Finite Group Databasebeta

← #147343

Subgroups — #147343

Order 512.

810 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 162 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×#1857C211yesmaximal, characteristic
2#13826C211yesmaximal, characteristic
3C2×#1394C211yesmaximal, characteristic
4C2×#1394C212yesmaximal
5#10750C212yesmaximal
6C2×#519C2211yescharacteristic
7C2×#428—21no—
8C22×#101C2212yes—
9C22×#101C2211yescharacteristic
10C2×#391C2211yescharacteristic
11C2×#391C2212yes—
12#1857—22no—
13#1857C414yescomplemented
14#1736—22no—
15C2×#339—21no—
16C2×#339—22no—
17#1436—21no—
18#1436—22no—
19C4×#95—21no—
20C4×#95—22no—
21#1394—22no—
22#1394C418yescomplemented
23C23×Q8C2311yescharacteristic
24C22×M16—21no—
25C22×M16—22no—
26#519—21no—
27#519—22no—
28#519—42no—
29#519C2×C412yes—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×#117—21no—
33C2×#117—22no—
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4D412yes—
36C22×C42—21no—
37C22×C42—22no—
38C22×C42D411yescharacteristic
39C22×C42D412yes—
40#428—24no—
41#428—42no—
42C2×#107—41no—
43C2×#106—41no—
44C2×#101—21no—
45C2×#101—22no—
46C2×#101—22no—
47C2×#101—22no—
48C2×#101—22no—
49C2×#101—22no—
50C2×#101—22no—
51C2×#101C2×C414yes—
52C2×#101C2×C412yes—
53C2×#101C2×C414yes—
54#391—22no—
55#391—22no—
56#391—22no—
57#391C2×C412yes—
58#391C2×C414yes—
59#391D412yes—
60#391Q814yes—
61C2×#95—21no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95—21no—
68#340—42no—
69#340—42no—
70#339—24no—
71#339—44no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—42no—
75C22×Q8C22×C411yescharacteristic
76C22×Q8C22×C412yes—
77C22×Q8C2×D411yescharacteristic
78C22×Q8C2×Q812yes—
79C22×Q8#4112yes—
80C22×Q8#4112yes—
81C22×Q8#4112yes—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D412yes—
88C2×M16—22no—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—42no—
92C2×M16—41no—
93C2×M16—42no—
94C2×M16—42no—
95C2×M16—42no—
96C2×M16—42no—
97C2×M16—42no—
98C22×C8—41no—
99C22×C8—42no—
100C4×Q8—42no—
101C4×Q8—42no—
102C4×Q8—42no—
103C4×Q8—42no—
104C4×Q8—44no—
105C4×Q8—44no—
106#117—24no—
107#117—22no—
108#117—22no—
109#117—24no—
110#117—42no—
111#117—41no—
112#117—42no—
113#117—41no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4C22⋊C412yes—
125C2×C4⋊C4C4⋊C414yes—
126C2×C22⋊C4—41no—
127C2×C42—22no—
128C2×C42—21no—
129C2×C42—41no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—42no—
134C2×C42—44no—
135C2×C42—44no—
136C2×C42—44no—
137C2×C42—81no—
138C2×C42—82no—
139C2×C42C22⋊C412yes—
140C2×C42C4⋊C414yes—
141#107—44no—
142#106—44no—
143#101—22no—
144#101—24no—
145#101—24no—
146#101—24no—
147#101—42no—
148#101—42no—
149#101C4214yescomplemented
150#95—24no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—42no—
158#95—44no—
159#95—81no—
160C24#12011yescharacteristic
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8C2×C4211yescharacteristic
176C2×Q8#11711yescharacteristic
177C2×Q8#11712yes—
178C2×Q8C4×D412yes—
179C2×Q8C4×Q812yes—
180C2×Q8C4×Q812yes—
181C2×Q8#12411yescharacteristic
182C2×Q8#12512yes—
183C2×Q8#12612yes—
184C2×Q8C4⋊Q811yescharacteristic
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C4⋊C412yes—
220C22×C4C4×D412yes—
221C22×C4C4×D412yes—
222C22×C4C4×D412yes—
223C22×C4#12112yes—
224C22×C4C22⋊Q812yes—
225C22×C4C22⋊Q812yes—
226C22×C4#12311yescharacteristic
227C22×C4#12312yes—
228M16—24no—
229M16—44no—
230M16—42no—
231M16—44no—
232M16—44no—
233M16—82no—
234M16—82no—
235C2×C8—42no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—82no—
239C2×C8—82no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—82no—
251C4⋊C4—81no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—42no—
255C42—44no—
256C42—42no—
257C42—44no—
258C42—42no—
259C42—44no—
260C42—82no—
261C42—81no—
262C42—84no—
263C42—88no—
264C23—22no—
265C23—21no—
266C23—21no—
267C23#38512yes—
268C23#38611yescharacteristic
269C23B(4,𝔽2)11yescharacteristic
270C23B(4,𝔽2)11yescharacteristic
271C23#45812yes—
272Q8—41no—
273Q8—41no—
274Q8—42no—
275Q8—42no—
276Q8—42no—
277Q8—42no—
278Q8—41no—
279Q8—42no—
280Q8—42no—
281Q8—41no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—28no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4C4×C22⋊C411yescharacteristic
332C2×C4C4×C4⋊C412yes—
333C2×C4#38712yes—
334C2×C4#38712yes—
335C2×C4#38812yes—
336C2×C4#38912yes—
337C2×C4#39111yescharacteristic
338C8—44no—
339C8—84no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22#143611yescharacteristic
349C22#186712yes—
350C22#186712yes—
351C22#188911yescharacteristic
352C22#189011yescharacteristic
353C22#220412yes—
354C22#220411yescharacteristic
355C22#220511yescharacteristic
356C4—21no—
357C4—22no—
358C4—22no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—21no—
363C4—48no—
364C4—44no—
365C4—82no—
366C4—82no—
367C4—82no—
368C4—84no—
369C2—22no—
370C2—21no—
371C2—21no—
372C2#1091811yescharacteristic
373C2#1297212yes—
374C2#1297312yes—
375C2#1305711yescharacteristic
376C2#1305811yescharacteristic
subgroupG/H
center, socleC23B(4,𝔽2)
commutatorC2×Q8C2×C42
Frattini, cosocleC23×Q8C23
prime radicalC24#120
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1857C4semidirect
22. #1394C4semidirect
149. #101C42semidirect

Maximal subgroups

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

subgroupindex
C2×#18572
#138262
C2×#13942
#107502