Finite Group Databasebeta

← #135334

Subgroups — #135334

Order 512.

1002 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 116 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×#1441C211yesmaximal, characteristic
2#10727C211yesmaximal, characteristic
3#8299C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C23×C4⋊C4C411yescharacteristic
6C23×C4⋊C4C2211yescharacteristic
7C2×#407—21no—
8C22×#95—21no—
9C22×#95—21no—
10C22×#95C411yescharacteristic
11#1779—21no—
12C2×#343—21no—
13C2×#343C2212yes—
14C2×#324—21no—
15C2×#324C2212yes—
16#1441—21no—
17#1441—21no—
18#1441C412yes—
19#1441C2212yes—
20C24×C4—21no—
21C24×C4—21no—
22C24×C4C2×C411yescharacteristic
23C24×C4D411yescharacteristic
24C24×C4D411yescharacteristic
25C22×M16—22no—
26C22×C4⋊C4—24no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—42no—
29C22×C4⋊C4—41no—
30C22×C4⋊C4—41no—
31C22×C4⋊C4—41no—
32C22×C4⋊C4—41no—
33C22×C4⋊C4—44no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4D411yescharacteristic
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4C2311yescharacteristic
39#407—24no—
40#407—42no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—41no—
43C2×C22⋊C8—41no—
44C2×C22⋊C8—41no—
45C2×C22⋊C8—41no—
46C2×#95—21no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—44no—
52C2×#95—44no—
53C2×#95—42no—
54C2×#95C2×C412yes—
55#343—24no—
56#343—42no—
57#324—24no—
58#324—42no—
59C25C22⋊C411yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—24no—
70C23×C4—24no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—41no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—41no—
94C23×C4—41no—
95C23×C4—42no—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4SD1612yes—
100C23×C4Q1612yes—
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C2×M16—24no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—42no—
108C2×M16—42no—
109C2×M16—42no—
110C22×C8—42no—
111C22×C8—42no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—48no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—48no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4#4114yes—
133C22⋊C8—44no—
134C22⋊C8—44no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—82no—
138C22⋊C8—82no—
139#95—42no—
140#95—44no—
141#95—44no—
142#95—44no—
143#95—42no—
144#95—44no—
145C24—22no—
146C24—22no—
147C24—22no—
148C24—22no—
149C24—21no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24#9911yescharacteristic
154C24#9911yescharacteristic
155C24#10312yes—
156C24#10412yes—
157C24C2×C22⋊C411yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
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—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—48no—
199C22×C4—48no—
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—42no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—48no—
215C22×C4—48no—
216C22×C4—48no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—42no—
220C22×C4—48no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4—84no—
239C22×C4—84no—
240C22×C4—84no—
241C22×C4—82no—
242C22×C4—84no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4#9912yes—
247C22×C4#10312yes—
248C22×C4#11712yes—
249C22×C4#12312yes—
250C22×C4#12312yes—
251C22×C4C2×SD1611yescharacteristic
252C22×C4C2×Q1611yescharacteristic
253C22×C4AGL(1,ℤ/8)12yes—
254M16—44no—
255M16—44no—
256M16—82no—
257M16—82no—
258C2×C8—44no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—82no—
262C2×C8—82no—
263C2×C8—82no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—48no—
271C23—24no—
272C23—24no—
273C23—24no—
274C23—24no—
275C23—24no—
276C23—24no—
277C23—24no—
278C23—24no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—24no—
288C23—24no—
289C23—24no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—22no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23#32812yes—
300C23#32812yes—
301C23AΣL(1,𝔽16)12yes—
302C23#35212yes—
303C23#35512yes—
304C23#35612yes—
305C23#38111yescharacteristic
306C23C2×#9911yescharacteristic
307C23C2×#9911yescharacteristic
308C23C2×#10311yescharacteristic
309C23#41811yescharacteristic
310C23C2×#10411yescharacteristic
311C23#42111yescharacteristic
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—816no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—88no—
367C2×C4—88no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4#48112yes—
373C2×C4#48412yes—
374C8—84no—
375C8—162no—
376C22—28no—
377C22—28no—
378C22—28no—
379C22—28no—
380C22—22no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—21no—
387C22—22no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—22no—
394C22—21no—
395C22—22no—
396C22—22no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—21no—
404C22#135314yes—
405C22#135414yes—
406C22C2×#32811yescharacteristic
407C22#150711yescharacteristic
408C22#151011yescharacteristic
409C22#152011yescharacteristic
410C22#181911yescharacteristic
411C22#182411yescharacteristic
412C22#182611yescharacteristic
413C22C2×AΣL(1,𝔽16)11yescharacteristic
414C22C2×#35211yescharacteristic
415C22C2×#35511yescharacteristic
416C22C2×#35611yescharacteristic
417C4—22no—
418C4—22no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—84no—
424C4—84no—
425C4—81no—
426C2—21no—
427C2—21no—
428C2—21no—
429C2—24no—
430C2—24no—
431C2—21no—
432C2#871311yescharacteristic
433C2C2×#135311yescharacteristic
434C2C2×#135411yescharacteristic
435C2#1180211yescharacteristic
436C2#1180311yescharacteristic
437C2#1285211yescharacteristic
438C2#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

Maximal subgroups

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

subgroupindex
C2×#14412
#107272
#82992
#72512