Finite Group Databasebeta

← #244111

Subgroups — #244111

Order 512.

461 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#21954C211yesmaximal, characteristic
3#10689C212yesmaximal, complemented
4#10689C212yesmaximal, complemented
5#10688C212yesmaximal, complemented
6#10687C212yesmaximal, complemented
7#8095C211yesmaximal, characteristic
8#8094C211yesmaximal, characteristic
9#8081C211yesmaximal, characteristic
10#8076C211yesmaximal, characteristic
11#7863C211yesmaximal, characteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18#2285C2211yescharacteristic
19#1950—21no—
20#1950—21no—
21#1940C2212yes—
22#1940C2212yes—
23#1939—21no—
24#1939—21no—
25#1855—22no—
26#1849C2212yes—
27#1838C2212yes—
28#1798—22no—
29#1303C2212yes—
30#1303C2212yes—
31#1303C2212yes—
32#1303C2212yes—
33#1301C2211yescharacteristic
34#1301C2211yescharacteristic
35#1301C2211yescharacteristic
36#1301C2211yescharacteristic
37#1301C2212yes—
38#1301C2212yes—
39#1275C2212yes—
40#1275C2212yes—
41C22×M16—21no—
42C2×#117—21no—
43C22×C42C2311yescharacteristic
44#432—21no—
45#432—21no—
46#432—21no—
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2311yescharacteristic
63#431C2311yescharacteristic
64#431C2312yes—
65C2×#108—22no—
66C2×#108—22no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423C2311yescharacteristic
72#423C2312yes—
73#423C2311yescharacteristic
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8D411yescharacteristic
80C2×C4⋊C8D411yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83#407—21no—
84#407—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8D412yes—
88C4×M16—21no—
89C4×M16—21no—
90C2×#97C2312yes—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93#384—21no—
94#382Q812yes—
95#380—21no—
96#380Q812yes—
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2312yes—
99#341—22no—
100#341—22no—
101#336—22no—
102#336—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×Q811yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—21no—
108C2×M16—21no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—81no—
112C2×M16—81no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8C2×D412yes—
118C22×C8C2×D412yes—
119#117—81no—
120#117—81no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C42—22no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×Q811yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42#4111yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4111yescharacteristic
134C2×C42#4111yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42#4111yescharacteristic
137C2×C42C2411yescharacteristic
138#108—82no—
139#108—82no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—22no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161C4⋊C8—82no—
162C4⋊C8—82no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—21no—
166C22⋊C8—21no—
167C22⋊C8—22no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174#97—22no—
175#97—22no—
176#97—41no—
177#97—41no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—22no—
181C4×C8—41no—
182C4×C8—41no—
183C4×C8—41no—
184C4×C8—41no—
185#95—22no—
186#95C2×Q814yes—
187C24C22⋊Q811yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—24no—
193C22×C4—42no—
194C22×C4—81no—
195C22×C4C22⋊Q811yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4#12311yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12411yescharacteristic
201C22×C4#12411yescharacteristic
202C22×C4#12511yescharacteristic
203C22×C4#12511yescharacteristic
204C22×C4#12511yescharacteristic
205C22×C4#12511yescharacteristic
206C22×C4#12611yescharacteristic
207C22×C4#12611yescharacteristic
208C22×C4#12611yescharacteristic
209C22×C4#12611yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C22×Q811yescharacteristic
212C22×C4C2×#4111yescharacteristic
213M16—41no—
214M16—42no—
215M16—41no—
216M16—42no—
217M16—41no—
218M16—41no—
219M16—82no—
220M16—82no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—82no—
240C2×C8—82no—
241C4⋊C4—44no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C22⋊C4—82no—
245C42—42no—
246C42—42no—
247C42#12311yescharacteristic
248C42#12311yescharacteristic
249C42#12411yescharacteristic
250C42#12411yescharacteristic
251C42#12511yescharacteristic
252C42#12511yescharacteristic
253C42#12511yescharacteristic
254C42#12511yescharacteristic
255C42#12611yescharacteristic
256C42#12611yescharacteristic
257C42#12611yescharacteristic
258C42#12611yescharacteristic
259C42C2×#4111yescharacteristic
260C42C2×#4111yescharacteristic
261C42C2×#4111yescharacteristic
262C42C2×#4111yescharacteristic
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23#40011yescharacteristic
268C23#40011yescharacteristic
269C23C2×C22⋊Q811yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4#40011yescharacteristic
296C2×C4#40011yescharacteristic
297C2×C4#40011yescharacteristic
298C2×C4#40011yescharacteristic
299C2×C4#40011yescharacteristic
300C2×C4#40011yescharacteristic
301C2×C4C2×#12311yescharacteristic
302C2×C4C2×#12411yescharacteristic
303C2×C4C2×#12511yescharacteristic
304C2×C4C2×#12511yescharacteristic
305C2×C4C2×#12611yescharacteristic
306C2×C4C2×#12611yescharacteristic
307C2×C4#57611yescharacteristic
308C2×C4#57711yescharacteristic
309C2×C4#57711yescharacteristic
310C2×C4#57811yescharacteristic
311C8—42no—
312C8—42no—
313C8—81no—
314C8—81no—
315C8—82no—
316C8—81no—
317C8—81no—
318C8—82no—
319C8—81no—
320C8—81no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—81no—
330C22—81no—
331C22C2×#40011yescharacteristic
332C22#308111yescharacteristic
333C22#308211yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—84no—
344C4#210312yes—
345C4#309811yescharacteristic
346C4#309911yescharacteristic
347C4#313611yescharacteristic
348C4#313711yescharacteristic
349C2—21no—
350C2—21no—
351C2—41no—
352C2—41no—
353C2#2374911yescharacteristic
354C2#2375111yescharacteristic
355C2C2×#210311yescharacteristic
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22C2×#400
prime radicalC24C22⋊Q8
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
3. #10689C2semidirect
4. #10689C2semidirect
5. #10688C2semidirect
6. #10687C2semidirect

Maximal subgroups

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

subgroupindex
#219942
#219542
#106892
#106882
#106872
#80952
#80942
#80812
#80762
#78632