Finite Group Databasebeta

← #100446

Subgroups — #100446

Order 512.

612 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32957C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic
3#10789C211yesmaximal, characteristic
4#9884C211yesmaximal, characteristic
5#7667C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#477C411yescharacteristic, complemented
10C2×#477C411yescharacteristic, complemented
11C2×#446—21no—
12C22×#103—21no—
13#2857C2211yescharacteristic
14#1985—22no—
15#1984—21no—
16#1863—21no—
17#1863—21no—
18#1852—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
29C4⋊#95C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1272—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#477—42no—
51#477—42no—
52#446—42no—
53#446—42no—
54C2×#107—21no—
55C2×#107—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—
91C22×Q8—21no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—21no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8C22⋊C411yescharacteristic
102C22×Q8C22⋊C411yescharacteristic
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—41no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C22⋊Q8—81no—
124C22⋊Q8—81no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—44no—
134C4×Q8—44no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C22⋊C4—41no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—44no—
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42#4111yescharacteristic
169#107—42no—
170#107—42no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—44no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176#103—42no—
177#103—48no—
178#103—42no—
179#103—42no—
180#103—42no—
181C22⋊C8—21no—
182C22⋊C8—21no—
183C22⋊C8—21no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8Q1612yes—
189#97—44no—
190C4×C8—41no—
191C4×C8—41no—
192#95—41no—
193#95—41no—
194#95—41no—
195#95—41no—
196#95—81no—
197C24#12111yescharacteristic
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—81no—
222C2×Q8—81no—
223C2×Q8#9912yes—
224C2×Q8#10412yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4C2×Q1611yescharacteristic
263C22×C4#13511yescharacteristic
264C22×C4AGL(1,ℤ/8)11yescharacteristic
265C22×C4#13711yescharacteristic
266C22×C4#13711yescharacteristic
267C22×C4#13711yescharacteristic
268C22×C4#13711yescharacteristic
269C22×C4#13711yescharacteristic
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—81no—
289C2×C8—81no—
290C2×C8—81no—
291C2×C8—81no—
292C4⋊C4—24no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—82no—
312C4⋊C4#9912yes—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C42—24no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—41no—
322C42—42no—
323C42—42no—
324C42—42no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23#38611yescharacteristic
330C23#46211yescharacteristic
331C23#46411yescharacteristic
332C23#46811yescharacteristic
333C23#47011yescharacteristic
334C23#47111yescharacteristic
335C23#47211yescharacteristic
336Q8—22no—
337Q8—22no—
338Q8—22no—
339Q8—22no—
340Q8—44no—
341Q8—44no—
342Q8—41no—
343Q8—41no—
344Q8—41no—
345Q8—41no—
346Q8—162no—
347C2×C4—21no—
348C2×C4—21no—
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—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—161no—
399C2×C4—161no—
400C2×C4—161no—
401C2×C4—161no—
402C2×C4C2×#9911yescharacteristic
403C2×C4C2×#9911yescharacteristic
404C2×C4#41711yescharacteristic
405C2×C4#41911yescharacteristic
406C2×C4C2×#10411yescharacteristic
407C2×C4#42111yescharacteristic
408C2×C4C4×Q1611yescharacteristic
409C2×C4#44111yescharacteristic
410C2×C4#45111yescharacteristic
411C2×C4#45211yescharacteristic
412C2×C4#46811yescharacteristic
413C2×C4#47011yescharacteristic
414C8—42no—
415C8—81no—
416C8—82no—
417C8—82no—
418C8—81no—
419C8—81no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22#169211yescharacteristic
433C22#169611yescharacteristic
434C22#169811yescharacteristic
435C22#170411yescharacteristic
436C22#187111yescharacteristic
437C22#187811yescharacteristic
438C22#188111yescharacteristic
439C22#189411yescharacteristic
440C22#189811yescharacteristic
441C22#190511yescharacteristic
442C22#190711yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—41no—
450C4—41no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—81no—
455C4—162no—
456C4—161no—
457C4—161no—
458C4#150411yescharacteristic
459C4#150811yescharacteristic
460C4#151111yescharacteristic
461C4#151311yescharacteristic
462C2—21no—
463C2—21no—
464C2—21no—
465C2—21no—
466C2#899011yescharacteristic
467C2#911511yescharacteristic
468C2#916411yescharacteristic
469C2#922411yescharacteristic
470C2#925811yescharacteristic
471C2#930811yescharacteristic
472C2#934711yescharacteristic
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×#477C4semidirect
10. C2×#477C4semidirect
19. #1803C4semidirect
20. #1803C4semidirect

Maximal subgroups

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

subgroupindex
#329572
#225102
#107892
#98842
#76672
#70392
#70292