Finite Group Databasebeta

← #140966

Subgroups — #140966

Order 512.

520 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 90 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#15164C211yesmaximal, characteristic
2C2×#1354C212yesmaximal
3C2×#1353C212yesmaximal
4#10802C211yesmaximal, characteristic
5#8153C211yesmaximal, characteristic
6C2×#400C2211yescharacteristic
7C2×#395C2211yescharacteristic
8C2×#391C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1749—21no—
11C2×#343—21no—
12C2×#343C2212yes—
13#1728—21no—
14C2×#324C2212yes—
15#1446C411yescharacteristic
16#1445C411yescharacteristic
17#1441—21no—
18#1441—21no—
19#1436—21no—
20C4⋊#95C411yescharacteristic
21#1434—21no—
22#1354—24no—
23#1353—24no—
24#1280—21no—
25C23×Q8C2×C411yescharacteristic
26C22×M16—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C4⋊C4C2311yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C22⋊C8—22no—
36C2×#97—41no—
37#400—22no—
38#400—22no—
39#395—22no—
40#395—22no—
41#391—41no—
42#391—41no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95D411yescharacteristic
59#343—24no—
60#343—22no—
61#343—22no—
62#336—41no—
63#336—41no—
64#336—41no—
65#324—22no—
66#324D414yes—
67C22×Q8—22no—
68C22×Q8—42no—
69C22×Q8—41no—
70C22×Q8—41no—
71C22×Q8—81no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—24no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—42no—
89C22×C8—42no—
90C22×C8—42no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—81no—
104C2×C4⋊C4—81no—
105C2×C4⋊C4C2×D412yes—
106C2×C4⋊C4#4112yes—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—22no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—41no—
115C2×C42—81no—
116C2×C42—81no—
117C22⋊C8—24no—
118C22⋊C8—42no—
119#97—82no—
120#95—22no—
121#95—22no—
122#95—41no—
123#95—41no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—41no—
130#95—41no—
131#95—42no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95—81no—
137C24C2×C22⋊C411yescharacteristic
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—82no—
141C2×Q8—84no—
142C2×Q8—84no—
143C2×Q8—82no—
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—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
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—41no—
201C22×C4—41no—
202C22×C4#9911yescharacteristic
203C22×C4#9911yescharacteristic
204C22×C4#10412yes—
205C22×C4C4×D412yes—
206C22×C4#12112yes—
207C22×C4#12411yescharacteristic
208C22×C4#12711yescharacteristic
209C22×C4#13712yes—
210C22×C4#13712yes—
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—44no—
224C4⋊C4—82no—
225C4⋊C4—84no—
226C4⋊C4—84no—
227C42—42no—
228C42—42no—
229C42—42no—
230C42—81no—
231C42—81no—
232C42—81no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23#39011yescharacteristic
238C23C2×#9911yescharacteristic
239C23#41011yescharacteristic
240C23#41911yescharacteristic
241C23#41911yescharacteristic
242C23C2×#10411yescharacteristic
243C23#42111yescharacteristic
244Q8—84no—
245Q8—84no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—81no—
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#49112yes—
316C2×C4#49712yes—
317C8—84no—
318C8—84no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22#150611yescharacteristic
332C22#150811yescharacteristic
333C22#151111yescharacteristic
334C22#151511yescharacteristic
335C22#196011yescharacteristic
336C22#196411yescharacteristic
337C22#198011yescharacteristic
338C22#212011yescharacteristic
339C22#212111yescharacteristic
340C22#213211yescharacteristic
341C22#213311yescharacteristic
342C4—22no—
343C4—22no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—42no—
348C4—42no—
349C4—82no—
350C4—81no—
351C4—81no—
352C4—82no—
353C4—161no—
354C4—161no—
355C2—21no—
356C2—21no—
357C2—21no—
358C2—21no—
359C2#968311yescharacteristic
360C2#1179211yescharacteristic
361C2#1179311yescharacteristic
362C2#1181611yescharacteristic
363C2#1181711yescharacteristic
364C2#1322211yescharacteristic
365C2#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
#151642
C2×#13542
C2×#13532
#108022
#81532