Finite Group Databasebeta

← #475572

Subgroups — #475572

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1916C211yesmaximal, characteristic
2C2×#1844C211yesmaximal, characteristic
3#15196C211yesmaximal, characteristic
4#15151C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11327C212yesmaximal
9#11257C212yesmaximal
10#11231C212yesmaximal
11#10820C211yesmaximal, characteristic
12C22×#106C2212yes—
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#398—21no—
17C2×#398—21no—
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1916—24no—
24#1844—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1445—21no—
36#1445—21no—
37#1444—21no—
38#1444—21no—
39#1441—21no—
40#1441—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—42no—
67C2×#106—42no—
68C2×#106—42no—
69C2×#106—42no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—24no—
72C2×C4×C8—24no—
73C2×C4×C8D414yes—
74#398—44no—
75#398—44no—
76#389—42no—
77#389—42no—
78#389—42no—
79#389—42no—
80#384—22no—
81#384—22no—
82#384—22no—
83#384—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98#340—24no—
99#340—42no—
100#340—42no—
101#340—42no—
102#340—42no—
103#340Q818yes—
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8C2×D412yes—
136C22×C8C2×D412yes—
137C22×C8C2×Q814yes—
138C22×C8#4114yes—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
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—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4SD1614yes—
183C2×C4⋊C4C2×Q814yes—
184C2×C4⋊C4C2×Q814yes—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42C2×D412yes—
198C2×C42#4112yes—
199#106—48no—
200#106—48no—
201C4⋊C8—44no—
202C4×C8—28no—
203#95—22no—
204#95—22no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—42no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—42no—
213#95—82no—
214#95—82no—
215C24C22×D411yescharacteristic
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4#12112yes—
269C22×C4#12112yes—
270C22×C4C2×SD1612yes—
271C22×C4C2×SD1612yes—
272C22×C4C2×SD1612yes—
273C22×C4#13512yes—
274C22×C4AGL(1,ℤ/8)12yes—
275C22×C4#13712yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×Q812yes—
278C22×C4C2×#4112yes—
279C22×C4C2×#4111yescharacteristic
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C4H(5,𝔽2)11yescharacteristic
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C42−1+412yes—
285C22×C42−1+412yes—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8#12314yes—
305C2×C8C4⋊Q814yes—
306C4⋊C4—28no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—88no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C4⋊C4—88no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C42—22no—
331C42—22no—
332C42—24no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—42no—
340C23C2×#12111yescharacteristic
341C23#53511yescharacteristic
342C23#53511yescharacteristic
343C23#54712yes—
344C23D4×Q812yes—
345C23C22×SD1611yescharacteristic
346C23C2×#13511yescharacteristic
347C23C2×AGL(1,ℤ/8)11yescharacteristic
348C23C2×#13711yescharacteristic
349C23#57712yes—
350C23#57712yes—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#46514yes—
410C2×C4#46814yes—
411C2×C4C2×#12311yescharacteristic
412C2×C4C2×C4⋊Q811yescharacteristic
413C2×C4#53511yescharacteristic
414C2×C4#53511yescharacteristic
415C2×C4#53611yescharacteristic
416C2×C4#53611yescharacteristic
417C2×C4#53912yes—
418C2×C4#53912yes—
419C2×C4#54011yescharacteristic
420C2×C4#54011yescharacteristic
421C2×C4#54412yes—
422C2×C4#54412yes—
423C2×C4#55212yes—
424C2×C4#55412yes—
425C2×C4#55412yes—
426C2×C4#55712yes—
427C2×C4#56312yes—
428C2×C4#56312yes—
429C8—24no—
430C8—24no—
431C8—81no—
432C8—81no—
433C22#248611yescharacteristic
434C22#254611yescharacteristic
435C22#261911yescharacteristic
436C22#272012yes—
437C22#272012yes—
438C22C2×#46511yescharacteristic
439C22C2×#46811yescharacteristic
440C22#305711yescharacteristic
441C22#305711yescharacteristic
442C22#320611yescharacteristic
443C22#320711yescharacteristic
444C22#320811yescharacteristic
445C22#321011yescharacteristic
446C22#322112yes—
447C22#322212yes—
448C22#333512yes—
449C22#333712yes—
450C22#334412yes—
451C22#334512yes—
452C22Q8×SD1612yes—
453C22#338112yes—
454C22#338412yes—
455C22#338712yes—
456C4—24no—
457C4—24no—
458C4—42no—
459C4—42no—
460C4—81no—
461C4—84no—
462C4—84no—
463C4—81no—
464C4—162no—
465C4—162no—
466C4#250211yescharacteristic
467C4#262311yescharacteristic
468C4#266011yescharacteristic
469C4#279112yes—
470C4#279112yes—
471C4#282111yescharacteristic
472C2#1706111yescharacteristic
473C2#2765811yescharacteristic
474C2#2766311yescharacteristic
475C2#2867711yescharacteristic
476C2#2868411yescharacteristic
477C2#2980711yescharacteristic
478C2#2981111yescharacteristic
479C2#3234312yes—
480C2#3234412yes—
481C2#3234612yes—
482C2#3234912yes—
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×#19162
C2×#18442
#151962
#151512
#113732
#113712
#113272
#112572
#112312
#108202