Finite Group Databasebeta

← #124308

Subgroups — #124308

Order 512.

1180 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 116 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#13922C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7131C214yesmaximal, complemented
5C24×Q8C411yescharacteristic
6C2×#391—21no—
7C2×#391—21no—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C4×C22⋊C8—21no—
12C2×#323—21no—
13C2×#323—21no—
14#1446C412yes—
15#1446C2212yes—
16C22⋊#95—21no—
17C22⋊#95—21no—
18#1280C2212yes—
19#1280C2212yes—
20C23×Q8—21no—
21C23×Q8—22no—
22C23×Q8—41no—
23C23×Q8—41no—
24C23×Q8—42no—
25C23×Q8—44no—
26C23×Q8—44no—
27C23×Q8C814yescomplemented
28C23×Q8C2×C412yes—
29C24×C4—21no—
30C24×C4C2×C411yescharacteristic
31C22×C4⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4D411yescharacteristic
34C22×C22⋊C4D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C2×C4⋊C8—41no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C4×C8—41no—
41#391—44no—
42#391—44no—
43#380—42no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95C2×C412yes—
51C2×#95C2×C411yescharacteristic
52C2×#95C2311yescharacteristic
53#336—41no—
54#336—41no—
55#323—24no—
56#323—42no—
57C25C22⋊C411yescharacteristic
58C22×Q8—24no—
59C22×Q8—24no—
60C22×Q8—24no—
61C22×Q8—22no—
62C22×Q8—44no—
63C22×Q8—44no—
64C22×Q8—44no—
65C22×Q8—44no—
66C22×Q8—42no—
67C22×Q8—42no—
68C22×Q8—44no—
69C22×Q8—44no—
70C22×Q8—42no—
71C22×Q8—42no—
72C22×Q8—416no—
73C22×Q8—416no—
74C22×Q8—44no—
75C22×Q8—44no—
76C22×Q8—432no—
77C22×Q8—48no—
78C22×Q8—48no—
79C22×Q8—44no—
80C22×Q8—44no—
81C22×Q8—41no—
82C22×Q8—41no—
83C22×Q8—42no—
84C22×Q8C2×C814yes—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—24no—
90C23×C4—22no—
91C23×C4—44no—
92C23×C4—44no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C2×C812yes—
101C23×C4M1612yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C22×C8—42no—
106C22×C8—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4SD1612yes—
119C2×C22⋊C4Q1612yes—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—42no—
125C2×C42—42no—
126C4⋊C8—82no—
127C22⋊C8—24no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C4×C8—82no—
132#95—41no—
133#95—41no—
134#95—42no—
135#95—44no—
136#95—44no—
137#95—44no—
138#95—44no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24C22⋊C812yes—
148C24#10312yes—
149C24#10412yes—
150C24C2×C22⋊C411yescharacteristic
151C2×Q8—216no—
152C2×Q8—48no—
153C2×Q8—44no—
154C2×Q8—44no—
155C2×Q8—48no—
156C2×Q8—48no—
157C2×Q8—48no—
158C2×Q8—48no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—416no—
162C2×Q8—432no—
163C2×Q8—432no—
164C2×Q8—416no—
165C2×Q8—416no—
166C2×Q8—416no—
167C2×Q8—416no—
168C2×Q8—44no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—48no—
199C22×C4—416no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—48no—
205C22×C4—416no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C22⋊C812yes—
230C22×C4#9911yescharacteristic
231C22×C4#9911yescharacteristic
232C22×C4#10012yes—
233C22×C4#10312yes—
234C22×C4C22×C811yescharacteristic
235C22×C4C2×M1611yescharacteristic
236C22×C4C2×SD1611yescharacteristic
237C22×C4C2×Q1611yescharacteristic
238C22×C4#13712yes—
239C2×C8—44no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—84no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C22⋊C4—84no—
250C42—42no—
251C42—42no—
252C42—81no—
253C42—81no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23Q8⋊C814yes—
281C23C2×C22⋊C811yescharacteristic
282C23C2×#9911yescharacteristic
283C23C2×#10011yescharacteristic
284C23C2×#10311yescharacteristic
285C23#41911yescharacteristic
286C23C2×#10411yescharacteristic
287C23#42111yescharacteristic
288Q8—416no—
289Q8—48no—
290Q8—48no—
291Q8—48no—
292Q8—48no—
293Q8—416no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—28no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—432no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4#32312yes—
347C2×C4#32312yes—
348C2×C4AΣL(1,𝔽16)12yes—
349C2×C4AΣL(1,𝔽16)12yes—
350C8—84no—
351C8—162no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—24no—
365C22—44no—
366C22—44no—
367C22—44no—
368C22—44no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22C2×#32311yescharacteristic
376C22C2×Q8⋊C811yescharacteristic
377C22#147911yescharacteristic
378C22#150811yescharacteristic
379C22#152011yescharacteristic
380C22#152711yescharacteristic
381C22#154811yescharacteristic
382C22C2×AΣL(1,𝔽16)11yescharacteristic
383C22C2×AΣL(1,𝔽16)11yescharacteristic
384C22#212911yescharacteristic
385C22#213511yescharacteristic
386C4—22no—
387C4—42no—
388C4—41no—
389C4—41no—
390C4—42no—
391C4—48no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4—162no—
396C4#131614yes—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—44no—
402C2#757511yescharacteristic
403C2C2×#131611yescharacteristic
404C2#1143411yescharacteristic
405C2#1178511yescharacteristic
406C2#1179111yescharacteristic
407C2#1189811yescharacteristic
408C2#1190411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7131C2semidirect
27. C23×Q8C8semidirect

Maximal subgroups

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

subgroupindex
#139222
#85572
C4×#3232
#71312