Finite Group Databasebeta

← #192695

Subgroups — #192695

Order 512.

954 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34374C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5C2×#1485C211yesmaximal, characteristic
6#8895C214yesmaximal, complemented
7#8892C214yesmaximal, complemented
8#8038C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2382—22no—
15#2365—21no—
16#2365—21no—
17#2355—21no—
18#2355C2212yes—
19Q8×C4⋊C4—21no—
20Q8×C4⋊C4C2212yes—
21#2323—22no—
22#2323—22no—
23#2314—22no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#1990—21no—
28#1956—21no—
29#1947C2214yes—
30#1872—22no—
31#1852—22no—
32#1845—21no—
33#1836—22no—
34#1834—21no—
35#1834C2212yes—
36#1834C2212yes—
37#1803C2214yes—
38#1736—22no—
39#1485C418yescomplemented
40C2×Q8⋊C8C2214yes—
41#1296C2212yes—
42#1296C2212yes—
43#1279C2214yes—
44C22×M16—21no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48#533Q814yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8D411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8D412yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2312yes—
58C2×#117—21no—
59C2×#117—21no—
60C2×#117D412yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431—22no—
67#431—22no—
68#431C2×C414yes—
69C2×#107—24no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×#103—24no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C4×M16—22no—
76C4×M16C2×C414yes—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8C2312yes—
81#391—22no—
82#391—22no—
83#389—22no—
84#389—24no—
85#389—22no—
86#389—24no—
87#388—24no—
88#388—24no—
89#387—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#384—22no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384D411yescharacteristic
98#384D411yescharacteristic
99#384C2312yes—
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#382—22no—
103#382—21no—
104#382D412yes—
105#380—21no—
106#380—21no—
107#380—21no—
108#380D412yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2312yes—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121Q8⋊C8C2×C4116yescomplemented
122C22×Q8—21no—
123C22×Q8C2×D412yes—
124C23×C4—41no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×M16—24no—
129C2×M16—42no—
130C2×M16—42no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8C4⋊C414yes—
137C4⋊Q8C4⋊C414yes—
138#125—24no—
139#125—24no—
140C22⋊Q8—24no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8C4⋊C418yes—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—84no—
149C4×Q8—84no—
150C4×Q8C22×C414yes—
151C4×Q8C2×Q814yes—
152#117—24no—
153#117—24no—
154#117—44no—
155#117—82no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42C22×C412yes—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×Q812yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200C2×C42C2411yescharacteristic
201#107—84no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8C22×C414yes—
206#103—84no—
207C22⋊C8—24no—
208C22⋊C8—42no—
209#97—24no—
210C4×C8—24no—
211C4×C8—22no—
212C4×C8—42no—
213C4×C8C22×C414yes—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222C24#12011yescharacteristic
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8C2×C4⋊C414yes—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4C2×C4⋊C412yes—
247C22×C4C4×D412yes—
248C22×C4C4×D412yes—
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4C22⋊Q812yes—
253C22×C4C22⋊Q812yes—
254C22×C4#12312yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258M16—48no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—28no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C4⋊C4—82no—
295C4⋊C4C2×C4⋊C414yes—
296C22⋊C4—28no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C42—24no—
300C42—24no—
301C42—44no—
302C42—44no—
303C42—84no—
304C42—84no—
305C42C2×C4⋊C412yes—
306C42C4×D412yes—
307C42C4×D412yes—
308C42C22⋊Q812yes—
309C42C22⋊Q812yes—
310C42#12312yes—
311C42C23×C411yescharacteristic
312C42C22×Q811yescharacteristic
313C42C2×#4111yescharacteristic
314C42C2×#4111yescharacteristic
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23#38512yes—
322C23C2×#12011yescharacteristic
323Q8—28no—
324Q8—44no—
325Q8—48no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—164no—
360C2×C4#38512yes—
361C2×C4#38512yes—
362C2×C4#38512yes—
363C2×C4#45412yes—
364C2×C4#45512yes—
365C2×C4C22×C4⋊C411yescharacteristic
366C2×C4C2×C4×D411yescharacteristic
367C2×C4C2×C4×D411yescharacteristic
368C2×C4C2×C22⋊Q811yescharacteristic
369C2×C4C2×C22⋊Q811yescharacteristic
370C2×C4C2×#12311yescharacteristic
371C2×C4#57612yes—
372C2×C4#57712yes—
373C2×C4#57812yes—
374C2×C4#57812yes—
375C8—44no—
376C8—44no—
377C8—82no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22C2×#38511yescharacteristic
386C22#300811yescharacteristic
387C22#301011yescharacteristic
388C22#301111yescharacteristic
389C22#301111yescharacteristic
390C22C2×#45511yescharacteristic
391C22C2×#45411yescharacteristic
392C4—22no—
393C4—21no—
394C4—21no—
395C4—22no—
396C4—44no—
397C4—44no—
398C4—82no—
399C4—82no—
400C4—82no—
401C4—82no—
402C4—162no—
403C4#186014yes—
404C4#291312yes—
405C4#294712yes—
406C4#308212yes—
407C4#309812yes—
408C2—42no—
409C2#1872411yescharacteristic
410C2#1872511yescharacteristic
411C2#1872911yescharacteristic
412C2#1873211yescharacteristic
413C2#2084711yescharacteristic
414C2C2×#186011yescharacteristic
415C2#2087711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8895C2semidirect
7. #8892C2semidirect
39. #1485C4semidirect
121. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343742
#219822
#218602
#207642
C2×#14852
#88952
#88922
#80382
#78962