Finite Group Databasebeta

← #475830

Subgroups — #475830

Order 512.

990 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2088C212yesmaximal
2C2×#1941C211yesmaximal, characteristic
3C2×#1939C211yesmaximal, characteristic
4#15194C211yesmaximal, characteristic
5#15149C211yesmaximal, characteristic
6#11231C212yesmaximal
7#11228C211yesmaximal, characteristic
8#11227C211yesmaximal, characteristic
9#11220C212yesmaximal
10#11220C212yesmaximal
11#10861C211yesmaximal, characteristic
12C22×#106C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#398—21no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#2088—28no—
25#1941—24no—
26#1939—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1445—21no—
36#1444—21no—
37#1442—22no—
38#1441—22no—
39#1441—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—42no—
65C2×#106—42no—
66C2×#106—42no—
67C2×#106D414yes—
68C2×#106Q818yes—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—42no—
74C2×C4×C8—42no—
75C2×C4×C8D414yes—
76#398—44no—
77#389—24no—
78#389—24no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—42no—
84#389—42no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95D412yes—
100#340—24no—
101#340—22no—
102#340—22no—
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#336—24no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8C2×D412yes—
132C22×C8C2×D412yes—
133C22×C8C2×D412yes—
134C22×C8C2×Q814yes—
135C22×C8#4112yes—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4SD1614yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×Q814yes—
183C2×C4⋊C4C2×Q814yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C2×D412yes—
193C2×C42#4112yes—
194#106—28no—
195#106—48no—
196#106—44no—
197C4⋊C8—44no—
198C4⋊C8—48no—
199C4×C8—44no—
200#95—24no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—44no—
208#95—42no—
209#95—81no—
210#95—82no—
211#95—81no—
212#95—82no—
213C24C22×D411yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4#12112yes—
270C22×C4#12112yes—
271C22×C4C2×SD1614yes—
272C22×C4C2×SD1612yes—
273C22×C4#13512yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×Q812yes—
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4H(5,𝔽2)12yes—
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C42−1+412yes—
282C22×C42−1+412yes—
283C22×C42−1+411yescharacteristic
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8#12114yes—
300C2×C8C4⋊Q814yes—
301C4⋊C4—28no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—88no—
326C4⋊C4—84no—
327C4⋊C4—88no—
328C4⋊C4—84no—
329C42—22no—
330C42—22no—
331C42—24no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—42no—
336C23C2×#12111yescharacteristic
337C23#53712yes—
338C23D4211yescharacteristic
339C23#54711yescharacteristic
340C23D4×Q812yes—
341C23C22×SD1611yescharacteristic
342C23C2×#13511yescharacteristic
343C23#57512yes—
344C23#57711yescharacteristic
345C23#57711yescharacteristic
346C23#57712yes—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
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—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4#46514yes—
407C2×C4C2×#12111yescharacteristic
408C2×C4C2×C4⋊Q811yescharacteristic
409C2×C4#53411yescharacteristic
410C2×C4#53611yescharacteristic
411C2×C4#53611yescharacteristic
412C2×C4#53711yescharacteristic
413C2×C4#53712yes—
414C2×C4#53912yes—
415C2×C4#53912yes—
416C2×C4#54312yes—
417C2×C4#54412yes—
418C2×C4#54711yescharacteristic
419C2×C4D4×Q812yes—
420C2×C4#55511yescharacteristic
421C2×C4#55712yes—
422C2×C4#55712yes—
423C2×C4#56312yes—
424C2×C4#56412yes—
425C8—24no—
426C8—24no—
427C8—82no—
428C22#253711yescharacteristic
429C22#254611yescharacteristic
430C22#261811yescharacteristic
431C22#268312yes—
432C22#272612yes—
433C22C2×#46511yescharacteristic
434C22#305411yescharacteristic
435C22#305711yescharacteristic
436C22#306011yescharacteristic
437C22#324612yes—
438C22#325012yes—
439C22#326012yes—
440C22#326312yes—
441C22#327512yes—
442C22D4×SD1611yescharacteristic
443C22#328411yescharacteristic
444C22#333511yescharacteristic
445C22#334011yescharacteristic
446C22#335112yes—
447C22Q8×SD1612yes—
448C22#338112yes—
449C22#338812yes—
450C22#338812yes—
451C4—24no—
452C4—44no—
453C4—42no—
454C4—84no—
455C4—84no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4—161no—
460C4—161no—
461C4#254611yescharacteristic
462C4#261811yescharacteristic
463C4#265911yescharacteristic
464C4#272612yes—
465C4#280912yes—
466C4#282211yescharacteristic
467C2#1707011yescharacteristic
468C2#2822311yescharacteristic
469C2#2822911yescharacteristic
470C2#2867711yescharacteristic
471C2#2868111yescharacteristic
472C2#2977411yescharacteristic
473C2#2978111yescharacteristic
474C2#3124012yes—
475C2#3124312yes—
476C2#3250512yes—
477C2#3250712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20882
C2×#19412
C2×#19392
#151942
#151492
#112312
#112282
#112272
#112202
#108612