Finite Group Databasebeta

← #121591

Subgroups — #121591

Order 512.

766 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21982C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic
3#13910C211yesmaximal, characteristic
4C2×#1303C211yesmaximal, characteristic
5#8043C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#428—21no—
11C2×#423C2211yescharacteristic
12C2×#389—22no—
13#2285C2211yescharacteristic
14C2×#384—21no—
15C2×#384C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#1949—22no—
19#1948—21no—
20#1947—22no—
21#1947—21no—
22#1852—22no—
23#1845—21no—
24#1834—21no—
25#1736—21no—
26#1444—22no—
27C4⋊#95—21no—
28#1303C414yescomplemented
29#1296—22no—
30#1294—22no—
31#1279—22no—
32C22×M16—21no—
33C2×#117—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—24no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D412yes—
41C22×C42—21no—
42C22×C42—21no—
43C22×C42C2311yescharacteristic
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#431C2×C412yes—
50#431Q812yes—
51#431Q812yes—
52#428—42no—
53#428—42no—
54C2×#107—22no—
55C2×#107—24no—
56C2×#106—22no—
57#423—21no—
58#423—21no—
59#423C2×C412yes—
60#423Q812yes—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8D412yes—
64C2×C4⋊C8D412yes—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69#389—48no—
70#384—21no—
71#384—21no—
72#384—21no—
73#384—21no—
74#384—42no—
75#384—42no—
76#384D411yescharacteristic
77#384D411yescharacteristic
78#382—21no—
79#380—21no—
80#380—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2×C412yes—
90C4×C4⋊C4Q812yes—
91C2×#95—24no—
92C2×#95—21no—
93C2×#95D412yes—
94#340—24no—
95#340—22no—
96#340—22no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—41no—
113#117—41no—
114#117—41no—
115#117—41no—
116#117—41no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—48no—
141C2×C4⋊C4—48no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—84no—
152C2×C4⋊C4D812yes—
153C2×C4⋊C4Q1612yes—
154C2×C22⋊C4—41no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—44no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42C2×Q811yescharacteristic
171C2×C42C2×Q811yescharacteristic
172C2×C42C2×Q811yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177#107—44no—
178#107—84no—
179#106—44no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186C4⋊C8C4⋊C414yescomplemented
187C4⋊C8C4⋊C414yescomplemented
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—41no—
196#95—22no—
197#95—48no—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95C4⋊C414yes—
203C24#12011yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
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—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4C2×C4⋊C411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4C22⋊Q811yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12711yescharacteristic
247C22×C4C4⋊Q811yescharacteristic
248C22×C4C4⋊Q811yescharacteristic
249C22×C4C4⋊Q811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×Q1611yescharacteristic
252C22×C4#13511yescharacteristic
253C22×C4#13511yescharacteristic
254C22×C4AGL(1,ℤ/8)11yescharacteristic
255C22×C4AGL(1,ℤ/8)11yescharacteristic
256C22×C4#13711yescharacteristic
257C22×C4#13711yescharacteristic
258M16—42no—
259M16—42no—
260M16—42no—
261M16—42no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—81no—
270C2×C8—81no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—816no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C4⋊C4—81no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—82no—
305C42—82no—
306C42—81no—
307C42—81no—
308C42C2×C4⋊C411yescharacteristic
309C42C2×C4⋊C411yescharacteristic
310C42C4×D411yescharacteristic
311C42C4×Q811yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42#12511yescharacteristic
319C42#12511yescharacteristic
320C42#12511yescharacteristic
321C42C4⋊Q811yescharacteristic
322C42C4⋊Q811yescharacteristic
323C42C4⋊Q811yescharacteristic
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23#38511yescharacteristic
329C23#39211yescharacteristic
330C23#39511yescharacteristic
331C23#44711yescharacteristic
332C23#44911yescharacteristic
333C23#45111yescharacteristic
334C23#45211yescharacteristic
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—28no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4#38411yescharacteristic
378C2×C4#38511yescharacteristic
379C2×C4#38511yescharacteristic
380C2×C4#38911yescharacteristic
381C2×C4#38911yescharacteristic
382C2×C4#38911yescharacteristic
383C2×C4#39511yescharacteristic
384C2×C4#39511yescharacteristic
385C2×C4#39811yescharacteristic
386C2×C4#39811yescharacteristic
387C2×C4#39811yescharacteristic
388C2×C4#39911yescharacteristic
389C2×C4C4×D811yescharacteristic
390C2×C4C4×Q1611yescharacteristic
391C2×C4#44011yescharacteristic
392C2×C4#44011yescharacteristic
393C2×C4#44711yescharacteristic
394C2×C4#44911yescharacteristic
395C2×C4#45111yescharacteristic
396C2×C4#45211yescharacteristic
397C2×C4#45312yes—
398C2×C4#45612yes—
399C2×C4#46611yescharacteristic
400C2×C4#46711yescharacteristic
401C2×C4#46911yescharacteristic
402C2×C4#47011yescharacteristic
403C2×C4#47411yescharacteristic
404C2×C4#47411yescharacteristic
405C2×C4#47711yescharacteristic
406C2×C4#47711yescharacteristic
407C2×C4#47811yescharacteristic
408C2×C4#47811yescharacteristic
409C2×C4#47911yescharacteristic
410C2×C4#47911yescharacteristic
411C2×C4#49911yescharacteristic
412C2×C4#50011yescharacteristic
413C2×C4#50111yescharacteristic
414C2×C4#50111yescharacteristic
415C8—44no—
416C8—44no—
417C8—82no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22#144411yescharacteristic
431C22#186211yescharacteristic
432C22#186311yescharacteristic
433C22#199911yescharacteristic
434C22#200111yescharacteristic
435C22#200211yescharacteristic
436C22#200511yescharacteristic
437C22#200811yescharacteristic
438C22#200911yescharacteristic
439C22#205711yescharacteristic
440C22#205911yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—48no—
448C4—82no—
449C4—82no—
450C4—161no—
451C4—161no—
452C4—162no—
453C4#185912yes—
454C4#186211yescharacteristic
455C4#186311yescharacteristic
456C4#194211yescharacteristic
457C4#194411yescharacteristic
458C4#205711yescharacteristic
459C4#205911yescharacteristic
460C4#206012yes—
461C4#208611yescharacteristic
462C4#208711yescharacteristic
463C2—21no—
464C2—21no—
465C2—21no—
466C2—21no—
467C2#1023811yescharacteristic
468C2#1024311yescharacteristic
469C2#1024811yescharacteristic
470C2#1025311yescharacteristic
471C2#1132811yescharacteristic
472C2#1133311yescharacteristic
473C2#1134011yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #1303C4semidirect
186. C4⋊C8C4⋊C4semidirect
187. C4⋊C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#219822
#211702
#139102
C2×#13032
#80432
#80142
#78922