Finite Group Databasebeta

← #193321

Subgroups — #193321

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34374C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4#21322C211yesmaximal, characteristic
5C2×#1544C211yesmaximal, characteristic
6#8926C214yesmaximal, complemented
7#8923C214yesmaximal, complemented
8#7896C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2382—22no—
15#2365C2212yes—
16#2365C2212yes—
17#2355—21no—
18#2355—21no—
19Q8×C4⋊C4—21no—
20Q8×C4⋊C4—21no—
21#2323—22no—
22#2323—22no—
23#2314—22no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#1992—22no—
28#1990C2212yes—
29#1990C2212yes—
30#1948C2214yes—
31#1947C2214yes—
32#1853—22no—
33#1852—22no—
34#1845—21no—
35#1844—21no—
36#1834—21no—
37#1834—21no—
38#1736—22no—
39#1544C418yescomplemented
40C2×#332C2212yes—
41C2×#332C2212yes—
42#1279C2214yes—
43#1279C2214yes—
44C22×M16—21no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48#533Q814yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×#125D412yes—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
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#431C2×C414yes—
68#431C2×C414yes—
69C2×#107—24no—
70C2×#106—24no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C4×M16—22no—
76C4×M16—22no—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8C2312yes—
81#391—22no—
82#391—22no—
83#389—24no—
84#389—24no—
85#389—22no—
86#389—22no—
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#384D411yescharacteristic
100#384D411yescharacteristic
101#384C2312yes—
102#382—22no—
103#382—21no—
104#382—21no—
105#380—21no—
106#380—21no—
107#380D412yes—
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⋊C4C2312yes—
116C4×C4⋊C4C2312yes—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#332C2×C418yescomplemented
122#332C2×C418yescomplemented
123C22×Q8—21no—
124C22×Q8—21no—
125C23×C4—41no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×M16—24no—
130C2×M16—42no—
131C2×M16—42no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8C22×C412yes—
140C4⋊Q8C22×C412yes—
141C4⋊Q8C2×Q812yes—
142C4⋊Q8C2×Q812yes—
143#125—24no—
144#125C4⋊C418yes—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—84no—
154C4×Q8—84no—
155#117—24no—
156#117—82no—
157#117—82no—
158#117C4⋊C418yes—
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—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42C22×C412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×Q812yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203C2×C42C2411yescharacteristic
204#107—84no—
205#106—84no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8C22×C414yes—
209C4⋊C8C22×C414yes—
210C22⋊C8—24no—
211C22⋊C8—24no—
212#97—42no—
213C4×C8—24no—
214C4×C8—24no—
215C4×C8—22no—
216C4×C8—22no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225C24#12011yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
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—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4C2×C4⋊C412yes—
251C22×C4C4×D412yes—
252C22×C4C4×D412yes—
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4C22⋊Q812yes—
257C22×C4C22⋊Q812yes—
258C22×C4#12312yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262M16—48no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—84no—
298C4⋊C4—84no—
299C4⋊C4—82no—
300C4⋊C4C2×C4⋊C414yes—
301C4⋊C4C2×C4⋊C414yes—
302C22⋊C4—28no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C42—28no—
306C42—44no—
307C42—44no—
308C42—84no—
309C42—84no—
310C42C2×C4⋊C412yes—
311C42C4×D412yes—
312C42C4×D412yes—
313C42C22⋊Q812yes—
314C42C22⋊Q812yes—
315C42#12312yes—
316C42C23×C411yescharacteristic
317C42C22×Q811yescharacteristic
318C42C2×#4111yescharacteristic
319C42C2×#4111yescharacteristic
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23#38512yes—
327C23C2×#12011yescharacteristic
328Q8—44no—
329Q8—44no—
330Q8—44no—
331Q8—44no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
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—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—164no—
366C2×C4#38512yes—
367C2×C4#38512yes—
368C2×C4#38512yes—
369C2×C4#45412yes—
370C2×C4#45612yes—
371C2×C4C22×C4⋊C411yescharacteristic
372C2×C4C2×C4×D411yescharacteristic
373C2×C4C2×C4×D411yescharacteristic
374C2×C4C2×C22⋊Q811yescharacteristic
375C2×C4C2×C22⋊Q811yescharacteristic
376C2×C4C2×#12311yescharacteristic
377C2×C4#57612yes—
378C2×C4#57712yes—
379C2×C4#57712yes—
380C2×C4#57812yes—
381C8—44no—
382C8—44no—
383C8—44no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22C2×#38511yescharacteristic
392C22#300811yescharacteristic
393C22#300911yescharacteristic
394C22#301011yescharacteristic
395C22#301111yescharacteristic
396C22C2×#45411yescharacteristic
397C22C2×#45611yescharacteristic
398C4—21no—
399C4—21no—
400C4—22no—
401C4—22no—
402C4—44no—
403C4—44no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4—162no—
409C4#185714yes—
410C4#295112yes—
411C4#295212yes—
412C4#308312yes—
413C4#308412yes—
414C2—42no—
415C2#1872911yescharacteristic
416C2#1873011yescharacteristic
417C2#1873411yescharacteristic
418C2#1873711yescharacteristic
419C2C2×#185711yescharacteristic
420C2#2088111yescharacteristic
421C2#2088211yescharacteristic
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. #8926C2semidirect
7. #8923C2semidirect
39. #1544C4semidirect
121. #332C2×C4semidirect
122. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343742
#219822
#219812
#213222
C2×#15442
#89262
#89232
#78962