Finite Group Databasebeta

← #149283

Subgroups — #149283

Order 512.

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

normal subgroups only

This group has 768 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×#1857C212yesmaximal
2#14501C211yesmaximal, characteristic
3C2×#1394C212yesmaximal
4C2×#1394C211yesmaximal, characteristic
5#10750C211yesmaximal, characteristic
6C2×#519C2211yescharacteristic
7C2×#428—22no—
8C22×#101C2211yescharacteristic
9C22×#101C2212yes—
10C2×#391C2212yes—
11C2×#391C2211yescharacteristic
12#1857—24no—
13#1857—24no—
14#1736—21no—
15C2×#339—22no—
16C2×#339—21no—
17#1442—22no—
18#1441—21no—
19#1436—22no—
20#1434—21no—
21#1394—24no—
22#1394C414yescomplemented
23C23×Q8C2311yescharacteristic
24C22×M16—22no—
25C22×M16—21no—
26#519—22no—
27#519—21no—
28#519—42no—
29#519D412yes—
30C2×C4×Q8—42no—
31C2×#117—22no—
32C2×#117—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4D412yes—
35C22×C4⋊C4D411yescharacteristic
36C22×C42D412yes—
37C22×C42D411yescharacteristic
38#428—28no—
39#428—44no—
40C2×#107—42no—
41C2×#106—42no—
42C2×#101—22no—
43C2×#101—22no—
44C2×#101—22no—
45C2×#101—22no—
46C2×#101—22no—
47C2×#101—21no—
48C2×#101C2×C414yes—
49C2×#101D414yes—
50C2×#101Q812yes—
51C2×#101Q814yes—
52#391—22no—
53#391—22no—
54#391—22no—
55#391—22no—
56#391—22no—
57#391C2×C412yes—
58#391Q812yes—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67#340—42no—
68#339—28no—
69#339—42no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8C22×C411yescharacteristic
75C22×Q8C2×D411yescharacteristic
76C22×Q8C2×D412yes—
77C22×Q8C2×Q811yescharacteristic
78C22×Q8C2×Q812yes—
79C22×Q8C2×Q811yescharacteristic
80C22×Q8#4112yes—
81C22×Q8#4111yescharacteristic
82C22×Q8#4111yescharacteristic
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D412yes—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—24no—
92C2×M16—22no—
93C2×M16—42no—
94C2×M16—42no—
95C2×M16—42no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C22×C8—42no—
102C22×C8—41no—
103C4×Q8—44no—
104C4×Q8—44no—
105C4×Q8—44no—
106C4×Q8—44no—
107#117—22no—
108#117—24no—
109#117—24no—
110#117—22no—
111#117—41no—
112#117—42no—
113#117—42no—
114#117—41no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—81no—
127C2×C4⋊C4C4⋊C412yes—
128C2×C22⋊C4—41no—
129C2×C42—22no—
130C2×C42—21no—
131C2×C42—22no—
132C2×C42—42no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42C4⋊C412yes—
139#107—48no—
140#106—48no—
141#101—24no—
142#101—24no—
143#101—24no—
144#101—24no—
145#101—42no—
146#101C4⋊C418yescomplemented
147#95—24no—
148#95—24no—
149#95—44no—
150#95—44no—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—81no—
156#95—81no—
157#95—81no—
158C24#12011yescharacteristic
159C2×Q8—41no—
160C2×Q8—42no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—41no—
168C2×Q8—42no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—41no—
177C2×Q8C2×C4⋊C412yes—
178C2×Q8C4×D411yescharacteristic
179C2×Q8C4×Q811yescharacteristic
180C2×Q8C22⋊Q812yes—
181C2×Q8C22⋊Q812yes—
182C2×Q8#12312yes—
183C2×Q8#12511yescharacteristic
184C2×Q8#12512yes—
185C2×Q8C4⋊Q812yes—
186C2×Q8C4⋊Q811yescharacteristic
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4C2×C4⋊C411yescharacteristic
221C22×C4C4×D412yes—
222C22×C4#12012yes—
223C22×C4#12111yescharacteristic
224C22×C4#12112yes—
225C22×C4C22⋊Q812yes—
226C22×C4C22⋊Q811yescharacteristic
227C22×C4C22⋊Q812yes—
228C22×C4#12311yescharacteristic
229C22×C4#12711yescharacteristic
230C22×C4C4⋊Q811yescharacteristic
231C22×C4C4⋊Q812yes—
232M16—28no—
233M16—44no—
234M16—44no—
235M16—44no—
236M16—42no—
237M16—42no—
238M16—81no—
239M16—81no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—81no—
245C2×C8—81no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—81no—
255C4⋊C4—82no—
256C4⋊C4—84no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C42—42no—
260C42—44no—
261C42—42no—
262C42—44no—
263C42—42no—
264C42—44no—
265C42—81no—
266C42—82no—
267C23—22no—
268C23—22no—
269C23#38511yescharacteristic
270C23#39211yescharacteristic
271C23#39511yescharacteristic
272C23B(4,𝔽2)11yescharacteristic
273C23B(4,𝔽2)11yescharacteristic
274C23#45811yescharacteristic
275C23#45811yescharacteristic
276Q8—42no—
277Q8—41no—
278Q8—41no—
279Q8—42no—
280Q8—42no—
281Q8—42no—
282Q8—41no—
283Q8—42no—
284Q8—42no—
285Q8—41no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#38411yescharacteristic
338C2×C4#38512yes—
339C2×C4#38911yescharacteristic
340C2×C4#38912yes—
341C2×C4#39512yes—
342C2×C4#39812yes—
343C2×C4#39811yescharacteristic
344C2×C4#39911yescharacteristic
345C8—48no—
346C8—82no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22#144411yescharacteristic
356C22#186711yescharacteristic
357C22#186711yescharacteristic
358C22#202111yescharacteristic
359C22#202211yescharacteristic
360C22#206211yescharacteristic
361C22#206211yescharacteristic
362C22#220312yes—
363C22#220412yes—
364C4—22no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—22no—
369C4—22no—
370C4—22no—
371C4—21no—
372C4—84no—
373C4—82no—
374C4—84no—
375C4—82no—
376C4—162no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2#1134411yescharacteristic
382C2#1297111yescharacteristic
383C2#1297211yescharacteristic
384C2#1335812yes—
385C2#1346512yes—
subgroupG/H
center, socleC23B(4,𝔽2)
commutatorC22×Q8C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24#120
LCSG ▷ C22×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
22. #1394C4semidirect
146. #101C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18572
#145012
C2×#13942
#107502