Finite Group Databasebeta

← #242942

Subgroups — #242942

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21979C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic
3#17734C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5#10630C214yesmaximal, complemented
6#10628C214yesmaximal, complemented
7#8080C211yesmaximal, characteristic
8#8078C211yesmaximal, characteristic
9#7858C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2283C2211yescharacteristic
17#1995C2214yes—
18#1986—22no—
19#1940—22no—
20#1851—22no—
21#1843C2212yes—
22#1832C2212yes—
23#1826—22no—
24#1559—22no—
25#1558—21no—
26#1556—21no—
27#1550—22no—
28#1549—21no—
29#1548—21no—
30#1542C2214yes—
31#1542C2212yes—
32#1542C2212yes—
33#1301C2212yes—
34#1301C2214yes—
35#1301C2212yes—
36#1274C2214yes—
37C22×M16—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533D412yes—
45#533C2312yes—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431C2312yes—
70#431C2312yes—
71#423—21no—
72#423—21no—
73#423—21no—
74#423—21no—
75#423—21no—
76#423C2312yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2311yescharacteristic
83#407—21no—
84#407—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C4×M16—21no—
88C4×M16—21no—
89C2×#97—21no—
90C2×C4×C8—21no—
91#383—21no—
92#382—21no—
93#381—21no—
94#380—21no—
95C4×C4⋊C4D412yes—
96C4×C22⋊C4D412yes—
97#376C2312yes—
98#339—28no—
99#336—24no—
100#333—24no—
101#333—22no—
102#333—22no—
103#332—24no—
104#332—22no—
105#332—22no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C2×M16—22no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—82no—
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—41no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C2×D412yes—
127C4⋊Q8C2×D412yes—
128#125—22no—
129#125—22no—
130#125—41no—
131#125—41no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148#117—22no—
149#117—22no—
150#117—41no—
151#117—41no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—41no—
161C2×C42—41no—
162C2×C42C2×D414yes—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42C2411yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—84no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198#97—41no—
199#97—41no—
200#97—41no—
201#97—41no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—41no—
205C4×C8—41no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210C24#12311yescharacteristic
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—82no—
216C2×Q8—82no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12611yescharacteristic
236C22×C4#12611yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12611yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C2×#4111yescharacteristic
241C22×C4C2×#4111yescharacteristic
242M16—42no—
243M16—42no—
244M16—82no—
245M16—82no—
246M16—82no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—82no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—82no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—82no—
273C42—24no—
274C42—24no—
275C42—42no—
276C42—42no—
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12311yescharacteristic
280C42#12311yescharacteristic
281C42#12311yescharacteristic
282C42#12311yescharacteristic
283C42#12411yescharacteristic
284C42#12411yescharacteristic
285C42#12611yescharacteristic
286C42#12611yescharacteristic
287C42#12611yescharacteristic
288C42#12611yescharacteristic
289C42C22×D411yescharacteristic
290C42C2×#4111yescharacteristic
291C42C2×#4111yescharacteristic
292C42C2×#4111yescharacteristic
293C23—22no—
294C23—41no—
295C23—41no—
296C23#39711yescharacteristic
297C23#39711yescharacteristic
298C23C2×#12311yescharacteristic
299Q8—82no—
300Q8—82no—
301Q8—82no—
302Q8—82no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#39711yescharacteristic
330C2×C4#39711yescharacteristic
331C2×C4#39711yescharacteristic
332C2×C4#39711yescharacteristic
333C2×C4#39711yescharacteristic
334C2×C4#39711yescharacteristic
335C2×C4C2×#12111yescharacteristic
336C2×C4C2×#12311yescharacteristic
337C2×C4C2×#12311yescharacteristic
338C2×C4C2×#12411yescharacteristic
339C2×C4C2×#12611yescharacteristic
340C2×C4C2×#12611yescharacteristic
341C2×C4#57712yes—
342C2×C4#57812yes—
343C8—82no—
344C8—82no—
345C8—82no—
346C8—81no—
347C8—82no—
348C8—81no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—82no—
358C22C2×#39711yescharacteristic
359C22#309712yes—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—84no—
371C4—82no—
372C4—82no—
373C4#304311yescharacteristic
374C4#304411yescharacteristic
375C4#313712yes—
376C2—22no—
377C2—41no—
378C2—41no—
379C2#2363311yescharacteristic
380C2#2364911yescharacteristic
381C2#2364911yescharacteristic
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22C2×#397
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10630C2semidirect
6. #10628C2semidirect

Maximal subgroups

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

subgroupindex
#219792
#218452
#177342
#177082
#106302
#106282
#80802
#80782
#78582