Finite Group Databasebeta

← #119473

Subgroups — #119473

Order 512.

690 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 134 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#21822C211yesmaximal, characteristic
2C2×#1467C211yesmaximal, characteristic
3#13942C211yesmaximal, characteristic
4#8030C212yesmaximal
5C2×#1285C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2212yes—
9C2×#391C412yes—
10C2×#382C2211yescharacteristic
11C2×C4×C4⋊C4C2211yescharacteristic
12#1985—24no—
13#1817—21no—
14#1817—21no—
15#1467—24no—
16C2×#324—21no—
17C2×#324—21no—
18#1441—21no—
19#1441—21no—
20C4⋊#95C412yes—
21#1434—21no—
22#1434—21no—
23#1295—24no—
24#1285—24no—
25C23×Q8C2×C411yescharacteristic
26C2×C4×Q8—22no—
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4D411yescharacteristic
37C22×C42C2×C412yes—
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42C2311yescharacteristic
41#432—22no—
42#432—22no—
43#432—22no—
44#432—22no—
45#432—22no—
46#432—22no—
47#432—22no—
48#432D414yes—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×#97—22no—
54C2×#97D414yes—
55#391—44no—
56#382—22no—
57#382—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67#336—24no—
68#336—24no—
69#324—24no—
70#324—24no—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—21no—
74C22×Q8—21no—
75C22×Q8—22no—
76C22×Q8—22no—
77C22×Q8—41no—
78C22×Q8—41no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C22×C8—24no—
89C22×C8—42no—
90C4×Q8—42no—
91C4×Q8—42no—
92C4×Q8—42no—
93C4×Q8—42no—
94C4×Q8—42no—
95C4×Q8—42no—
96C4×Q8—42no—
97C4×Q8—42no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—24no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42C2×D412yes—
128C2×C42C2×D412yes—
129C2×C42#4112yes—
130C2×C42#4112yes—
131C2×C42#4112yes—
132C2×C42#4112yes—
133C4⋊C8—24no—
134C4⋊C8—42no—
135C4⋊C8—42no—
136C4⋊C8—42no—
137C22⋊C8—24no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140C22⋊C8—42no—
141#97—24no—
142#97—24no—
143#97—24no—
144#97—42no—
145#95—44no—
146#95—44no—
147#95—44no—
148#95—44no—
149#95—41no—
150#95—42no—
151#95—42no—
152#95—41no—
153C24C2×C22⋊C411yescharacteristic
154C2×Q8—24no—
155C2×Q8—24no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—24no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4#10012yes—
205C22×C4#10112yes—
206C22×C4#11711yescharacteristic
207C22×C4#11711yescharacteristic
208C22×C4C4×D412yes—
209C22×C4C4×D412yes—
210C22×C4#12011yescharacteristic
211C22×C4#12011yescharacteristic
212C22×C4#12112yes—
213C22×C4#12112yes—
214C22×C4#12311yescharacteristic
215C22×C4#12312yes—
216C22×C4#12312yes—
217C22×C4#12311yescharacteristic
218C2×C8—48no—
219C2×C8—44no—
220C2×C8—44no—
221C2×C8—84no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—84no—
231C42—42no—
232C42—42no—
233C42—42no—
234C42—42no—
235C42—84no—
236C42#12112yes—
237C42#12112yes—
238C42#12312yes—
239C42#12312yes—
240C42#12412yes—
241C42#12412yes—
242C42#12612yes—
243C42#12612yes—
244C23—22no—
245C23—22no—
246C23#38111yescharacteristic
247C23#38611yescharacteristic
248C23#38611yescharacteristic
249C23C2×#10011yescharacteristic
250C23C2×#10111yescharacteristic
251C23#41311yescharacteristic
252C23#41311yescharacteristic
253Q8—48no—
254Q8—48no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4#38812yes—
308C2×C4#38812yes—
309C2×C4#39412yes—
310C2×C4#39412yes—
311C2×C4#39712yes—
312C2×C4#39712yes—
313C8—84no—
314C8—84no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22#144011yescharacteristic
324C22#182111yescharacteristic
325C22#182211yescharacteristic
326C22#188412yes—
327C22#190011yescharacteristic
328C22#190011yescharacteristic
329C22#190111yescharacteristic
330C22#190111yescharacteristic
331C22#191212yes—
332C4—22no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—42no—
339C4—42no—
340C4—84no—
341C4—81no—
342C4—81no—
343C4—162no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2#944812yes—
349C2#945012yes—
350C2#1115611yescharacteristic
351C2#1116011yescharacteristic
352C2#1119811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#218222
C2×#14672
#139422
#80302
C2×#12852