Finite Group Databasebeta

← #100194

Subgroups — #100194

Order 512.

780 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 106 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#21902C211yesmaximal, characteristic
2#14039C211yesmaximal, characteristic
3#8700C211yesmaximal, characteristic
4#8299C212yesmaximal
5#7032C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C2×#387—21no—
8C2×#385C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1923—24no—
11#1797—21no—
12#1779C2212yes—
13#1779C2212yes—
14C4×C22⋊C8—21no—
15C2×#324—24no—
16#1442—21no—
17#1441—21no—
18#1436—21no—
19C22⋊#95C411yescharacteristic
20C22⋊#95C411yescharacteristic
21C22⋊#95C411yescharacteristic
22C22⋊#95C2211yescharacteristic
23#1280—21no—
24#1280—21no—
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2311yescharacteristic
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—44no—
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C2×C4⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8D414yes—
47C2×C4×C8—22no—
48#387—44no—
49#385—22no—
50#385C814yescomplemented
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95D411yescharacteristic
63C2×#95D411yescharacteristic
64#336—22no—
65#336—22no—
66#324—48no—
67C25C22×C411yescharacteristic
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—21no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—81no—
85C23×C4—81no—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C2×C812yes—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4#4112yes—
95C22×C8—24no—
96C22×C8—42no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4C2×C812yes—
123C2×C22⋊C4C2×C812yes—
124C2×C22⋊C4M1612yes—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—42no—
130C2×C42—81no—
131C4⋊C8—82no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C22⋊C8—44no—
135C22⋊C8—44no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138C22⋊C8—82no—
139C22⋊C8—82no—
140C4×C8—82no—
141#95—24no—
142#95—22no—
143#95—22no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—81no—
150#95—81no—
151#95—81no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24C2×C22⋊C411yescharacteristic
161C24C4×D412yes—
162C24C22×C811yescharacteristic
163C24C2×M1611yescharacteristic
164C24#13112yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—21no—
189C22×C4—44no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
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—42no—
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—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4—84no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4C22⋊C812yes—
242C22×C4C22⋊C812yes—
243C22×C4#10112yes—
244C22×C4#12011yescharacteristic
245C22×C4#12112yes—
246C22×C4#12311yescharacteristic
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C4⋊C4—48no—
254C4⋊C4—44no—
255C4⋊C4—48no—
256C4⋊C4—44no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—82no—
263C42—42no—
264C42—42no—
265C42—81no—
266C42—81no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—41no—
287C23—41no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23#38611yescharacteristic
298C23C2×C22⋊C811yescharacteristic
299C23#40811yescharacteristic
300C23#41011yescharacteristic
301C23#41011yescharacteristic
302C23C2×#10111yescharacteristic
303C23#41311yescharacteristic
304C23C8×D412yes—
305C23#43512yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—28no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—84no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—84no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—84no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—84no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4B(4,𝔽2)12yes—
375C8—84no—
376C8—162no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—42no—
396C22—42no—
397C22—44no—
398C22—42no—
399C22—42no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22#145811yescharacteristic
406C22#146111yescharacteristic
407C22#147011yescharacteristic
408C22#147311yescharacteristic
409C22#186911yescharacteristic
410C22#188911yescharacteristic
411C22#189111yescharacteristic
412C22#189911yescharacteristic
413C22#190111yescharacteristic
414C22#191311yescharacteristic
415C22#191511yescharacteristic
416C4—22no—
417C4—44no—
418C4—42no—
419C4—42no—
420C4—41no—
421C4—42no—
422C4—42no—
423C4—82no—
424C4—82no—
425C4—81no—
426C4—161no—
427C2—21no—
428C2—21no—
429C2—21no—
430C2—21no—
431C2—42no—
432C2—42no—
433C2#896511yescharacteristic
434C2#908211yescharacteristic
435C2#910811yescharacteristic
436C2#949811yescharacteristic
437C2#950611yescharacteristic
438C2#951411yescharacteristic
439C2#951711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
50. #385C8semidirect

Maximal subgroups

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

subgroupindex
#219022
#140392
#87002
#82992
#70322