Finite Group Databasebeta

← #128790

Subgroups — #128790

Order 512.

542 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 124 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#14318C211yesmaximal, characteristic
2C2×#1353C212yesmaximal
3C4×#343C211yesmaximal, characteristic
4C4×#324C211yesmaximal, characteristic
5#7251C212yesmaximal
6C2×C4×M16—21no—
7C2×#400C2211yescharacteristic
8C2×#389C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#343—21no—
12C2×#343C2212yes—
13C2×#324C2212yes—
14#1445C411yescharacteristic
15#1444—21no—
16#1444C411yescharacteristic
17#1442—21no—
18#1441C2211yescharacteristic
19#1436—21no—
20#1434—21no—
21#1434C411yescharacteristic
22#1353C418yescomplemented
23#1280—21no—
24C22×M16—22no—
25C22×C4⋊C4—21no—
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
34C2×C22⋊C8—22no—
35C4×M16—24no—
36C4×M16—42no—
37C2×#97—41no—
38C2×C4×C8—41no—
39C2×C4×C8—41no—
40#400—22no—
41#400C2×C414yes—
42#389—41no—
43#389—41no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—41no—
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×#95—21no—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60#343—24no—
61#343—22no—
62#343C2×C414yes—
63#336—41no—
64#324—22no—
65#324C2×C414yes—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×M16—24no—
78C2×M16—24no—
79C2×M16—24no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C22×C8—42no—
84C22×C8—42no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4—81no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4—81no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4C4212yes—
102C2×C4⋊C4C22×C412yes—
103C2×C4⋊C4#4112yes—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—22no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—81no—
112C2×C42—81no—
113C2×C42SD1612yes—
114C2×C42Q1612yes—
115C22⋊C8—24no—
116C22⋊C8—42no—
117#97—44no—
118C4×C8—44no—
119C4×C8—82no—
120#95—21no—
121#95—24no—
122#95—24no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—42no—
129#95—42no—
130#95—81no—
131#95—81no—
132#95—81no—
133#95—81no—
134#95—81no—
135#95C4212yes—
136C24C2×C22⋊C411yescharacteristic
137C22×C4—21no—
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—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
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—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#9911yescharacteristic
193C22×C4#9911yescharacteristic
194C22×C4#10312yes—
195C22×C4#10312yes—
196C22×C4#10412yes—
197C22×C4#10412yes—
198C22×C4C2×C4211yescharacteristic
199C22×C4#11711yescharacteristic
200C22×C4C4×D412yes—
201C22×C4C4×D412yes—
202C22×C4C2×SD1611yescharacteristic
203C22×C4C2×Q1611yescharacteristic
204C22×C4#13512yes—
205M16—28no—
206M16—44no—
207M16—44no—
208M16—44no—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—82no—
214C2×C8—82no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C42—22no—
224C42—22no—
225C42—22no—
226C42—22no—
227C42—42no—
228C42—81no—
229C42—81no—
230C42—81no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23C4×C22⋊C411yescharacteristic
236C23C2×#9911yescharacteristic
237C23#41011yescharacteristic
238C23C2×#10311yescharacteristic
239C23#41611yescharacteristic
240C23C2×#10411yescharacteristic
241C23C2×#10411yescharacteristic
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
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#32812yes—
312C2×C4#32812yes—
313C2×C4C4×SD1612yes—
314C2×C4C4×Q1612yes—
315C8—48no—
316C8—84no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22C2×#32811yescharacteristic
330C22#150011yescharacteristic
331C22#150111yescharacteristic
332C22#150311yescharacteristic
333C22C4×#9911yescharacteristic
334C22C4×#10411yescharacteristic
335C22C4×#10311yescharacteristic
336C22#212111yescharacteristic
337C22#212311yescharacteristic
338C22#213311yescharacteristic
339C22#213411yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—42no—
348C4—42no—
349C4—82no—
350C4—82no—
351C4—81no—
352C4—161no—
353C4—161no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2—21no—
358C2C4×#32811yescharacteristic
359C2#1172211yescharacteristic
360C2#1172311yescharacteristic
361C2#1173711yescharacteristic
362C2#1174011yescharacteristic
363C2#1262111yescharacteristic
364C2#1263211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1353C4semidirect

Maximal subgroups

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

subgroupindex
#143182
C2×#13532
C4×#3432
C4×#3242
#72512