Finite Group Databasebeta

← #241198

Subgroups — #241198

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4#13910C211yesmaximal, characteristic
5#10572C218yesmaximal
6#8014C211yesmaximal, characteristic
7#8014C211yesmaximal, characteristic
8#7908C211yesmaximal, characteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11#2851—22no—
12#2646C2214yes—
13#2571—21no—
14#2571—21no—
15#2570—22no—
16#2570—22no—
17#2552—21no—
18#2552—21no—
19#2317—21no—
20#2317—21no—
21C2×#389—22no—
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×#384—21no—
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1948—22no—
29#1947—22no—
30#1853C2214yes—
31#1852C2214yes—
32#1845—21no—
33#1844—21no—
34#1834—21no—
35#1834—21no—
36#1444C2214yes—
37C4⋊#95—21no—
38#1294C2214yes—
39#1294C2214yes—
40#1280C2214yes—
41C2×#125—22no—
42C2×#117—21no—
43C2×#117—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C42—21no—
52C22×C42D412yes—
53C22×C42C2311yescharacteristic
54#431—22no—
55#431—22no—
56#431Q814yes—
57#431Q814yes—
58C2×#107—24no—
59C2×#106—24no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C22⋊C8C2312yes—
63C2×C22⋊C8C2312yes—
64C2×C4×C8—21no—
65C2×C4×C8—21no—
66#400—24no—
67#400—24no—
68#398—24no—
69#398—24no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—48no—
77#387—24no—
78#387—24no—
79#385—24no—
80#385—24no—
81#384—22no—
82#384—22no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—44no—
88#384D411yescharacteristic
89#384D411yescharacteristic
90#384D411yescharacteristic
91#384D411yescharacteristic
92#382—22no—
93#382—22no—
94#382—22no—
95#382—21no—
96#382—21no—
97#380—21no—
98#380—21no—
99#380C2312yes—
100#380C2312yes—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4Q814yes—
111C2×#95—24no—
112C2×#95C2312yes—
113C2×#95C2312yes—
114#340—24no—
115#340—24no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125#125—84no—
126#117—42no—
127#117—42no—
128#117—42no—
129#117—42no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
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—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42—22no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—44no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×Q812yes—
185C2×C42C2×Q812yes—
186C2×C42C2×Q812yes—
187C2×C42#4112yes—
188#107—84no—
189#106—84no—
190C4⋊C8—24no—
191C4⋊C8—24no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8C2×Q814yes—
195C22⋊C8C2×Q814yes—
196C4×C8—24no—
197C4×C8—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—48no—
210#95C2×Q814yes—
211C24C22×D411yescharacteristic
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4C22⋊Q812yes—
247C22×C4C22⋊Q812yes—
248C22×C4C4⋊Q812yes—
249C22×C4C4⋊Q812yes—
250C22×C4#13512yes—
251C22×C4#13512yes—
252C22×C4AGL(1,ℤ/8)12yes—
253C22×C4#13712yes—
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×Q811yescharacteristic
257C22×C4C22×Q811yescharacteristic
258C22×C4C22×Q811yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C4⋊C4—216no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—82no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C4⋊C4—816no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C42—44no—
299C42—44no—
300C42—44no—
301C42—48no—
302C42—82no—
303C42C22⋊Q812yes—
304C42C22⋊Q812yes—
305C42C22⋊Q812yes—
306C42C22⋊Q812yes—
307C42C4⋊Q812yes—
308C42C4⋊Q812yes—
309C42C4⋊Q812yes—
310C42C4⋊Q812yes—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23C2×#12011yescharacteristic
316C23C2×C22⋊Q811yescharacteristic
317C23C2×C4⋊Q811yescharacteristic
318C23C2×#13511yescharacteristic
319C23C2×#13511yescharacteristic
320C23C2×AGL(1,ℤ/8)11yescharacteristic
321C23C2×#13711yescharacteristic
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—28no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4#39512yes—
369C2×C4#39512yes—
370C2×C4#39512yes—
371C2×C4#39512yes—
372C2×C4#44912yes—
373C2×C4#44912yes—
374C2×C4#45212yes—
375C2×C4#45212yes—
376C2×C4#45312yes—
377C2×C4#45612yes—
378C2×C4#47814yes—
379C2×C4#47914yes—
380C2×C4#49914yes—
381C2×C4#50114yes—
382C2×C4C2×C22⋊Q811yescharacteristic
383C2×C4C2×C22⋊Q811yescharacteristic
384C2×C4C2×C4⋊Q811yescharacteristic
385C2×C4C2×C4⋊Q811yescharacteristic
386C8—44no—
387C8—44no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22C2×#39511yescharacteristic
401C22C2×#44911yescharacteristic
402C22C2×#45211yescharacteristic
403C22C2×#45311yescharacteristic
404C22C2×#45611yescharacteristic
405C22#302711yescharacteristic
406C22#302811yescharacteristic
407C22C2×#47811yescharacteristic
408C22C2×#47911yescharacteristic
409C22C2×#49911yescharacteristic
410C22C2×#50111yescharacteristic
411C4—22no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—48no—
417C4—82no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4—82no—
423C4—162no—
424C4#205914yes—
425C4#205914yes—
426C4#206014yes—
427C2—21no—
428C2—21no—
429C2—21no—
430C2—21no—
431C2#1839211yescharacteristic
432C2#1839811yescharacteristic
433C2#1840511yescharacteristic
434C2#1841311yescharacteristic
435C2C2×#205911yescharacteristic
436C2C2×#206011yescharacteristic
437C2#2356611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#351812
#219822
#219812
#139102
#105722
#80142
#79082