Finite Group Databasebeta

← #120045

Subgroups — #120045

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14347C211yesmaximal, characteristic
2#14275C211yesmaximal, characteristic
3#8023C212yesmaximal
4C2×#1289C212yesmaximal
5C2×#1280C211yesmaximal, characteristic
6C22×C22⋊C8C2212yes—
7C2×#385—21no—
8C2×#380C2211yescharacteristic
9C2×#380C2212yes—
10C2×#380C2211yescharacteristic
11C22×#95C2211yescharacteristic
12#1444C412yescomplemented
13#1442—24no—
14#1442C412yescomplemented
15#1441—22no—
16C22⋊#95—21no—
17C22⋊#95—21no—
18C22⋊#95—21no—
19#1294—24no—
20#1289—24no—
21#1280—22no—
22C24×C4D411yescharacteristic
23C24×C4D411yescharacteristic
24C24×C4C2311yescharacteristic
25C23×C8—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—22no—
33C22×C22⋊C4—21no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8Q814yes—
40#385—44no—
41#380—21no—
42#380—21no—
43#380—22no—
44#380—22no—
45#380—22no—
46#380—22no—
47#380—22no—
48#380—22no—
49#380D412yes—
50#380Q812yes—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95C2×C412yes—
62C2×#95C2×C412yes—
63C2×#95D414yes—
64C2×#95D412yes—
65C25C2×D411yescharacteristic
66C23×C4—22no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—24no—
78C23×C4—81no—
79C23×C4—81no—
80C23×C4—81no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4—81no—
84C23×C4C22⋊C412yes—
85C23×C4C22⋊C412yes—
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D412yes—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×Q811yescharacteristic
91C23×C4C2×Q812yes—
92C23×C4#4112yes—
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4112yes—
96C22×C8—24no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—82no—
100C22×C8—82no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—81no—
109C2×C4⋊C4—81no—
110C2×C4⋊C4—82no—
111C2×C4⋊C4—82no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4—82no—
114C2×C4⋊C4—81no—
115C2×C4⋊C4—81no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C22⋊C8—24no—
131C22⋊C8—24no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134#95—22no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—44no—
140#95—42no—
141#95—82no—
142#95—82no—
143#95—82no—
144#95—82no—
145#95—82no—
146#95—82no—
147#95C4⋊C414yes—
148C24—41no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24C2×C22⋊C411yescharacteristic
153C24#12011yescharacteristic
154C24#12111yescharacteristic
155C24C22⋊Q811yescharacteristic
156C24#12311yescharacteristic
157C24#12411yescharacteristic
158C24C4⋊Q811yescharacteristic
159C24#13512yes—
160C24#13512yes—
161C24AGL(1,ℤ/8)12yes—
162C24#13712yes—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—44no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—84no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4C2×C4⋊C411yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4C4×D412yes—
230C22×C4C4×Q812yes—
231C22×C4#12112yes—
232C22×C4#12112yes—
233C22×C4#12111yescharacteristic
234C22×C4C22⋊Q812yes—
235C22×C4C22⋊Q812yes—
236C22×C4C22⋊Q812yes—
237C22×C4C22⋊Q811yescharacteristic
238C22×C4#12312yes—
239C22×C4#12512yes—
240C22×C4#12512yes—
241C22×C4#12612yes—
242C22×C4C4⋊Q812yes—
243C2×C8—44no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—84no—
247C2×C8—84no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—84no—
253C4⋊C4—84no—
254C4⋊C4—88no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23#38011yescharacteristic
292C23#38611yescharacteristic
293C23#39111yescharacteristic
294C23#39411yescharacteristic
295C23#39511yescharacteristic
296C23#39811yescharacteristic
297C23#40011yescharacteristic
298C23#41611yescharacteristic
299C23#41611yescharacteristic
300C23#41811yescharacteristic
301C23#41911yescharacteristic
302C23#44912yes—
303C23#45212yes—
304C23#46312yes—
305C23#46412yes—
306C23#47812yes—
307C23#47912yes—
308C23#48212yes—
309C23#48512yes—
310C23#48812yes—
311C23#48912yes—
312C23#49912yes—
313C23#50112yes—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—84no—
370C2×C4#38712yes—
371C2×C4#38912yes—
372C2×C4#39612yes—
373C2×C4#39812yes—
374C8—84no—
375C22—21no—
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—22no—
390C22—22no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22#144111yescharacteristic
412C22#180411yescharacteristic
413C22#180511yescharacteristic
414C22#187311yescharacteristic
415C22#187511yescharacteristic
416C22#191111yescharacteristic
417C22#191211yescharacteristic
418C22#191311yescharacteristic
419C22#191311yescharacteristic
420C22#199111yescharacteristic
421C22#199311yescharacteristic
422C22#204212yes—
423C22#205912yes—
424C22#208512yes—
425C22#210112yes—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—84no—
431C4—82no—
432C4—82no—
433C4—164no—
434C4—161no—
435C4—161no—
436C2—21no—
437C2—22no—
438C2—22no—
439C2—21no—
440C2—22no—
441C2—22no—
442C2—21no—
443C2—21no—
444C2#920111yescharacteristic
445C2#920211yescharacteristic
446C2#920511yescharacteristic
447C2#920611yescharacteristic
448C2#1120511yescharacteristic
449C2#1120811yescharacteristic
450C2#1122411yescharacteristic
subgroupG/H
center, socleC23#386
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
12. #1444C4semidirect
14. #1442C4semidirect

Maximal subgroups

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

subgroupindex
#143472
#142752
#80232
C2×#12892
C2×#12802