Finite Group Databasebeta

← #127577

Subgroups — #127577

Order 512.

960 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 154 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×#1800C211yesmaximal, characteristic
2C2×#1465C211yesmaximal, characteristic
3#13839C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C22×#117C411yescharacteristic
7C2×#423—21no—
8C22×#100—21no—
9C2×#407—21no—
10C2×C4×C22⋊C4—21no—
11C2×C4×C22⋊C4C411yescharacteristic
12#1800—22no—
13#1800C414yescomplemented
14C2×#343C2212yes—
15#1465—22no—
16#1465C414yescomplemented
17C2×#324C2212yes—
18C2×#323—21no—
19#1441C2212yes—
20#1440—21no—
21#1439C412yes—
22#1438—21no—
23C4×#95—21no—
24C23×D4D411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C22×M16—22no—
28C22×M16—41no—
29C2×C4×D4—41no—
30C2×C4×D4—41no—
31C2×C4×D4—42no—
32C2×C4×D4C2×C412yes—
33C2×C4×D4C2×C412yes—
34C2×C4×D4D412yes—
35C2×C4×D4Q812yes—
36C2×#117—44no—
37C2×#117—41no—
38C2×#117—41no—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4D411yescharacteristic
46C22×C42—21no—
47C22×C42—21no—
48C22×C42D411yescharacteristic
49#423—24no—
50#423—42no—
51C2×C4⋊C8—41no—
52C2×C4⋊C8—41no—
53C2×#100—24no—
54C2×#100—44no—
55#407—24no—
56#407—42no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—41no—
59C4×C22⋊C4—44no—
60C4×C22⋊C4—42no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95C2×C412yes—
65#343—22no—
66#343C2×C414yes—
67#324—22no—
68#324C2×C414yes—
69#323—24no—
70C25C22⋊C411yescharacteristic
71C25C22⋊C411yescharacteristic
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4—42no—
77C22×D4C22⋊C412yes—
78C22×D4C4⋊C412yes—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4C4212yes—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C4⋊C412yes—
96C23×C4C4⋊C412yes—
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—24no—
101C2×M16—42no—
102C2×M16—44no—
103C2×M16—44no—
104C2×M16—44no—
105C2×M16—82no—
106C2×M16—82no—
107C22×C8—42no—
108C22×C8—42no—
109C4×D4—42no—
110C4×D4—42no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117#117—48no—
118#117—44no—
119#117—44no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4C22×C411yescharacteristic
125C2×C4⋊C4C22×C411yescharacteristic
126C2×C4⋊C4C2×D411yescharacteristic
127C2×C4⋊C4C2×Q811yescharacteristic
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—44no—
133C2×C22⋊C4—44no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4C4212yes—
140C2×C22⋊C4C22⋊C412yes—
141C2×C22⋊C4C4⋊C412yes—
142C2×C22⋊C4C4⋊C412yes—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—22no—
146C2×C42—24no—
147C2×C42—42no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42—42no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42C22⋊C412yes—
154C2×C42C4⋊C412yes—
155C4⋊C8—44no—
156C4⋊C8—44no—
157#100—48no—
158C22⋊C8—24no—
159C22⋊C8—42no—
160C22⋊C8—44no—
161#95—42no—
162#95—44no—
163#95—44no—
164#95—42no—
165#95—42no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—22no—
170C24—21no—
171C24—22no—
172C24—22no—
173C24—22no—
174C24—41no—
175C24—42no—
176C24—41no—
177C24—41no—
178C24—42no—
179C24—41no—
180C24#9512yes—
181C24#9512yes—
182C24C2×C22⋊C411yescharacteristic
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—44no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—48no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—84no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4#9512yes—
247C22×C4#9512yes—
248C22×C4#9912yes—
249C22×C4#9911yescharacteristic
250C22×C4#9911yescharacteristic
251C22×C4C2×C4211yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4AGL(1,ℤ/8)12yes—
258C22×C4#13712yes—
259M16—44no—
260M16—44no—
261M16—84no—
262M16—88no—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—84no—
266C2×C8—82no—
267C2×C8—82no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—48no—
279C22⋊C4—416no—
280C22⋊C4—48no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—44no—
286C42—44no—
287C42—84no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—44no—
301C23—42no—
302C23—44no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—82no—
318C23—82no—
319C23—82no—
320C23—82no—
321C23—82no—
322C23—82no—
323C23C2×#9511yescharacteristic
324C23C2×#9911yescharacteristic
325C23C2×#9911yescharacteristic
326C23#41712yes—
327C23#42112yes—
328D4—44no—
329D4—44no—
330D4—48no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4#34212yes—
384C2×C4#34212yes—
385C2×C4AΣL(1,𝔽16)12yes—
386C2×C4#35212yes—
387C2×C4#41811yescharacteristic
388C2×C4#41911yescharacteristic
389C2×C4#42812yes—
390C8—84no—
391C8—84no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22—84no—
411C22—84no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22#151312yes—
417C22#151312yes—
418C22#173311yescharacteristic
419C22#173611yescharacteristic
420C22C2×#34211yescharacteristic
421C22C2×AΣL(1,𝔽16)11yescharacteristic
422C22C2×#35211yescharacteristic
423C22#213212yes—
424C4—22no—
425C4—21no—
426C4—21no—
427C4—22no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—42no—
433C4—84no—
434C4—84no—
435C4#139014yes—
436C2—22no—
437C2—21no—
438C2—21no—
439C2—82no—
440C2—82no—
441C2#801611yescharacteristic
442C2#1182812yes—
443C2#1183012yes—
444C2C2×#139011yescharacteristic
445C2#1258511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7251C2semidirect
13. #1800C4semidirect
16. #1465C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18002
C2×#14652
#138392
#72512