Finite Group Databasebeta

← #238182

Subgroups — #238182

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4C2×#1956C211yesmaximal, characteristic
5#10355C214yesmaximal, complemented
6#10352C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7897C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×#103—22no—
14C22×C4×C8C2211yescharacteristic
15#2859—21no—
16#2859—21no—
17#2647—22no—
18#2571—21no—
19#2571C2212yes—
20#2552—21no—
21#2552C2212yes—
22#2442—22no—
23#2442—22no—
24#2317—21no—
25#2316—21no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1990—21no—
30#1990—21no—
31#1988—21no—
32#1988—21no—
33#1987C2212yes—
34#1987C2212yes—
35#1987C2214yes—
36#1956—22no—
37#1956C2214yes—
38#1824—22no—
39#1736—21no—
40#1736—21no—
41#1296C2214yes—
42#1279C2212yes—
43#1279C2214yes—
44#1279C2212yes—
45C23×Q8—21no—
46C22×M16—21no—
47C23×C8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8Q814yes—
55C2×C4⋊Q8C2312yes—
56C2×#125—22no—
57C2×#125—22no—
58C2×#117—21no—
59C2×#117D411yescharacteristic
60C2×#117D411yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D412yes—
66C22×C42C2311yescharacteristic
67#431—22no—
68#431—22no—
69#423—22no—
70#423—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×#103—28no—
75C2×#103—48no—
76C2×C22⋊C8—21no—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8C2312yes—
82#400—24no—
83#400—24no—
84#398—24no—
85#398—24no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#385—24no—
97#384—22no—
98#384—21no—
99#384—21no—
100#384D411yescharacteristic
101#384D411yescharacteristic
102#384D412yes—
103#384C2312yes—
104#382—22no—
105#382—22no—
106#382—22no—
107#382—21no—
108#380—24no—
109#380—21no—
110#380D412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4C2311yescharacteristic
115C4×C4⋊C4C2312yes—
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376—21no—
120C43—21no—
121#340—24no—
122#340—24no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127C22×Q8—24no—
128C22×Q8—22no—
129C22×Q8—44no—
130C22×Q8—44no—
131C22×Q8—41no—
132C22×Q8—42no—
133C23×C4—22no—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—42no—
139C2×M16—42no—
140C2×M16—42no—
141C22×C8—24no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—42no—
146C22×C8—42no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8C2×Q814yes—
157#125—84no—
158#125—84no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—22no—
163#117—42no—
164#117—42no—
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—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D414yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×Q812yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—42no—
220#103—416no—
221C22⋊C8—42no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—42no—
225C4×C8—42no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—22no—
236C24#12011yescharacteristic
237C2×Q8—24no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—416no—
241C2×Q8—42no—
242C2×Q8—44no—
243C2×Q8—88no—
244C2×Q8—82no—
245C2×Q8—82no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4#12011yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4C22⋊Q812yes—
268C22×C4C22⋊Q812yes—
269C22×C4C22⋊Q812yes—
270C22×C4#12412yes—
271C22×C4#12412yes—
272C22×C4#12412yes—
273C22×C4#13512yes—
274C22×C4#13512yes—
275C22×C4#13712yes—
276C22×C4#13712yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—84no—
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—48no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—82no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42C22⋊Q812yes—
334C42C22⋊Q812yes—
335C42C22⋊Q812yes—
336C42#12412yes—
337C42#12412yes—
338C42#12412yes—
339C42C22×Q811yescharacteristic
340C42C2×#4111yescharacteristic
341C42C2×#4111yescharacteristic
342C42C2×#4111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23#39312yes—
348C23#45212yes—
349C23#45212yes—
350C23C2×#12011yescharacteristic
351Q8—48no—
352Q8—84no—
353Q8—88no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—88no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—164no—
388C2×C4#39312yes—
389C2×C4#39312yes—
390C2×C4#39312yes—
391C2×C4#45212yes—
392C2×C4#45212yes—
393C2×C4#45312yes—
394C2×C4#45412yes—
395C2×C4#47914yes—
396C2×C4#47914yes—
397C2×C4#48814yes—
398C2×C4#49114yes—
399C2×C4C2×C22⋊Q811yescharacteristic
400C2×C4C2×C22⋊Q811yescharacteristic
401C2×C4C2×C22⋊Q811yescharacteristic
402C2×C4C2×#12411yescharacteristic
403C2×C4C2×#12411yescharacteristic
404C2×C4C2×#12411yescharacteristic
405C2×C4C2×#13511yescharacteristic
406C2×C4C2×#13511yescharacteristic
407C2×C4C2×#13711yescharacteristic
408C2×C4C2×#13711yescharacteristic
409C2×C4#57712yes—
410C2×C4#57812yes—
411C8—44no—
412C8—82no—
413C8—82no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22#202614yes—
421C22C2×#39311yescharacteristic
422C22C2×#45211yescharacteristic
423C22C2×#45211yescharacteristic
424C22#300911yescharacteristic
425C22#301111yescharacteristic
426C22C2×#45311yescharacteristic
427C22C2×#45411yescharacteristic
428C4—22no—
429C4—21no—
430C4—21no—
431C4—84no—
432C4—81no—
433C4—84no—
434C4—81no—
435C4—82no—
436C4—81no—
437C4—81no—
438C4—82no—
439C4#202614yes—
440C4#202714yes—
441C4C2×#47911yescharacteristic
442C4C2×#47911yescharacteristic
443C4#308212yes—
444C4C2×#48811yescharacteristic
445C4C2×#49111yescharacteristic
446C4#313712yes—
447C2—24no—
448C2#1862511yescharacteristic
449C2#1862711yescharacteristic
450C2#1864511yescharacteristic
451C2#1864711yescharacteristic
452C2C2×#202611yescharacteristic
453C2#2345711yescharacteristic
454C2C2×#202711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
5. #10355C2semidirect
6. #10352C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#218342
#213132
C2×#19562
#103552
#103522
#80432
#78972
#78902