Finite Group Databasebeta

← #138331

Subgroups — #138331

Order 512.

660 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 90 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×#1822C211yesmaximal, characteristic
2#14893C211yesmaximal, characteristic
3C2×#1353C212yesmaximal, complemented
4#8300C211yesmaximal, characteristic
5#7251C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7C2×#407—21no—
8C2×#400C2211yescharacteristic
9C2×#399C411yescharacteristic
10C2×#389C411yescharacteristic
11#1822—24no—
12#1817—21no—
13C2×#343—21no—
14C2×#343C2212yes—
15C2×#324C2212yes—
16C2×#323—21no—
17#1444—21no—
18#1441C2211yescharacteristic
19#1439—21no—
20#1439C411yescharacteristic
21#1436—21no—
22#1434C411yescharacteristic
23C22⋊#95—21no—
24#1353—24no—
25C24×C4D411yescharacteristic
26C22×M16—22no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—42no—
30C2×#117—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4C2311yescharacteristic
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C22⋊C4C2×C411yescharacteristic
40C22×C22⋊C4D411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42D411yescharacteristic
43#407—24no—
44#407—42no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—42no—
47#400—22no—
48#400—22no—
49#399—42no—
50#389—42no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61#343—24no—
62#343—22no—
63#343D414yes—
64#336—42no—
65#324—22no—
66#324—22no—
67#323—42no—
68C25C22⋊C411yescharacteristic
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C2×M16—24no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C2×M16—42no—
93C22×C8—42no—
94C22×C8—42no—
95#117—48no—
96#117—48no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—81no—
112C2×C4⋊C4—81no—
113C2×C4⋊C4—81no—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4#4112yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—81no—
127C2×C22⋊C4—81no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—41no—
133C2×C42—41no—
134C22⋊C8—48no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137#95—22no—
138#95—41no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—41no—
144#95—42no—
145#95—41no—
146#95—42no—
147#95—81no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—41no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24#9912yes—
159C24C2×C22⋊C411yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
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—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4#9912yes—
227C22×C4C4×D412yes—
228C22×C4#12011yescharacteristic
229C22×C4#12112yes—
230C22×C4#12311yescharacteristic
231C22×C4#13512yes—
232C22×C4#13712yes—
233M16—44no—
234M16—44no—
235M16—82no—
236M16—82no—
237C2×C8—44no—
238C2×C8—82no—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—82no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—84no—
255C22⋊C4—84no—
256C42—44no—
257C42—44no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23#38611yescharacteristic
286C23C2×#9911yescharacteristic
287C23C2×#9911yescharacteristic
288C23#41611yescharacteristic
289C23#41911yescharacteristic
290C23#42111yescharacteristic
291C23#42111yescharacteristic
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
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—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4#45212yes—
358C2×C4#48512yes—
359C8—84no—
360C8—162no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—44no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#151111yescharacteristic
385C22#151111yescharacteristic
386C22#151611yescharacteristic
387C22#151611yescharacteristic
388C22#187311yescharacteristic
389C22#189711yescharacteristic
390C22#189811yescharacteristic
391C22#212011yescharacteristic
392C22#212311yescharacteristic
393C22#213211yescharacteristic
394C22#213411yescharacteristic
395C4—22no—
396C4—22no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—82no—
401C4—81no—
402C4—81no—
403C4—82no—
404C4—161no—
405C4—161no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—82no—
411C2#915411yescharacteristic
412C2#1181611yescharacteristic
413C2#1181811yescharacteristic
414C2#1184711yescharacteristic
415C2#1185311yescharacteristic
416C2#1308111yescharacteristic
417C2#1309311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1353C2semidirect
5. #7251C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18222
#148932
C2×#13532
#83002
#72512