Finite Group Databasebeta

← #99552

Subgroups — #99552

Order 512.

864 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 120 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#21912C211yesmaximal, characteristic, complemented
2C4×#394C211yesmaximal, characteristic
3#8310C212yesmaximal
4C4×#324C211yesmaximal, characteristic, complemented
5#7028C212yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7C2×#394C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4—21no—
11#1916—24no—
12#1817—21no—
13#1797C2212yes—
14C4×C22⋊C8—21no—
15C4×C22⋊C8C2212yes—
16C2×D4⋊C8—24no—
17C2×#324—21no—
18#1442C411yescharacteristic
19#1441C411yescharacteristic
20#1440—21no—
21#1439—21no—
22#1438C411yescharacteristic
23C4×#95C2211yescharacteristic
24#1280—21no—
25C23×D4C2×C411yescharacteristic
26C24×C4—21no—
27C2×C4×D4—22no—
28C2×C4×D4—22no—
29C2×C4×D4—44no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C42—21no—
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42C2311yescharacteristic
42C2×C4⋊C8D414yes—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C4×C8—22no—
46C2×C4×C8—22no—
47#394—22no—
48#394C814yescomplemented
49C4×C4⋊C4—22no—
50C4×C4⋊C4—22no—
51C4×C22⋊C4—44no—
52C4×C22⋊C4—44no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95D411yescharacteristic
62#336—24no—
63#336—22no—
64#336—22no—
65#336—22no—
66D4⋊C8—48no—
67#324—24no—
68C25—21no—
69C22×D4—22no—
70C22×D4—21no—
71C22×D4—22no—
72C22×D4—42no—
73C22×D4—42no—
74C22×D4C2×C812yes—
75C23×C4—24no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—24no—
94C22×C8—42no—
95C4×D4—28no—
96C4×D4—44no—
97C4×D4—44no—
98C4×D4—44no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2×C812yes—
107C2×C4⋊C4M1612yes—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—21no—
129C2×C42—42no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—42no—
134C2×C42C2×D412yes—
135C2×C42#4112yes—
136C4⋊C8—44no—
137C22⋊C8—24no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140C22⋊C8—42no—
141C4×C8—44no—
142C4×C8—82no—
143#95—22no—
144#95—22no—
145#95—21no—
146#95—22no—
147#95—42no—
148#95—42no—
149#95—42no—
150#95—42no—
151#95—42no—
152#95C2×C812yes—
153C24—22no—
154C24—22no—
155C24—22no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24C2×C22⋊C411yescharacteristic
169C2×D4—24no—
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—44no—
175C2×D4—48no—
176C2×D4—44no—
177C2×D4—42no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—44no—
203C22×C4—48no—
204C22×C4—48no—
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—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—44no—
230C22×C4—82no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4C22⋊C812yes—
237C22×C4C22⋊C812yes—
238C22×C4#10412yes—
239C22×C4C4×D412yes—
240C22×C4#12011yescharacteristic
241C22×C4#12112yes—
242C22×C4#12311yescharacteristic
243C22×C4C22×C811yescharacteristic
244C22×C4C2×M1611yescharacteristic
245C22×C4#13112yes—
246C22×C4#13512yes—
247C22×C4AGL(1,ℤ/8)12yes—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—44no—
252C2×C8—82no—
253C2×C8—82no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C22⋊C4—48no—
260C22⋊C4—48no—
261C22⋊C4—48no—
262C22⋊C4—48no—
263C22⋊C4—48no—
264C42—24no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—81no—
270C42—84no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—24no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23#38611yescharacteristic
311C23C2×C22⋊C811yescharacteristic
312C23#40811yescharacteristic
313C23#41611yescharacteristic
314C23#41811yescharacteristic
315C23C2×#10411yescharacteristic
316C23#42111yescharacteristic
317D4—28no—
318D4—44no—
319D4—44no—
320D4—44no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
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—42no—
347C2×C4—42no—
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—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—88no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
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—88no—
379C2×C4—81no—
380C2×C4C8×D412yes—
381C2×C4#43512yes—
382C2×C4#44912yes—
383C2×C4#45412yes—
384C2×C4B(4,𝔽2)12yes—
385C2×C4#48212yes—
386C8—84no—
387C8—84no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22#147611yescharacteristic
416C22#147811yescharacteristic
417C22#148211yescharacteristic
418C22#148411yescharacteristic
419C22#186911yescharacteristic
420C22#187111yescharacteristic
421C22#187511yescharacteristic
422C22#188411yescharacteristic
423C22#188811yescharacteristic
424C22#188911yescharacteristic
425C22#189111yescharacteristic
426C4—22no—
427C4—22no—
428C4—22no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—82no—
434C4—82no—
435C4—82no—
436C4—81no—
437C4#159714yes—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—84no—
443C2#894011yescharacteristic
444C2#896011yescharacteristic
445C2#896511yescharacteristic
446C2#900911yescharacteristic
447C2#901811yescharacteristic
448C2#903211yescharacteristic
449C2#903611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21912C2semidirect
4. C4×#324C2semidirect
5. #7028C2semidirect
48. #394C8semidirect

Maximal subgroups

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

subgroupindex
#219122
C4×#3942
#83102
C4×#3242
#70282