Finite Group Databasebeta

← #127579

Subgroups — #127579

Order 512.

822 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 146 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#14344C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#423—21no—
7C22×#100—21no—
8C2×#407—21no—
9C2×#381C411yescharacteristic
10C2×#380C411yescharacteristic
11#1800—22no—
12#1800C414yescomplemented
13C2×#343C2212yes—
14#1465—22no—
15#1465C414yescomplemented
16C2×#324C2212yes—
17C2×#323—21no—
18#1441C2212yes—
19#1440—21no—
20#1439C412yes—
21#1438—21no—
22#1434—21no—
23C22⋊#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—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2311yescharacteristic
38C22×C22⋊C4—22no—
39C22×C22⋊C4—21no—
40C22×C22⋊C4—21no—
41C22×C22⋊C4C2×C411yescharacteristic
42C22×C22⋊C4D411yescharacteristic
43C22×C42D411yescharacteristic
44#423—24no—
45#423—42no—
46C2×C4⋊C8—41no—
47C2×C4⋊C8—41no—
48C2×#100—24no—
49C2×#100—44no—
50#407—24no—
51#407—42no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—41no—
54#381—42no—
55#380—41no—
56#380—41no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95C2×C412yes—
63#343—22no—
64#343C2×C414yes—
65#324—22no—
66#324C2×C414yes—
67#323—24no—
68C25C22⋊C411yescharacteristic
69C25C22⋊C411yescharacteristic
70C22×D4—22no—
71C22×D4—22no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—42no—
75C22×D4C22⋊C412yes—
76C22×D4C4⋊C412yes—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—82no—
86C23×C4—82no—
87C23×C4C4212yes—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C4⋊C412yes—
92C23×C4C4⋊C412yes—
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C2×M16—24no—
97C2×M16—42no—
98C2×M16—44no—
99C2×M16—44no—
100C2×M16—44no—
101C2×M16—84no—
102C22×C8—42no—
103C22×C8—42no—
104C4×D4—42no—
105C4×D4—42no—
106C4×D4—42no—
107C4×D4—42no—
108C4×D4—42no—
109C4×D4—42no—
110C4×D4—42no—
111C4×D4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4C22×C411yescharacteristic
117C2×C4⋊C4C22×C411yescharacteristic
118C2×C4⋊C4C2×D411yescharacteristic
119C2×C4⋊C4C2×Q811yescharacteristic
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—82no—
128C2×C22⋊C4—82no—
129C2×C22⋊C4—82no—
130C2×C22⋊C4—82no—
131C2×C22⋊C4C4212yes—
132C2×C22⋊C4C22⋊C412yes—
133C2×C22⋊C4C4⋊C412yes—
134C2×C22⋊C4C4⋊C412yes—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—22no—
138C2×C42C22⋊C412yes—
139C2×C42C4⋊C412yes—
140C4⋊C8—44no—
141C4⋊C8—44no—
142#100—48no—
143C22⋊C8—24no—
144C22⋊C8—42no—
145C22⋊C8—44no—
146#95—42no—
147#95—41no—
148#95—41no—
149#95—42no—
150#95—42no—
151#95—42no—
152#95—82no—
153#95—82no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—22no—
158C24—21no—
159C24—22no—
160C24—22no—
161C24—22no—
162C24—41no—
163C24—42no—
164C24—41no—
165C24—41no—
166C24—42no—
167C24—41no—
168C24#9512yes—
169C24#9512yes—
170C24C2×C22⋊C411yescharacteristic
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—24no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4#9512yes—
229C22×C4#9512yes—
230C22×C4#9911yescharacteristic
231C22×C4#9911yescharacteristic
232C22×C4#9912yes—
233C22×C4C2×C4211yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C4⋊C411yescharacteristic
237C22×C4C2×C4⋊C411yescharacteristic
238C22×C4C2×C4⋊C411yescharacteristic
239C22×C4AGL(1,ℤ/8)12yes—
240C22×C4#13712yes—
241M16—44no—
242M16—44no—
243M16—84no—
244M16—88no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—84no—
248C2×C8—84no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—84no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—84no—
259C22⋊C4—88no—
260C22⋊C4—84no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—44no—
278C23—42no—
279C23—44no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—81no—
293C23—81no—
294C23—82no—
295C23—82no—
296C23—82no—
297C23—82no—
298C23—82no—
299C23—82no—
300C23C2×#9511yescharacteristic
301C23C2×#9911yescharacteristic
302C23C2×#9911yescharacteristic
303C23#41712yes—
304C23#42112yes—
305D4—44no—
306D4—44no—
307D4—48no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4#34212yes—
360C2×C4#34212yes—
361C2×C4#41811yescharacteristic
362C2×C4#41911yescharacteristic
363C2×C4#42812yes—
364C8—84no—
365C8—84no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—44no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—84no—
385C22—84no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22#151312yes—
391C22#151312yes—
392C22#173311yescharacteristic
393C22#173611yescharacteristic
394C22C2×#34211yescharacteristic
395C22#212112yes—
396C22#213411yescharacteristic
397C22#213511yescharacteristic
398C4—22no—
399C4—22no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—42no—
408C4—162no—
409C4—162no—
410C2—22no—
411C2—21no—
412C2—21no—
413C2—82no—
414C2—82no—
415C2#801611yescharacteristic
416C2#1183212yes—
417C2#1183412yes—
418C2#1257411yescharacteristic
419C2#1258711yescharacteristic
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
12. #1800C4semidirect
15. #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
#143442
#72512