Finite Group Databasebeta

← #140987

Subgroups — #140987

Order 512.

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

normal subgroups only

This group has 648 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#15150C211yesmaximal, characteristic
2C2×#1344C214yesmaximal
3#10801C211yesmaximal, characteristic
4#8152C211yesmaximal, characteristic
5C2×#397—21no—
6C2×#397C2212yes—
7C2×#384C411yescharacteristic, complemented
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1749—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1445—21no—
15#1445C412yescomplemented
16C4⋊#95—21no—
17C4⋊#95C411yescharacteristic, complemented
18C22⋊#95—21no—
19#1344—28no—
20#1280—21no—
21C24×C4D411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
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#397—24no—
35#397—24no—
36#397—44no—
37#384—41no—
38#384—41no—
39#380—44no—
40C4×C22⋊C4—41no—
41C4×C22⋊C4—41no—
42C2×#95—22no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—41no—
47C2×#95—41no—
48C2×#95C2×C412yes—
49C2×#95C2×C411yescharacteristic
50C2×#95D411yescharacteristic
51#342—24no—
52#342—24no—
53#336—41no—
54#323—22no—
55#323D414yes—
56C25C2×D411yescharacteristic
57C23×C4—22no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—41no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C22×C8—42no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
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—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—44no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4C2×D412yes—
102C2×C22⋊C4#4112yes—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—42no—
108C2×C42—42no—
109C22⋊C8—24no—
110C22⋊C8—42no—
111#97—82no—
112#95—24no—
113#95—41no—
114#95—41no—
115#95—44no—
116#95—42no—
117#95—44no—
118#95—44no—
119#95—82no—
120#95—82no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—41no—
126C24—41no—
127C24—41no—
128C24—41no—
129C24C2×C22⋊C411yescharacteristic
130C24#12411yescharacteristic
131C24#12711yescharacteristic
132C24AGL(1,ℤ/8)12yes—
133C24AGL(1,ℤ/8)12yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
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—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—48no—
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—41no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—84no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4#9911yescharacteristic
190C22×C4#9911yescharacteristic
191C22×C4#10412yes—
192C22×C4C4×D412yes—
193C22×C4#12112yes—
194C2×C8—44no—
195C2×C8—82no—
196C2×C8—82no—
197C4⋊C4—84no—
198C4⋊C4—84no—
199C4⋊C4—82no—
200C4⋊C4—82no—
201C4⋊C4—84no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—44no—
207C22⋊C4—82no—
208C22⋊C4—84no—
209C22⋊C4—82no—
210C22⋊C4—84no—
211C42—42no—
212C42—42no—
213C42—42no—
214C42—81no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—42no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23—81no—
245C23#39011yescharacteristic
246C23C2×#9911yescharacteristic
247C23#41011yescharacteristic
248C23#41811yescharacteristic
249C23#41811yescharacteristic
250C23C2×#10411yescharacteristic
251C23#42111yescharacteristic
252C23#49012yes—
253C23#49612yes—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—84no—
283C2×C4—84no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—84no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—162no—
305C2×C4—162no—
306C2×C4—162no—
307C2×C4—162no—
308C2×C4#35212yes—
309C2×C4#35412yes—
310C8—84no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—42no—
335C22—81no—
336C22—81no—
337C22—82no—
338C22—82no—
339C22#150511yescharacteristic
340C22#150711yescharacteristic
341C22#151011yescharacteristic
342C22#151411yescharacteristic
343C22#196111yescharacteristic
344C22#196411yescharacteristic
345C22#198011yescharacteristic
346C22C2×#35211yescharacteristic
347C22#212311yescharacteristic
348C22C2×#35411yescharacteristic
349C22#212911yescharacteristic
350C4—22no—
351C4—22no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—81no—
356C4—81no—
357C4—162no—
358C4—164no—
359C4—162no—
360C2—21no—
361C2—21no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—42no—
366C2#968411yescharacteristic
367C2#1177311yescharacteristic
368C2#1177711yescharacteristic
369C2#1179711yescharacteristic
370C2#1180111yescharacteristic
371C2#1321711yescharacteristic
372C2#1322411yescharacteristic
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. #1445C4semidirect
17. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151502
C2×#13442
#108012
#81522