Finite Group Databasebeta

← #109679

Subgroups — #109679

Order 512.

701 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21932C211yesmaximal, characteristic
2#20523C211yesmaximal, characteristic
3#8057C211yesmaximal, characteristic
4#8051C211yesmaximal, characteristic
5C2×#1297C211yesmaximal, characteristic
6#8012C211yesmaximal, characteristic
7#7927C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×C4×C22⋊C4C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1851C414yescomplemented
16#1850—22no—
17#1797—22no—
18C8×C22⋊C4—22no—
19C8×C22⋊C4—22no—
20C4×C22⋊C8—22no—
21#1298—22no—
22#1297—22no—
23#1297C412yescomplemented
24#1297C412yescomplemented
25#1294C414yescomplemented
26#1282—22no—
27C24×C4C2311yescharacteristic
28C22×M16—21no—
29C22×M16C2×C412yes—
30C23×C8—21no—
31C23×C8D412yes—
32C2×#117—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C42—21no—
36C22×C42—21no—
37C2×C4⋊C8—24no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407C2×C412yes—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8C2×C412yes—
58C2×C22⋊C8C2×C412yes—
59C2×C22⋊C8D412yes—
60C2×C22⋊C8D412yes—
61C2×C22⋊C8Q812yes—
62C2×C4×C8—24no—
63#381—21no—
64#380—21no—
65#380C2×C412yes—
66#379—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C4×C22⋊C4C2×C411yescharacteristic
74C4×C22⋊C4C2×C411yescharacteristic
75C4×C22⋊C4C2×C412yes—
76C4×C22⋊C4C2×C412yes—
77C4×C22⋊C4D412yes—
78C2×#95C2×C412yes—
79C2×#95Q812yes—
80#336—24no—
81#336—24no—
82C25#4111yescharacteristic
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—82no—
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×Q811yescharacteristic
94C23×C4C2×Q811yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—41no—
104C2×M16—82no—
105C2×M16—82no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—41no—
113C22×C8—82no—
114C22×C8—82no—
115C22×C8C22⋊C414yes—
116#117—82no—
117C2×C4⋊C4—22no—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—21no—
131C2×C42—22no—
132C2×C42—21no—
133C2×C42—41no—
134C2×C42—82no—
135C2×C42—82no—
136C4⋊C8—84no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8C4⋊C414yes—
153C4×C8—84no—
154#95—22no—
155#95—22no—
156#95—22no—
157#95—41no—
158#95C4212yes—
159#95C4212yes—
160#95C4⋊C414yes—
161C24—24no—
162C24—42no—
163C24—81no—
164C24#11711yescharacteristic
165C24C4×D411yescharacteristic
166C24C4×D411yescharacteristic
167C24C22⋊Q811yescharacteristic
168C24#12311yescharacteristic
169C24#12511yescharacteristic
170C24#12611yescharacteristic
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4C2×C4211yescharacteristic
196C22×C4C2×C22⋊C411yescharacteristic
197C22×C4C2×C4⋊C411yescharacteristic
198C22×C4C2×C4⋊C411yescharacteristic
199C22×C4#11711yescharacteristic
200C22×C4#11711yescharacteristic
201C22×C4C4×D411yescharacteristic
202C22×C4C4×D411yescharacteristic
203C22×C4C4×D411yescharacteristic
204C22×C4C4×D411yescharacteristic
205C22×C4C4×D411yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4C4×Q811yescharacteristic
208C22×C4C4×Q811yescharacteristic
209C22×C4C4×Q811yescharacteristic
210C22×C4C4×Q811yescharacteristic
211C22×C4#12011yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4C22⋊Q811yescharacteristic
215C22×C4C22⋊Q811yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12411yescharacteristic
220C22×C4#12511yescharacteristic
221C22×C4#12611yescharacteristic
222C22×C4C4⋊Q811yescharacteristic
223M16—82no—
224M16—82no—
225M16—82no—
226M16—82no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—44no—
230C2×C8—44no—
231C2×C8—44no—
232C2×C8—44no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—82no—
236C2×C8—82no—
237C2×C8—82no—
238C2×C8—82no—
239C4⋊C4—82no—
240C22⋊C4—28no—
241C22⋊C4—28no—
242C22⋊C4—24no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C42—44no—
249C42—44no—
250C42—82no—
251C42—82no—
252C42—81no—
253C42—81no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—88no—
275C23C4×C22⋊C411yescharacteristic
276C23#38511yescharacteristic
277C23#38711yescharacteristic
278C23#38711yescharacteristic
279C23#38711yescharacteristic
280C23#38811yescharacteristic
281C23#38911yescharacteristic
282C23#44314yes—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—28no—
291C2×C4—24no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—88no—
321C2×C4—88no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4C4×C4⋊C411yescharacteristic
326C2×C4C4×C4⋊C411yescharacteristic
327C2×C4#38511yescharacteristic
328C2×C4#38611yescharacteristic
329C2×C4#38711yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#38911yescharacteristic
332C2×C4#39111yescharacteristic
333C8—44no—
334C8—44no—
335C8—82no—
336C8—82no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#143611yescharacteristic
362C22#176412yes—
363C22#186112yes—
364C22#193712yes—
365C22#194512yes—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—48no—
374C4—44no—
375C4—82no—
376C4—162no—
377C4#191312yes—
378C2—22no—
379C2—22no—
380C2—22no—
381C2—22no—
382C2—41no—
383C2—41no—
384C2—41no—
385C2#1088211yescharacteristic
386C2#1089411yescharacteristic
387C2#1091511yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22#1436
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1851C4semidirect
23. #1297C4semidirect
24. #1297C4semidirect
25. #1294C4semidirect

Maximal subgroups

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

subgroupindex
#219322
#205232
#80572
#80512
C2×#12972
#80122
#79272