Finite Group Databasebeta

← #140950

Subgroups — #140950

Order 512.

506 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 92 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#15085C211yesmaximal, characteristic
2C2×#1353C212yesmaximal
3C2×#1346C212yesmaximal
4#10847C211yesmaximal, characteristic
5#8153C211yesmaximal, characteristic
6C2×#423—21no—
7C2×#400C2211yescharacteristic
8C2×#398C2211yescharacteristic
9C2×#382C411yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1749—21no—
12C2×#343—21no—
13C2×#343C2212yes—
14C2×#324C2212yes—
15#1445C411yescharacteristic
16#1445C411yescharacteristic
17#1443—21no—
18#1441—21no—
19#1436—21no—
20#1434—21no—
21#1434C411yescharacteristic
22#1353—24no—
23#1346—24no—
24#1280—21no—
25C22×M16—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4C2311yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34#423—24no—
35#423—42no—
36C2×C4⋊C8—41no—
37C2×C4⋊C8—41no—
38C2×C22⋊C8—22no—
39C2×#97—41no—
40#400—22no—
41#400—22no—
42#398—22no—
43#398—22no—
44#382—41no—
45#382—41no—
46C4×C4⋊C4—41no—
47C4×C4⋊C4—41no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95D411yescharacteristic
63#343—22no—
64#343—22no—
65#343—42no—
66#336—41no—
67#324—22no—
68#324D414yes—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C2×M16—24no—
81C2×M16—42no—
82C2×M16—42no—
83C2×M16—42no—
84C2×M16—42no—
85C2×M16—42no—
86C22×C8—42no—
87C22×C8—42no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2×D412yes—
107C2×C4⋊C4#4112yes—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—22no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—42no—
118C2×C42—42no—
119C4⋊C8—44no—
120C4⋊C8—44no—
121C22⋊C8—24no—
122C22⋊C8—42no—
123#97—82no—
124#95—22no—
125#95—22no—
126#95—41no—
127#95—41no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—41no—
135#95—41no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—41no—
140#95—41no—
141C24C2×C22⋊C411yescharacteristic
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—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
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—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4#9911yescharacteristic
201C22×C4#9911yescharacteristic
202C22×C4#10412yes—
203C22×C4C4×D412yes—
204C22×C4#12112yes—
205C22×C4#12411yescharacteristic
206C22×C4#12711yescharacteristic
207C22×C4AGL(1,ℤ/8)11yescharacteristic
208C22×C4AGL(1,ℤ/8)11yescharacteristic
209C22×C4#13711yescharacteristic
210C22×C4#13711yescharacteristic
211M16—44no—
212M16—44no—
213M16—82no—
214M16—82no—
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—42no—
236C42—42no—
237C42—42no—
238C42—81no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23#39011yescharacteristic
244C23C2×#9911yescharacteristic
245C23#41011yescharacteristic
246C23#41711yescharacteristic
247C23#41711yescharacteristic
248C23C2×#10411yescharacteristic
249C23#42111yescharacteristic
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4#35212yes—
323C2×C4#48911yescharacteristic
324C2×C4#48911yescharacteristic
325C2×C4#49611yescharacteristic
326C2×C4#49711yescharacteristic
327C8—84no—
328C8—84no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22#150411yescharacteristic
342C22#150911yescharacteristic
343C22#151211yescharacteristic
344C22#151311yescharacteristic
345C22#195911yescharacteristic
346C22#196411yescharacteristic
347C22#198011yescharacteristic
348C22C2×#35211yescharacteristic
349C22#212311yescharacteristic
350C22#213211yescharacteristic
351C22#213311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—22no—
357C4—42no—
358C4—42no—
359C4—41no—
360C4—42no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4—161no—
365C4—161no—
366C4—161no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2#968211yescharacteristic
372C2#1176411yescharacteristic
373C2#1176711yescharacteristic
374C2#1183011yescharacteristic
375C2#1183511yescharacteristic
376C2#1322011yescharacteristic
377C2#1322811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#150852
C2×#13532
C2×#13462
#108472
#81532