Finite Group Databasebeta

← #142053

Subgroups — #142053

Order 512.

658 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 94 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#15135C211yesmaximal, characteristic
2#10801C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7250C214yesmaximal
5C2×#399—21no—
6C2×#389—21no—
7C2×#384C411yescharacteristic, complemented
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1749—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1445—21no—
15#1444C412yescomplemented
16#1439C2212yes—
17C4⋊#95C411yescharacteristic, complemented
18C22⋊#95—21no—
19#1280—21no—
20C24×C4D411yescharacteristic
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4C2×C412yes—
26C22×C22⋊C4—22no—
27C22×C22⋊C4—22no—
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4C2311yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D411yescharacteristic
32C2×C22⋊C8—22no—
33C2×#97—41no—
34#399—44no—
35#389—44no—
36#384—41no—
37#384—41no—
38#380—44no—
39C4×C22⋊C4—41no—
40C4×C22⋊C4—41no—
41C2×#95—21no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—41no—
45C2×#95—41no—
46C2×#95C2×C412yes—
47C2×#95C2×C411yescharacteristic
48C2×#95D411yescharacteristic
49#342—24no—
50#342—24no—
51#336—41no—
52#323—22no—
53#323Q814yes—
54C25C2×D411yescharacteristic
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—41no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C22×C8—42no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—44no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—82no—
85C2×C4⋊C4—82no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—44no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—44no—
99C2×C22⋊C4—82no—
100C2×C22⋊C4C2×Q812yes—
101C2×C22⋊C4#4112yes—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—42no—
107C2×C42—42no—
108C22⋊C8—24no—
109C22⋊C8—42no—
110#97—82no—
111#95—41no—
112#95—41no—
113#95—42no—
114#95—44no—
115#95—44no—
116#95—42no—
117#95—82no—
118#95—82no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—41no—
124C24—41no—
125C24—41no—
126C24—41no—
127C24C2×C22⋊C411yescharacteristic
128C24#12411yescharacteristic
129C24C4⋊Q811yescharacteristic
130C24AGL(1,ℤ/8)12yes—
131C24#13712yes—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—48no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—84no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4#9911yescharacteristic
188C22×C4#9911yescharacteristic
189C22×C4#10412yes—
190C22×C4C4×Q812yes—
191C22×C4C22⋊Q812yes—
192C2×C8—44no—
193C2×C8—82no—
194C2×C8—82no—
195C4⋊C4—48no—
196C4⋊C4—82no—
197C4⋊C4—84no—
198C4⋊C4—88no—
199C4⋊C4—84no—
200C4⋊C4—82no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—44no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—88no—
207C22⋊C4—82no—
208C22⋊C4—82no—
209C42—42no—
210C42—42no—
211C42—42no—
212C42—81no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—21no—
223C23—22no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23#39111yescharacteristic
244C23C2×#9911yescharacteristic
245C23#41011yescharacteristic
246C23#41811yescharacteristic
247C23#41911yescharacteristic
248C23C2×#10411yescharacteristic
249C23#42111yescharacteristic
250C23#48912yes—
251C23#50112yes—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—84no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—162no—
303C2×C4—162no—
304C2×C4—162no—
305C2×C4—162no—
306C2×C4#35212yes—
307C2×C4#35312yes—
308C8—84no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22—82no—
334C22—81no—
335C22—81no—
336C22—82no—
337C22#150611yescharacteristic
338C22#150711yescharacteristic
339C22#151111yescharacteristic
340C22#151411yescharacteristic
341C22#198611yescharacteristic
342C22#199311yescharacteristic
343C22#199611yescharacteristic
344C22C2×#35211yescharacteristic
345C22#212211yescharacteristic
346C22C2×#35311yescharacteristic
347C22#212911yescharacteristic
348C4—22no—
349C4—22no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—84no—
354C4—81no—
355C4—81no—
356C4—164no—
357C4—162no—
358C2—21no—
359C2—21no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—42no—
364C2#988811yescharacteristic
365C2#1177311yescharacteristic
366C2#1177611yescharacteristic
367C2#1180911yescharacteristic
368C2#1181211yescharacteristic
369C2#1329211yescharacteristic
370C2#1329811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#384C4semidirect
15. #1444C4semidirect
17. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151352
#108012
#81522
#72502