Finite Group Databasebeta

← #109620

Subgroups — #109620

Order 512.

776 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20837C211yesmaximal, characteristic
2#20386C211yesmaximal, characteristic
3#10729C211yesmaximal, characteristic
4#8055C211yesmaximal, characteristic
5#8049C211yesmaximal, characteristic
6#7942C211yesmaximal, characteristic
7#7942C211yesmaximal, characteristic
8C23×M16C2211yescharacteristic
9C2×#423C412yescomplemented
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C2×#405—21no—
14#2280C2211yescharacteristic
15#2280C2211yescharacteristic
16C2×C4×C22⋊C4C2211yescharacteristic
17#1851—22no—
18#1850—21no—
19#1850—21no—
20#1753—22no—
21#1753—22no—
22C2×#344—22no—
23#1729—21no—
24#1728C412yescomplemented
25#1298—22no—
26#1297—22no—
27#1284C414yescomplemented
28#1284C412yescomplemented
29#1284C412yescomplemented
30C24×C4C2311yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16—21no—
36C22×M16—21no—
37C22×M16—44no—
38C22×M16C2×C411yescharacteristic
39C22×M16C2×C411yescharacteristic
40C22×M16C2×C412yes—
41C22×M16D412yes—
42C23×C8D411yescharacteristic
43C23×C8D411yescharacteristic
44C2×#117—21no—
45C2×#117—21no—
46C22×C22⋊C4—21no—
47C22×C42C2×C412yes—
48#423—44no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51#407—21no—
52#407—21no—
53#407—21no—
54#407—21no—
55#407—21no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407C2×C412yes—
61#407C2×C412yes—
62#407Q812yes—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8D412yes—
69#405—44no—
70C2×#97—22no—
71C2×C4×C8—22no—
72#381C2×C412yes—
73#381C2×C412yes—
74#380—21no—
75#380—21no—
76#380—21no—
77#380Q812yes—
78#379—21no—
79#379—21no—
80C4×C22⋊C4—21no—
81C4×C22⋊C4—21no—
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4C2×C411yescharacteristic
86C4×C22⋊C4C2×C412yes—
87C4×C22⋊C4C2×C411yescharacteristic
88C4×C22⋊C4D412yes—
89C2×#95—21no—
90#344—44no—
91#336—24no—
92#336—22no—
93#336—22no—
94C25#4111yescharacteristic
95C23×C4—22no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×Q811yescharacteristic
106C23×C4C2×Q811yescharacteristic
107C23×C4#4111yescharacteristic
108C23×C4#4111yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—22no—
121C2×M16—21no—
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—42no—
126C2×M16—42no—
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—44no—
132C2×M16—44no—
133C2×M16—42no—
134C2×M16—42no—
135C2×M16—41no—
136C2×M16—42no—
137C2×M16—42no—
138C2×M16—42no—
139C2×M16—42no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—24no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—42no—
149C22×C8—41no—
150C22×C8C22⋊C412yes—
151C22×C8C22⋊C412yes—
152#117—42no—
153#117—42no—
154#117—42no—
155#117—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—41no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—41no—
173C2×C42C4212yes—
174C2×C42C4212yes—
175C4⋊C8—44no—
176C4⋊C8—44no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8C4⋊C414yes—
189#97—44no—
190C4×C8—44no—
191#95—21no—
192#95—22no—
193#95—21no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95C4⋊C414yes—
198C24—42no—
199C24—42no—
200C24—81no—
201C24#11711yescharacteristic
202C24C4×D411yescharacteristic
203C24C4×D411yescharacteristic
204C24C22⋊Q811yescharacteristic
205C24#12311yescharacteristic
206C24#12511yescharacteristic
207C24#12611yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4C2×C4211yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C4⋊C411yescharacteristic
239C22×C4C2×C4⋊C411yescharacteristic
240C22×C4#11711yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×Q811yescharacteristic
249C22×C4C4×Q811yescharacteristic
250C22×C4C4×Q811yescharacteristic
251C22×C4C4×Q811yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4#12511yescharacteristic
262C22×C4#12611yescharacteristic
263C22×C4C4⋊Q811yescharacteristic
264M16—44no—
265M16—44no—
266M16—44no—
267M16—44no—
268M16—44no—
269M16—44no—
270M16—44no—
271M16—82no—
272M16—81no—
273M16—81no—
274M16—81no—
275M16—81no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—81no—
296C2×C8—81no—
297C2×C8#9914yes—
298C4⋊C4—44no—
299C4⋊C4—82no—
300C22⋊C4—24no—
301C22⋊C4—44no—
302C22⋊C4—42no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—81no—
316C42—81no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—88no—
339C23C4×C22⋊C411yescharacteristic
340C23#38511yescharacteristic
341C23#38711yescharacteristic
342C23#38711yescharacteristic
343C23#38711yescharacteristic
344C23#38811yescharacteristic
345C23#38911yescharacteristic
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—88no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4C4×C4⋊C411yescharacteristic
393C2×C4C4×C4⋊C411yescharacteristic
394C2×C4#38511yescharacteristic
395C2×C4#38611yescharacteristic
396C2×C4#38711yescharacteristic
397C2×C4#38811yescharacteristic
398C2×C4#38911yescharacteristic
399C2×C4#39111yescharacteristic
400C2×C4C2×#9912yes—
401C2×C4#41012yes—
402C8—24no—
403C8—44no—
404C8—42no—
405C8—82no—
406C8—81no—
407C8—81no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—44no—
428C22—44no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22#143611yescharacteristic
435C22C4×#9912yes—
436C22#186112yes—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—44no—
444C4—84no—
445C4—82no—
446C4—162no—
447C4#189811yescharacteristic
448C4#189911yescharacteristic
449C4#199612yes—
450C2—22no—
451C2—22no—
452C2—41no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—41no—
457C2#1089011yescharacteristic
458C2#1089411yescharacteristic
459C2#1091211yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22#1436
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×#423C4semidirect
24. #1728C4semidirect
27. #1284C4semidirect
28. #1284C4semidirect
29. #1284C4semidirect

Maximal subgroups

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

subgroupindex
#208372
#203862
#107292
#80552
#80492
#79422