Finite Group Databasebeta

← #99659

Subgroups — #99659

Order 512.

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

normal subgroups only

This group has 865 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
1C2×#2094C211yesmaximal, characteristic
2#14440C211yesmaximal, characteristic
3#11371C212yesmaximal, complemented
4C2×#1270C211yesmaximal, characteristic
5#7025C212yesmaximal, complemented
6C22×#106—21no—
7C22×#102—21no—
8C2×#390C2211yescharacteristic
9C2×#386—21no—
10C2×#385—22no—
11C2×#384C2211yescharacteristic
12#2094—22no—
13#2094C414yescomplemented
14C2×#340—22no—
15C2×#340—22no—
16C2×#336C2212yes—
17C2×#336C2211yescharacteristic
18#1441—21no—
19#1439—22no—
20C4⋊#95C2212yes—
21#1270—22no—
22C23×D4D411yescharacteristic
23C24×C4—21no—
24C23×C8—21no—
25C23×C8—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4D412yes—
30C22×C4⋊C4D411yescharacteristic
31C22×C4⋊C4D412yes—
32C22×C22⋊C4—22no—
33C22×C22⋊C4—22no—
34C22×C22⋊C4—21no—
35C22×C42C2311yescharacteristic
36C2×#106—24no—
37C2×#106—42no—
38C2×#106—42no—
39C2×#102—24no—
40C2×#102—42no—
41C2×#102—42no—
42#390—22no—
43#390C2×C412yes—
44#390D412yes—
45#386—42no—
46#386—42no—
47#385—44no—
48#385—44no—
49#384—21no—
50#384—22no—
51#384C2×C412yescomplemented
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—21no—
56#340—44no—
57#340—44no—
58#336—21no—
59#336—22no—
60#336—22no—
61#336C2×C412yescomplemented
62C25—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—41no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4C22⋊C412yes—
71C22×D4SD1612yes—
72C22×D4Q1612yes—
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—44no—
77C23×C4—42no—
78C23×C4—41no—
79C23×C4—82no—
80C23×C4—81no—
81C23×C4—81no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—24no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8—42no—
95C22×C8—42no—
96C22×C8—42no—
97C22×C8—42no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—81no—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4SD1614yes—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—44no—
131C2×C22⋊C4—44no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—82no—
134C2×C22⋊C4—81no—
135C2×C22⋊C4—81no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42C22×C411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4112yes—
141#106—48no—
142#102—48no—
143#95—44no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—81no—
149C24—22no—
150C24—22no—
151C24—22no—
152C24—42no—
153C24—42no—
154C24—42no—
155C24—41no—
156C24—41no—
157C24—42no—
158C24—41no—
159C24—41no—
160C24#12011yescharacteristic
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4#10314yes—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—84no—
206C22×C4—88no—
207C22×C4—88no—
208C22×C4—84no—
209C22×C4—84no—
210C22×C4—82no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—82no—
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C4×D412yes—
216C22×C4#12112yes—
217C22×C4#12311yescharacteristic
218C22×C4C2×SD1611yescharacteristic
219C22×C4C2×SD1612yes—
220C22×C4C2×Q1611yescharacteristic
221C22×C4#13512yes—
222C22×C4AGL(1,ℤ/8)11yescharacteristic
223C22×C4AGL(1,ℤ/8)12yes—
224C22×C4#13711yescharacteristic
225C22×C4#13712yes—
226C2×C8—24no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—82no—
237C2×C8—82no—
238C2×C8—82no—
239C2×C8—82no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—82no—
248C4⋊C4—81no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C4⋊C4—84no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—88no—
260C22⋊C4—84no—
261C42—41no—
262C42—41no—
263C42—42no—
264C23—24no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—44no—
270C23—44no—
271C23—44no—
272C23—42no—
273C23—44no—
274C23—42no—
275C23—44no—
276C23—44no—
277C23—82no—
278C23—82no—
279C23—82no—
280C23—82no—
281C23—81no—
282C23—81no—
283C23—82no—
284C23—81no—
285C23—82no—
286C23—81no—
287C23#38611yescharacteristic
288C23#45011yescharacteristic
289C23#45012yes—
290C23#45111yescharacteristic
291C23#45212yes—
292C23#45311yescharacteristic
293C23#45311yescharacteristic
294C23#45312yes—
295C23#45412yes—
296C23#45811yescharacteristic
297C23#45811yescharacteristic
298D4—44no—
299D4—82no—
300D4—82no—
301D4—82no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—816no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—84no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—161no—
349C2×C4—162no—
350C2×C4—161no—
351C2×C4C2×#10311yescharacteristic
352C2×C4#41711yescharacteristic
353C2×C4C4×SD1612yes—
354C2×C4#44012yes—
355C2×C4#46112yes—
356C2×C4#46312yes—
357C2×C4#48311yescharacteristic
358C2×C4#48411yescharacteristic
359C8—44no—
360C8—84no—
361C22—48no—
362C22—44no—
363C22—82no—
364C22—84no—
365C22—82no—
366C22—82no—
367C22—84no—
368C22—82no—
369C22—84no—
370C22—82no—
371C22—82no—
372C22#161311yescharacteristic
373C22#161311yescharacteristic
374C22#161312yes—
375C22#161411yescharacteristic
376C22#161411yescharacteristic
377C22#161512yes—
378C22#161612yes—
379C22#161812yes—
380C22#162912yes—
381C22#162911yescharacteristic
382C22#163011yescharacteristic
383C22#163112yes—
384C22#163212yes—
385C22#163512yes—
386C22#163712yes—
387C22#165511yescharacteristic
388C22#165612yes—
389C22#165711yescharacteristic
390C22#187211yescharacteristic
391C22#187712yes—
392C22#188312yes—
393C22#188711yescharacteristic
394C22#189011yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—82no—
404C4—84no—
405C4—162no—
406C4—162no—
407C2—84no—
408C2#782211yescharacteristic
409C2#782312yes—
410C2#782411yescharacteristic
411C2#782512yes—
412C2#782611yescharacteristic
413C2#783311yescharacteristic
414C2#897312yes—
415C2#900111yescharacteristic
416C2#902111yescharacteristic
417C2#904512yes—
418C2#906411yescharacteristic
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
3. #11371C2semidirect
5. #7025C2semidirect
13. #2094C4semidirect
51. #384C2×C4semidirect
61. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20942
#144402
#113712
C2×#12702
#70252