Finite Group Databasebeta

← #189740

Subgroups — #189740

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2343C211yesmaximal, characteristic
2#20717C212yesmaximal
3#8703C212yesmaximal, complemented
4#8702C212yesmaximal
5#8699C212yesmaximal, complemented
6#8698C212yesmaximal
7#7916C212yesmaximal, complemented
8#7912C211yesmaximal, characteristic, complemented
9#7908C211yesmaximal, characteristic, complemented
10C22×#117C2211yescharacteristic
11C2×#432C2212yes—
12C2×#431C2212yes—
13#2343—24no—
14#2343—24no—
15C2×#387C412yes—
16C2×#387C412yes—
17C2×#385C412yes—
18C2×#383C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#381C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C412yes—
23C2×#376C2211yescharacteristic
24C22×#95C2211yescharacteristic
25#1817C2214yes—
26#1798—24no—
27#1779C2214yes—
28#1750—24no—
29#1280C2212yes—
30#1280C2212yes—
31#1280C2212yes—
32#1280C2212yes—
33#1280C2214yes—
34#1280C2214yes—
35C24×C4C2×C412yes—
36C24×C4C2311yescharacteristic
37C2×#117—22no—
38C2×#117—24no—
39C2×#117—22no—
40C2×#117D414yes—
41C2×#117D414yes—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C412yes—
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4C2×C412yes—
49C22×C42C2×C412yes—
50C22×C42C2×C412yes—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53#432—24no—
54#432—24no—
55#431—24no—
56#431—24no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C22⋊C8—28no—
60C2×C22⋊C8C2314yes—
61C2×C22⋊C8C2314yes—
62C2×#97—22no—
63C2×C4×C8—22no—
64#387—24no—
65#387—24no—
66#387—24no—
67#387—24no—
68#385—24no—
69#385—24no—
70#383—22no—
71#383—22no—
72#382—24no—
73#381—24no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C22⋊C4—22no—
77C4×C22⋊C4—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80#376—24no—
81C2×#95—24no—
82C2×#95C2×C412yes—
83C2×#95C2×C412yes—
84C2×#95C2×C412yes—
85C2×#95C2×C412yes—
86C2×#95C2×C412yes—
87C2×#95C2×C412yes—
88C2×#95C2311yescharacteristic
89C2×#95C2311yescharacteristic
90C2×#95C2311yescharacteristic
91C2×#95C2311yescharacteristic
92#336—28no—
93C25C22×C411yescharacteristic
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—24no—
100C23×C4—44no—
101C23×C4—42no—
102C23×C4—44no—
103C23×C4—82no—
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C412yes—
107C23×C4C22×C412yes—
108C23×C4C22×C412yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C412yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D414yes—
114C23×C4C2411yescharacteristic
115C22×C8—24no—
116C22×C8—24no—
117#117—28no—
118#117—28no—
119#117—28no—
120#117—28no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—82no—
130C2×C4⋊C4C22⋊C414yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C22⋊C412yes—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4C22⋊C412yes—
141C2×C22⋊C4C22⋊C412yes—
142C2×C22⋊C4C22⋊C414yes—
143C2×C42—22no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—24no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—42no—
152C2×C42—82no—
153C2×C42C2×D414yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42#4112yes—
157C2×C42#4114yes—
158C2×C42#4112yes—
159C4⋊C8—44no—
160C4⋊C8—44no—
161C22⋊C8—24no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C22⋊C8—24no—
165C22⋊C8—416no—
166#97—44no—
167C4×C8—44no—
168#95—24no—
169#95—24no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174#95—24no—
175#95—24no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180C24—21no—
181C24—21no—
182C24—24no—
183C24—22no—
184C24—22no—
185C24—41no—
186C24—41no—
187C24C2×C22⋊C412yes—
188C24C23×C411yescharacteristic
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—28no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—48no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—88no—
238C22×C4—88no—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C412yes—
245C22×C4#11712yes—
246C22×C4#11712yes—
247C22×C4#11712yes—
248C22×C4#11712yes—
249C22×C4#13114yes—
250C22×C4C22×D412yes—
251C22×C4C2×#4112yes—
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C2×C8—48no—
255C2×C8—48no—
256C2×C8—44no—
257C2×C8—44no—
258C4⋊C4—28no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—48no—
262C4⋊C4—84no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—28no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C42—24no—
269C42—24no—
270C42—44no—
271C42—44no—
272C42—82no—
273C42—82no—
274C42#12314yes—
275C42#12314yes—
276C42#12314yes—
277C42#12314yes—
278C23—22no—
279C23—21no—
280C23—21no—
281C23—24no—
282C23—24no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23—44no—
293C23—42no—
294C23—42no—
295C23—41no—
296C23—41no—
297C23—42no—
298C23#41014yes—
299C23C22×C22⋊C411yescharacteristic
300C23C2×#11711yescharacteristic
301C23C2×#11711yescharacteristic
302C23C2×#13112yes—
303C23#56812yes—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—88no—
350C2×C4—88no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#38114yes—
358C2×C4#38112yes—
359C2×C4#38112yes—
360C2×C4C2×#12312yes—
361C2×C4C2×#12312yes—
362C8—84no—
363C8—84no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—24no—
373C22—24no—
374C22—24no—
375C22—44no—
376C22—44no—
377C22—48no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22C2×#38111yescharacteristic
383C22#288012yes—
384C22C2×#41012yes—
385C22#288811yescharacteristic
386C22#288911yescharacteristic
387C22#292211yescharacteristic
388C22#292311yescharacteristic
389C22#292512yes—
390C4—24no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—44no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—82no—
400C4—82no—
401C4—82no—
402C2—21no—
403C2—21no—
404C2—22no—
405C2—44no—
406C2#1741212yes—
407C2#1741712yes—
408C2#2068411yescharacteristic
409C2#2068811yescharacteristic
410C2#2069511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8703C2semidirect
5. #8699C2semidirect
7. #7916C2semidirect
8. #7912C2semidirect
9. #7908C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23432
#207172
#87032
#87022
#86992
#86982
#79162
#79122
#79082