Finite Group Databasebeta

← #99669

Subgroups — #99669

Order 512.

849 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 849 conjugacy classes of subgroups — too many to draw. Only its 146 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
1C2×#2016C211yesmaximal, characteristic
2#14471C211yesmaximal, characteristic
3#11384C212yesmaximal
4C2×#1272C211yesmaximal, characteristic
5#7025C212yesmaximal
6C23×Q16—21no—
7C22×C4⋊Q8C2211yescharacteristic
8C22×#103—21no—
9C2×#391C2211yescharacteristic
10#2016—22no—
11#2016C414yescomplemented
12C2×#340—22no—
13C2×#340—22no—
14C2×#336C2212yes—
15C2×#336C2211yescharacteristic
16#1446—21no—
17#1442—22no—
18#1436—21no—
19#1436—22no—
20C4⋊#95C2212yes—
21#1272—22no—
22C23×Q8—21no—
23C23×Q8D411yescharacteristic
24C22×Q16—24no—
25C22×Q16—42no—
26C22×Q16—42no—
27C22×Q16—44no—
28C22×Q16—44no—
29C23×C8—21no—
30C23×C8—22no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—44no—
34C2×C4⋊Q8C2×C412yes—
35C22×C4⋊C4—22no—
36C22×C4⋊C4D412yes—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D412yes—
39C22×C42—21no—
40C22×C42C2311yescharacteristic
41C2×#103—24no—
42C2×#103—42no—
43C2×#103—42no—
44#391—22no—
45#391C2×C412yes—
46#391D412yes—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—22no—
54C2×#95—21no—
55#340—44no—
56#340—44no—
57#336—21no—
58#336—22no—
59#336—22no—
60#336C2×C412yes—
61C22×Q8—21no—
62C22×Q8—24no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—42no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8—42no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8C22⋊C412yes—
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—41no—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D412yes—
85C2×Q16—48no—
86C2×Q16—48no—
87C2×Q16—48no—
88C2×Q16—416no—
89C2×Q16—416no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—24no—
94C22×C8—42no—
95C22×C8—42no—
96C22×C8—42no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8—42no—
102C4⋊Q8—44no—
103C4⋊Q8—48no—
104C4⋊Q8—44no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4C22⋊C412yes—
122C2×C4⋊C4Q1614yescomplemented
123C2×C42—42no—
124C2×C42—81no—
125C2×C42—82no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42#4112yes—
129#103—48no—
130#95—24no—
131#95—24no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—81no—
137#95—81no—
138#95—81no—
139#95—82no—
140#95—82no—
141C24#12011yescharacteristic
142C2×Q8—24no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—42no—
146C2×Q8—44no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—81no—
161C2×Q8—81no—
162C2×Q8—82no—
163C2×Q8—82no—
164C2×Q8—82no—
165C2×Q8—82no—
166C2×Q8—81no—
167C2×Q8—81no—
168C2×Q8—82no—
169C2×Q8—82no—
170C2×Q8—82no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—82no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C4×D412yes—
209C22×C4#12112yes—
210C22×C4#12311yescharacteristic
211C22×C4C2×Q1612yes—
212C22×C4#13512yes—
213C22×C4#13512yes—
214C22×C4AGL(1,ℤ/8)12yes—
215C22×C4#13712yes—
216C22×C4#13712yes—
217Q16—432no—
218C2×C8—24no—
219C2×C8—44no—
220C2×C8—44no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C4⋊C4—28no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—82no—
238C4⋊C4—81no—
239C4⋊C4—82no—
240C42—41no—
241C42—41no—
242C42—42no—
243C42—84no—
244C23#38611yescharacteristic
245C23#44912yes—
246C23#45112yes—
247C23#45212yes—
248C23#45311yescharacteristic
249C23#45311yescharacteristic
250C23#45512yes—
251C23#45612yes—
252C23B(4,𝔽2)12yes—
253Q8—44no—
254Q8—44no—
255Q8—44no—
256Q8—82no—
257Q8—82no—
258Q8—82no—
259Q8—84no—
260Q8—84no—
261Q8—84no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
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—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—81no—
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#41611yescharacteristic
312C2×C4#41811yescharacteristic
313C2×C4C4×Q1612yes—
314C2×C4#44112yes—
315C2×C4#46212yes—
316C2×C4#46412yes—
317C2×C4#48212yes—
318C8—44no—
319C8—84no—
320C22#160012yes—
321C22#160012yes—
322C22#160212yes—
323C22#160312yes—
324C22#160512yes—
325C22#160612yes—
326C22#163212yes—
327C22#163311yescharacteristic
328C22#163411yescharacteristic
329C22#164012yes—
330C22#164412yes—
331C22#164812yes—
332C22#165212yes—
333C22#165911yescharacteristic
334C22#166011yescharacteristic
335C22#166112yes—
336C22#187511yescharacteristic
337C22#187812yes—
338C22#188212yes—
339C22#188711yescharacteristic
340C22#188911yescharacteristic
341C4—21no—
342C4—21no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—22no—
347C4—22no—
348C4—82no—
349C4—82no—
350C4—84no—
351C4—84no—
352C4—162no—
353C2#778212yes—
354C2#778312yes—
355C2#778412yes—
356C2#778512yes—
357C2#898612yes—
358C2#900311yescharacteristic
359C2#902011yescharacteristic
360C2#904012yes—
361C2#906611yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2016C4semidirect
122. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#20162
#144712
#113842
C2×#12722
#70252