Finite Group Databasebeta

← #124255

Subgroups — #124255

Order 512.

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

normal subgroups only

This group has 708 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#13904C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7104C214yesmaximal, complemented
5C23×C4⋊C4C411yescharacteristic
6C2×#381—21no—
7C2×#381—21no—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C4×C22⋊C8—21no—
12C2×#323—21no—
13C2×#323—21no—
14#1445C412yes—
15#1445C2212yes—
16C4⋊#95—21no—
17C4⋊#95—21no—
18#1280C2212yes—
19#1280C2212yes—
20C24×C4—21no—
21C24×C4C2×C411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—41no—
26C22×C4⋊C4—41no—
27C22×C4⋊C4—42no—
28C22×C4⋊C4C814yescomplemented
29C22×C4⋊C4C2×C412yes—
30C22×C22⋊C4—21no—
31C22×C22⋊C4D411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C4⋊C8—41no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C4×C8—41no—
39#381—44no—
40#381—42no—
41#381—42no—
42#380—42no—
43C4×C22⋊C4—41no—
44C4×C22⋊C4—41no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95C2×C412yes—
50C2×#95C2×C411yescharacteristic
51C2×#95C2311yescharacteristic
52#336—41no—
53#336—41no—
54#323—24no—
55#323—42no—
56C25C22⋊C411yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—24no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C2×C812yes—
75C23×C4M1612yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C22×C8—42no—
80C22×C8—42no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—44no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4C2×C814yes—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4D812yes—
105C2×C22⋊C4SD1612yes—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—42no—
111C2×C42—42no—
112C4⋊C8—82no—
113C22⋊C8—24no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C4×C8—82no—
118#95—41no—
119#95—41no—
120#95—42no—
121#95—44no—
122#95—44no—
123#95—42no—
124#95—42no—
125#95—44no—
126C24—21no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—41no—
131C24—41no—
132C24—41no—
133C24—41no—
134C24C22⋊C812yes—
135C24#10212yes—
136C24#10412yes—
137C24C2×C22⋊C411yescharacteristic
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—48no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—42no—
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—41no—
172C22×C4—41no—
173C22×C4—48no—
174C22×C4—48no—
175C22×C4—48no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4—84no—
192C22×C4—84no—
193C22×C4C22⋊C812yes—
194C22×C4#9911yescharacteristic
195C22×C4#9911yescharacteristic
196C22×C4#10012yes—
197C22×C4#10212yes—
198C22×C4C22×C811yescharacteristic
199C22×C4C2×M1611yescharacteristic
200C22×C4C2×D811yescharacteristic
201C22×C4C2×SD1611yescharacteristic
202C22×C4AGL(1,ℤ/8)12yes—
203C2×C8—44no—
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—82no—
207C2×C8—82no—
208C2×C8—82no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—84no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—44no—
217C22⋊C4—84no—
218C42—42no—
219C42—42no—
220C42—81no—
221C42—81no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23D4⋊C814yes—
249C23C2×C22⋊C811yescharacteristic
250C23C2×#9911yescharacteristic
251C23C2×#10011yescharacteristic
252C23C2×#10211yescharacteristic
253C23#41811yescharacteristic
254C23C2×#10411yescharacteristic
255C23#42111yescharacteristic
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—816no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4#32312yes—
309C2×C4#32312yes—
310C2×C4#35212yes—
311C2×C4#35212yes—
312C8—84no—
313C8—162no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—44no—
328C22—44no—
329C22—44no—
330C22—44no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22C2×#32311yescharacteristic
338C22C2×D4⋊C811yescharacteristic
339C22#147811yescharacteristic
340C22#150711yescharacteristic
341C22#151911yescharacteristic
342C22#152711yescharacteristic
343C22#154511yescharacteristic
344C22C2×#35211yescharacteristic
345C22C2×#35211yescharacteristic
346C22#212911yescharacteristic
347C22#213511yescharacteristic
348C4—22no—
349C4—41no—
350C4—41no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—42no—
355C4—81no—
356C4—84no—
357C4—162no—
358C4#131714yes—
359C2—21no—
360C2—21no—
361C2—21no—
362C2—21no—
363C2—44no—
364C2#757211yescharacteristic
365C2C2×#131711yescharacteristic
366C2#1143411yescharacteristic
367C2#1177311yescharacteristic
368C2#1178011yescharacteristic
369C2#1187411yescharacteristic
370C2#1188111yescharacteristic
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. #7104C2semidirect
28. C22×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
#139042
#85572
C4×#3232
#71042