Finite Group Databasebeta

← #101030

Subgroups — #101030

Order 512.

810 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 104 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#21906C211yesmaximal, characteristic
2#15178C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#8340C212yesmaximal, complemented
5#7039C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C22×C22⋊Q8—21no—
8C2×#391—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1924—24no—
11#1797C2212yes—
12#1797C2212yes—
13#1779—21no—
14#1749—21no—
15C2×#332—24no—
16C2×#323—21no—
17#1444—21no—
18#1444C412yes—
19#1439—21no—
20#1439C412yes—
21C4⋊#95C2211yescharacteristic
22#1280—21no—
23C23×Q8—21no—
24C24×C4D411yescharacteristic
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—44no—
28C2×C22⋊Q8—44no—
29C2×C22⋊Q8—44no—
30C2×C22⋊Q8—44no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4C2×C412yes—
35C22×C22⋊C4—22no—
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42C2311yescharacteristic
40C2×C4⋊C8—22no—
41C2×C4⋊C8D414yes—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×#97—22no—
47#391—44no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—21no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C412yes—
55C2×#95D411yescharacteristic
56#336—22no—
57#336—22no—
58#336—22no—
59#332—48no—
60#323—42no—
61C25C22⋊C411yescharacteristic
62C22×Q8—24no—
63C22×Q8—42no—
64C22×Q8—42no—
65C22×Q8—42no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—42no—
69C22×Q8—42no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—24no—
85C22×C8—42no—
86C4⋊Q8—28no—
87C4⋊Q8—44no—
88C4⋊Q8—48no—
89C22⋊Q8—432no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—44no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—44no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42C2×D412yes—
115C2×C42#4112yes—
116C4⋊C8—44no—
117C4⋊C8—44no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—82no—
123C22⋊C8—82no—
124#97—82no—
125#95—22no—
126#95—22no—
127#95—42no—
128#95—44no—
129#95—44no—
130#95—41no—
131#95—41no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—41no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24#10112yes—
143C24C2×C22⋊C411yescharacteristic
144C2×Q8—48no—
145C2×Q8—44no—
146C2×Q8—48no—
147C2×Q8—44no—
148C2×Q8—82no—
149C2×Q8—84no—
150C2×Q8—84no—
151C2×Q8—82no—
152C2×Q8—82no—
153C2×Q8—82no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—48no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
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—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—84no—
199C22×C4C4×D412yes—
200C22×C4#12011yescharacteristic
201C22×C4#12112yes—
202C22×C4#12311yescharacteristic
203C22×C4#13512yes—
204C22×C4AGL(1,ℤ/8)12yes—
205C22×C4AGL(1,ℤ/8)11yescharacteristic
206C22×C4#13712yes—
207C22×C4#13711yescharacteristic
208C2×C8—44no—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—82no—
212C2×C8—82no—
213C2×C8—82no—
214C4⋊C4—28no—
215C4⋊C4—48no—
216C4⋊C4—44no—
217C4⋊C4—48no—
218C4⋊C4—416no—
219C22⋊C4—44no—
220C22⋊C4—416no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C42—24no—
224C42—42no—
225C42—42no—
226C42—81no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—81no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23#38611yescharacteristic
255C23C2×#10111yescharacteristic
256C23#41311yescharacteristic
257C23#41611yescharacteristic
258C23#41711yescharacteristic
259C23#41811yescharacteristic
260C23#41911yescharacteristic
261Q8—88no—
262Q8—88no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—44no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—84no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—84no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4#44912yes—
309C2×C4#45312yes—
310C2×C4B(4,𝔽2)12yes—
311C2×C4#46812yes—
312C2×C4#47012yes—
313C2×C4#48212yes—
314C8—84no—
315C8—162no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—44no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#155211yescharacteristic
340C22#155411yescharacteristic
341C22#155611yescharacteristic
342C22#155711yescharacteristic
343C22#187511yescharacteristic
344C22#188711yescharacteristic
345C22#188911yescharacteristic
346C22#189411yescharacteristic
347C22#190111yescharacteristic
348C22#190911yescharacteristic
349C22#191511yescharacteristic
350C4—22no—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—84no—
355C4—81no—
356C4—84no—
357C4—81no—
358C4—81no—
359C4#160014yes—
360C2—21no—
361C2—21no—
362C2—21no—
363C2—21no—
364C2—82no—
365C2#902011yescharacteristic
366C2#914011yescharacteristic
367C2#919611yescharacteristic
368C2#940311yescharacteristic
369C2#945511yescharacteristic
370C2#947111yescharacteristic
371C2#950511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8340C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219062
#151782
#85572
#83402
#70392