Finite Group Databasebeta

← #124381

Subgroups — #124381

Order 512.

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

normal subgroups only

This group has 734 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
1#14176C211yesmaximal, characteristic
2#8701C211yesmaximal, characteristic, complemented
3C4×#324C211yesmaximal, characteristic, complemented
4#7123C212yesmaximal, complemented
5#7103C212yesmaximal
6C22×C4×D4C411yescharacteristic
7C2×#394C411yescharacteristic
8C2×#388—21no—
9C2×#385—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1817—21no—
12C4×C22⋊C8—21no—
13C2×#324—21no—
14#1444C2211yescharacteristic
15#1441C411yescharacteristic
16#1438C2211yescharacteristic
17#1437—21no—
18#1434C411yescharacteristic
19C22⋊#95—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×D4C2×C411yescharacteristic
24C24×C4—21no—
25C2×C4×D4—41no—
26C2×C4×D4—41no—
27C2×C4×D4—44no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4D411yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C4×C8—41no—
43#394—22no—
44#394C814yescomplemented
45#388—44no—
46#385—44no—
47C4×C4⋊C4—41no—
48C4×C4⋊C4—41no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95D411yescharacteristic
57C2×#95C2311yescharacteristic
58#336—41no—
59#336—41no—
60#336—41no—
61#324—24no—
62C25—21no—
63C22×D4—21no—
64C22×D4—41no—
65C22×D4—41no—
66C22×D4—42no—
67C22×D4—42no—
68C22×D4C2×C812yes—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—42no—
87C22×C8—42no—
88C4×D4—44no—
89C4×D4—44no—
90C4×D4—44no—
91C4×D4—44no—
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—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4C2×C812yes—
103C2×C4⋊C4M1612yes—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—81no—
119C2×C22⋊C4—81no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42—21no—
127C2×C42—41no—
128C2×C42—42no—
129C2×C42—42no—
130C22⋊C8—24no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C4×C8—82no—
135#95—21no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—81no—
144#95C2×C812yes—
145C24—22no—
146C24—22no—
147C24—22no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24—41no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24C2×C22⋊C411yescharacteristic
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—44no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—48no—
168C2×D4—42no—
169C2×D4—44no—
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—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
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—48no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4—84no—
228C22×C4C22⋊C812yes—
229C22×C4C22⋊C812yes—
230C22×C4#9911yescharacteristic
231C22×C4#9911yescharacteristic
232C22×C4#10412yes—
233C22×C4C22×C811yescharacteristic
234C22×C4C2×M1611yescharacteristic
235C22×C4#13512yes—
236C22×C4AGL(1,ℤ/8)12yes—
237C2×C8—44no—
238C2×C8—82no—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—84no—
249C22⋊C4—48no—
250C22⋊C4—44no—
251C22⋊C4—48no—
252C22⋊C4—44no—
253C22⋊C4—84no—
254C22⋊C4—84no—
255C42—42no—
256C42—42no—
257C42—42no—
258C42—42no—
259C42—42no—
260C42—81no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—24no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23C2×C22⋊C811yescharacteristic
301C23C2×#9911yescharacteristic
302C23#41311yescharacteristic
303C23#41611yescharacteristic
304C23#41811yescharacteristic
305C23C2×#10411yescharacteristic
306C23#42111yescharacteristic
307D4—44no—
308D4—44no—
309D4—44no—
310D4—44no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—816no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4AΣL(1,𝔽16)12yes—
375C8—84no—
376C8—162no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22#145911yescharacteristic
405C22#147611yescharacteristic
406C22#147811yescharacteristic
407C22#150711yescharacteristic
408C22#151611yescharacteristic
409C22#152911yescharacteristic
410C22#155311yescharacteristic
411C22C2×AΣL(1,𝔽16)11yescharacteristic
412C22#212011yescharacteristic
413C22#212811yescharacteristic
414C22#213311yescharacteristic
415C4—22no—
416C4—42no—
417C4—41no—
418C4—42no—
419C4—41no—
420C4—41no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—82no—
425C4—82no—
426C4—161no—
427C2—21no—
428C2—21no—
429C2—21no—
430C2—21no—
431C2—84no—
432C2#758211yescharacteristic
433C2#1147111yescharacteristic
434C2#1148011yescharacteristic
435C2#1177211yescharacteristic
436C2#1177811yescharacteristic
437C2#1184011yescharacteristic
438C2#1184811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8701C2semidirect
3. C4×#324C2semidirect
4. #7123C2semidirect
44. #394C8semidirect

Maximal subgroups

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

subgroupindex
#141762
#87012
C4×#3242
#71232
#71032