Finite Group Databasebeta

← #499388

Subgroups — #499388

Order 512.

694 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2155C211yesmaximal, characteristic, complemented
2C2×#2115C211yesmaximal, characteristic
3#20668C211yesmaximal, characteristic
4#20663C211yesmaximal, characteristic
5#13501C218yesmaximal, complemented
6C2×#1387C211yesmaximal, characteristic, complemented
7C2×#1387C211yesmaximal, characteristic, complemented
8#12563C211yesmaximal, characteristic, complemented
9C22×#131C2211yescharacteristic
10C22×M32C2211yescharacteristic
11C22×M32C2211yescharacteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#427C2211yescharacteristic
15C22×#107C2211yescharacteristic
16#2879—21no—
17#2311—21no—
18#2155—24no—
19C2×#365C2214yes—
20#2115—24no—
21C2×#349C2214yes—
22#1811C2214yes—
23#1810—22no—
24#1809C2214yes—
25#1808—22no—
26#1738—21no—
27#1736—21no—
28#1387C2214yes—
29#1387C2214yes—
30#1383C2214yes—
31C22×#41D411yescharacteristic
32C2×#131—22no—
33C2×#131—22no—
34C2×#131—21no—
35C2×#131—21no—
36C2×#131—22no—
37C2×#131—21no—
38C2×#131—21no—
39C2×#131—22no—
40C22×M16—21no—
41C22×M16—21no—
42C22×M16D411yescharacteristic
43C22×M16C2312yes—
44C23×C8D412yes—
45C23×C8C2311yescharacteristic
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C2×M32—22no—
49C2×M32—21no—
50C2×M32—21no—
51C2×M32—22no—
52C2×M32—22no—
53C2×M32D414yes—
54C2×M32C2312yes—
55C2×M32C2312yes—
56C22×C16—21no—
57C22×C16—21no—
58C2×#108—22no—
59C2×#108—22no—
60C2×#108—22no—
61C2×#108—22no—
62C2×#108—21no—
63C2×#108—21no—
64C2×#108C2312yes—
65C2×#108C2312yes—
66#427—22no—
67#427—22no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107C2312yes—
75C2×#106C2312yes—
76#408—42no—
77#408—42no—
78#407—24no—
79#391—41no—
80#390—41no—
81#386—42no—
82#380—24no—
83#365—28no—
84#349—28no—
85#341—24no—
86#340—24no—
87C2×#41—22no—
88C2×#41—21no—
89C2×#41—21no—
90C2×#41—22no—
91C2×#41—22no—
92C2×#41—21no—
93C2×#41—21no—
94C2×#41—22no—
95C22×Q8SD1611yescharacteristic
96C22×D4SD1612yes—
97C22×D4SD1611yescharacteristic
98C23×C4—41no—
99C23×C4C2×D412yes—
100C23×C4C2×D411yescharacteristic
101#131—42no—
102#131—42no—
103#131—42no—
104#131—42no—
105#131—42no—
106#131—42no—
107#131—42no—
108#131—42no—
109C2×M16—22no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—42no—
124C2×M16—42no—
125C2×M16—42no—
126C2×M16—42no—
127C22×C8—24no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—24no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8C2×D412yes—
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D412yes—
136C22×C8#4112yes—
137C22×C8#4111yescharacteristic
138C22×C8#4111yescharacteristic
139C22×C8#4112yes—
140C22×C8#4111yescharacteristic
141C22×C8#4111yescharacteristic
142C22×C8#4112yes—
143C22×C8C2411yescharacteristic
144#117—42no—
145#117—42no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—44no—
155C2×C42—41no—
156M32—24no—
157M32—22no—
158M32—22no—
159M32—22no—
160M32—42no—
161M32C2×D414yes—
162C2×C16—24no—
163C2×C16—22no—
164C2×C16—22no—
165C2×C16—42no—
166#108—22no—
167#108—22no—
168#108—22no—
169#108—22no—
170#108—42no—
171#108—42no—
172#107—22no—
173#107—22no—
174#107—42no—
175#107—41no—
176#107—41no—
177#106—22no—
178#106—22no—
179C22⋊C8—48no—
180#95—44no—
181C24C2×SD1612yes—
182C24C2×SD1611yescharacteristic
183#41—42no—
184#41—42no—
185#41—42no—
186#41—42no—
187#41—42no—
188#41—42no—
189#41—42no—
190#41—42no—
191C2×Q8—22no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—24no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—162no—
214C22×C4#12112yes—
215C22×C4#12311yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4#12312yes—
218C22×C4C2×SD1612yes—
219C22×C4C2×SD1611yescharacteristic
220C22×C4#13512yes—
221C22×C4AGL(1,ℤ/8)11yescharacteristic
222C22×C4AGL(1,ℤ/8)11yescharacteristic
223C22×C4#13712yes—
224C22×C4C22×D411yescharacteristic
225M16—42no—
226M16—42no—
227M16—42no—
228M16—42no—
229M16—42no—
230M16—42no—
231M16—42no—
232M16—42no—
233M16—42no—
234M16—42no—
235M16—82no—
236M16—82no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8#12112yes—
247C2×C8#12312yes—
248C2×C8#12312yes—
249C2×C8#12311yescharacteristic
250C2×C8#12311yescharacteristic
251C2×C8#12312yes—
252C2×C8#12412yes—
253C2×C8#12412yes—
254C2×C8#12612yes—
255C2×C8#12612yes—
256C2×C8#12612yes—
257C2×C8#12612yes—
258C2×C8C22×D411yescharacteristic
259C2×C8C2×#4111yescharacteristic
260C2×C8C2×#4111yescharacteristic
261C2×C8C2×#4111yescharacteristic
262C2×C8C2×#4111yescharacteristic
263C2×C8C2×#4111yescharacteristic
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C22⋊C4—82no—
271C22⋊C4—164no—
272C42—82no—
273C16—24no—
274C16—42no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23#46512yes—
283C23#48111yescharacteristic
284C23#48111yescharacteristic
285C23#48312yes—
286C23C22×SD1611yescharacteristic
287Q8—42no—
288Q8—42no—
289D4—44no—
290D4—44no—
291D4—42no—
292D4—42no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—84no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—162no—
306C2×C4—162no—
307C2×C4#39712yes—
308C2×C4#39712yes—
309C2×C4#46512yes—
310C2×C4#46812yes—
311C2×C4#46812yes—
312C2×C4#48111yescharacteristic
313C2×C4#48111yescharacteristic
314C2×C4#48212yes—
315C2×C4#48212yes—
316C2×C4#48312yes—
317C2×C4#48512yes—
318C2×C4#48512yes—
319C2×C4C2×#12111yescharacteristic
320C2×C4C2×#12311yescharacteristic
321C2×C4C2×#12311yescharacteristic
322C2×C4C2×#13511yescharacteristic
323C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
324C2×C4C2×#13711yescharacteristic
325C8—42no—
326C8—82no—
327C8—82no—
328C8#39712yes—
329C8#39712yes—
330C8C2×#12311yescharacteristic
331C8C2×#12411yescharacteristic
332C8C2×#12611yescharacteristic
333C8C2×#12611yescharacteristic
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—44no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—82no—
347C22#207712yes—
348C22#207712yes—
349C22C2×#46511yescharacteristic
350C22C2×#48311yescharacteristic
351C22C2×#48111yescharacteristic
352C4—42no—
353C4—162no—
354C4—162no—
355C4#207712yes—
356C4#207712yes—
357C4C2×#39711yescharacteristic
358C4C2×#46811yescharacteristic
359C4C2×#48211yescharacteristic
360C4C2×#48511yescharacteristic
361C2—21no—
362C2—21no—
363C2—22no—
364C2—22no—
365C2—82no—
366C2C2×#207711yescharacteristic
367C2#3365711yescharacteristic
368C2#3365811yescharacteristic
subgroupG/H
center, socleC22C2×#481
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×#41D4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2155C2semidirect
5. #13501C2semidirect
6. C2×#1387C2semidirect
7. C2×#1387C2semidirect
8. #12563C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21552
C2×#21152
#206682
#206632
#135012
C2×#13872
#125632