Finite Group Databasebeta

← #100185

Subgroups — #100185

Order 512.

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

normal subgroups only

This group has 674 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#20524C211yesmaximal, characteristic
2#14225C211yesmaximal, characteristic
3#8706C212yesmaximal, complemented
4#8559C211yesmaximal, characteristic
5#7024C212yesmaximal, complemented
6C2×#407—21no—
7C2×#397C411yescharacteristic
8C2×#388—21no—
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11#1797C2212yes—
12#1753—24no—
13#1749—21no—
14C4×C22⋊C8C2212yes—
15C2×#323—21no—
16#1445C411yescharacteristic
17#1444C411yescharacteristic
18#1439C411yescharacteristic
19#1436—21no—
20#1434—21no—
21#1434C2211yescharacteristic
22C22⋊#95—21no—
23#1280—21no—
24#1279—24no—
25C24×C4D411yescharacteristic
26C22×M16—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4D411yescharacteristic
34C22×C42—21no—
35C22×C42D411yescharacteristic
36C22×C42C2311yescharacteristic
37C2×C4⋊C8—22no—
38#407—24no—
39#407—24no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×#97—22no—
44C2×C4×C8D414yes—
45#397—22no—
46#397—22no—
47#388—44no—
48C4×C4⋊C4—22no—
49C4×C4⋊C4—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—22no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95C2×C411yescharacteristic
63C2×#95D411yescharacteristic
64#336—22no—
65#336—22no—
66#336—22no—
67#323—42no—
68C25C22⋊C411yescharacteristic
69C23×C4—22no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—42no—
78C23×C4—81no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C2×M16—28no—
86C2×M16—82no—
87C2×M16—82no—
88C22×C8—24no—
89C22×C8—42no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—81no—
98C2×C4⋊C4M1612yescomplemented
99C2×C22⋊C4—24no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—81no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—81no—
121C2×C42C2×D412yes—
122C2×C42#4112yes—
123C4⋊C8—44no—
124C22⋊C8—28no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#97—82no—
130C4×C8—44no—
131#95—24no—
132#95—22no—
133#95—22no—
134#95—21no—
135#95—22no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—41no—
140#95—41no—
141#95—42no—
142#95—81no—
143#95—81no—
144#95M1612yescomplemented
145C24—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24—41no—
155C24#10412yes—
156C24C2×C22⋊C411yescharacteristic
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—24no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4#10412yes—
217C22×C4C4×D412yes—
218C22×C4#12011yescharacteristic
219C22×C4#12112yes—
220C22×C4#12311yescharacteristic
221C22×C4C2×M1611yescharacteristic
222C22×C4C2×M1611yescharacteristic
223C22×C4#13112yes—
224C22×C4AGL(1,ℤ/8)12yes—
225C22×C4AGL(1,ℤ/8)11yescharacteristic
226C22×C4#13711yescharacteristic
227M16—88no—
228C2×C8—28no—
229C2×C8—44no—
230C2×C8—44no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—84no—
239C22⋊C4—48no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—84no—
245C42—24no—
246C42—42no—
247C42—42no—
248C42—81no—
249C42—82no—
250C42—82no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23#38611yescharacteristic
279C23#40711yescharacteristic
280C23#40811yescharacteristic
281C23#41711yescharacteristic
282C23#41811yescharacteristic
283C23C2×#10411yescharacteristic
284C23C2×#10411yescharacteristic
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—81no—
326C2×C4—84no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4#43512yes—
349C2×C4#43612yes—
350C2×C4#45812yes—
351C2×C4#46812yes—
352C2×C4#46912yes—
353C8—48no—
354C8—162no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#148311yescharacteristic
379C22#148411yescharacteristic
380C22#148911yescharacteristic
381C22#149011yescharacteristic
382C22#187011yescharacteristic
383C22#189011yescharacteristic
384C22#189111yescharacteristic
385C22#189511yescharacteristic
386C22#189611yescharacteristic
387C22#190911yescharacteristic
388C22#191111yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—42no—
396C4—82no—
397C4—81no—
398C4—82no—
399C4—82no—
400C4—81no—
401C4—161no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—82no—
407C2#896811yescharacteristic
408C2#907811yescharacteristic
409C2#910311yescharacteristic
410C2#947411yescharacteristic
411C2#947711yescharacteristic
412C2#948911yescharacteristic
413C2#949111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ 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
3. #8706C2semidirect
5. #7024C2semidirect
98. C2×C4⋊C4M16semidirect
144. #95M16semidirect

Maximal subgroups

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

subgroupindex
#205242
#142252
#87062
#85592
#70242