Finite Group Databasebeta

← #191739

Subgroups — #191739

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2925C211yesmaximal, characteristic
2#21373C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic
4C2×#1570C211yesmaximal, characteristic, complemented
5C2×#1567C211yesmaximal, characteristic, complemented
6#8834C214yesmaximal, complemented
7#8831C214yesmaximal, complemented
8#7877C211yesmaximal, characteristic, complemented
9C2×#1277C211yesmaximal, characteristic, complemented
10C2×C4×#41C2211yescharacteristic
11#2925—24no—
12#2925—24no—
13C2×#433—22no—
14C2×#432C2212yes—
15C2×#431C2212yes—
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#408—21no—
22C2×#408—21no—
23C2×#405—21no—
24C22×C4×C8C2211yescharacteristic
25#1984C411yescharacteristic
26#1984C411yescharacteristic
27#1954C411yescharacteristic
28#1954C411yescharacteristic
29#1869C414yes—
30#1798—22no—
31#1797C2214yes—
32C4×C22⋊C8C2214yes—
33#1729—21no—
34#1570—24no—
35#1567—24no—
36C2×#335C2214yes—
37C2×#334C2212yes—
38C2×#334C2212yes—
39#1277C2212yes—
40#1277C2214yes—
41#1277C2212yes—
42C22×#41D411yescharacteristic
43C22×M16—21no—
44C22×M16—21no—
45C23×C8—21no—
46C23×C8—21no—
47C23×C8C2×C412yes—
48C4×#41—22no—
49C4×#41—22no—
50C4×#41—22no—
51C4×#41—22no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8Q811yescharacteristic
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4Q812yes—
57C2×C4×D4Q811yescharacteristic
58C2×#117D412yes—
59C2×#117D411yescharacteristic
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62#433—216no—
63#432—24no—
64#432—24no—
65#431—22no—
66#431—22no—
67#431—24no—
68#431—24no—
69#423—22no—
70#423—22no—
71#423—22no—
72#423D414yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2×C411yescharacteristic
81C2×C4⋊C8C2×C411yescharacteristic
82C2×C4⋊C8D414yes—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2312yes—
88#408—28no—
89#408—44no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8C2×C412yes—
94C2×C22⋊C8C2×C414yes—
95C2×C22⋊C8C2×C414yes—
96#405—28no—
97C2×#97—22no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8C2×C411yescharacteristic
102C2×C4×C8C2×C411yescharacteristic
103C2×C4×C8C2312yes—
104C2×C4×C8C2312yes—
105#336—24no—
106#336C2×C414yes—
107#336C2×C414yes—
108#335—28no—
109#334—24no—
110#334—24no—
111C2×#41—22no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—22no—
116C2×#41—22no—
117C22×Q8C4⋊C411yescharacteristic
118C22×D4C4⋊C412yes—
119C22×D4C4⋊C411yescharacteristic
120C23×C4C22×C412yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×Q812yes—
125C23×C4C2×Q811yescharacteristic
126C2×M16—24no—
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C22×C8—24no—
133C22×C8—24no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—24no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8C22×C412yes—
147C22×C8C22×C412yes—
148C22×C8C22×C412yes—
149C22×C8C22×C412yes—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162#117—22no—
163#117—22no—
164#117—24no—
165#117—24no—
166C2×C4⋊C4C4⋊C412yes—
167C2×C4⋊C4C4⋊C411yescharacteristic
168C2×C4⋊C4C4⋊C411yescharacteristic
169C2×C4⋊C4C4⋊C412yes—
170C2×C22⋊C4C4⋊C412yes—
171C2×C22⋊C4C4⋊C412yes—
172C2×C22⋊C4C4⋊C412yes—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42C22×C412yes—
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×Q812yes—
184C2×C42C2×Q811yescharacteristic
185C2×C42#4112yes—
186C2×C42#4112yes—
187C2×C42C2411yescharacteristic
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—24no—
191C4⋊C8—24no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201C4⋊C8C2×D412yes—
202C4⋊C8C2×D412yes—
203C4⋊C8C2×D414yes—
204C22⋊C8—28no—
205C22⋊C8—28no—
206C22⋊C8—24no—
207C22⋊C8—28no—
208C22⋊C8—44no—
209C22⋊C8—48no—
210#97—28no—
211C4×C8—28no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—22no—
218C24C2×C4⋊C412yes—
219C24C2×C4⋊C411yescharacteristic
220#41—44no—
221#41—44no—
222#41—44no—
223#41—44no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4C2×C4⋊C412yes—
250C22×C4C2×C4⋊C412yes—
251C22×C4C2×C4⋊C412yes—
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4C4×D412yes—
256C22×C4C4×D412yes—
257C22×C4#12012yes—
258C22×C4C22⋊Q812yes—
259C22×C4C22⋊Q812yes—
260C22×C4#12312yes—
261C22×C4#13512yes—
262C22×C4#13512yes—
263C22×C4AGL(1,ℤ/8)12yes—
264C22×C4#13712yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×Q811yescharacteristic
268M16—48no—
269M16—84no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—48no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—48no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—84no—
297C2×C8#13114yes—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42C4×D412yes—
312C42C4×D412yes—
313C42#12012yes—
314C42C22⋊Q812yes—
315C42C22⋊Q812yes—
316C42#12312yes—
317C42C22×D411yescharacteristic
318C42C22×D411yescharacteristic
319C42C2×#4111yescharacteristic
320C42C2×#4111yescharacteristic
321C23—22no—
322C23—22no—
323C23—24no—
324C23—22no—
325C23—22no—
326C23—24no—
327C23—22no—
328C23#38512yes—
329C23#42712yes—
330C23#42812yes—
331C23C22×C4⋊C411yescharacteristic
332Q8—44no—
333D4—48no—
334D4—44no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
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—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4#38512yes—
360C2×C4#38512yes—
361C2×C4#38512yes—
362C2×C4#42712yes—
363C2×C4#42812yes—
364C2×C4#44914yes—
365C2×C4#45214yes—
366C2×C4#48214yes—
367C2×C4#48514yes—
368C2×C4C2×C4×D411yescharacteristic
369C2×C4C2×C4×D411yescharacteristic
370C2×C4C2×#12011yescharacteristic
371C2×C4C2×C22⋊Q811yescharacteristic
372C2×C4C2×C22⋊Q811yescharacteristic
373C2×C4C2×#12311yescharacteristic
374C2×C4C2×#13112yes—
375C2×C4#56812yes—
376C2×C4C2×#13511yescharacteristic
377C2×C4C2×#13511yescharacteristic
378C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
379C2×C4C2×#13711yescharacteristic
380C8—24no—
381C8—42no—
382C8—42no—
383C8—42no—
384C8—44no—
385C8—42no—
386C8—42no—
387C8—82no—
388C8—82no—
389C22—22no—
390C22—22no—
391C22—24no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—44no—
397C22—44no—
398C22#185414yes—
399C22C2×#38511yescharacteristic
400C22#290612yes—
401C22C2×#42711yescharacteristic
402C22C2×#42811yescharacteristic
403C22#291612yes—
404C4—22no—
405C4—21no—
406C4—21no—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4#185414yes—
412C4#293111yescharacteristic
413C4#293211yescharacteristic
414C4#293312yes—
415C4C2×#44911yescharacteristic
416C4C2×#45211yescharacteristic
417C4C2×#48211yescharacteristic
418C4C2×#48511yescharacteristic
419C2—24no—
420C2—82no—
421C2#1782912yes—
422C2#1783012yes—
423C2#2083911yescharacteristic
424C2C2×#185411yescharacteristic
425C2#2085211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C4⋊C4
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1570C2semidirect
5. C2×#1567C2semidirect
6. #8834C2semidirect
7. #8831C2semidirect
8. #7877C2semidirect
9. C2×#1277C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29252
#213732
#206292
C2×#15702
C2×#15672
#88342
#88312
#78772
C2×#12772