Finite Group Databasebeta

← #99557

Subgroups — #99557

Order 512.

584 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 132 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#21951C211yesmaximal, characteristic
2C4×#400C211yesmaximal, characteristic
3#8707C212yesmaximal
4C4×#324C211yesmaximal, characteristic
5#7028C212yesmaximal
6C2×#431—21no—
7C2×#400C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1939—24no—
11#1797—21no—
12#1797C2212yes—
13C4×C22⋊C8C2212yes—
14C2×#324—21no—
15#1445C411yescharacteristic
16#1443—21no—
17#1442—21no—
18#1442C411yescharacteristic
19#1441C411yescharacteristic
20C4×#95—21no—
21C4×#95—21no—
22C4×#95C2211yescharacteristic
23#1280—21no—
24#1279—24no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42—21no—
31C22×C42D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42C2311yescharacteristic
34#431—42no—
35#431—42no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8D414yes—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C4×C8—22no—
42C2×C4×C8—22no—
43#400—22no—
44#400C814yescomplemented
45C4×C4⋊C4—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—22no—
48C4×C4⋊C4—22no—
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#336—22no—
64#336—22no—
65#336—22no—
66#324—24no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C22×C8—24no—
79C22×C8—42no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4C2×C812yes—
91C2×C4⋊C4C2×C812yes—
92C2×C4⋊C4M1612yes—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—42no—
100C2×C42—42no—
101C2×C42—42no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42—41no—
108C2×C42—41no—
109C2×C42SD1612yes—
110C2×C42Q1612yes—
111C2×C42C2×D412yes—
112C2×C42#4112yes—
113C4⋊C8—44no—
114C4⋊C8—44no—
115C4⋊C8—82no—
116C22⋊C8—24no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C4×C8—44no—
121C4×C8—44no—
122#95—24no—
123#95—24no—
124#95—22no—
125#95—22no—
126#95—21no—
127#95—22no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95C2×C812yes—
137C24C2×C22⋊C411yescharacteristic
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
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×C4C22⋊C812yes—
192C22×C4C22⋊C812yes—
193C22×C4#10312yes—
194C22×C4#10312yes—
195C22×C4#10412yes—
196C22×C4C4×D412yes—
197C22×C4#12011yescharacteristic
198C22×C4#12112yes—
199C22×C4#12311yescharacteristic
200C22×C4C22×C811yescharacteristic
201C22×C4C2×M1611yescharacteristic
202C22×C4#13112yes—
203C22×C4C2×SD1611yescharacteristic
204C22×C4C2×Q1611yescharacteristic
205C22×C4AGL(1,ℤ/8)11yescharacteristic
206C22×C4#13711yescharacteristic
207C2×C8—44no—
208C2×C8—44no—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—82no—
212C2×C8—82no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C42—22no—
221C42—22no—
222C42—24no—
223C42—22no—
224C42—22no—
225C42—42no—
226C42—42no—
227C42—81no—
228C42—82no—
229C42—82no—
230C42—82no—
231C42—82no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23#38611yescharacteristic
237C23C2×C22⋊C811yescharacteristic
238C23#40811yescharacteristic
239C23C2×#10311yescharacteristic
240C23#41711yescharacteristic
241C23C2×#10411yescharacteristic
242C23#42111yescharacteristic
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
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—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
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×C4Q8⋊C814yes—
309C2×C4C8×D412yes—
310C2×C4#43512yes—
311C2×C4#45011yescharacteristic
312C2×C4#45111yescharacteristic
313C2×C4#45412yes—
314C2×C4#45812yes—
315C2×C4#48311yescharacteristic
316C2×C4#48411yescharacteristic
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—
331C22C2×Q8⋊C811yescharacteristic
332C22#147711yescharacteristic
333C22#148111yescharacteristic
334C22#148311yescharacteristic
335C22#186911yescharacteristic
336C22#187111yescharacteristic
337C22#187211yescharacteristic
338C22#188411yescharacteristic
339C22#188811yescharacteristic
340C22#189011yescharacteristic
341C22#189111yescharacteristic
342C4—22no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—24no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—82no—
351C4—82no—
352C4—82no—
353C4—81no—
354C4—82no—
355C4#161612yes—
356C4#161712yes—
357C2—21no—
358C2—21no—
359C2—21no—
360C2—21no—
361C2#893711yescharacteristic
362C2#896011yescharacteristic
363C2#896611yescharacteristic
364C2#901211yescharacteristic
365C2#902311yescharacteristic
366C2#903211yescharacteristic
367C2#903311yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
44. #400C8semidirect

Maximal subgroups

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

subgroupindex
#219512
C4×#4002
#87072
C4×#3242
#70282