Finite Group Databasebeta

← #100454

Subgroups — #100454

Order 512.

620 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32954C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic
3#10789C211yesmaximal, characteristic
4#9679C211yesmaximal, characteristic
5#7665C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#475C411yescharacteristic, complemented
10C2×#475C411yescharacteristic, complemented
11C2×#446—21no—
12C22×#103—21no—
13#2857C2211yescharacteristic
14#1985—22no—
15#1984—21no—
16#1865—21no—
17#1865—21no—
18#1853—21no—
19#1803C411yescharacteristic, complemented
20#1803C411yescharacteristic, complemented
21#1797—21no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24#1787—21no—
25#1749—21no—
26C4×C22⋊C8C2211yescharacteristic
27C2×#340—21no—
28C2×#336C2211yescharacteristic
29C2×#328—21no—
30C2×#328—21no—
31C4⋊#95C2211yescharacteristic
32#1271—22no—
33C23×Q8D411yescharacteristic
34C23×C8—21no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C22⋊Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—42no—
40C2×C4×Q8—42no—
41C2×C4×Q8C2×C411yescharacteristic
42C2×C4×Q8C2×C411yescharacteristic
43C2×C4×Q8D411yescharacteristic
44C2×C4×Q8D411yescharacteristic
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4D411yescharacteristic
48C22×C42—21no—
49C22×C42C2311yescharacteristic
50#475—42no—
51#475—42no—
52#446—42no—
53#446—42no—
54C2×#106—21no—
55C2×#106—21no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8C2×C411yescharacteristic
59C2×C4⋊C8C2×C411yescharacteristic
60C2×#103—24no—
61C2×#103—21no—
62C2×#103—21no—
63C2×#103—21no—
64C2×#103—21no—
65C2×#103—42no—
66C2×#103—42no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8D411yescharacteristic
70C2×C22⋊C8D411yescharacteristic
71C2×#97—22no—
72C2×C4×C8—21no—
73#391—21no—
74#389—21no—
75#389—21no—
76#384C2×C411yescharacteristic
77#380—21no—
78#380—21no—
79C2×#95—21no—
80C2×#95—21no—
81#340—21no—
82#340—21no—
83#340—42no—
84#336—22no—
85#336—22no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336D412yes—
91#328—42no—
92#328—42no—
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
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—44no—
138C4×Q8—44no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C22⋊C4—41no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—44no—
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173#106—42no—
174#106—42no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—44no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180#103—42no—
181#103—48no—
182#103—42no—
183#103—42no—
184#103—42no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8SD1612yes—
193#97—44no—
194C4×C8—41no—
195C4×C8—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—81no—
201C24#12111yescharacteristic
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—81no—
227C2×Q8—81no—
228C2×Q8#9912yes—
229C2×Q8#10412yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4#12011yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4C2×SD1611yescharacteristic
268C22×C4#13511yescharacteristic
269C22×C4AGL(1,ℤ/8)11yescharacteristic
270C22×C4AGL(1,ℤ/8)11yescharacteristic
271C22×C4#13711yescharacteristic
272C22×C4#13711yescharacteristic
273C22×C4#13711yescharacteristic
274C22×C4#13711yescharacteristic
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—81no—
294C2×C8—81no—
295C2×C8—81no—
296C2×C8—81no—
297C4⋊C4—24no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—81no—
315C4⋊C4—81no—
316C4⋊C4—81no—
317C4⋊C4—81no—
318C4⋊C4—81no—
319C4⋊C4#9912yes—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—24no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—41no—
328C42—41no—
329C42—42no—
330C42—42no—
331C42—42no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23#38611yescharacteristic
337C23#46011yescharacteristic
338C23#46411yescharacteristic
339C23#46811yescharacteristic
340C23#47011yescharacteristic
341C23#47111yescharacteristic
342C23#47311yescharacteristic
343Q8—22no—
344Q8—22no—
345Q8—22no—
346Q8—22no—
347Q8—44no—
348Q8—44no—
349Q8—41no—
350Q8—41no—
351Q8—41no—
352Q8—41no—
353Q8—161no—
354Q8—161no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—161no—
407C2×C4—161no—
408C2×C4—161no—
409C2×C4—161no—
410C2×C4C2×#9911yescharacteristic
411C2×C4C2×#9911yescharacteristic
412C2×C4#41711yescharacteristic
413C2×C4#41911yescharacteristic
414C2×C4C2×#10411yescharacteristic
415C2×C4#42111yescharacteristic
416C2×C4C4×SD1611yescharacteristic
417C2×C4#44011yescharacteristic
418C2×C4#44811yescharacteristic
419C2×C4#44911yescharacteristic
420C2×C4#46811yescharacteristic
421C2×C4#47011yescharacteristic
422C8—42no—
423C8—81no—
424C8—82no—
425C8—82no—
426C8—81no—
427C8—81no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22#168611yescharacteristic
441C22#169011yescharacteristic
442C22#170011yescharacteristic
443C22#170411yescharacteristic
444C22#187111yescharacteristic
445C22#188011yescharacteristic
446C22#188111yescharacteristic
447C22#189411yescharacteristic
448C22#189811yescharacteristic
449C22#190511yescharacteristic
450C22#190811yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—41no—
458C4—41no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—81no—
463C4—162no—
464C4—161no—
465C4—161no—
466C4#150411yescharacteristic
467C4#150811yescharacteristic
468C4#151111yescharacteristic
469C4#151311yescharacteristic
470C2—21no—
471C2—21no—
472C2—21no—
473C2—21no—
474C2#899211yescharacteristic
475C2#911511yescharacteristic
476C2#916611yescharacteristic
477C2#922811yescharacteristic
478C2#926211yescharacteristic
479C2#930811yescharacteristic
480C2#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×#475C4semidirect
10. C2×#475C4semidirect
19. #1803C4semidirect
20. #1803C4semidirect

Maximal subgroups

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

subgroupindex
#329542
#225102
#107892
#96792
#76652
#70392
#70292