Finite Group Databasebeta

← #99575

Subgroups — #99575

Order 512.

650 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 108 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#21906C211yesmaximal, characteristic
2C4×#397C211yesmaximal, characteristic
3#8706C212yesmaximal, complemented
4C4×#323C211yesmaximal, characteristic
5#7028C212yesmaximal, complemented
6C2×#397C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8C2×C4×C22⋊C4—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1924—24no—
11#1797C2212yes—
12#1779—21no—
13#1749—21no—
14C4×C22⋊C8C2212yes—
15C2×#323—21no—
16#1445C411yescharacteristic
17#1443—21no—
18#1442C411yescharacteristic
19#1440—21no—
20#1439C411yescharacteristic
21C4×#95—21no—
22C4×#95C2211yescharacteristic
23#1280—21no—
24#1279—24no—
25C24×C4D411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42—21no—
33C22×C42—21no—
34C22×C42D411yescharacteristic
35C22×C42C2311yescharacteristic
36C2×C4⋊C8D414yes—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×#97—22no—
42C2×C4×C8—22no—
43#397—22no—
44#397—22no—
45C4×C4⋊C4—22no—
46C4×C4⋊C4—22no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—44no—
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×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95D411yescharacteristic
62#336—22no—
63#336—22no—
64#336—22no—
65#323—24no—
66C25C22⋊C411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C22×C8—24no—
84C22×C8—42no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4M1612yes—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42—42no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42C2×D412yes—
116C2×C42#4112yes—
117C4⋊C8—44no—
118C22⋊C8—24no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—82no—
123C22⋊C8—82no—
124#97—82no—
125C4×C8—44no—
126#95—24no—
127#95—22no—
128#95—22no—
129#95—21no—
130#95—22no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95M1612yes—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—21no—
145C24—41no—
146C24—41no—
147C24—41no—
148C24—41no—
149C24#10412yes—
150C24C2×C22⋊C411yescharacteristic
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—48no—
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—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—84no—
210C22×C4C4×D412yes—
211C22×C4#12011yescharacteristic
212C22×C4#12112yes—
213C22×C4#12311yescharacteristic
214C22×C4C2×M1611yescharacteristic
215C22×C4C2×M1611yescharacteristic
216C22×C4#13112yes—
217C22×C4#13512yes—
218C22×C4AGL(1,ℤ/8)12yes—
219C2×C8—44no—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8—44no—
223C2×C8—82no—
224C2×C8—82no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C22⋊C4—44no—
230C22⋊C4—48no—
231C22⋊C4—48no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C42—24no—
235C42—42no—
236C42—42no—
237C42—81no—
238C42—84no—
239C42—82no—
240C42—82no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23#38611yescharacteristic
269C23#40711yescharacteristic
270C23#40811yescharacteristic
271C23#41611yescharacteristic
272C23#41811yescharacteristic
273C23C2×#10411yescharacteristic
274C23#42111yescharacteristic
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—84no—
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—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—84no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4#43512yes—
335C2×C4#43612yes—
336C2×C4#44912yes—
337C2×C4#45512yes—
338C2×C4#45812yes—
339C2×C4#48212yes—
340C8—84no—
341C8—84no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22#148811yescharacteristic
366C22#149011yescharacteristic
367C22#149411yescharacteristic
368C22#149611yescharacteristic
369C22#187011yescharacteristic
370C22#187111yescharacteristic
371C22#187511yescharacteristic
372C22#188411yescharacteristic
373C22#188511yescharacteristic
374C22#189011yescharacteristic
375C22#189111yescharacteristic
376C4—22no—
377C4—22no—
378C4—22no—
379C4—42no—
380C4—82no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—82no—
386C4—82no—
387C4#161214yes—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—21no—
392C2—82no—
393C2#894311yescharacteristic
394C2#896111yescharacteristic
395C2#896811yescharacteristic
396C2#901111yescharacteristic
397C2#902811yescharacteristic
398C2#902911yescharacteristic
399C2#903611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC25C22⋊C4
LCSG ▷ C23×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
3. #8706C2semidirect
5. #7028C2semidirect

Maximal subgroups

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

subgroupindex
#219062
C4×#3972
#87062
C4×#3232
#70282