Finite Group Databasebeta

← #193160

Subgroups — #193160

Order 512.

776 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60249C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic
3#17823C211yesmaximal, characteristic
4#17795C211yesmaximal, characteristic
5#17648C211yesmaximal, characteristic
6#8919C218yesmaximal, complemented
7#8074C211yesmaximal, characteristic
8#8070C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10#2987C2214yes—
11#2986—22no—
12#2985—22no—
13#2985—22no—
14Q8×M16—22no—
15Q8×M16—22no—
16#2921—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2283C2211yescharacteristic
23#1995C2214yes—
24#1986—22no—
25#1826—22no—
26#1574—22no—
27#1572—22no—
28#1569—22no—
29#1569C2214yes—
30#1569C2214yes—
31#1564—22no—
32#1531C412yescomplemented
33#1531C412yescomplemented
34#1528C414yescomplemented
35#1526C2214yes—
36#1300C2214yes—
37#1300C2214yes—
38C22×M16—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533D412yes—
45#533Q812yes—
46#533C2312yes—
47C2×C4⋊Q8Q811yescharacteristic
48C2×C4⋊Q8Q811yescharacteristic
49C2×#125C2×C412yes—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×#117D412yes—
55C22×C42C2311yescharacteristic
56#446—24no—
57#446—24no—
58#446—28no—
59#446—28no—
60C8×Q8—24no—
61C8×Q8—24no—
62#433—28no—
63#433—28no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432C2×C412yes—
69#432C2×C412yes—
70#432C2312yes—
71#432C2312yes—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431D412yes—
79#431D412yes—
80#423—24no—
81#423—21no—
82#423D412yes—
83#423D412yes—
84#423C2312yes—
85#407—21no—
86#407—21no—
87#407—21no—
88#407—21no—
89C4×M16—24no—
90C4×M16—21no—
91C4×M16—21no—
92C4×M16—21no—
93C4×M16—21no—
94C4×M16—21no—
95C4×M16C2×C412yes—
96C4×M16C2312yes—
97C4×M16C2312yes—
98C2×#97C2×C411yescharacteristic
99C2×#97C2×C411yescharacteristic
100C2×C4×C8—21no—
101#383—21no—
102#382—21no—
103#381—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C22⋊C4—21no—
107#376C2312yes—
108#339—28no—
109#335—24no—
110#335—24no—
111#334—24no—
112#334—24no—
113#330C2×C418yes—
114#330C2×C418yes—
115C22×Q8—21no—
116C22×Q8—21no—
117C23×C4C2×D411yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—44no—
127C22×C8—21no—
128C22×C8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8C4⋊C412yes—
134C4⋊Q8C4⋊C412yes—
135C4⋊Q8C2×Q812yes—
136C4⋊Q8C2×Q812yes—
137#125—22no—
138#125—22no—
139#125C4⋊C414yes—
140#125C22×C414yes—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157#117—22no—
158#117—22no—
159#117C4⋊C414yes—
160#117C2×D414yes—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4C4⋊C412yes—
167C2×C4⋊C4C4⋊C412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—41no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×Q811yescharacteristic
180C2×C42C2×Q811yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42C2411yescharacteristic
184C4⋊C8—28no—
185C4⋊C8—28no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8C2×D414yes—
193C4⋊C8C2×D414yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202#97—28no—
203#97—21no—
204#97—21no—
205#97—22no—
206#97—22no—
207#97—41no—
208#97C22×C412yes—
209#97C22×C412yes—
210#97C22×C414yes—
211C4×C8—28no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—41no—
218#95—22no—
219#95—22no—
220#95—41no—
221#95—41no—
222C24#12311yescharacteristic
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—82no—
228C2×Q8—82no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—81no—
235C22×C4C2×C4⋊C411yescharacteristic
236C22×C4C2×C4⋊C411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4C22⋊Q811yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253M16—44no—
254M16—44no—
255M16—44no—
256M16—44no—
257M16—44no—
258M16—82no—
259C2×C8—24no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—42no—
266C2×C8—42no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—82no—
278C4⋊C4C2×C4⋊C414yes—
279C4⋊C4C2×C4⋊C414yes—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—82no—
283C42—24no—
284C42—24no—
285C42—44no—
286C42C2×C4⋊C411yescharacteristic
287C42C2×C4⋊C411yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42#12011yescharacteristic
293C42#12011yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42C22⋊Q811yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42C23×C411yescharacteristic
299C42C22×D411yescharacteristic
300C42C22×D411yescharacteristic
301C42C22×Q811yescharacteristic
302C23—41no—
303C23—41no—
304C23—41no—
305C23#38511yescharacteristic
306C23#38511yescharacteristic
307C23C2×#12311yescharacteristic
308Q8—44no—
309Q8—44no—
310Q8—82no—
311Q8—82no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#38511yescharacteristic
338C2×C4#38511yescharacteristic
339C2×C4#38511yescharacteristic
340C2×C4#38511yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4C22×C4⋊C411yescharacteristic
344C2×C4C2×C4×D411yescharacteristic
345C2×C4C2×C4×D411yescharacteristic
346C2×C4C2×#12011yescharacteristic
347C2×C4C2×C22⋊Q811yescharacteristic
348C2×C4C2×C22⋊Q811yescharacteristic
349C8—24no—
350C8—42no—
351C8—44no—
352C8—44no—
353C8—44no—
354C8—82no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—82no—
365C22C2×#38511yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—44no—
377C4—44no—
378C4—162no—
379C4#186112yes—
380C4#302012yes—
381C2—41no—
382C2—41no—
383C2—41no—
384C2#2085711yescharacteristic
385C2C2×#186111yescharacteristic
386C2#2086711yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8919C2semidirect
32. #1531C4semidirect
33. #1531C4semidirect
34. #1528C4semidirect

Maximal subgroups

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

subgroupindex
#602492
#218452
#178232
#177952
#176482
#89192
#80742
#80702