Finite Group Databasebeta

← #100166

Subgroups — #100166

Order 512.

820 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 112 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
2#14176C211yesmaximal, characteristic
3#8560C211yesmaximal, characteristic, complemented
4#8310C212yesmaximal
5#7024C212yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7C2×#394C411yescharacteristic
8C2×#388—21no—
9C2×#385—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1916—24no—
12#1817—21no—
13#1797—21no—
14#1797C2212yes—
15C4×C22⋊C8C2212yes—
16C2×D4⋊C8—24no—
17C2×#324—21no—
18#1444C411yescharacteristic
19#1441C411yescharacteristic
20#1438C411yescharacteristic
21#1437—21no—
22#1434C2211yescharacteristic
23C22⋊#95—21no—
24#1280—21no—
25C23×D4C2×C411yescharacteristic
26C24×C4—21no—
27C2×C4×D4—22no—
28C2×C4×D4—22no—
29C2×C4×D4—44no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42C2311yescharacteristic
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C4×C8D414yes—
47#394—22no—
48#394C814yescomplemented
49#388—44no—
50#385—44no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—22no—
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—22no—
63#336—24no—
64#336—22no—
65#336—22no—
66D4⋊C8—48no—
67#324—42no—
68C25—21no—
69C22×D4—22no—
70C22×D4—21no—
71C22×D4—22no—
72C22×D4—42no—
73C22×D4—42no—
74C22×D4C2×C812yes—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—41no—
82C23×C4—42no—
83C23×C4—41no—
84C23×C4—81no—
85C23×C4—81no—
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—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—81no—
110C2×C4⋊C4C2×C812yes—
111C2×C4⋊C4M1612yes—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—81no—
127C2×C22⋊C4—81no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—42no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42—42no—
136C2×C42C2×D412yes—
137C2×C42#4112yes—
138C4⋊C8—44no—
139C4⋊C8—82no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142C22⋊C8—42no—
143C22⋊C8—42no—
144C4×C8—44no—
145#95—22no—
146#95—22no—
147#95—21no—
148#95—22no—
149#95—42no—
150#95—42no—
151#95—42no—
152#95—42no—
153#95—81no—
154#95C2×C812yes—
155C24—22no—
156C24—22no—
157C24—22no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24C2×C22⋊C411yescharacteristic
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—24no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—44no—
177C2×D4—48no—
178C2×D4—44no—
179C2×D4—42no—
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—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
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—41no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—84no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4C22⋊C812yes—
240C22×C4C22⋊C812yes—
241C22×C4C4×D412yes—
242C22×C4#12011yescharacteristic
243C22×C4#12112yes—
244C22×C4#12311yescharacteristic
245C22×C4C22×C811yescharacteristic
246C22×C4C2×M1611yescharacteristic
247C22×C4#13112yes—
248C22×C4AGL(1,ℤ/8)12yes—
249C22×C4AGL(1,ℤ/8)11yescharacteristic
250C22×C4#13711yescharacteristic
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—82no—
255C2×C8—82no—
256C2×C8—82no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—84no—
264C22⋊C4—48no—
265C22⋊C4—44no—
266C22⋊C4—48no—
267C22⋊C4—44no—
268C22⋊C4—84no—
269C22⋊C4—84no—
270C42—24no—
271C42—42no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—81no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—24no—
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—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23#38611yescharacteristic
316C23C2×C22⋊C811yescharacteristic
317C23#40811yescharacteristic
318C23#41711yescharacteristic
319C23#41811yescharacteristic
320C23#42111yescharacteristic
321C23#42111yescharacteristic
322D4—28no—
323D4—44no—
324D4—44no—
325D4—44no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—88no—
369C2×C4—88no—
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×C4C8×D412yes—
389C2×C4#43512yes—
390C2×C4B(4,𝔽2)12yes—
391C2×C4#46812yes—
392C2×C4#46912yes—
393C8—84no—
394C8—162no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—44no—
408C22—44no—
409C22—44no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22#147711yescharacteristic
423C22#147811yescharacteristic
424C22#149511yescharacteristic
425C22#149611yescharacteristic
426C22#186911yescharacteristic
427C22#188911yescharacteristic
428C22#189111yescharacteristic
429C22#189511yescharacteristic
430C22#189711yescharacteristic
431C22#191011yescharacteristic
432C22#191211yescharacteristic
433C4—22no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—42no—
439C4—41no—
440C4—42no—
441C4—82no—
442C4—81no—
443C4—82no—
444C4—81no—
445C4—161no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—84no—
451C2#896511yescharacteristic
452C2#907411yescharacteristic
453C2#910211yescharacteristic
454C2#947211yescharacteristic
455C2#948011yescharacteristic
456C2#949011yescharacteristic
457C2#949411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21912C2semidirect
3. #8560C2semidirect
5. #7024C2semidirect
48. #394C8semidirect

Maximal subgroups

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

subgroupindex
#219122
#141762
#85602
#83102
#70242