Finite Group Databasebeta

← #475931

Subgroups — #475931

Order 512.

920 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2085C212yesmaximal
2#15221C212yesmaximal
3#11327C212yesmaximal
4#11325C212yesmaximal
5#11232C212yesmaximal
6#11220C212yesmaximal
7#10861C211yesmaximal, characteristic
8#10861C211yesmaximal, characteristic
9#10834C211yesmaximal, characteristic
10C22×#107C2212yes—
11C22×#106C2211yescharacteristic
12C22×#106C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#382C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21#2085—28no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29#1445—22no—
30#1444—22no—
31#1442—22no—
32#1442—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C23×C8C2312yes—
37C23×C8C2311yescharacteristic
38C23×C8C2311yescharacteristic
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4D412yes—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C42C2312yes—
47C22×C42C2311yescharacteristic
48C2×#107—24no—
49C2×#107—42no—
50C2×#107—42no—
51C2×#106—22no—
52C2×#106—22no—
53C2×#106—41no—
54C2×#106—41no—
55C2×#106—41no—
56C2×#106—41no—
57C2×#106—41no—
58C2×#106—41no—
59C2×#106—41no—
60C2×#106—41no—
61C2×#106D414yes—
62C2×#106Q814yes—
63C2×C4⋊C8—41no—
64C2×C4⋊C8—41no—
65C2×C4⋊C8Q814yes—
66C2×C4×C8—41no—
67C2×C4×C8—41no—
68C2×C4×C8Q814yes—
69#389—24no—
70#389—42no—
71#389—42no—
72#389—42no—
73#382—24no—
74#382—42no—
75C4×C4⋊C4—41no—
76C4×C4⋊C4—41no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D411yescharacteristic
88C2×#95D411yescharacteristic
89#340—24no—
90#340—24no—
91#340—24no—
92#340—42no—
93#340—42no—
94#336—24no—
95#336—24no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2411yescharacteristic
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8C2×D412yes—
120C22×C8C2×Q814yes—
121C22×C8C2×Q812yes—
122C22×C8C2×Q812yes—
123C22×C8C2×Q812yes—
124C22×C8#4112yes—
125C22×C8#4112yes—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
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—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4C2×D414yes—
158C2×C4⋊C4C2×Q814yes—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42C2×Q812yes—
170C2×C42C2×Q812yes—
171#107—48no—
172#106—44no—
173#106—44no—
174#106—44no—
175#106—44no—
176C4⋊C8—44no—
177C4×C8—44no—
178#95—41no—
179#95—41no—
180#95—44no—
181#95—44no—
182#95—44no—
183#95—44no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—82no—
188#95—82no—
189#95—82no—
190C24C22×D411yescharacteristic
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
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—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C4⋊Q812yes—
241C22×C4C4⋊Q812yes—
242C22×C4#13514yes—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×Q811yescharacteristic
245C22×C4C22×Q811yescharacteristic
246C22×C4C22×Q811yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C4H(5,𝔽2)11yescharacteristic
250C22×C4H(5,𝔽2)11yescharacteristic
251C22×C4H(5,𝔽2)11yescharacteristic
252C22×C4H(5,𝔽2)11yescharacteristic
253C22×C4H(5,𝔽2)11yescharacteristic
254C22×C42−1+411yescharacteristic
255C22×C42−1+411yescharacteristic
256C22×C42−1+411yescharacteristic
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8#12514yes—
277C2×C8C4⋊Q814yes—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—48no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—88no—
296C4⋊C4—88no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C23C2×C4⋊Q811yescharacteristic
306C23#53411yescharacteristic
307C23#53711yescharacteristic
308C23D4211yescharacteristic
309C23#54711yescharacteristic
310C23D4×Q811yescharacteristic
311C23#55011yescharacteristic
312C23C2×#13512yes—
313C23#57512yes—
314C23#57712yes—
315C23#57712yes—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4#49914yes—
368C2×C4C2×#12511yescharacteristic
369C2×C4C2×C4⋊Q811yescharacteristic
370C2×C4#53411yescharacteristic
371C2×C4#53711yescharacteristic
372C2×C4#53811yescharacteristic
373C2×C4#53811yescharacteristic
374C2×C4#53911yescharacteristic
375C2×C4#53911yescharacteristic
376C2×C4#54311yescharacteristic
377C2×C4#54311yescharacteristic
378C2×C4#54311yescharacteristic
379C2×C4#54411yescharacteristic
380C2×C4#54411yescharacteristic
381C2×C4#54411yescharacteristic
382C2×C4#54411yescharacteristic
383C2×C4#54411yescharacteristic
384C2×C4#54711yescharacteristic
385C2×C4D4×Q811yescharacteristic
386C2×C4#55011yescharacteristic
387C2×C4#55411yescharacteristic
388C2×C4#55411yescharacteristic
389C2×C4#55711yescharacteristic
390C2×C4#55711yescharacteristic
391C2×C4Q8211yescharacteristic
392C2×C4#56211yescharacteristic
393C2×C4#56211yescharacteristic
394C2×C4#56211yescharacteristic
395C2×C4#56411yescharacteristic
396C8—24no—
397C8—24no—
398C8—44no—
399C22#255111yescharacteristic
400C22#255211yescharacteristic
401C22#265911yescharacteristic
402C22#268211yescharacteristic
403C22#268211yescharacteristic
404C22#272611yescharacteristic
405C22#273211yescharacteristic
406C22C2×#49911yescharacteristic
407C22#316211yescharacteristic
408C22#316311yescharacteristic
409C22#316311yescharacteristic
410C22#319112yes—
411C22#320012yes—
412C22#325012yes—
413C22#326312yes—
414C22#327512yes—
415C22#328412yes—
416C22#334012yes—
417C22#335112yes—
418C22#338112yes—
419C22#338812yes—
420C22#339412yes—
421C22#340112yes—
422C4—24no—
423C4—42no—
424C4—84no—
425C4—82no—
426C4—82no—
427C4—84no—
428C4—162no—
429C4—162no—
430C4#255211yescharacteristic
431C4#264611yescharacteristic
432C4#265911yescharacteristic
433C4#272611yescharacteristic
434C4#273211yescharacteristic
435C4#280611yescharacteristic
436C4#280611yescharacteristic
437C4#284511yescharacteristic
438C2#1707311yescharacteristic
439C2#2881111yescharacteristic
440C2#2881511yescharacteristic
441C2#2883011yescharacteristic
442C2#2883411yescharacteristic
443C2#3029211yescharacteristic
444C2#3030211yescharacteristic
445C2#3119112yes—
446C2#3121712yes—
447C2#3250712yes—
448C2#3265112yes—
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×#20852
#152212
#113272
#113252
#112322
#112202
#108612
#108342