Finite Group Databasebeta

← #135367

Subgroups — #135367

Order 512.

732 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 92 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#13923C211yesmaximal, characteristic
2#10727C211yesmaximal, characteristic
3#8299C211yesmaximal, characteristic
4#7251C212yesmaximal, complemented
5#7251C212yesmaximal, complemented
6C23×C4⋊C4C2211yescharacteristic
7C2×#407—21no—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C411yescharacteristic
10#1779—21no—
11C2×#343—21no—
12C2×#343C2212yes—
13C2×#324—21no—
14C2×#324C2212yes—
15#1441C411yescharacteristic
16#1441C411yescharacteristic
17#1441C2211yescharacteristic
18#1441C2211yescharacteristic
19#1436—21no—
20#1436—21no—
21C22⋊#95—21no—
22C22⋊#95—21no—
23C24×C4C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4D411yescharacteristic
26C22×M16—22no—
27C22×C4⋊C4—24no—
28C22×C4⋊C4—42no—
29C22×C4⋊C4—41no—
30C22×C4⋊C4—41no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4C2311yescharacteristic
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C42—21no—
40#407—24no—
41#407—42no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—41no—
44C2×C22⋊C8—41no—
45C2×C22⋊C8—41no—
46C2×C22⋊C8—41no—
47#380—41no—
48#380—41no—
49C4×C22⋊C4—42no—
50C2×#95—21no—
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—42no—
63#324—24no—
64#324—42no—
65C25C22⋊C411yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4—41no—
87C23×C4—81no—
88C23×C4—81no—
89C23×C4—81no—
90C23×C4—81no—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C2×M16—24no—
98C2×M16—42no—
99C2×M16—42no—
100C2×M16—42no—
101C2×M16—42no—
102C2×M16—42no—
103C22×C8—42no—
104C22×C8—42no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—48no—
117C2×C4⋊C4—48no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—81no—
121C2×C4⋊C4—81no—
122C2×C4⋊C4#4114yes—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C42—42no—
131C2×C42—81no—
132C2×C42—81no—
133C22⋊C8—44no—
134C22⋊C8—44no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—82no—
138C22⋊C8—82no—
139#95—41no—
140#95—41no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—41no—
146#95—41no—
147#95—81no—
148#95—81no—
149#95—81no—
150#95—81no—
151C24—22no—
152C24—22no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—41no—
160C24—41no—
161C24#9911yescharacteristic
162C24#9911yescharacteristic
163C24C2×C22⋊C411yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
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—42no—
216C22×C4—42no—
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—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—84no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4#9912yes—
250C22×C4#10412yes—
251C22×C4#11712yes—
252C22×C4#12312yes—
253C22×C4#12312yes—
254C22×C4#13512yes—
255C22×C4AGL(1,ℤ/8)11yescharacteristic
256C22×C4#13711yescharacteristic
257M16—44no—
258M16—44no—
259M16—82no—
260M16—82no—
261C2×C8—44no—
262C2×C8—82no—
263C2×C8—82no—
264C2×C8—82no—
265C2×C8—82no—
266C2×C8—82no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C22⋊C4—48no—
274C22⋊C4—82no—
275C22⋊C4—82no—
276C42—84no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—24no—
282C23—24no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#35512yes—
310C23#35612yes—
311C23#38111yescharacteristic
312C23C2×#9911yescharacteristic
313C23C2×#9911yescharacteristic
314C23#41611yescharacteristic
315C23#41711yescharacteristic
316C23C2×#10411yescharacteristic
317C23#42111yescharacteristic
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4#48212yes—
390C2×C4#48512yes—
391C8—84no—
392C8—162no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—44no—
412C22—44no—
413C22—44no—
414C22—44no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22#150111yescharacteristic
424C22#150411yescharacteristic
425C22#151311yescharacteristic
426C22#151611yescharacteristic
427C22#181911yescharacteristic
428C22#182511yescharacteristic
429C22#182611yescharacteristic
430C22#212111yescharacteristic
431C22#212111yescharacteristic
432C22C2×#35511yescharacteristic
433C22C2×#35611yescharacteristic
434C4—22no—
435C4—22no—
436C4—42no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—82no—
441C4—82no—
442C4—81no—
443C4—161no—
444C4—161no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—42no—
450C2—42no—
451C2#871511yescharacteristic
452C2#1176611yescharacteristic
453C2#1176811yescharacteristic
454C2#1185211yescharacteristic
455C2#1185411yescharacteristic
456C2#1285411yescharacteristic
457C2#1286011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7251C2semidirect
5. #7251C2semidirect

Maximal subgroups

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

subgroupindex
#139232
#107272
#82992
#72512