Finite Group Databasebeta

← #142041

Subgroups — #142041

Order 512.

612 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 92 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#15045C211yesmaximal, characteristic
2#10846C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7250C212yesmaximal
5#7250C212yesmaximal
6C2×#389C411yescharacteristic, complemented
7C2×#380—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9#1749—21no—
10C2×#342—21no—
11C2×#342C2212yes—
12C2×#323C2212yes—
13#1442—21no—
14#1442C411yescharacteristic, complemented
15#1442C411yescharacteristic, complemented
16#1439—21no—
17#1439C2211yescharacteristic
18#1439C2211yescharacteristic
19#1434—21no—
20#1434C411yescharacteristic, complemented
21C22⋊#95—21no—
22#1280—21no—
23C24×C4D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—41no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—22no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4D411yescharacteristic
32C22×C22⋊C4C2311yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C22⋊C8—22no—
36C2×#97—41no—
37#389—41no—
38#389—41no—
39#380—24no—
40#380—42no—
41C4×C22⋊C4—41no—
42C4×C22⋊C4—41no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—41no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95D411yescharacteristic
57#342—24no—
58#342—42no—
59#336—41no—
60#323—22no—
61#323Q814yes—
62C25C2×D411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—22no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4—81no—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—81no—
88C2×C4⋊C4—81no—
89C2×C4⋊C4—82no—
90C2×C4⋊C4—82no—
91C2×C4⋊C4—81no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—81no—
110C2×C22⋊C4—81no—
111C2×C22⋊C4—81no—
112C2×C22⋊C4C2×Q812yes—
113C2×C22⋊C4#4112yes—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—42no—
119C2×C42—81no—
120C22⋊C8—24no—
121C22⋊C8—42no—
122#97—82no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—44no—
131#95—41no—
132#95—41no—
133#95—42no—
134#95—81no—
135#95—81no—
136#95—81no—
137#95—81no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—41no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24C2×C22⋊C411yescharacteristic
147C24#12411yescharacteristic
148C24C4⋊Q811yescharacteristic
149C24AGL(1,ℤ/8)11yescharacteristic
150C24AGL(1,ℤ/8)11yescharacteristic
151C24#13711yescharacteristic
152C24#13711yescharacteristic
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—44no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
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—81no—
215C22×C4—81no—
216C22×C4—84no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4#9911yescharacteristic
226C22×C4#9911yescharacteristic
227C22×C4#10412yes—
228C22×C4C4×Q812yes—
229C22×C4C22⋊Q812yes—
230C2×C8—44no—
231C2×C8—82no—
232C2×C8—82no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—82no—
236C4⋊C4—88no—
237C4⋊C4—84no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C42—42no—
251C42—81no—
252C42—81no—
253C42—81no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23#39111yescharacteristic
293C23C2×#9911yescharacteristic
294C23#41011yescharacteristic
295C23#41711yescharacteristic
296C23#41711yescharacteristic
297C23C2×#10411yescharacteristic
298C23#42111yescharacteristic
299C23#49011yescharacteristic
300C23#49111yescharacteristic
301C23#50111yescharacteristic
302C23#50111yescharacteristic
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—84no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—162no—
371C2×C4—162no—
372C2×C4—162no—
373C2×C4—162no—
374C2×C4#35412yes—
375C8—84no—
376C22—21no—
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—21no—
390C22—21no—
391C22—21no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22#150411yescharacteristic
415C22#150911yescharacteristic
416C22#151211yescharacteristic
417C22#151311yescharacteristic
418C22#198611yescharacteristic
419C22#199211yescharacteristic
420C22#199611yescharacteristic
421C22#212011yescharacteristic
422C22#212111yescharacteristic
423C22C2×#35411yescharacteristic
424C22#212811yescharacteristic
425C4—22no—
426C4—42no—
427C4—41no—
428C4—42no—
429C4—42no—
430C4—81no—
431C4—81no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4—164no—
436C4—161no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—41no—
445C2#988711yescharacteristic
446C2#1175411yescharacteristic
447C2#1175911yescharacteristic
448C2#1182311yescharacteristic
449C2#1182411yescharacteristic
450C2#1329411yescharacteristic
451C2#1329911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#389C4semidirect
14. #1442C4semidirect
15. #1442C4semidirect
20. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#150452
#108462
#81522
#72502