Finite Group Databasebeta

← #115702

Subgroups — #115702

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21930C211yesmaximal, characteristic
2C22×#324C212yesmaximal
3#13776C211yesmaximal, characteristic
4C2×#1282C212yesmaximal
5#7907C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C22×C22⋊C8C2212yes—
9C2×#381C411yescharacteristic
10C2×#381C411yescharacteristic
11C2×C4×C22⋊C4C2211yescharacteristic
12C2×C4×C22⋊C4C2211yescharacteristic
13#1850—24no—
14#1797—22no—
15C2×#324—24no—
16C2×#324—24no—
17C2×#324—24no—
18#1436—22no—
19#1436—22no—
20C22⋊#95C411yescharacteristic
21C22⋊#95C411yescharacteristic
22#1282—24no—
23#1280—22no—
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2311yescharacteristic
27C23×C8—22no—
28C23×C8—22no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4D412yes—
38C22×C4⋊C4Q812yes—
39C22×C22⋊C4C2×C411yescharacteristic
40C22×C22⋊C4C2×C411yescharacteristic
41C22×C22⋊C4C2×C411yescharacteristic
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C42—21no—
44C22×C42D412yes—
45C2×C4⋊C8—24no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8D414yes—
57C2×C22⋊C8D414yes—
58#381—22no—
59#381—22no—
60#381—22no—
61#381C814yescomplemented
62C4×C22⋊C4—22no—
63C4×C22⋊C4—22no—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—21no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95D412yes—
72#336—24no—
73#324—216no—
74C25C22×C411yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—82no—
90C23×C4—82no—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C412yes—
93C23×C4C4⋊C412yes—
94C23×C4C4⋊C412yes—
95C23×C4C2×C812yes—
96C23×C4M1612yes—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×Q811yescharacteristic
103C23×C4#4112yes—
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C23×C4#4112yes—
107C22×C8—24no—
108C22×C8—24no—
109C22×C8—42no—
110C22×C8—42no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—84no—
114C22×C8—84no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4C2×C812yes—
146C2×C22⋊C4C2×C812yes—
147C2×C22⋊C4M1612yes—
148C2×C22⋊C4M1612yes—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—42no—
152C2×C42—82no—
153C2×C42—82no—
154C4⋊C8—84no—
155C22⋊C8—24no—
156C22⋊C8—24no—
157C22⋊C8—24no—
158C22⋊C8—24no—
159C22⋊C8—42no—
160C22⋊C8—42no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163#95—28no—
164#95—28no—
165#95—24no—
166#95—41no—
167#95—41no—
168#95—82no—
169#95—82no—
170C24—42no—
171C24—42no—
172C24C2×C22⋊C411yescharacteristic
173C24C2×C4⋊C411yescharacteristic
174C24#11711yescharacteristic
175C24C4×D412yes—
176C24C4×D412yes—
177C24C22×C811yescharacteristic
178C24C2×M1611yescharacteristic
179C24C2×M1611yescharacteristic
180C24C2×M1611yescharacteristic
181C24#13112yes—
182C24#13112yes—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—82no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—82no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—84no—
244C22×C4—84no—
245C22×C4C22⋊C814yes—
246C22×C4C4⋊C814yescomplemented
247C22×C4#12011yescharacteristic
248C22×C4#12012yes—
249C22×C4#12112yes—
250C22×C4#12112yes—
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12112yes—
254C22×C4C22⋊Q812yes—
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q812yes—
258C22×C4#12311yescharacteristic
259C22×C4#12312yes—
260C22×C4#12312yes—
261C22×C4#12412yes—
262C22×C4#12411yescharacteristic
263C22×C4#12612yes—
264C22×C4#12711yescharacteristic
265C2×C8—48no—
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—84no—
269C2×C8—84no—
270C2×C8—84no—
271C2×C8—84no—
272C2×C8—84no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C42—44no—
288C42—82no—
289C42—84no—
290C23—22no—
291C23—24no—
292C23—22no—
293C23—24no—
294C23—41no—
295C23—41no—
296C23—42no—
297C23—41no—
298C23—41no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—44no—
305C23—44no—
306C23#38011yescharacteristic
307C23#38512yes—
308C23#38611yescharacteristic
309C23#38812yes—
310C23#39011yescharacteristic
311C23C2×C22⋊C811yescharacteristic
312C23#40711yescharacteristic
313C23#40811yescharacteristic
314C23#40811yescharacteristic
315C23C2×C4⋊C811yescharacteristic
316C23#42311yescharacteristic
317C23#42412yes—
318C23#43111yescharacteristic
319C23#43211yescharacteristic
320C23#43312yes—
321C23C8×D412yes—
322C23C8×D412yes—
323C23#43512yes—
324C23#43512yes—
325C23#43512yes—
326C23#43512yes—
327C23#43612yes—
328C23#43612yes—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—28no—
334C2×C4—24no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—84no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—84no—
385C2×C4#39211yescharacteristic
386C2×C4#39311yescharacteristic
387C2×C4#39412yes—
388C2×C4#39612yes—
389C2×C4#39711yescharacteristic
390C2×C4#39911yescharacteristic
391C8—84no—
392C8—84no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—22no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—24no—
404C22—24no—
405C22—22no—
406C22—22no—
407C22—24no—
408C22—24no—
409C22—41no—
410C22—41no—
411C22—42no—
412C22—41no—
413C22—41no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22#143911yescharacteristic
420C22#179711yescharacteristic
421C22#179811yescharacteristic
422C22#185012yes—
423C22#185112yes—
424C22#186911yescharacteristic
425C22#187011yescharacteristic
426C22#191312yes—
427C22#191512yes—
428C22#192312yes—
429C22#192412yes—
430C22#195411yescharacteristic
431C22#195511yescharacteristic
432C4—21no—
433C4—22no—
434C4—21no—
435C4—22no—
436C4—21no—
437C4—21no—
438C4—44no—
439C4—44no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—82no—
444C2—21no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—22no—
449C2—22no—
450C2—22no—
451C2—22no—
452C2#910812yes—
453C2#911012yes—
454C2#1104211yescharacteristic
455C2#1104611yescharacteristic
456C2#1104811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ 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
61. #381C8semidirect
246. C22×C4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
#219302
C22×#3242
#137762
C2×#12822
#79072