Finite Group Databasebeta

← #99375

Subgroups — #99375

Order 512.

877 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 877 conjugacy classes of subgroups — too many to draw. Only its 154 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#14440C211yesmaximal, characteristic
2#11323C212yesmaximal, complemented
3#11049C211yesmaximal, characteristic
4C2×#1270C211yesmaximal, characteristic
5#7025C212yesmaximal, complemented
6C22×#106—22no—
7C22×#102—21no—
8C2×#390C2211yescharacteristic
9C2×#386—21no—
10C2×#385—22no—
11C2×#384C2211yescharacteristic
12C2×#340—21no—
13C2×#340—22no—
14C2×#336C2211yescharacteristic
15C2×#336C2212yes—
16#1441—21no—
17#1439—22no—
18C4⋊#95C2212yes—
19#1270C414yescomplemented
20C23×D4D411yescharacteristic
21C24×C4—21no—
22C23×C8—22no—
23C23×C8—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4D412yes—
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4D412yes—
30C22×C22⋊C4—22no—
31C22×C22⋊C4—22no—
32C22×C22⋊C4—21no—
33C22×C42C2311yescharacteristic
34C2×#106—44no—
35C2×#106—44no—
36C2×#106—44no—
37C2×#102—42no—
38C2×#102—42no—
39C2×#102—42no—
40#390—22no—
41#390C2×C412yes—
42#390Q812yes—
43#386—42no—
44#386—42no—
45#385—44no—
46#385—44no—
47#384—42no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—21no—
52#340—44no—
53#340—42no—
54#336—22no—
55#336—22no—
56#336C2×C414yescomplemented
57C25—21no—
58C22×D4—21no—
59C22×D4—21no—
60C22×D4—21no—
61C22×D4—41no—
62C22×D4—41no—
63C22×D4—41no—
64C22×D4—41no—
65C22×D4C4⋊C412yes—
66C22×D4SD1612yes—
67C22×D4SD1612yes—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—44no—
72C23×C4—42no—
73C23×C4—41no—
74C23×C4—82no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—24no—
82C22×C8—22no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—41no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—81no—
117C2×C4⋊C4C4⋊C412yes—
118C2×C4⋊C4SD1614yes—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4—44no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—82no—
129C2×C22⋊C4—81no—
130C2×C22⋊C4—81no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42C22×C411yescharacteristic
134C2×C42C2×Q811yescharacteristic
135C2×C42#4112yes—
136#106—416no—
137#102—48no—
138#95—44no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—81no—
144C24—22no—
145C24—22no—
146C24—22no—
147C24—42no—
148C24—42no—
149C24—42no—
150C24—41no—
151C24—41no—
152C24—42no—
153C24—41no—
154C24—41no—
155C24#12011yescharacteristic
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—81no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4#10614yescomplemented
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—84no—
201C22×C4—88no—
202C22×C4—88no—
203C22×C4—84no—
204C22×C4—84no—
205C22×C4—81no—
206C22×C4—82no—
207C22×C4—81no—
208C22×C4—82no—
209C22×C4C2×C4⋊C411yescharacteristic
210C22×C4C4×D412yes—
211C22×C4C22⋊Q812yes—
212C22×C4#12311yescharacteristic
213C22×C4C2×SD1611yescharacteristic
214C22×C4C2×SD1611yescharacteristic
215C22×C4C2×SD1612yes—
216C22×C4#13512yes—
217C22×C4AGL(1,ℤ/8)11yescharacteristic
218C22×C4AGL(1,ℤ/8)12yes—
219C22×C4AGL(1,ℤ/8)12yes—
220C22×C4#13711yescharacteristic
221C2×C8—28no—
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—81no—
232C2×C8—81no—
233C2×C8—81no—
234C2×C8—81no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—81no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—88no—
255C22⋊C4—84no—
256C42—41no—
257C42—41no—
258C42—42no—
259C23—24no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—44no—
264C23—42no—
265C23—44no—
266C23—42no—
267C23—42no—
268C23—44no—
269C23—44no—
270C23—44no—
271C23—44no—
272C23—82no—
273C23—82no—
274C23—82no—
275C23—82no—
276C23—81no—
277C23—81no—
278C23—82no—
279C23—81no—
280C23—81no—
281C23—82no—
282C23#38511yescharacteristic
283C23#44811yescharacteristic
284C23#44912yes—
285C23#45011yescharacteristic
286C23#45012yes—
287C23#45311yescharacteristic
288C23#45312yes—
289C23#45511yescharacteristic
290C23#45612yes—
291C23B(4,𝔽2)11yescharacteristic
292C23#45811yescharacteristic
293D4—44no—
294D4—82no—
295D4—82no—
296D4—82no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—816no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—84no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—161no—
344C2×C4—162no—
345C2×C4—161no—
346C2×C4C2×#10611yescharacteristic
347C2×C4#42811yescharacteristic
348C2×C4C4×SD1612yes—
349C2×C4#44212yes—
350C2×C4#47612yes—
351C2×C4#47812yes—
352C2×C4#48111yescharacteristic
353C2×C4#48311yescharacteristic
354C8—48no—
355C8—82no—
356C22—44no—
357C22—48no—
358C22—82no—
359C22—82no—
360C22—84no—
361C22—84no—
362C22—82no—
363C22—82no—
364C22—84no—
365C22—82no—
366C22—82no—
367C22#159611yescharacteristic
368C22#160012yes—
369C22#160111yescharacteristic
370C22#160412yes—
371C22#160811yescharacteristic
372C22#160912yes—
373C22#161312yes—
374C22#161311yescharacteristic
375C22#162011yescharacteristic
376C22#162212yes—
377C22#162911yescharacteristic
378C22#162912yes—
379C22#164212yes—
380C22#165112yes—
381C22#165412yes—
382C22#165511yescharacteristic
383C22#166011yescharacteristic
384C22#166112yes—
385C22#185311yescharacteristic
386C22#185611yescharacteristic
387C22#185912yes—
388C22#186412yes—
389C22#186711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—84no—
399C4—82no—
400C4—162no—
401C4—162no—
402C2—84no—
403C2#774511yescharacteristic
404C2#775411yescharacteristic
405C2#777812yes—
406C2#780112yes—
407C2#781911yescharacteristic
408C2#782211yescharacteristic
409C2#885411yescharacteristic
410C2#887312yes—
411C2#888612yes—
412C2#891011yescharacteristic
413C2#892411yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11323C2semidirect
5. #7025C2semidirect
19. #1270C4semidirect
56. #336C2×C4semidirect
171. C2×D4#106semidirect

Maximal subgroups

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

subgroupindex
#144402
#113232
#110492
C2×#12702
#70252