Finite Group Databasebeta

← #192103

Subgroups — #192103

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34186C211yesmaximal, characteristic
2#21359C211yesmaximal, characteristic, complemented
3#21358C211yesmaximal, characteristic, complemented
4C2×#1460C211yesmaximal, characteristic
5#13941C211yesmaximal, characteristic
6#8858C218yesmaximal, complemented
7#8026C211yesmaximal, characteristic, complemented
8#8026C211yesmaximal, characteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2357—22no—
13#2357—22no—
14#2354—22no—
15#2354—22no—
16#2312C2214yes—
17#2309—21no—
18C2×#390—21no—
19#2285C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×#382C2211yescharacteristic
22#2280—22no—
23#2274—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1948—22no—
26#1947—22no—
27#1853C2214yes—
28#1852C2214yes—
29C4×#107—22no—
30C4×#106—22no—
31#1460C418yescomplemented
32C2×#323C2214yes—
33#1439C2214yes—
34C4⋊#95—21no—
35C22⋊#95—22no—
36#1294C2214yes—
37#1294C2214yes—
38C23×D4D411yescharacteristic
39C24×C4D412yes—
40C2×C4×D4—21no—
41C2×C4×D4—21no—
42C2×C4×D4—21no—
43C2×C4×D4—21no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4C2×C414yes—
47C2×C4×D4Q814yes—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—24no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C22⋊C4—24no—
55C22×C22⋊C4C2312yes—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58#431—22no—
59#431—22no—
60#431C2×C414yes—
61#431C2×C414yes—
62C2×#107—24no—
63C2×#106—24no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C22⋊C8C2312yes—
67C2×C22⋊C8C2312yes—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70#390—24no—
71#390—24no—
72#390—44no—
73#388—28no—
74#388—28no—
75#384D412yes—
76#384D412yes—
77#382—21no—
78#382—21no—
79#382—22no—
80#382—22no—
81#381—24no—
82#380—24no—
83#380—21no—
84#380—21no—
85#380C2312yes—
86#380C2312yes—
87#379—24no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4D412yes—
97C4×C4⋊C4D412yes—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—24no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—24no—
103C2×#95C2312yes—
104#340—24no—
105#340—24no—
106#323C2×C4116yes—
107C25C2×D412yes—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4C4⋊C414yes—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—84no—
122C23×C4C4⋊C418yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—24no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—24no—
136C4×D4—24no—
137#117—42no—
138#117—42no—
139#117—42no—
140#117—42no—
141#117—88no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C4⋊C414yes—
156C2×C4⋊C4C2×D414yes—
157C2×C4⋊C4C2×D414yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—48no—
168C2×C22⋊C4—84no—
169C2×C22⋊C4C22×C414yes—
170C2×C22⋊C4C2×Q814yes—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—28no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42C22×C412yes—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×Q812yes—
184C2×C42#4112yes—
185C2×C42#4112yes—
186#107—48no—
187#106—48no—
188C4⋊C8—24no—
189C4⋊C8—24no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8C22×C414yes—
193C22⋊C8C22×C414yes—
194C4×C8—24no—
195C4×C8—24no—
196#95—28no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—41no—
204#95—41no—
205#95—42no—
206#95—84no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—22no—
213C24—44no—
214C24—44no—
215C24C2×C4⋊C414yes—
216C24C22×D411yescharacteristic
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—28no—
220C2×D4—28no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—44no—
226C2×D4—44no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—84no—
250C22×C4—88no—
251C22×C4—88no—
252C22×C4C2×C4⋊C414yes—
253C22×C4C2×C4⋊C412yes—
254C22×C4C2×C4⋊C412yes—
255C22×C4#12011yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12312yes—
260C22×C4#12312yes—
261C22×C4#13512yes—
262C22×C4#13512yes—
263C22×C4AGL(1,ℤ/8)12yes—
264C22×C4#13712yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—44no—
274C2×C8—44no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—84no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—88no—
297C22⋊C4—88no—
298C22⋊C4—816no—
299C42—44no—
300C42—44no—
301C42—88no—
302C42—82no—
303C42—82no—
304C42C4×D412yes—
305C42C4×D412yes—
306C42C4×D412yes—
307C42C4×D412yes—
308C42C22⋊Q812yes—
309C42C22⋊Q812yes—
310C42C22⋊Q812yes—
311C42C22⋊Q812yes—
312C23—22no—
313C23—22no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—44no—
318C23—44no—
319C23—48no—
320C23—48no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—44no—
326C23—44no—
327C23—82no—
328C23—82no—
329C23—84no—
330C23—84no—
331C23C22×C4⋊C411yescharacteristic
332C23C2×#12011yescharacteristic
333C23C2×#12311yescharacteristic
334C23C2×#13511yescharacteristic
335C23C2×#13511yescharacteristic
336C23C2×AGL(1,ℤ/8)11yescharacteristic
337C23C2×#13711yescharacteristic
338D4—48no—
339D4—48no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—416no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—88no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4#38512yes—
375C2×C4#38512yes—
376C2×C4#38512yes—
377C2×C4#38512yes—
378C2×C4#42714yes—
379C2×C4#42814yes—
380C2×C4#44912yes—
381C2×C4#44912yes—
382C2×C4#45212yes—
383C2×C4#45212yes—
384C2×C4#48214yes—
385C2×C4#48514yes—
386C2×C4C2×C4×D411yescharacteristic
387C2×C4C2×C4×D411yescharacteristic
388C2×C4C2×C22⋊Q811yescharacteristic
389C2×C4C2×C22⋊Q811yescharacteristic
390C8—44no—
391C8—44no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—48no—
401C22—44no—
402C22—44no—
403C22—44no—
404C22—84no—
405C22—84no—
406C22—84no—
407C22—84no—
408C22—84no—
409C22C2×#38511yescharacteristic
410C22C2×#42711yescharacteristic
411C22C2×#42811yescharacteristic
412C22C2×#44911yescharacteristic
413C22C2×#45211yescharacteristic
414C22#301912yes—
415C22#302512yes—
416C22C2×#48211yescharacteristic
417C22C2×#48511yescharacteristic
418C4—22no—
419C4—22no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—44no—
425C4—84no—
426C4—84no—
427C4—84no—
428C4—162no—
429C4#185414yes—
430C4#185414yes—
431C2—21no—
432C2—21no—
433C2—21no—
434C2—21no—
435C2—84no—
436C2#1822212yes—
437C2#1822312yes—
438C2C2×#185411yescharacteristic
439C2#2085611yescharacteristic
440C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21359C2semidirect
3. #21358C2semidirect
6. #8858C2semidirect
7. #8026C2semidirect
8. #8026C2semidirect
31. #1460C4semidirect

Maximal subgroups

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

subgroupindex
#341862
#213592
#213582
C2×#14602
#139412
#88582
#80262