Finite Group Databasebeta

← #100457

Subgroups — #100457

Order 512.

527 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33107C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic, complemented
3#10807C211yesmaximal, characteristic, complemented
4#9680C211yesmaximal, characteristic
5#7664C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic, complemented
7#7029C211yesmaximal, characteristic, complemented
8C2×#481—21no—
9#2856C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1982C411yescharacteristic, complemented
12#1982C411yescharacteristic, complemented
13#1948C411yescharacteristic, complemented
14#1948C411yescharacteristic, complemented
15#1939—21no—
16#1926—21no—
17#1926—21no—
18#1917—22no—
19#1874—21no—
20#1797—21no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23#1770—21no—
24#1749—21no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—21no—
27C2×#340—21no—
28C2×#336C2211yescharacteristic
29C2×#327—21no—
30C2×#327—21no—
31C4⋊#95C2211yescharacteristic
32#1270—22no—
33C23×C8—21no—
34C2×#127C2×C411yescharacteristic
35C2×#121—21no—
36C2×#121—21no—
37C2×#121—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C42—21no—
42C22×C42C2311yescharacteristic
43#481—42no—
44#481—42no—
45C2×#106—21no—
46C2×#106—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2×C411yescharacteristic
50C2×C4⋊C8C2×C411yescharacteristic
51C2×#102—21no—
52C2×#102—21no—
53C2×#102—21no—
54C2×#102—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8D411yescharacteristic
58C2×C22⋊C8D411yescharacteristic
59C2×#97—22no—
60C2×C4×C8—21no—
61#390—21no—
62#390—21no—
63#390—21no—
64#384C2×C411yescharacteristic
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4C2×C411yescharacteristic
68C4×C4⋊C4C2×C411yescharacteristic
69C4×C4⋊C4D411yescharacteristic
70C4×C4⋊C4D411yescharacteristic
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74#340—24no—
75#340—21no—
76#340—21no—
77#340—42no—
78#336—22no—
79#336—22no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336D412yes—
85#327—42no—
86#327—42no—
87C22×D4—21no—
88C22×D4—21no—
89C22×D4—21no—
90C22×D4—41no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—41no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4#4111yescharacteristic
97C22×C8—21no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109#127—41no—
110#127—41no—
111#121—41no—
112#121—41no—
113#121—41no—
114#121—41no—
115#121—41no—
116#121—41no—
117#121—41no—
118#121—41no—
119#121—81no—
120#121—81no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4C22⋊C411yescharacteristic
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C22⋊C411yescharacteristic
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42—82no—
152C2×C42C22×C411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42#4111yescharacteristic
156#106—42no—
157#106—42no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161C4⋊C8—41no—
162C4⋊C8—82no—
163#102—42no—
164#102—42no—
165#102—42no—
166#102—42no—
167C22⋊C8—21no—
168C22⋊C8—21no—
169C22⋊C8—21no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8SD1612yes—
175#97—82no—
176C4×C8—41no—
177C4×C8—41no—
178#95—21no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—41no—
183#95—41no—
184#95—81no—
185C24—41no—
186C24#12111yescharacteristic
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—81no—
197C2×D4—81no—
198C2×D4—81no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4—81no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4C2×SD1611yescharacteristic
243C22×C4#13511yescharacteristic
244C22×C4AGL(1,ℤ/8)11yescharacteristic
245C22×C4AGL(1,ℤ/8)11yescharacteristic
246C22×C4AGL(1,ℤ/8)11yescharacteristic
247C22×C4AGL(1,ℤ/8)11yescharacteristic
248C22×C4#13711yescharacteristic
249C22×C4#13711yescharacteristic
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—81no—
269C2×C8—81no—
270C2×C8—81no—
271C2×C8—81no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C4⋊C4#10412yes—
292C22⋊C4—82no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—41no—
302C42—41no—
303C42—42no—
304C42—42no—
305C42—81no—
306C42—81no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23#38611yescharacteristic
319C23#46011yescharacteristic
320C23#46411yescharacteristic
321C23#46811yescharacteristic
322C23#46911yescharacteristic
323C23#47111yescharacteristic
324C23#47211yescharacteristic
325D4—81no—
326D4—81no—
327D4—81no—
328D4—81no—
329D4—161no—
330D4—161no—
331D4—162no—
332D4—161no—
333D4—161no—
334D4—161no—
335D4—161no—
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—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—161no—
385C2×C4—161no—
386C2×C4—161no—
387C2×C4—161no—
388C2×C4#41011yescharacteristic
389C2×C4#41011yescharacteristic
390C2×C4#41711yescharacteristic
391C2×C4#41811yescharacteristic
392C2×C4C2×#10411yescharacteristic
393C2×C4#42111yescharacteristic
394C2×C4C4×SD1611yescharacteristic
395C2×C4#44011yescharacteristic
396C2×C4#44811yescharacteristic
397C2×C4#44911yescharacteristic
398C2×C4#46811yescharacteristic
399C2×C4#46911yescharacteristic
400C8—42no—
401C8—81no—
402C8—82no—
403C8—82no—
404C8—81no—
405C8—81no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22—161no—
425C22#168611yescharacteristic
426C22#168811yescharacteristic
427C22#169811yescharacteristic
428C22#170211yescharacteristic
429C22#187111yescharacteristic
430C22#188011yescharacteristic
431C22#188111yescharacteristic
432C22#189511yescharacteristic
433C22#189911yescharacteristic
434C22#190411yescharacteristic
435C22#190711yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—41no—
444C4—42no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—162no—
449C4—161no—
450C4#150511yescharacteristic
451C4#150911yescharacteristic
452C4#151211yescharacteristic
453C4#151411yescharacteristic
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—321no—
459C2#899211yescharacteristic
460C2#911811yescharacteristic
461C2#916311yescharacteristic
462C2#923011yescharacteristic
463C2#926111yescharacteristic
464C2#930911yescharacteristic
465C2#934811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
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
2. #22629C2semidirect
3. #10807C2semidirect
6. #7039C2semidirect
7. #7029C2semidirect
11. #1982C4semidirect
12. #1982C4semidirect
13. #1948C4semidirect
14. #1948C4semidirect

Maximal subgroups

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

subgroupindex
#331072
#226292
#108072
#96802
#76642
#70392
#70292