Finite Group Databasebeta

← #193171

Subgroups — #193171

Order 512.

753 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic
3#17823C211yesmaximal, characteristic
4#17821C211yesmaximal, characteristic
5#17664C211yesmaximal, characteristic
6#8919C218yesmaximal, complemented
7#8070C211yesmaximal, characteristic
8#8069C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10#2987C2214yes—
11#2985—22no—
12#2984—22no—
13#2983—22no—
14#2962—22no—
15#2961—22no—
16#2924—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#432C2211yescharacteristic
22#2283C2211yescharacteristic
23#1995C2214yes—
24#1986—22no—
25#1826—22no—
26#1574—22no—
27#1573—22no—
28#1572—22no—
29#1572—22no—
30#1569C2214yes—
31#1569C2214yes—
32#1537C414yescomplemented
33#1534C414yescomplemented
34#1526C2214yes—
35#1300C2214yes—
36#1300C2214yes—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533D412yes—
43#533Q812yes—
44#533C2312yes—
45C2×C4⋊Q8Q811yescharacteristic
46C2×C4⋊Q8Q811yescharacteristic
47C2×#125C2×C412yes—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117D412yes—
53C22×C42C2311yescharacteristic
54#446—28no—
55#446—24no—
56#446—28no—
57#446—28no—
58C8×Q8—24no—
59#433—28no—
60#432—24no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432C2×C412yes—
67#432C2×C412yes—
68#432C2×C412yes—
69#432C2×C412yes—
70#432D412yes—
71#432C2312yes—
72#432C2312yes—
73#431—24no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431D412yes—
82#424—24no—
83#423—21no—
84#423—21no—
85#423D412yes—
86#423C2312yes—
87C2×C4⋊C8D412yes—
88#407—21no—
89#407—21no—
90#407—21no—
91#407—21no—
92C2×C22⋊C8—21no—
93#405—24no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16C2312yes—
97C4×M16C2312yes—
98C2×#97—21no—
99#383—21no—
100#382—21no—
101#381—21no—
102#380—21no—
103C4×C4⋊C4—21no—
104C4×C22⋊C4—21no—
105#376C2312yes—
106#339—28no—
107#335—24no—
108#335—24no—
109#334—24no—
110#334—24no—
111#330C2×C4116yes—
112C22×Q8—21no—
113C22×Q8—21no—
114C23×C4C2×D411yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C22×C8—22no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C4⋊C412yes—
127C4⋊Q8C4⋊C412yes—
128C4⋊Q8C2×Q812yes—
129C4⋊Q8C2×Q812yes—
130#125—22no—
131#125—22no—
132#125C4⋊C414yes—
133#125C22×C414yes—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150#117—22no—
151#117—22no—
152#117C4⋊C414yes—
153#117C2×D414yes—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4C4⋊C412yes—
160C2×C4⋊C4C4⋊C412yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—41no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×Q811yescharacteristic
173C2×C42C2×Q811yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42C2411yescharacteristic
177C4⋊C8—28no—
178C4⋊C8—28no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8C2×D414yes—
188C4⋊C8C2×D414yes—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199#97—28no—
200#97—22no—
201#97—22no—
202#97—22no—
203#97—41no—
204#97C22×C414yes—
205#97C22×C414yes—
206C4×C8—28no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—41no—
211#95—22no—
212#95—22no—
213#95—41no—
214#95—41no—
215C24#12311yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—82no—
221C2×Q8—82no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4C2×C4⋊C411yescharacteristic
229C22×C4C2×C4⋊C411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4C22⋊Q811yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C2×#4111yescharacteristic
245C22×C4C2×#4111yescharacteristic
246M16—44no—
247M16—44no—
248M16—82no—
249M16—82no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—44no—
255C2×C8—42no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—82no—
267C4⋊C4C2×C4⋊C414yes—
268C4⋊C4C2×C4⋊C414yes—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—82no—
272C42—24no—
273C42—24no—
274C42—44no—
275C42C2×C4⋊C411yescharacteristic
276C42C2×C4⋊C411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42#12011yescharacteristic
282C42#12011yescharacteristic
283C42C22⋊Q811yescharacteristic
284C42C22⋊Q811yescharacteristic
285C42C22⋊Q811yescharacteristic
286C42C22⋊Q811yescharacteristic
287C42C23×C411yescharacteristic
288C42C22×D411yescharacteristic
289C42C22×D411yescharacteristic
290C42C22×Q811yescharacteristic
291C23—41no—
292C23—41no—
293C23—41no—
294C23#38511yescharacteristic
295C23#38511yescharacteristic
296C23C2×#12311yescharacteristic
297Q8—44no—
298Q8—82no—
299Q8—82no—
300Q8—82no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4#38511yescharacteristic
327C2×C4#38511yescharacteristic
328C2×C4#38511yescharacteristic
329C2×C4#38511yescharacteristic
330C2×C4#38511yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4C22×C4⋊C411yescharacteristic
333C2×C4C2×C4×D411yescharacteristic
334C2×C4C2×C4×D411yescharacteristic
335C2×C4C2×#12011yescharacteristic
336C2×C4C2×C22⋊Q811yescharacteristic
337C2×C4C2×C22⋊Q811yescharacteristic
338C8—44no—
339C8—44no—
340C8—44no—
341C8—44no—
342C8—82no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—82no—
353C22C2×#38511yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—44no—
366C4—44no—
367C4—162no—
368C4#302012yes—
369C2—41no—
370C2—41no—
371C2—41no—
372C2#2085711yescharacteristic
373C2#2086711yescharacteristic
374C2#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. #1537C4semidirect
33. #1534C4semidirect

Maximal subgroups

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

subgroupindex
#602512
#218452
#178232
#178212
#176642
#89192
#80702
#80692