Finite Group Databasebeta

← #101015

Subgroups — #101015

Order 512.

668 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21930C211yesmaximal, characteristic
2#15150C211yesmaximal, characteristic
3#8723C212yesmaximal, complemented
4#8557C211yesmaximal, characteristic
5#7039C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#397—21no—
8C2×#397C412yes—
9C2×#384C2211yescharacteristic
10C2×#380—21no—
11C2×C4×C22⋊C4C2211yescharacteristic
12#1850—24no—
13#1797—21no—
14#1797C2212yes—
15#1797C2212yes—
16C2×#323—21no—
17#1445—21no—
18#1445C412yes—
19C4⋊#95—21no—
20C4⋊#95C2211yescharacteristic
21#1296—24no—
22#1280—21no—
23C24×C4D411yescharacteristic
24C23×C8—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C412yes—
29C22×C22⋊C4—22no—
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42C2311yescharacteristic
34C2×C4⋊C8—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8D414yes—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—42no—
41C2×C22⋊C8—42no—
42#397—44no—
43#397—42no—
44#397—42no—
45#384—22no—
46#384—22no—
47#380—42no—
48#380—42no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C412yes—
57C2×#95D411yescharacteristic
58#336—22no—
59#336—22no—
60#336—22no—
61#323—42no—
62C25C22⋊C411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4D812yes—
75C23×C4SD1612yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C22×C8—24no—
80C22×C8—44no—
81C22×C8—42no—
82C22×C8—82no—
83C22×C8—82no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—44no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4D812yes—
110C2×C22⋊C4SD1612yes—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42C2×D412yes—
116C2×C42#4112yes—
117C4⋊C8—44no—
118C4⋊C8—44no—
119C4⋊C8—82no—
120C22⋊C8—44no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125#95—22no—
126#95—22no—
127#95—42no—
128#95—44no—
129#95—44no—
130#95—44no—
131#95—41no—
132#95—41no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—41no—
138C24—41no—
139C24—41no—
140C24—41no—
141C24#10012yes—
142C24#10212yes—
143C24#10212yes—
144C24C2×C22⋊C411yescharacteristic
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4#10212yes—
193C22×C4#10212yes—
194C22×C4C4×D412yes—
195C22×C4#12011yescharacteristic
196C22×C4#12112yes—
197C22×C4#12311yescharacteristic
198C22×C4C2×D811yescharacteristic
199C22×C4C2×D811yescharacteristic
200C22×C4C2×SD1611yescharacteristic
201C22×C4C2×SD1611yescharacteristic
202C22×C4#13512yes—
203C22×C4AGL(1,ℤ/8)11yescharacteristic
204C22×C4#13711yescharacteristic
205C2×C8—44no—
206C2×C8—44no—
207C2×C8—44no—
208C2×C8—88no—
209C2×C8—82no—
210C2×C8—82no—
211C2×C8—82no—
212C4⋊C4—84no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—44no—
224C22⋊C4—88no—
225C42—24no—
226C42—42no—
227C42—42no—
228C42—81no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23#33114yes—
256C23#38611yescharacteristic
257C23C2×#10011yescharacteristic
258C23#41311yescharacteristic
259C23C2×#10211yescharacteristic
260C23C2×#10211yescharacteristic
261C23#41611yescharacteristic
262C23#41711yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4#44711yescharacteristic
313C2×C4#44811yescharacteristic
314C2×C4#45512yes—
315C2×C4#45812yes—
316C2×C4#46512yes—
317C2×C4#46612yes—
318C2×C4#48011yescharacteristic
319C2×C4#48111yescharacteristic
320C8—84no—
321C8—162no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—24no—
335C22—44no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—82no—
345C22C2×#33111yescharacteristic
346C22#154011yescharacteristic
347C22#154711yescharacteristic
348C22#155111yescharacteristic
349C22#187411yescharacteristic
350C22#188511yescharacteristic
351C22#189011yescharacteristic
352C22#189311yescharacteristic
353C22#190011yescharacteristic
354C22#190911yescharacteristic
355C22#191511yescharacteristic
356C4—21no—
357C4—21no—
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—81no—
364C4—162no—
365C4—162no—
366C4#160712yes—
367C4#160812yes—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—21no—
372C2—44no—
373C2#902511yescharacteristic
374C2#913311yescharacteristic
375C2#919511yescharacteristic
376C2#939311yescharacteristic
377C2#945111yescharacteristic
378C2#947011yescharacteristic
379C2#950711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8723C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219302
#151502
#87232
#85572
#70392