Finite Group Databasebeta

← #238228

Subgroups — #238228

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5#10356C214yesmaximal, complemented
6#10353C214yesmaximal, complemented
7#8039C211yesmaximal, characteristic
8#7893C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2851—22no—
15#2646—22no—
16#2571—21no—
17#2571C2212yes—
18#2570—22no—
19#2570—22no—
20#2552—21no—
21#2552C2212yes—
22#2317—21no—
23#2317—21no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990C2212yes—
28#1990C2212yes—
29#1988—22no—
30#1988C2214yes—
31#1987—22no—
32#1981C2214yes—
33#1956—21no—
34#1956—21no—
35#1824—22no—
36#1787—22no—
37C2×#340—21no—
38C2×#340—21no—
39#1296C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×Q8—21no—
44C23×C8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8Q814yes—
52C2×C4⋊Q8C2312yes—
53C2×#125—22no—
54C2×#117—21no—
55C2×#117—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×#103—24no—
73C2×#103—24no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8C2312yes—
78#400—24no—
79#400—24no—
80#398—24no—
81#398—24no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#387—24no—
89#387—24no—
90#385—24no—
91#385—24no—
92#384—22no—
93#384—22no—
94#384D411yescharacteristic
95#384D411yescharacteristic
96#384C2311yescharacteristic
97#384C2311yescharacteristic
98#382—22no—
99#382—22no—
100#382—22no—
101#382—21no—
102#382—21no—
103#380—21no—
104#380—21no—
105#380D412yes—
106#380D412yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C4×C4⋊C4C2312yes—
115C2×#95—21no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8—41no—
123C22×Q8—42no—
124C22×Q8—82no—
125C22×Q8—82no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147C4⋊Q8C2×Q814yes—
148#125—84no—
149#117—42no—
150#117—42no—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
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—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4C2×D412yes—
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×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—82no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×Q812yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199C2×C42C2411yescharacteristic
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—42no—
205C4⋊C8—42no—
206#103—84no—
207#103—84no—
208C22⋊C8—42no—
209C22⋊C8—42no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—42no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223C24#12011yescharacteristic
224C2×Q8—24no—
225C2×Q8—42no—
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—88no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—88no—
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—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4C22⋊Q812yes—
257C22×C4C22⋊Q812yes—
258C22×C4C22⋊Q812yes—
259C22×C4#12412yes—
260C22×C4#12412yes—
261C22×C4#12412yes—
262C22×C4#13512yes—
263C22×C4#13512yes—
264C22×C4#13712yes—
265C22×C4#13712yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—84no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C42—42no—
312C42—42no—
313C42—44no—
314C42—44no—
315C42—82no—
316C42C22⋊Q812yes—
317C42C22⋊Q812yes—
318C42C22⋊Q812yes—
319C42#12412yes—
320C42#12412yes—
321C42#12412yes—
322C42C22×Q811yescharacteristic
323C42C2×#4111yescharacteristic
324C42C2×#4111yescharacteristic
325C42C2×#4111yescharacteristic
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23#39312yes—
331C23#45212yes—
332C23#45212yes—
333C23C2×#12011yescharacteristic
334Q8—48no—
335Q8—88no—
336Q8—84no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
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—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—88no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4#39312yes—
372C2×C4#39312yes—
373C2×C4#39312yes—
374C2×C4#45212yes—
375C2×C4#45212yes—
376C2×C4#47914yes—
377C2×C4#47914yes—
378C2×C4#48814yes—
379C2×C4#49114yes—
380C2×C4C2×C22⋊Q811yescharacteristic
381C2×C4C2×C22⋊Q811yescharacteristic
382C2×C4C2×C22⋊Q811yescharacteristic
383C2×C4C2×#12411yescharacteristic
384C2×C4C2×#12411yescharacteristic
385C2×C4C2×#12411yescharacteristic
386C2×C4C2×#13511yescharacteristic
387C2×C4C2×#13511yescharacteristic
388C2×C4C2×#13711yescharacteristic
389C2×C4C2×#13711yescharacteristic
390C2×C4#57612yes—
391C2×C4#57712yes—
392C8—82no—
393C8—82no—
394C8—82no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22#202614yes—
402C22C2×#39311yescharacteristic
403C22C2×#45211yescharacteristic
404C22C2×#45211yescharacteristic
405C22#300811yescharacteristic
406C22#301011yescharacteristic
407C22#302211yescharacteristic
408C22#302211yescharacteristic
409C4—22no—
410C4—21no—
411C4—21no—
412C4—22no—
413C4—84no—
414C4—82no—
415C4—82no—
416C4—82no—
417C4—82no—
418C4—82no—
419C4—82no—
420C4#202614yes—
421C4C2×#47911yescharacteristic
422C4C2×#47911yescharacteristic
423C4#308112yes—
424C4C2×#48811yescharacteristic
425C4C2×#49111yescharacteristic
426C4#313612yes—
427C2—24no—
428C2#1890711yescharacteristic
429C2#1890911yescharacteristic
430C2#1892711yescharacteristic
431C2#1892911yescharacteristic
432C2C2×#202611yescharacteristic
433C2#2345611yescharacteristic
434C2#2346411yescharacteristic
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. #10356C2semidirect
6. #10353C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#218342
#218042
C2×#19882
#103562
#103532
#80392
#78932
#78922