Finite Group Databasebeta

← #211908

Subgroups — #211908

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22130C212yesmaximal
2#19158C211yesmaximal, characteristic
3#17961C211yesmaximal, characteristic, complemented
4#17331C211yesmaximal, characteristic
5#9718C211yesmaximal, characteristic, complemented
6#9623C211yesmaximal, characteristic, complemented
7#9591C212yesmaximal, complemented
8#9207C212yesmaximal
9#8668C211yesmaximal, characteristic
10#8284C211yesmaximal, characteristic, complemented
11#8252C212yesmaximal, complemented
12#3170C2211yescharacteristic
13#3076C2212yes—
14#2965C2211yescharacteristic
15C8×#41C2211yescharacteristic
16#2960C2212yes—
17#2922C2211yescharacteristic
18#1986C2212yes—
19#1950C2212yes—
20#1945—22no—
21#1937—22no—
22#1929—22no—
23#1922—22no—
24#1806C2212yes—
25C4×#107C2211yescharacteristic
26C8×C4⋊C4C2212yes—
27#1704C412yescomplemented
28#1703C412yescomplemented
29#1696C411yescharacteristic, complemented
30#1695C411yescharacteristic, complemented
31#1690C411yescharacteristic, complemented
32#1689C411yescharacteristic, complemented
33#1593—21no—
34#1593—21no—
35#1579—21no—
36#1579—21no—
37#1566C2211yescharacteristic
38C2×#334C2211yescharacteristic
39#1542C2212yes—
40#1503C2212yes—
41#1493C2212yes—
42#1454—21no—
43#1453—21no—
44#1452—21no—
45#1451C2211yescharacteristic
46#1450C2211yescharacteristic
47C8×M16—21no—
48C8×M16C2211yescharacteristic
49C2×#322C2211yescharacteristic
50#1303C2212yes—
51#1301C2212yes—
52#1279C2212yes—
53#533C2312yes—
54C4×#41C2311yescharacteristic
55#500C2×C411yescharacteristic
56#494C2×C411yescharacteristic
57#487C2×C412yes—
58#479—22no—
59#478—22no—
60#477C2×C412yes—
61#476C2×C412yes—
62#475C2×C412yes—
63#474C2×C412yes—
64#467—21no—
65#467—21no—
66#446—22no—
67#446—22no—
68#446C2×C411yescharacteristic
69#446C2×C411yescharacteristic
70C8×Q8—21no—
71C8×Q8—22no—
72C8×Q8—22no—
73C4×Q16—21no—
74C4×Q16—21no—
75#436C2×C411yescharacteristic
76#436C2×C411yescharacteristic
77#435—21no—
78#435—21no—
79C8×D4—21no—
80C8×D4—22no—
81#433—22no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431C2311yescharacteristic
87#431C2311yescharacteristic
88C2×#108—22no—
89#427—22no—
90C2×#107—21no—
91#424—22no—
92#424D412yes—
93#423C2312yes—
94C2×C4⋊C8C2312yes—
95C2×#104—22no—
96#416—22no—
97#408D412yes—
98#405—22no—
99C4×M16—21no—
100C4×M16C2311yescharacteristic
101C2×C4×C8—21no—
102C2×C4×C8—22no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C4×C4⋊C4C2312yes—
107#341—22no—
108#340—21no—
109#339—22no—
110#339—22no—
111#339—22no—
112#336—22no—
113#334—21no—
114#334—21no—
115#334C2×C411yescharacteristic
116#334C2×C411yescharacteristic
117#334C2×C412yes—
118#333C2×C412yes—
119#333C2×C412yes—
120#332C2×C412yes—
121#332C2×C412yes—
122#328—24no—
123Q8⋊C8—22no—
124Q8⋊C8—22no—
125#322—21no—
126#322—21no—
127#322—21no—
128#322—21no—
129#322D411yescharacteristic
130#322D411yescharacteristic
131#322D411yescharacteristic
132#322D411yescharacteristic
133C82—21no—
134C82—21no—
135C2×#41C2×D411yescharacteristic
136C2×Q16—21no—
137C2×Q16—21no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—22no—
141C2×M16C2×D412yes—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8C2×D412yes—
146C22×C8#4111yescharacteristic
147C22×C8#4111yescharacteristic
148C4⋊Q8C22×C412yes—
149C4⋊Q8C22×C412yes—
150#125—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8C22×C411yescharacteristic
156C4×Q8#4111yescharacteristic
157C4×D4—21no—
158C4×D4—21no—
159C4×D4C22×C411yescharacteristic
160C4×D4#4111yescharacteristic
161#117—21no—
162#117—22no—
163#117C2×D411yescharacteristic
164C2×C4⋊C4—22no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42C2411yescharacteristic
168#108—82no—
169#107—24no—
170#107—21no—
171#107—21no—
172#107—22no—
173#107—22no—
174#106—22no—
175#106—22no—
176C4⋊C8—22no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—22no—
185C4⋊C8—42no—
186C4⋊C8—42no—
187C4⋊C8C22×C411yescharacteristic
188C4⋊C8C22×C411yescharacteristic
189C4⋊C8C22×C412yes—
190C4⋊C8C22×C412yes—
191#104—82no—
192#103—22no—
193#103—22no—
194#103—22no—
195#103—22no—
196#102—22no—
197#102—22no—
198C22⋊C8—22no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8#4112yes—
205#97—21no—
206#97—21no—
207#97—21no—
208#97—42no—
209C4×C8—22no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—42no—
215C4×C8—82no—
216C4×C8C22×C411yescharacteristic
217C4×C8C22×C411yescharacteristic
218C4×C8C2×D411yescharacteristic
219C4×C8C2×D411yescharacteristic
220C4×C8C2×D411yescharacteristic
221C4×C8C2×D411yescharacteristic
222C4×C8C2×D411yescharacteristic
223C4×C8C2×D411yescharacteristic
224C4×C8#4112yes—
225#95—22no—
226#41—22no—
227#41—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8C4×D411yescharacteristic
231C2×D4—21no—
232C2×D4C4×D411yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4C22×D411yescharacteristic
238C22×C4C2×#4111yescharacteristic
239C22×C4C2×#4111yescharacteristic
240Q16—42no—
241Q16—42no—
242M16—22no—
243M16—22no—
244M16—42no—
245M16—42no—
246M16—42no—
247M16—42no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—21no—
257C2×C8—22no—
258C2×C8—21no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—81no—
264C2×C8—82no—
265C2×C8C4×D411yescharacteristic
266C2×C8C4×D411yescharacteristic
267C2×C8C4×D411yescharacteristic
268C2×C8C4×D411yescharacteristic
269C2×C8#12112yes—
270C2×C8#12112yes—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4C4×D411yescharacteristic
282C4⋊C4C4×D411yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—42no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42C23×C411yescharacteristic
294C42C22×D411yescharacteristic
295C42C2×#4111yescharacteristic
296C42C2×#4111yescharacteristic
297C23—21no—
298C23#53211yescharacteristic
299Q8—22no—
300Q8—42no—
301Q8—42no—
302Q8—82no—
303D4—22no—
304D4—41no—
305D4—41no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4C2×C4×D411yescharacteristic
322C2×C4C2×C4×D411yescharacteristic
323C2×C4C4×#4111yescharacteristic
324C2×C4C2×#12111yescharacteristic
325C2×C4#52511yescharacteristic
326C2×C4#52911yescharacteristic
327C2×C4#57712yes—
328C2×C4#57812yes—
329C8—21no—
330C8—21no—
331C8—21no—
332C8—22no—
333C8—22no—
334C8—42no—
335C8—41no—
336C8—41no—
337C8—42no—
338C8—42no—
339C8—41no—
340C8—41no—
341C8—42no—
342C8—82no—
343C8#44312yes—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22C4×#12111yescharacteristic
350C22#317311yescharacteristic
351C22#317711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—84no—
359C4—82no—
360C4—82no—
361C4#294712yes—
362C4C2×#44311yescharacteristic
363C4#295511yescharacteristic
364C4#306312yes—
365C2—21no—
366C2—21no—
367C2—81no—
368C2#2151711yescharacteristic
369C2#2154211yescharacteristic
370C2#2155911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC2×#41C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17961C2semidirect
5. #9718C2semidirect
6. #9623C2semidirect
7. #9591C2semidirect
10. #8284C2semidirect
11. #8252C2semidirect
27. #1704C4semidirect
28. #1703C4semidirect
29. #1696C4semidirect
30. #1695C4semidirect
31. #1690C4semidirect
32. #1689C4semidirect

Maximal subgroups

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

subgroupindex
#221302
#191582
#179612
#173312
#97182
#96232
#95912
#92072
#86682
#82842
#82522