Finite Group Databasebeta

← #193301

Subgroups — #193301

Order 512.

960 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#34365C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic
3#21963C211yesmaximal, characteristic
4#21804C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#8929C212yesmaximal, complemented
7#8928C212yesmaximal, complemented
8#8927C212yesmaximal, complemented
9#8927C212yesmaximal, complemented
10#7899C211yesmaximal, characteristic
11#7894C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2381—22no—
17#2369C2212yes—
18#2369C2212yes—
19#2344—22no—
20#2344—22no—
21#2331—21no—
22#2331—21no—
23#2331—22no—
24#2310—22no—
25#2285C2211yescharacteristic
26#2283C2211yescharacteristic
27#2283C2211yescharacteristic
28#1981—22no—
29#1956C2212yes—
30#1956C2212yes—
31#1949—22no—
32#1948C2212yes—
33#1948C2212yes—
34#1947C2212yes—
35#1947C2212yes—
36#1854—22no—
37#1846—21no—
38#1835—21no—
39#1809—22no—
40#1787—22no—
41#1548C418yescomplemented
42C2×#333C2212yes—
43C2×#333C2212yes—
44#1279C2211yescharacteristic
45#1279C2211yescharacteristic
46#1279C2211yescharacteristic
47#1279C2211yescharacteristic
48#1279C2212yes—
49#1279C2212yes—
50C23×Q8D412yes—
51C22×M16—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D412yes—
58C2×C4⋊Q8Q814yes—
59C2×C4⋊Q8C2312yes—
60C2×#117—24no—
61C2×#117—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—22no—
70C2×#107—22no—
71C2×#106—22no—
72C2×#106—22no—
73#423—22no—
74#423C2×C414yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×#103—24no—
81C2×#103—24no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#391—24no—
88#391—24no—
89#391—24no—
90#391—24no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#387—24no—
96#387—24no—
97#387—24no—
98#387—24no—
99#384C2312yes—
100#383—21no—
101#383—21no—
102#382—21no—
103#382D411yescharacteristic
104#382D411yescharacteristic
105#382D412yes—
106#381—24no—
107#381—21no—
108#381—21no—
109#380—21no—
110#380—21no—
111#380D412yes—
112#380D412yes—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2312yes—
121C4×C4⋊C4C2311yescharacteristic
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124#376—21no—
125#376—21no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#333C2×C418yes—
131#333C2×C418yes—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—84no—
135C22×Q8C4⋊C418yes—
136C22×Q8C2×D414yes—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—24no—
142C2×M16—82no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8C4⋊C418yes—
156C4⋊Q8C22×C414yes—
157C4⋊Q8C2×Q814yes—
158#117—88no—
159#117—81no—
160#117—81no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—41no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—41no—
187C2×C42C22×C412yes—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q812yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194C2×C42C2411yescharacteristic
195#107—82no—
196#107—82no—
197#106—82no—
198#106—82no—
199C4⋊C8—24no—
200C4⋊C8—22no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—42no—
204C4⋊C8C22×C414yes—
205C4⋊C8C22×C412yes—
206C4⋊C8C22×C412yes—
207#103—84no—
208#103—84no—
209C22⋊C8—24no—
210C22⋊C8—42no—
211C4×C8—24no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—22no—
221#95—22no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95—21no—
231#95—41no—
232#95—41no—
233C24#12011yescharacteristic
234C2×Q8—24no—
235C2×Q8—28no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—84no—
239C2×Q8—88no—
240C2×Q8—88no—
241C2×Q8C2×C4⋊C418yes—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—48no—
256C22×C4—42no—
257C22×C4C2×C4⋊C412yes—
258C22×C4C4×D412yes—
259C22×C4C4×D412yes—
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4C22⋊Q812yes—
264C22×C4C22⋊Q812yes—
265C22×C4#12312yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269M16—84no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—24no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—41no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—28no—
288C4⋊C4—28no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—82no—
307C42—41no—
308C42—41no—
309C42—42no—
310C42—41no—
311C42—41no—
312C42—42no—
313C42—82no—
314C42—84no—
315C42—84no—
316C42C2×C4⋊C412yes—
317C42C4×D412yes—
318C42C4×D412yes—
319C42C22⋊Q812yes—
320C42C22⋊Q812yes—
321C42#12312yes—
322C42C23×C411yescharacteristic
323C42C22×Q811yescharacteristic
324C42C2×#4111yescharacteristic
325C42C2×#4111yescharacteristic
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23#38512yes—
331C23C2×#12011yescharacteristic
332Q8—48no—
333Q8—84no—
334Q8—88no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—28no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—416no—
351C2×C4—44no—
352C2×C4—44no—
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—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4#38512yes—
369C2×C4#38512yes—
370C2×C4#38512yes—
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#57711yescharacteristic
378C2×C4#57711yescharacteristic
379C2×C4#57712yes—
380C2×C4#57811yescharacteristic
381C2×C4#57811yescharacteristic
382C2×C4#57812yes—
383C8—42no—
384C8—42no—
385C8—44no—
386C8—81no—
387C8—81no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—42no—
392C22—42no—
393C22C2×#38511yescharacteristic
394C22#301011yescharacteristic
395C22#301111yescharacteristic
396C22#301312yes—
397C22#301912yes—
398C4—21no—
399C4—21no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—48no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4—162no—
409C4#294711yescharacteristic
410C4#294711yescharacteristic
411C4#294911yescharacteristic
412C4#295011yescharacteristic
413C4#308211yescharacteristic
414C4#308211yescharacteristic
415C4#308212yes—
416C2—42no—
417C2#1893212yes—
418C2#1893312yes—
419C2#2085611yescharacteristic
420C2#2087711yescharacteristic
421C2#2088011yescharacteristic
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. #8929C2semidirect
7. #8928C2semidirect
8. #8927C2semidirect
9. #8927C2semidirect
41. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#343652
#219852
#219632
#218042
C2×#15482
#89292
#89282
#89272
#78992
#78942