Finite Group Databasebeta

← #124281

Subgroups — #124281

Order 512.

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

normal subgroups only

This group has 730 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#14130C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7122C214yesmaximal, complemented
5C23×C4⋊C4C411yescharacteristic
6C2×#389—21no—
7C2×#385—21no—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C4×C22⋊C8—21no—
12C2×#323—21no—
13C2×#323—21no—
14#1444C412yes—
15#1442C2212yes—
16C4⋊#95—21no—
17C22⋊#95—21no—
18#1280C2212yes—
19#1280C2212yes—
20C24×C4—21no—
21C24×C4C2×C411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—44no—
26C22×C4⋊C4—41no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4C2×C412yes—
29C22×C22⋊C4—21no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C4×C8—41no—
38#389—44no—
39#385—42no—
40#385—42no—
41#380—22no—
42#380C814yescomplemented
43C4×C22⋊C4—41no—
44C4×C22⋊C4—41no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95C2×C412yes—
49C2×#95C2×C411yescharacteristic
50C2×#95C2311yescharacteristic
51#336—41no—
52#336—41no—
53#323—24no—
54#323—42no—
55C25C22⋊C411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4—44no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—82no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C2×C812yes—
72C23×C4M1612yes—
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C22×C8—42no—
77C22×C8—42no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—48no—
85C2×C4⋊C4—44no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—82no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4D812yes—
102C2×C22⋊C4SD1612yes—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—42no—
108C2×C42—42no—
109C4⋊C8—82no—
110C22⋊C8—24no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C4×C8—82no—
115#95—22no—
116#95—41no—
117#95—41no—
118#95—44no—
119#95—42no—
120#95—42no—
121#95—82no—
122#95C2×C814yes—
123C24—21no—
124C24—21no—
125C24—21no—
126C24—21no—
127C24—41no—
128C24—41no—
129C24—41no—
130C24—41no—
131C24C22⋊C812yes—
132C24#10212yes—
133C24#10412yes—
134C24C2×C22⋊C411yescharacteristic
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—48no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—48no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—88no—
185C22×C4—88no—
186C22×C4—84no—
187C22×C4—84no—
188C22×C4C22⋊C812yes—
189C22×C4#9911yescharacteristic
190C22×C4#9911yescharacteristic
191C22×C4#10112yes—
192C22×C4#10212yes—
193C22×C4C22×C811yescharacteristic
194C22×C4C2×M1611yescharacteristic
195C22×C4C2×D811yescharacteristic
196C22×C4C2×SD1611yescharacteristic
197C22×C4#13712yes—
198C2×C8—44no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8—82no—
203C2×C8—82no—
204C4⋊C4—48no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—88no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—44no—
213C22⋊C4—48no—
214C42—42no—
215C42—42no—
216C42—81no—
217C42—81no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23D4⋊C814yes—
245C23C2×C22⋊C811yescharacteristic
246C23C2×#9911yescharacteristic
247C23C2×#10111yescharacteristic
248C23C2×#10211yescharacteristic
249C23#41911yescharacteristic
250C23C2×#10411yescharacteristic
251C23#42111yescharacteristic
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—44no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—816no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4#32412yes—
305C2×C4#32412yes—
306C2×C4AΣL(1,𝔽16)12yes—
307C2×C4#35212yes—
308C8—84no—
309C8—162no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—24no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22C2×#32411yescharacteristic
334C22C2×D4⋊C811yescharacteristic
335C22#147911yescharacteristic
336C22#150811yescharacteristic
337C22#151911yescharacteristic
338C22#152811yescharacteristic
339C22#154611yescharacteristic
340C22C2×AΣL(1,𝔽16)11yescharacteristic
341C22C2×#35211yescharacteristic
342C22#212911yescharacteristic
343C22#213411yescharacteristic
344C4—22no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—41no—
351C4—84no—
352C4—81no—
353C4—84no—
354C4#132614yes—
355C2—21no—
356C2—21no—
357C2—21no—
358C2—21no—
359C2—44no—
360C2#757311yescharacteristic
361C2#1144911yescharacteristic
362C2C2×#132611yescharacteristic
363C2#1178511yescharacteristic
364C2#1179011yescharacteristic
365C2#1187411yescharacteristic
366C2#1188011yescharacteristic
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. #7122C2semidirect
42. #380C8semidirect

Maximal subgroups

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

subgroupindex
#141302
#85572
C4×#3232
#71222