Finite Group Databasebeta

← #121504

Subgroups — #121504

Order 512.

730 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21986C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic
3#13954C211yesmaximal, characteristic
4#10851C211yesmaximal, characteristic
5#8033C211yesmaximal, characteristic
6C2×#1285C211yesmaximal, characteristic
7#7904C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×C4×M16C2211yescharacteristic
11C2×#389—21no—
12C2×#389—21no—
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1949—22no—
18#1948—21no—
19#1947—21no—
20#1853—21no—
21#1853—21no—
22#1852—21no—
23#1852—21no—
24#1847—21no—
25#1836—21no—
26#1776—22no—
27C2×#339—22no—
28#1444—21no—
29#1444—21no—
30C4⋊#95—22no—
31#1295—22no—
32#1285C414yescomplemented
33#1279—22no—
34C22×M16—21no—
35C2×#117—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C42—21no—
44C22×C42D412yes—
45C22×C42C2311yescharacteristic
46#432—21no—
47#432—21no—
48#432—21no—
49#432—21no—
50#432C2×C412yes—
51#432C2×C412yes—
52#432Q812yes—
53#432Q812yes—
54C2×#107—22no—
55C2×#107—22no—
56C2×#106—22no—
57C2×#106—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8D412yes—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C4×M16—21no—
63C4×M16—21no—
64C4×M16—21no—
65C4×M16Q812yes—
66C2×#97—21no—
67C2×#97—21no—
68C2×#97D412yes—
69C2×C4×C8—21no—
70#389—44no—
71#389—44no—
72#384D411yescharacteristic
73#384D411yescharacteristic
74#382—21no—
75#382—21no—
76#382—21no—
77#382—21no—
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—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#339—44no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—41no—
119#117—42no—
120#117—42no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
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—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—82no—
149C2×C22⋊C4—41no—
150C2×C42—22no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42C4⋊C414yes—
159C2×C42C22×C411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×Q811yescharacteristic
164C2×C42C2×Q811yescharacteristic
165C2×C42C2×Q811yescharacteristic
166C2×C42C2×Q811yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171#107—44no—
172#107—82no—
173#106—44no—
174#106—82no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8C4⋊C414yescomplemented
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—41no—
188#97C4⋊C414yescomplemented
189C4×C8—22no—
190C4×C8—41no—
191#95—22no—
192#95—22no—
193#95—44no—
194#95—44no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203C24#12011yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—41no—
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—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12711yescharacteristic
260C22×C4C4⋊Q811yescharacteristic
261C22×C4C4⋊Q811yescharacteristic
262C22×C4C4⋊Q811yescharacteristic
263M16—24no—
264M16—42no—
265M16—42no—
266M16—42no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—88no—
297C4⋊C4—88no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C42—44no—
301C42—82no—
302C42—82no—
303C42—82no—
304C42—82no—
305C42C2×C4⋊C411yescharacteristic
306C42C2×C4⋊C411yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×Q811yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42C22⋊Q811yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42#12311yescharacteristic
314C42#12311yescharacteristic
315C42#12511yescharacteristic
316C42#12511yescharacteristic
317C42#12511yescharacteristic
318C42C4⋊Q811yescharacteristic
319C42C4⋊Q811yescharacteristic
320C42C4⋊Q811yescharacteristic
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23#38511yescharacteristic
326C23#39211yescharacteristic
327C23#39511yescharacteristic
328C23#45311yescharacteristic
329C23#45411yescharacteristic
330C23#45511yescharacteristic
331C23#45611yescharacteristic
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—28no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—44no—
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—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4#38411yescharacteristic
396C2×C4#38511yescharacteristic
397C2×C4#38511yescharacteristic
398C2×C4#38911yescharacteristic
399C2×C4#38911yescharacteristic
400C2×C4#38911yescharacteristic
401C2×C4#39511yescharacteristic
402C2×C4#39511yescharacteristic
403C2×C4#39811yescharacteristic
404C2×C4#39811yescharacteristic
405C2×C4#39811yescharacteristic
406C2×C4#39911yescharacteristic
407C2×C4#45312yes—
408C2×C4#45612yes—
409C8—44no—
410C8—44no—
411C8—44no—
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#144411yescharacteristic
425C22#185611yescharacteristic
426C22#185711yescharacteristic
427C22#200211yescharacteristic
428C22#200511yescharacteristic
429C22#200611yescharacteristic
430C22#200711yescharacteristic
431C22#201011yescharacteristic
432C22#201011yescharacteristic
433C22#206011yescharacteristic
434C22#206111yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—48no—
446C4—82no—
447C4—82no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4#185912yes—
453C4#206012yes—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2#1000411yescharacteristic
459C2#1000711yescharacteristic
460C2#1000811yescharacteristic
461C2#1001111yescharacteristic
462C2#1133511yescharacteristic
463C2#1134011yescharacteristic
464C2#1134211yescharacteristic
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
32. #1285C4semidirect
178. C4⋊C8C4⋊C4semidirect
188. #97C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#219862
#213652
#139542
#108512
#80332
C2×#12852
#79042