⋊C414yes—
135C22×Q8C2×Q814yes—
136C23×C4C2×Q811yescharacteristic
137C23×C4C2×Q812yes—
138C23×C4C2×Q811yescharacteristic
139C23×C4C2×Q812yes—
140C23×C4C2411yescharacteristic
141C2×M16—21no—
142C2×M16—24no—
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—42no—
146C2×M16—44no—
147C2×M16C2414yes—
148C2×M16C2412yes—
149C2×M16C2414yes—
150C2×M16C2414yes—
151C2×M16C2412yes—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8C2418yes—
156C4×Q8C4⋊C414yes—
157C4×Q8C4⋊C414yes—
158C4×Q8C4⋊C414yes—
159C4×Q8C4⋊C414yes—
160C4×Q8C4⋊C418yes—
161C4×Q8C4⋊C418yes—
162C4×Q8C22×C414yes—
163C4×Q8C22×C414yes—
164C4×Q8C22×C418yes—
165C4×Q8C2×D414yes—
166C4×Q8C2×D414yes—
167C4×Q8C2×D418yes—
168C2×C4⋊C4C4⋊C412yes—
169C2×C4⋊C4C4⋊C412yes—
170C2×C4⋊C4C4⋊C412yes—
171C2×C4⋊C4C4⋊C412yes—
172C2×C4⋊C4C4⋊C412yes—
173C2×C4⋊C4C4⋊C414yes—
174C2×C4⋊C4C4⋊C418yes—
175C2×C4⋊C4C4⋊C411yescharacteristic
176C2×C4⋊C4C4⋊C411yescharacteristic
177C2×C4⋊C4C2×Q814yes—
178C2×C4⋊C4C2×Q818yes—
179C2×C42C22×C412yes—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C22×C412yes—
184C2×C42C22×C414yes—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D414yes—
191C2×C42C2×Q812yes—
192C2×C42C2×Q814yes—
193C2×C42C2412yes—
194C2×C42C2414yes—
195#108—216no—
196#108—416no—
197#108—48no—
198C4⋊C8—24no—
199C4⋊C8—28no—
200C4⋊C8—28no—
201C4⋊C8—24no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—24no—
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8—24no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8C22×C418yes—
211C4⋊C8C22×C418yes—
212C4⋊C8C22×C4116yes—
213C4⋊C8C22×C4116yes—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8C22×C418yes—
217C22⋊C8C22×C418yes—
218#97—28no—
219#97—24no—
220#97—28no—
221#97—24no—
222#97C22×C4116yes—
223#97C22×C4116yes—
224C4×C8—24no—
225C4×C8—28no—
226C4×C8—24no—
227C4×C8—28no—
228C4×C8C22×C418yes—
229C4×C8C22×C418yes—
230C24C22×Q811yescharacteristic
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—44no—
234C2×Q8—48no—
235C2×Q8—48no—
236C2×Q8C2×C4⋊C414yes—
237C2×Q8C2×C4⋊C414yes—
238C2×Q8C2×C4⋊C418yes—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—44no—
243C22×C4—81no—
244C22×C4—82no—
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4C2×C4⋊C412yes—
247C22×C4C2×C4⋊C412yes—
248C22×C4C2×C4⋊C412yes—
249C22×C4C2×C4⋊C412yes—
250C22×C4C2×C4⋊C414yes—
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C2×C4⋊C412yes—
254C22×C4C2×C4⋊C412yes—
255C22×C4C2×C4⋊C414yes—
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4C23×C411yescharacteristic
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×Q811yescharacteristic
262C22×C4C22×Q812yes—
263C22×C4C22×Q812yes—
264C22×C4C22×Q814yes—
265C22×C4C22×Q814yes—
266C22×C4C2511yescharacteristic
267M16—24no—
268M16—22no—
269M16—28no—
270M16—28no—
271M16—28no—
272M16—24no—
273M16—42no—
274M16—44no—
275M162−1+414yes—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—28no—
279C2×C8—28no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—42no—
285C2×C8—44no—
286C2×C8C23×C4116yes—
287C4⋊C4C2×C4⋊C418yes—
288C4⋊C4C2×C4⋊C418yes—
289C4⋊C4C2×C4⋊C418yes—
290C4⋊C4C2×C4⋊C418yes—
291C4⋊C4C2×C4⋊C418yes—
292C4⋊C4C2×C4⋊C414yes—
293C4⋊C4C2×C4⋊C414yes—
294C42C2×C4⋊C412yes—
295C42C2×C4⋊C412yes—
296C42C2×C4⋊C414yes—
297C42C2×C4⋊C414yes—
298C42C2×C4⋊C414yes—
299C42C2×C4⋊C414yes—
300C42C2×C4⋊C412yes—
301C42C2×C4⋊C412yes—
302C42C23×C412yes—
303C42C23×C412yes—
304C42C23×C414yes—
305C42C23×C414yes—
306C42C22×D412yes—
307C42C22×D414yes—
308C42C22×D414yes—
309C42C22×D412yes—
310C23—22no—
311C23—22no—
312C23—41no—
313C23C22×C4⋊C411yescharacteristic
314C23C22×C4⋊C411yescharacteristic
315C23C23×Q811yescharacteristic
316Q8—28no—
317Q8—28no—
318Q8—48no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4#43014yes—
342C2×C4C22×C4⋊C411yescharacteristic
343C2×C4C22×C4⋊C411yescharacteristic
344C2×C4C22×C4⋊C412yes—
345C2×C4C22×C4⋊C412yes—
346C2×C4C22×C4⋊C414yes—
347C2×C4C22×C4⋊C414yes—
348C2×C4C22×C4⋊C412yes—
349C2×C4C22×C4⋊C414yes—
350C2×C4C22×C4⋊C414yes—
351C2×C4C22×C4⋊C412yes—
352C2×C4C24×C411yescharacteristic
353C2×C4C23×D411yescharacteristic
354C2×C4C2×2−1+412yes—
355C2×C4#58512yes—
356C8—216no—
357C8—24no—
358C8—44no—
359C8—42no—
360C8—44no—
361C8—44no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—44no—
370C22C2×#43012yes—
371C22C23×C4⋊C411yescharacteristic
372C22#350212yes—
373C4—21no—
374C4—21no—
375C4—22no—
376C4—24no—
377C4—24no—
378C4—22no—
379C4—22no—
380C4—24no—
381C4—24no—
382C4C2×#43014yes—
383C4C4×2−1+412yes—
384C4#348611yescharacteristic
385C4#348711yescharacteristic
386C2—22no—
387C2—22no—
388C2—41no—
389C2#3424911yescharacteristic
390C2C22×#43011yescharacteristic
391C2#6011311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C23×C4⋊C4
prime radicalC24C22×Q8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22747C2semidirect
5. #22733C2semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#29832
C2×Q8×M162
#227472
#227332
#219942
#219932
#213712
#213702