Finite Group Databasebeta

← #100450

Subgroups — #100450

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32906C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic
3#10789C211yesmaximal, characteristic
4#9885C211yesmaximal, characteristic
5#7668C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#462C411yescharacteristic, complemented
10C2×#462C411yescharacteristic, complemented
11C2×#451—21no—
12C2×#446—21no—
13C22×#103—21no—
14#2857C2211yescharacteristic
15#1985—22no—
16#1984—21no—
17#1878—21no—
18#1878—21no—
19#1872—21no—
20#1803C411yescharacteristic, complemented
21#1803C411yescharacteristic, complemented
22#1797—21no—
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25#1749—21no—
26C4×C22⋊C8C2211yescharacteristic
27C2×#340—21no—
28C2×#336C2211yescharacteristic
29C4⋊#95C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1271—22no—
33C23×Q8D411yescharacteristic
34C22×Q16—21no—
35C22×Q16—21no—
36C23×C8—21no—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×C22⋊Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—42no—
42C2×C4×Q8—42no—
43C2×C4×Q8C2×C411yescharacteristic
44C2×C4×Q8C2×C411yescharacteristic
45C2×C4×Q8D411yescharacteristic
46C2×C4×Q8D411yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D411yescharacteristic
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#462—42no—
53#462—42no—
54#451—42no—
55#451—42no—
56#446—42no—
57#446—42no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2×C411yescharacteristic
61C2×C4⋊C8C2×C411yescharacteristic
62C2×#103—24no—
63C2×#103—21no—
64C2×#103—21no—
65C2×#103—21no—
66C2×#103—21no—
67C2×#103—42no—
68C2×#103—42no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8D411yescharacteristic
72C2×C22⋊C8D411yescharacteristic
73C2×#97—22no—
74C2×C4×C8—21no—
75#391—21no—
76#389—21no—
77#389—21no—
78#384C2×C411yescharacteristic
79#380—21no—
80#380—21no—
81C2×#95—21no—
82C2×#95—21no—
83#340—21no—
84#340—21no—
85#340—42no—
86#336—22no—
87#336—22no—
88#336—21no—
89#336—21no—
90#336—21no—
91#336—21no—
92#336D412yes—
93C22×Q8—21no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8C22⋊C411yescharacteristic
104C22×Q8C22⋊C411yescharacteristic
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—41no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C2×Q16—42no—
112C2×Q16—42no—
113C2×Q16—42no—
114C2×Q16—42no—
115C2×Q16—42no—
116C2×Q16—42no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—41no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—44no—
144C4×Q8—44no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C22⋊C4—41no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42—44no—
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42#4111yescharacteristic
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—44no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184#103—42no—
185#103—48no—
186#103—42no—
187#103—42no—
188#103—42no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8Q1612yescomplemented
197#97—44no—
198C4×C8—41no—
199C4×C8—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—81no—
205C24#12111yescharacteristic
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8#9912yes—
235C2×Q8#10412yes—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4C4×D411yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4C2×Q1611yescharacteristic
274C22×C4#13511yescharacteristic
275C22×C4AGL(1,ℤ/8)11yescharacteristic
276C22×C4AGL(1,ℤ/8)11yescharacteristic
277C22×C4AGL(1,ℤ/8)11yescharacteristic
278C22×C4#13711yescharacteristic
279C22×C4#13711yescharacteristic
280C22×C4#13711yescharacteristic
281Q16—84no—
282Q16—84no—
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—81no—
302C2×C8—81no—
303C2×C8—81no—
304C2×C8—81no—
305C4⋊C4—24no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—81no—
319C4⋊C4—81no—
320C4⋊C4—81no—
321C4⋊C4—81no—
322C4⋊C4—81no—
323C4⋊C4—81no—
324C4⋊C4—81no—
325C4⋊C4#9912yes—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C42—24no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—41no—
334C42—41no—
335C42—42no—
336C42—42no—
337C42—42no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23#38611yescharacteristic
343C23#46211yescharacteristic
344C23#46411yescharacteristic
345C23#46811yescharacteristic
346C23#46911yescharacteristic
347C23#47111yescharacteristic
348C23#47311yescharacteristic
349Q8—22no—
350Q8—22no—
351Q8—22no—
352Q8—22no—
353Q8—44no—
354Q8—44no—
355Q8—41no—
356Q8—41no—
357Q8—41no—
358Q8—41no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362Q8—81no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—161no—
415C2×C4—161no—
416C2×C4—161no—
417C2×C4—161no—
418C2×C4C2×#9911yescharacteristic
419C2×C4C2×#9911yescharacteristic
420C2×C4#41711yescharacteristic
421C2×C4#41811yescharacteristic
422C2×C4C2×#10411yescharacteristic
423C2×C4#42111yescharacteristic
424C2×C4C4×Q1611yescharacteristic
425C2×C4#44111yescharacteristic
426C2×C4#45111yescharacteristic
427C2×C4#45211yescharacteristic
428C2×C4#46811yescharacteristic
429C2×C4#46911yescharacteristic
430C8—42no—
431C8—81no—
432C8—82no—
433C8—82no—
434C8—81no—
435C8—81no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22#169211yescharacteristic
449C22#169411yescharacteristic
450C22#170011yescharacteristic
451C22#170211yescharacteristic
452C22#187111yescharacteristic
453C22#187811yescharacteristic
454C22#188111yescharacteristic
455C22#189511yescharacteristic
456C22#189811yescharacteristic
457C22#190511yescharacteristic
458C22#190811yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—41no—
466C4—41no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—81no—
471C4—162no—
472C4—161no—
473C4—161no—
474C4#150411yescharacteristic
475C4#150711yescharacteristic
476C4#151011yescharacteristic
477C4#151311yescharacteristic
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2#899011yescharacteristic
483C2#911711yescharacteristic
484C2#916611yescharacteristic
485C2#922611yescharacteristic
486C2#925811yescharacteristic
487C2#931011yescharacteristic
488C2#934511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×#462C4semidirect
10. C2×#462C4semidirect
20. #1803C4semidirect
21. #1803C4semidirect
196. C22⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#329062
#225102
#107892
#98852
#76682
#70392
#70292