Finite Group Databasebeta

← #480217

Subgroups — #480217

Order 512.

804 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2990C211yesmaximal, characteristic
2#33488C212yesmaximal, complemented
3#11495C212yesmaximal, complemented
4#11467C214yesmaximal, complemented
5#11460C211yesmaximal, characteristic, complemented
6#11459C211yesmaximal, characteristic, complemented
7#11440C212yesmaximal, complemented
8C2×#1322C212yesmaximal, complemented
9C2×C4×#41C411yescharacteristic
10#2990—22no—
11#2990—24no—
12#2990—22no—
13C2×#446C411yescharacteristic
14C2×#446C2211yescharacteristic
15C2×#435C412yes—
16C2×#435C2212yes—
17C2×C8×D4C411yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#432C412yes—
20C2×#431C2211yescharacteristic
21C2×#423C411yescharacteristic
22C22×C4⋊C8C2212yes—
23#2169—22no—
24#2168—24no—
25D4×C16—22no—
26#2163C2212yes—
27#2162C2212yes—
28#2112C2214yes—
29C2×C22⋊C16C2214yes—
30#1322C2211yescharacteristic
31#1322C2212yes—
32#1322C2214yes—
33#1322C2211yescharacteristic
34#1322C2214yes—
35#1314C2212yes—
36#1314C2212yes—
37C22×#41C811yescharacteristic
38C22×M16C2×C412yes—
39C23×C8C2×C412yes—
40C23×C8C2312yes—
41C4×#41—22no—
42C4×#41—21no—
43C4×#41—21no—
44C4×#41—21no—
45C4×#41—22no—
46C4×#41—21no—
47C2×C4×Q8C811yescharacteristic
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×D4C811yescharacteristic
50C2×C4×D4C812yes—
51C2×C4×D4C2×C412yes—
52C2×C4×D4C2×C411yescharacteristic
53C2×#117C811yescharacteristic
54C2×#117C812yes—
55C22×C42C2×C412yes—
56C22×C42C2×C411yescharacteristic
57C2×M32—24no—
58C22×C16—24no—
59#446—22no—
60#446—22no—
61#446—22no—
62#446—22no—
63#435—24no—
64#435—24no—
65#435—24no—
66#435—24no—
67C8×D4—22no—
68C8×D4—21no—
69C8×D4—21no—
70C8×D4D412yes—
71C8×D4D414yes—
72C8×D4D412yes—
73#432—24no—
74#432—24no—
75#431—22no—
76#431—22no—
77#423—22no—
78#423—22no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8C2×C412yes—
82C2×C4⋊C8C2×C412yes—
83C2×C4⋊C8C2×C411yescharacteristic
84C2×C4⋊C8C2×C411yescharacteristic
85C2×C4⋊C8C2×C411yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2312yes—
88C2×C22⋊C8C2×C411yescharacteristic
89C2×C22⋊C8C2×C412yes—
90C2×C22⋊C8C2×C412yes—
91C2×C22⋊C8C2311yescharacteristic
92C2×#97C2×C412yes—
93C2×#97C2×C412yes—
94C2×C4×C8C2×C411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C4⋊C16—22no—
97C4⋊C16—22no—
98C22⋊C16—24no—
99C22⋊C16—24no—
100C22⋊C16C2312yes—
101C22⋊C16C2314yes—
102C22⋊C16C2312yes—
103#346—22no—
104C4×C16—22no—
105C2×#41—21no—
106C2×#41—22no—
107C2×#41—21no—
108C2×#41—44no—
109C22×Q8C2×C811yescharacteristic
110C22×D4C2×C811yescharacteristic
111C22×D4C2×C812yes—
112C23×C4C2×C812yes—
113C23×C4C2×C811yescharacteristic
114C23×C4C2×C812yes—
115C23×C4C2×C811yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C412yes—
118C2×M16—24no—
119C2×M16—82no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—82no—
123C22×C8—82no—
124C22×C8C22×C411yescharacteristic
125C22×C8C22×C412yes—
126C22×C8C22×C411yescharacteristic
127C22×C8C22×C412yes—
128C22×C8C22×C411yescharacteristic
129C22×C8C2×D414yes—
130C22×C8C2×D414yes—
131C22×C8C2411yescharacteristic
132C4×Q8—22no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—21no—
138C4×Q8—21no—
139C4×D4—22no—
140C4×D4—21no—
141C4×D4—21no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4C22⋊C412yes—
151C4×D4C22⋊C411yescharacteristic
152C4×D4C22⋊C411yescharacteristic
153C4×D4M1612yes—
154C4×D4M1611yescharacteristic
155C4×D4M1611yescharacteristic
156#117—22no—
157#117—21no—
158#117—21no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—22no—
163C2×C4⋊C4C2×C811yescharacteristic
164C2×C4⋊C4C2×C812yes—
165C2×C4⋊C4C2×C812yes—
166C2×C4⋊C4C2×C811yescharacteristic
167C2×C22⋊C4C2×C812yes—
168C2×C22⋊C4C2×C812yes—
169C2×C22⋊C4C2×C812yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C2×C812yes—
179C2×C42C2×C811yescharacteristic
180C2×C42C2×C811yescharacteristic
181C2×C42C2×C812yes—
182C2×C42C22×C412yes—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185M32—84no—
186C2×C16—22no—
187C2×C16—22no—
188C2×C16—24no—
189C2×C16—84no—
190C4⋊C8—24no—
191C4⋊C8—24no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—24no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8C22⋊C412yes—
198C4⋊C8C22⋊C412yes—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8C22⋊C414yes—
204C22⋊C8C22⋊C414yes—
205#97—24no—
206#97—24no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8C2×D412yes—
210C4×C8C2×D412yes—
211C24C22×C812yes—
212C24C22×C811yescharacteristic
213#41—48no—
214#41—44no—
215#41—44no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—42no—
219C2×D4—24no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—44no—
223C2×D4C22⋊C812yes—
224C2×D4C22⋊C811yescharacteristic
225C2×D4C22⋊C811yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—44no—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C22×C812yes—
249C22×C4C22×C811yescharacteristic
250C22×C4C22×C811yescharacteristic
251C22×C4C22×C811yescharacteristic
252C22×C4C22×C811yescharacteristic
253C22×C4C22×C811yescharacteristic
254C22×C4C22×C812yes—
255C22×C4C22×C812yes—
256C22×C4C2×M1612yes—
257C22×C4C2×M1612yes—
258C22×C4C23×C411yescharacteristic
259M16—84no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—22no—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—82no—
272C2×C8—82no—
273C2×C8C2×C22⋊C412yes—
274C2×C8C2×C22⋊C414yes—
275C2×C8C22×D412yes—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4C22⋊C812yes—
283C4⋊C4C22⋊C811yescharacteristic
284C4⋊C4C22⋊C811yescharacteristic
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4C22⋊C812yes—
290C22⋊C4C22⋊C812yes—
291C22⋊C4C22⋊C812yes—
292C22⋊C4C22⋊C812yes—
293C42—22no—
294C42—21no—
295C42—21no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—22no—
304C42—22no—
305C42C2×C22⋊C411yescharacteristic
306C42C2×C22⋊C411yescharacteristic
307C42C2×M1611yescharacteristic
308C42C2×M1611yescharacteristic
309C16—42no—
310C16—44no—
311C16—42no—
312C23—24no—
313C23—22no—
314C23—22no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—41no—
320C23—41no—
321C23C2×C22⋊C812yes—
322C23C2×C22⋊C812yes—
323C23C23×C811yescharacteristic
324Q8—42no—
325Q8—42no—
326D4—24no—
327D4—22no—
328D4—44no—
329D4—44no—
330D4#34914yes—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4C2×C22⋊C812yes—
366C2×C4C2×C22⋊C812yes—
367C2×C4C2×C22⋊C811yescharacteristic
368C2×C4C2×C22⋊C811yescharacteristic
369C2×C4C2×C22⋊C811yescharacteristic
370C2×C4C2×C22⋊C811yescharacteristic
371C2×C4#50412yes—
372C2×C4C22×C22⋊C411yescharacteristic
373C2×C4C22×M1611yescharacteristic
374C8—22no—
375C8—24no—
376C8—22no—
377C8—41no—
378C8—42no—
379C8—41no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—44no—
388C22—44no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—81no—
393C22—81no—
394C22C2×#34914yes—
395C22C22×C22⋊C811yescharacteristic
396C22C2×#50411yescharacteristic
397C22#340711yescharacteristic
398C4—22no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—22no—
404C4—22no—
405C4—42no—
406C4—41no—
407C4—42no—
408C4—41no—
409C4C2×#34912yes—
410C4#288311yescharacteristic
411C4#288911yescharacteristic
412C2—21no—
413C2—21no—
414C2—24no—
415C2—42no—
416C2—81no—
417C2#1735111yescharacteristic
418C2#3333411yescharacteristic
419C2C22×#34911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC22×C8C24
socleC22C22×C22⋊C8
prime radicalC22×#41C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33488C2semidirect
3. #11495C2semidirect
4. #11467C2semidirect
5. #11460C2semidirect
6. #11459C2semidirect
7. #11440C2semidirect
8. C2×#1322C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29902
#334882
#114952
#114672
#114602
#114592
#114402
C2×#13222