Finite Group Databasebeta

← #121776

Subgroups — #121776

Order 512.

756 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 134 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#13953C211yesmaximal, characteristic
2C4×#342C211yesmaximal, characteristic
3C2×#1297C212yesmaximal
4C2×#1284C212yesmaximal
5#7911C211yesmaximal, characteristic
6C2×#407C2212yes—
7C2×#381C2211yescharacteristic
8C2×C4×C22⋊C4C2212yes—
9C2×C4×C22⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11C2×#342—21no—
12C2×#342—21no—
13#1439—21no—
14#1439—21no—
15#1436—21no—
16#1436—21no—
17C4⋊#95C411yescharacteristic, complemented
18C4⋊#95C411yescharacteristic, complemented
19#1434C411yescharacteristic, complemented
20#1434C411yescharacteristic, complemented
21#1297—24no—
22#1284—24no—
23#1280—21no—
24#1280—21no—
25C24×C4C2311yescharacteristic
26C22×M16—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C22⋊C4—22no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4D411yescharacteristic
33C22×C22⋊C4D411yescharacteristic
34C22×C42—22no—
35C22×C42—21no—
36C22×C42C2×C411yescharacteristic
37C22×C42C2×C411yescharacteristic
38#407—22no—
39#407—22no—
40#407—22no—
41#407Q814yes—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44#381—22no—
45#381—22no—
46C4×C22⋊C4—22no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4D414yes—
54C2×#95—22no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95C2×C411yescharacteristic
63C2×#95C2×C411yescharacteristic
64C2×#95C2×C411yescharacteristic
65C2×#95C2×C411yescharacteristic
66C2×#95D411yescharacteristic
67C2×#95D411yescharacteristic
68#342—24no—
69#342—24no—
70C25C2×D411yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—42no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D412yes—
85C23×C4C2×Q812yes—
86C23×C4#4112yes—
87C23×C4#4112yes—
88C23×C4#4112yes—
89C23×C4#4112yes—
90C2×M16—24no—
91C2×M16—42no—
92C2×M16—82no—
93C2×M16—82no—
94C22×C8—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—81no—
100C2×C4⋊C4—81no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
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—81no—
118C2×C22⋊C4—81no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—24no—
124C2×C42—41no—
125C2×C42—42no—
126C2×C42—41no—
127C2×C42—81no—
128C2×C42—81no—
129C2×C42—82no—
130C2×C42—82no—
131C2×C42—81no—
132C2×C42—81no—
133C22⋊C8—24no—
134C22⋊C8—24no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—24no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—41no—
149#95—42no—
150#95—41no—
151#95—81no—
152#95—81no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24C2×C22⋊C411yescharacteristic
162C24#12112yes—
163C24#12312yes—
164C24#12411yescharacteristic
165C24C4⋊Q811yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—44no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
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—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4#10412yes—
232C22×C4#10412yes—
233C22×C4#11711yescharacteristic
234C22×C4#11711yescharacteristic
235C22×C4C4×D412yes—
236C22×C4C4×Q812yes—
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q812yes—
239C22×C4C22⋊Q812yes—
240C22×C4#12312yes—
241C22×C4#12312yes—
242C22×C4#12312yes—
243C22×C4#12411yescharacteristic
244C22×C4#12512yes—
245C22×C4#12512yes—
246C22×C4#12612yes—
247C22×C4#12612yes—
248C22×C4#12711yescharacteristic
249M16—84no—
250M16—84no—
251C2×C8—44no—
252C2×C8—82no—
253C2×C8—82no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C22⋊C4—28no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—48no—
263C22⋊C4—84no—
264C42—81no—
265C42—81no—
266C42—81no—
267C42—81no—
268C42—84no—
269C42—82no—
270C42—84no—
271C42—82no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—42no—
289C23—42no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23#38111yescharacteristic
303C23#39011yescharacteristic
304C23#39111yescharacteristic
305C23#39712yes—
306C23#39812yes—
307C23C2×#10411yescharacteristic
308C23C2×#10411yescharacteristic
309C23#42111yescharacteristic
310C23#42111yescharacteristic
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—48no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
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—81no—
354C2×C4—81no—
355C2×C4—84no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
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—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4#38712yes—
386C2×C4#38712yes—
387C2×C4#39912yes—
388C2×C4#40012yes—
389C8—84no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—82no—
413C22—82no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#144511yescharacteristic
419C22#182611yescharacteristic
420C22#182611yescharacteristic
421C22#182711yescharacteristic
422C22#182711yescharacteristic
423C22#182811yescharacteristic
424C22#182811yescharacteristic
425C22#196211yescharacteristic
426C22#196311yescharacteristic
427C22#198611yescharacteristic
428C22#198611yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—42no—
434C4—42no—
435C4—82no—
436C4—82no—
437C4—88no—
438C4—81no—
439C4—81no—
440C4—161no—
441C4—161no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—42no—
447C2—42no—
448C2#877611yescharacteristic
449C2#877711yescharacteristic
450C2#877811yescharacteristic
451C2#877911yescharacteristic
452C2#1134711yescharacteristic
453C2#1135111yescharacteristic
454C2#1137011yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C4⋊#95C4semidirect
18. C4⋊#95C4semidirect
19. #1434C4semidirect
20. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#139532
C4×#3422
C2×#12972
C2×#12842
#79112