Finite Group Databasebeta

← #124606

Subgroups — #124606

Order 512.

666 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C4×#393C211yesmaximal, characteristic
2#8700C211yesmaximal, characteristic
3#8559C211yesmaximal, characteristic
4#7131C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C22×C4×Q8C411yescharacteristic
7C2×#393C411yescharacteristic
8C2×C4×C22⋊C4—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11#1779—21no—
12C2×#323—21no—
13#1446C2211yescharacteristic
14#1442—21no—
15#1442C2211yescharacteristic
16#1439—21no—
17#1439C411yescharacteristic
18C4×#95—21no—
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C2×C4×Q8—44no—
26C2×C4×Q8—41no—
27C2×C4×Q8—41no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4D411yescharacteristic
34C22×C42—21no—
35C22×C42—21no—
36C22×C42C2×C411yescharacteristic
37C22×C42D411yescharacteristic
38C2×C4⋊C8—41no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—41no—
42C2×C22⋊C8—41no—
43#393—22no—
44#393C814yescomplemented
45C4×C22⋊C4—41no—
46C4×C22⋊C4—41no—
47C4×C22⋊C4—44no—
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×#95D411yescharacteristic
58C2×#95C2311yescharacteristic
59#336—41no—
60#323—42no—
61C25C22⋊C411yescharacteristic
62C22×Q8—21no—
63C22×Q8—42no—
64C22×Q8—42no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8C2×C812yes—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—41no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—42no—
85C22×C8—42no—
86C4×Q8—44no—
87C4×Q8—44no—
88C4×Q8—44no—
89C4×Q8—44no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4C2×C812yes—
108C2×C22⋊C4M1612yes—
109C2×C42—24no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—42no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42—42no—
122C4⋊C8—82no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—82no—
128C22⋊C8—82no—
129#95—24no—
130#95—21no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—41no—
140#95—41no—
141#95C2×C812yes—
142C24—21no—
143C24—21no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24C22⋊C812yes—
153C24C2×C22⋊C411yescharacteristic
154C2×Q8—44no—
155C2×Q8—44no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—48no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
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—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—48no—
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—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—84no—
224C22×C4C22⋊C812yes—
225C22×C4#9911yescharacteristic
226C22×C4#9911yescharacteristic
227C22×C4C22×C811yescharacteristic
228C22×C4C2×M1611yescharacteristic
229C22×C4AGL(1,ℤ/8)11yescharacteristic
230C22×C4#13711yescharacteristic
231C22×C4#13712yes—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C2×C8—82no—
237C2×C8—82no—
238C4⋊C4—42no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—48no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—48no—
247C42—44no—
248C42—42no—
249C42—44no—
250C42—42no—
251C42—42no—
252C42—81no—
253C42—84no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23C2×C22⋊C811yescharacteristic
282C23C2×#9911yescharacteristic
283C23#41311yescharacteristic
284C23#41711yescharacteristic
285C23#41911yescharacteristic
286C23#42111yescharacteristic
287C23#42111yescharacteristic
288Q8—44no—
289Q8—44no—
290Q8—44no—
291Q8—44no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4AΣL(1,𝔽16)12yes—
358C2×C4#35412yes—
359C8—162no—
360C8—162no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—44no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#145911yescharacteristic
385C22#147711yescharacteristic
386C22#147911yescharacteristic
387C22#151111yescharacteristic
388C22#151311yescharacteristic
389C22#153511yescharacteristic
390C22#155911yescharacteristic
391C22C2×AΣL(1,𝔽16)11yescharacteristic
392C22#212011yescharacteristic
393C22C2×#35411yescharacteristic
394C22#213211yescharacteristic
395C4—22no—
396C4—22no—
397C4—42no—
398C4—41no—
399C4—41no—
400C4—44no—
401C4—41no—
402C4—41no—
403C4—42no—
404C4—82no—
405C4—82no—
406C4—81no—
407C4—82no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2—82no—
413C2#760011yescharacteristic
414C2#1147111yescharacteristic
415C2#1147711yescharacteristic
416C2#1180811yescharacteristic
417C2#1181611yescharacteristic
418C2#1182311yescharacteristic
419C2#1182811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7131C2semidirect
5. #7122C2semidirect
44. #393C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3932
#87002
#85592
#71312
#71222