Finite Group Databasebeta

← #110605

Subgroups — #110605

Order 512.

958 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 174 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×#1826C212yesmaximal
2#20524C211yesmaximal, characteristic
3#13853C211yesmaximal, characteristic
4C4×#342C211yesmaximal, characteristic
5C2×#1297C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8C2×C4×C22⋊C4C2212yes—
9C2×C4×C22⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11#1826—24no—
12#1826C418yescomplemented
13#1753—24no—
14#1749—21no—
15C4×C22⋊C8—21no—
16C2×#342—22no—
17C2×#339—24no—
18#1445C412yescomplemented
19#1445C412yescomplemented
20#1440—22no—
21C4×#95—21no—
22C4×#95—21no—
23#1297—24no—
24C24×C4C2311yescharacteristic
25C22×M16D414yes—
26C2×#117—22no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117C2×C414yes—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C22⋊C4—22no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4D412yes—
38C22×C42—21no—
39C22×C42—22no—
40C22×C42—21no—
41C22×C42D411yescharacteristic
42C22×C42D411yescharacteristic
43#407—22no—
44#407—22no—
45#407C2×C414yes—
46#407D414yes—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×#97—22no—
50C2×C4×C8—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—22no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4—22no—
58C4×C22⋊C4C2×C414yes—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95C2×C412yes—
65C2×#95C2×C412yes—
66#342—28no—
67#339—216no—
68#336—22no—
69#336—22no—
70C25C2×D411yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4C22⋊C412yes—
78C23×C4C22⋊C412yes—
79C23×C4C22×C412yes—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D412yes—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C23×C4#4112yes—
85C23×C4#4112yes—
86C23×C4#4112yes—
87C2×M16—24no—
88C2×M16—44no—
89C2×M16—44no—
90C2×M16C22⋊C418yes—
91C22×C8—24no—
92#117—24no—
93#117—24no—
94#117—24no—
95#117—24no—
96#117—42no—
97#117—42no—
98#117—42no—
99#117—42no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4C4212yes—
107C2×C4⋊C4C4212yes—
108C2×C22⋊C4—28no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4—44no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—41no—
122C2×C42—24no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—24no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—41no—
129C2×C42—42no—
130C2×C42—41no—
131C2×C42—44no—
132C2×C42—44no—
133C2×C42—42no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42—42no—
139C22⋊C8—24no—
140C22⋊C8—24no—
141C22⋊C8—24no—
142C22⋊C8—24no—
143#97—82no—
144C4×C8—82no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—44no—
149#95—44no—
150#95—44no—
151#95—44no—
152#95—42no—
153C24—22no—
154C24—22no—
155C24—42no—
156C24—41no—
157C24—41no—
158C24C2×C22⋊C411yescharacteristic
159C24C4×D412yes—
160C24#12112yes—
161C24#12411yescharacteristic
162C24#12711yescharacteristic
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—48no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—84no—
206C22×C4#10414yes—
207C22×C4C2×C4211yescharacteristic
208C22×C4C2×C22⋊C412yes—
209C22×C4#11712yes—
210C22×C4#11711yescharacteristic
211C22×C4C4×D412yes—
212C22×C4C4×D412yes—
213C22×C4C4×D412yes—
214C22×C4C4×D412yes—
215C22×C4C4×D412yes—
216C22×C4#12011yescharacteristic
217C22×C4#12112yes—
218C22×C4#12112yes—
219C22×C4#12312yes—
220C22×C4#12311yescharacteristic
221C22×C4#12412yes—
222C22×C4#12612yes—
223M16—48no—
224M16—84no—
225C2×C8—28no—
226C2×C8—44no—
227C2×C8—44no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C22⋊C4—28no—
233C22⋊C4—48no—
234C22⋊C4—48no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C42—48no—
241C42—44no—
242C42—44no—
243C42—42no—
244C42—42no—
245C42—81no—
246C42—81no—
247C42—84no—
248C42—84no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—24no—
253C23—24no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—41no—
261C23—41no—
262C23—42no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—82no—
267C23—82no—
268C23—82no—
269C23—82no—
270C23C4×C22⋊C411yescharacteristic
271C23#38611yescharacteristic
272C23#38812yes—
273C23#39012yes—
274C23#39011yescharacteristic
275C23C2×#10412yes—
276C23#42112yes—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—84no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—88no—
320C2×C4—88no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4C4×C22⋊C412yes—
327C2×C4#38612yes—
328C2×C4#38812yes—
329C2×C4#38812yes—
330C2×C4#45812yes—
331C2×C4#45812yes—
332C8—48no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—44no—
348C22—44no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22—82no—
353C22#143711yescharacteristic
354C22C4×#10411yescharacteristic
355C22#175911yescharacteristic
356C22#187112yes—
357C22#188412yes—
358C22#189011yescharacteristic
359C22#189011yescharacteristic
360C22#198012yes—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—44no—
365C4—84no—
366C4—88no—
367C4—82no—
368C4—82no—
369C4—82no—
370C4#189014yes—
371C2—22no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#901112yes—
377C2#901212yes—
378C2#1093311yescharacteristic
379C2#1093611yescharacteristic
380C2#1095911yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC23#386
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1826C4semidirect
18. #1445C4semidirect
19. #1445C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18262
#205242
#138532
C4×#3422
C2×#12972