Finite Group Databasebeta

← #242715

Subgroups — #242715

Order 512.

671 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4#17803C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6C2×#1548C211yesmaximal, characteristic
7#10609C214yesmaximal, complemented
8#10605C214yesmaximal, complemented
9#8004C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1997—21no—
18#1997—21no—
19#1997C2214yes—
20#1950C2214yes—
21#1855—22no—
22#1848—21no—
23#1837—21no—
24#1821—22no—
25#1574—21no—
26#1574—21no—
27#1572—22no—
28#1571—21no—
29#1571C2212yes—
30#1570—21no—
31#1570C2212yes—
32#1565C2212yes—
33#1565C2212yes—
34#1564—22no—
35#1548—21no—
36#1548—21no—
37#1548—22no—
38#1548—22no—
39#1548C2212yes—
40#1548C2212yes—
41C2×#333—21no—
42C2×#333—21no—
43C2×#333C2214yes—
44#1292C2214yes—
45C23×Q8Q812yes—
46C22×M16—21no—
47C22×M16—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8Q812yes—
59C2×C4⋊Q8C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C42C2311yescharacteristic
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431D412yes—
76#431D412yes—
77#431D411yescharacteristic
78#431D411yescharacteristic
79#431C2311yescharacteristic
80#431C2311yescharacteristic
81C2×#108—24no—
82#423—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86#423—21no—
87#423—21no—
88#423—21no—
89#423D411yescharacteristic
90#423D411yescharacteristic
91#423C2312yes—
92#423C2312yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100#407—21no—
101#407D412yes—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C4×M16—21no—
105C4×M16C2312yes—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#343—24no—
109#343—24no—
110#341—24no—
111#335—24no—
112#335—22no—
113#335—22no—
114#334—24no—
115#334—22no—
116#334—22no—
117#333—22no—
118#333—22no—
119#333—22no—
120#333—22no—
121#333—28no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—84no—
128C22×Q8C2×Q814yes—
129C23×C4—22no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×Q811yescharacteristic
132C23×C4#4111yescharacteristic
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—41no—
136C2×M16—41no—
137C2×M16—41no—
138C2×M16—82no—
139C2×M16—82no—
140C2×M16C2×D412yes—
141C2×M16C2×D412yes—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—41no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—41no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×Q811yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42C2411yescharacteristic
187#108—84no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8C2×D411yescharacteristic
206C4⋊C8C2×D411yescharacteristic
207C4⋊C8C2×D414yes—
208C4⋊C8C2×D411yescharacteristic
209C4⋊C8C2×D411yescharacteristic
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220#97—22no—
221#97—41no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—22no—
228C4×C8—21no—
229C4×C8—41no—
230C24C22⋊Q811yescharacteristic
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—84no—
236C2×Q8—84no—
237C2×Q8—84no—
238C2×Q8—84no—
239C2×Q8—84no—
240C2×Q8—84no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—84no—
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4#12611yescharacteristic
264C22×C4#12611yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×Q811yescharacteristic
267C22×C4C2×#4111yescharacteristic
268M16—24no—
269M16—24no—
270M16—44no—
271M16—42no—
272M16—42no—
273M16—82no—
274M16—82no—
275M16—82no—
276C2×C8—24no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—82no—
291C2×C8—82no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42#12111yescharacteristic
305C42#12111yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42C22⋊Q811yescharacteristic
308C42C22⋊Q811yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42#12611yescharacteristic
311C42#12611yescharacteristic
312C42C22×D411yescharacteristic
313C42C22×D411yescharacteristic
314C42C2×#4111yescharacteristic
315C42C2×#4111yescharacteristic
316C23—22no—
317C23—22no—
318C23—41no—
319C23#39611yescharacteristic
320C23#39611yescharacteristic
321C23C2×C22⋊Q811yescharacteristic
322Q8—84no—
323Q8—84no—
324Q8—84no—
325Q8—84no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4#39611yescharacteristic
351C2×C4#39611yescharacteristic
352C2×C4#39611yescharacteristic
353C2×C4#39611yescharacteristic
354C2×C4#39611yescharacteristic
355C2×C4#39611yescharacteristic
356C2×C4#47112yes—
357C2×C4#47312yes—
358C2×C4C2×#12111yescharacteristic
359C2×C4C2×#12111yescharacteristic
360C2×C4C2×#12111yescharacteristic
361C2×C4C2×C22⋊Q811yescharacteristic
362C2×C4C2×C22⋊Q811yescharacteristic
363C2×C4C2×#12611yescharacteristic
364C2×C4#57712yes—
365C2×C4#57812yes—
366C8—44no—
367C8—42no—
368C8—42no—
369C8—41no—
370C8—41no—
371C8—41no—
372C8—41no—
373C8—82no—
374C8—82no—
375C8—82no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—44no—
384C22#206412yes—
385C22C2×#39611yescharacteristic
386C22#308212yes—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—84no—
395C4#206412yes—
396C4#206912yes—
397C4#304612yes—
398C4#305711yescharacteristic
399C4#305911yescharacteristic
400C4C2×#47111yescharacteristic
401C4C2×#47311yescharacteristic
402C2—22no—
403C2—22no—
404C2—41no—
405C2C2×#206411yescharacteristic
406C2#2361311yescharacteristic
407C2C2×#206911yescharacteristic
subgroupG/H
center, socleC22C2×#396
commutator, Frattini, cosocleC2×C42C24
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
7. #10609C2semidirect
8. #10605C2semidirect

Maximal subgroups

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

subgroupindex
#219932
#218292
#178092
#178032
C2×#15482
#106092
#106052
#80042