Finite Group Databasebeta

← #238139

Subgroups — #238139

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5#10355C214yesmaximal, complemented
6#10352C214yesmaximal, complemented
7#8045C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2571C2212yes—
15#2570—22no—
16#2569—22no—
17#2552C2212yes—
18#2442—22no—
19#2402—22no—
20C4×#125—22no—
21#2296—21no—
22#2285—21no—
23#2285C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1988—22no—
27#1987—22no—
28#1987C2212yes—
29#1987C2212yes—
30#1987C2214yes—
31#1981—22no—
32#1956C2214yes—
33#1787—22no—
34#1736—21no—
35#1736—21no—
36C2×#340—22no—
37#1296C2214yes—
38#1279C2212yes—
39#1279C2214yes—
40#1279C2212yes—
41C23×Q8—21no—
42C22×M16—21no—
43C23×C8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8Q814yes—
51C2×C4⋊Q8C2312yes—
52C2×#125—22no—
53C2×#117—21no—
54C2×#117D411yescharacteristic
55C2×#117D411yescharacteristic
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D412yes—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#431—22no—
62#431—22no—
63#423—22no—
64#423—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×#103—28no—
73C2×C22⋊C8—21no—
74C2×C4×C8C2312yes—
75#400—24no—
76#400—24no—
77#398—24no—
78#398—24no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#387—24no—
84#387—24no—
85#385—24no—
86#384—22no—
87#384C2312yes—
88#382—22no—
89#382—22no—
90#382—21no—
91#380—24no—
92#380—21no—
93#380D412yes—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4D412yes—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4C2311yescharacteristic
103C4×C4⋊C4C2312yes—
104C4×C4⋊C4C2311yescharacteristic
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107#376—21no—
108C2×#95—21no—
109C43—21no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114C22×Q8—24no—
115C22×Q8—22no—
116C22×Q8—41no—
117C22×Q8—42no—
118C22×Q8—84no—
119C23×C4—22no—
120C23×C4—41no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×M16—42no—
125C2×M16—42no—
126C2×M16—42no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—44no—
131C22×C8—42no—
132C4⋊Q8—24no—
133C4⋊Q8—24no—
134C4⋊Q8—24no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—42no—
141C4⋊Q8C2×Q814yes—
142#125—44no—
143#125—44no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—42no—
149#117—42no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D414yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×Q812yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42#4112yes—
192C2×C42C2411yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8—42no—
200#103—88no—
201C22⋊C8—42no—
202C4×C8—22no—
203C4×C8—22no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—22no—
215C24#12011yescharacteristic
216C2×Q8—24no—
217C2×Q8—48no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—84no—
221C2×Q8—88no—
222C2×Q8—88no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—82no—
236C22×C4—84no—
237C22×C4#12011yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4C22⋊Q812yes—
241C22×C4C22⋊Q812yes—
242C22×C4C22⋊Q812yes—
243C22×C4#12412yes—
244C22×C4#12412yes—
245C22×C4#12412yes—
246C22×C4#13512yes—
247C22×C4#13512yes—
248C22×C4#13712yes—
249C22×C4#13712yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D411yescharacteristic
253M16—84no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—44no—
261C2×C8—84no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—84no—
284C4⋊C4—84no—
285C4⋊C4#13714yes—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—44no—
289C42—41no—
290C42—41no—
291C42—41no—
292C42—41no—
293C42—42no—
294C42—42no—
295C42—44no—
296C42—42no—
297C42—42no—
298C42—44no—
299C42—82no—
300C42C22⋊Q812yes—
301C42C22⋊Q812yes—
302C42C22⋊Q812yes—
303C42#12412yes—
304C42#12412yes—
305C42#12412yes—
306C42C22×Q811yescharacteristic
307C42C2×#4111yescharacteristic
308C42C2×#4111yescharacteristic
309C42C2×#4111yescharacteristic
310C23—22no—
311C23—22no—
312C23#39312yes—
313C23#45212yes—
314C23#45212yes—
315C23C2×#12011yescharacteristic
316Q8—48no—
317Q8—84no—
318Q8—88no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—88no—
344C2×C4—88no—
345C2×C4—84no—
346C2×C4#39312yes—
347C2×C4#39312yes—
348C2×C4#39312yes—
349C2×C4#45212yes—
350C2×C4#45212yes—
351C2×C4#47914yes—
352C2×C4#47914yes—
353C2×C4#48814yes—
354C2×C4#49114yes—
355C2×C4C2×C22⋊Q811yescharacteristic
356C2×C4C2×C22⋊Q811yescharacteristic
357C2×C4C2×C22⋊Q811yescharacteristic
358C2×C4C2×#12411yescharacteristic
359C2×C4C2×#12411yescharacteristic
360C2×C4C2×#12411yescharacteristic
361C2×C4C2×#13511yescharacteristic
362C2×C4C2×#13511yescharacteristic
363C2×C4C2×#13711yescharacteristic
364C2×C4C2×#13711yescharacteristic
365C2×C4C2×#13712yes—
366C2×C4#57512yes—
367C8—82no—
368C8—82no—
369C8—82no—
370C22—24no—
371C22—24no—
372C22—22no—
373C22—22no—
374C22#202614yes—
375C22C2×#39311yescharacteristic
376C22C2×#45211yescharacteristic
377C22C2×#45211yescharacteristic
378C22#300712yes—
379C22#302112yes—
380C4—22no—
381C4—21no—
382C4—21no—
383C4—22no—
384C4—44no—
385C4—82no—
386C4—84no—
387C4—82no—
388C4—82no—
389C4—84no—
390C4#202614yes—
391C4C2×#47911yescharacteristic
392C4C2×#47911yescharacteristic
393C4#307812yes—
394C4C2×#48811yescharacteristic
395C4C2×#49111yescharacteristic
396C4#313911yescharacteristic
397C4#314011yescharacteristic
398C2—24no—
399C2#1879112yes—
400C2#1879312yes—
401C2C2×#202611yescharacteristic
402C2#2345511yescharacteristic
403C2#2346311yescharacteristic
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
#349482
#218042
#213132
C2×#19872
#103552
#103522
#80452
#79012
#78912