Finite Group Databasebeta

← #101104

Subgroups — #101104

Order 512.

720 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 102 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#15015C211yesmaximal, characteristic
2#14347C211yesmaximal, characteristic
3#8713C212yesmaximal
4#8698C211yesmaximal, characteristic
5#7035C212yesmaximal
6C2×#380—21no—
7C2×#380C2211yescharacteristic
8C22×#95C2211yescharacteristic
9#1779C2212yes—
10#1445C412yescomplemented
11#1443—22no—
12#1442—24no—
13#1442C412yescomplemented
14C22⋊#95—21no—
15C22⋊#95—21no—
16C22⋊#95—21no—
17C22⋊#95C2211yescharacteristic
18C22⋊#95C2212yes—
19#1294—24no—
20#1280—22no—
21C24×C4D411yescharacteristic
22C24×C4D411yescharacteristic
23C24×C4C2311yescharacteristic
24C22×C4⋊C4—22no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4—21no—
29C22×C22⋊C4—22no—
30C2×C22⋊C8—22no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33#380—22no—
34#380—22no—
35#380—42no—
36#380—42no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—21no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—21no—
49C2×#95C2×C412yes—
50C2×#95C2×C412yes—
51C2×#95D412yes—
52C2×#95D414yes—
53C25C2×D411yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—22no—
63C23×C4—24no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4—81no—
69C23×C4—81no—
70C23×C4—81no—
71C23×C4—81no—
72C23×C4C22⋊C412yes—
73C23×C4C22⋊C412yes—
74C23×C4SD1612yes—
75C23×C4Q1612yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D412yes—
78C23×C4C2×D411yescharacteristic
79C23×C4#4112yes—
80C22×C8—42no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—82no—
86C2×C4⋊C4—82no—
87C2×C4⋊C4—81no—
88C2×C22⋊C4—44no—
89C2×C22⋊C4—44no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—41no—
99C22⋊C8—44no—
100C22⋊C8—44no—
101C22⋊C8—82no—
102#95—24no—
103#95—24no—
104#95—42no—
105#95—44no—
106#95—44no—
107#95—44no—
108#95—42no—
109#95—81no—
110#95—82no—
111#95—82no—
112#95—82no—
113#95—82no—
114#95—82no—
115#95—82no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—41no—
121C24—41no—
122C24C2×C22⋊C411yescharacteristic
123C24#12011yescharacteristic
124C24#12311yescharacteristic
125C24C2×SD1611yescharacteristic
126C24C2×Q1611yescharacteristic
127C24AGL(1,ℤ/8)11yescharacteristic
128C24#13711yescharacteristic
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—88no—
191C22×C4—88no—
192C22×C4#10314yes—
193C22×C4C4×D412yes—
194C22×C4#12112yes—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—82no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4—88no—
201C22⋊C4—82no—
202C22⋊C4—82no—
203C22⋊C4—84no—
204C22⋊C4—84no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—44no—
234C23—44no—
235C23—44no—
236C23—44no—
237C23#38611yescharacteristic
238C23#41012yes—
239C23C2×#10311yescharacteristic
240C23#41711yescharacteristic
241C23#42112yes—
242C23#45011yescharacteristic
243C23#45111yescharacteristic
244C23#45412yes—
245C23#45812yes—
246C23#48311yescharacteristic
247C23#48411yescharacteristic
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—82no—
266C2×C4—84no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—84no—
276C2×C4—84no—
277C2×C4—82no—
278C2×C4—84no—
279C2×C4—82no—
280C2×C4—84no—
281C2×C4—82no—
282C2×C4—84no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C8—162no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—44no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#151212yes—
328C22#151812yes—
329C22#161612yes—
330C22#161712yes—
331C22#187211yescharacteristic
332C22#188811yescharacteristic
333C22#189011yescharacteristic
334C22#189711yescharacteristic
335C22#189911yescharacteristic
336C22#191111yescharacteristic
337C22#191311yescharacteristic
338C4—42no—
339C4—42no—
340C4—44no—
341C4—82no—
342C4—164no—
343C4—162no—
344C4—161no—
345C4—161no—
346C2—22no—
347C2—22no—
348C2—22no—
349C2—22no—
350C2—41no—
351C2—41no—
352C2#902311yescharacteristic
353C2#915211yescharacteristic
354C2#919911yescharacteristic
355C2#941411yescharacteristic
356C2#946511yescharacteristic
357C2#948111yescharacteristic
358C2#950011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #1445C4semidirect
13. #1442C4semidirect

Maximal subgroups

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

subgroupindex
#150152
#143472
#87132
#86982
#70352