Finite Group Databasebeta

← #401494

Subgroups — #401494

Order 512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2086C211yesmaximal, characteristic
2C2×#1942C211yesmaximal, characteristic
3C2×#1916C211yesmaximal, characteristic
4C2×C4×#107C211yesmaximal, characteristic
5#14046C212yesmaximal
6#11379C211yesmaximal, characteristic
7#11328C211yesmaximal, characteristic
8#11232C211yesmaximal, characteristic
9#11219C212yesmaximal
10#11219C212yesmaximal
11#10884C212yesmaximal
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#389—22no—
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#2086—24no—
24#1942—24no—
25#1916—24no—
26C4×#107—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1442—22no—
39#1441—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42—22no—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#107—22no—
61C2×#107—24no—
62C2×#107—24no—
63C2×#107—24no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#107Q818yes—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8D414yes—
74#389—24no—
75#389—24no—
76#389—24no—
77#389—24no—
78#389—24no—
79#389—42no—
80#389—48no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—42no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95D411yescharacteristic
94C2×#95D411yescharacteristic
95#340—24no—
96#340—22no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—42no—
101#340D414yes—
102#336—22no—
103#336—22no—
104#336—22no—
105#336—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2411yescharacteristic
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8C2×D412yes—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8C2×Q814yes—
136C22×C8#4112yes—
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—22no—
150C2×C4⋊C4—28no—
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—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4D814yes—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×Q814yes—
172C2×C4⋊C4C2×Q814yes—
173C2×C4⋊C4#4114yes—
174C2×C4⋊C4#4114yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—44no—
190C2×C42—44no—
191C2×C42—82no—
192C2×C42C2×D412yes—
193C2×C42#4112yes—
194#107—28no—
195#107—216no—
196#107—44no—
197C4⋊C8—44no—
198C4×C8—28no—
199#95—28no—
200#95—22no—
201#95—22no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—82no—
209C24C22×D411yescharacteristic
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
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#12112yes—
267C22×C4#12112yes—
268C22×C4C2×D812yes—
269C22×C4C2×D812yes—
270C22×C4C2×D812yes—
271C22×C4#13512yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×Q812yes—
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4112yes—
277C22×C4C2×#4112yes—
278C22×C4C2×#4111yescharacteristic
279C22×C4H(5,𝔽2)11yescharacteristic
280C22×C42−1+412yes—
281C22×C42−1+411yescharacteristic
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8#12114yes—
301C2×C8C4⋊Q814yes—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—416no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—88no—
321C4⋊C4—88no—
322C4⋊C4C22⋊Q818yes—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—41no—
328C42—41no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—88no—
335C23C2×#12111yescharacteristic
336C23#52512yes—
337C23D4211yescharacteristic
338C23#54711yescharacteristic
339C23D4×Q812yes—
340C23C22×D811yescharacteristic
341C23C2×#13511yescharacteristic
342C23#57512yes—
343C23#57611yescharacteristic
344C23#57811yescharacteristic
345C23#57812yes—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—28no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—41no—
384C2×C4—44no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
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#46614yes—
408C2×C4C2×#12111yescharacteristic
409C2×C4C2×C22⋊Q812yes—
410C2×C4#52511yescharacteristic
411C2×C4#52912yes—
412C2×C4C2×C4⋊Q811yescharacteristic
413C2×C4#53211yescharacteristic
414C2×C4#53312yes—
415C2×C4#53611yescharacteristic
416C2×C4#53711yescharacteristic
417C2×C4#53912yes—
418C2×C4#54112yes—
419C2×C4#54712yes—
420C2×C4#54811yescharacteristic
421C2×C4#55212yes—
422C2×C4#55311yescharacteristic
423C2×C4#55412yes—
424C2×C4#55712yes—
425C2×C4#55712yes—
426C8—24no—
427C8—24no—
428C8—81no—
429C8—81no—
430C22#239211yescharacteristic
431C22#241611yescharacteristic
432C22#244912yes—
433C22#246212yes—
434C22#254611yescharacteristic
435C22C2×#46611yescharacteristic
436C22#305411yescharacteristic
437C22#305911yescharacteristic
438C22#306111yescharacteristic
439C22#310012yes—
440C22#310212yes—
441C22#312412yes—
442C22#312712yes—
443C22#326912yes—
444C22D4×D811yescharacteristic
445C22#328511yescharacteristic
446C22#333411yescharacteristic
447C22#333911yescharacteristic
448C22#334912yes—
449C22Q8×D812yes—
450C22#338312yes—
451C22#338912yes—
452C22#339012yes—
453C4—24no—
454C4—48no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—84no—
459C4—84no—
460C4—81no—
461C4—81no—
462C4—162no—
463C4#241311yescharacteristic
464C4#245312yes—
465C4#258411yescharacteristic
466C4#258512yes—
467C4#266411yescharacteristic
468C4#285711yescharacteristic
469C2#1481811yescharacteristic
470C2#2369611yescharacteristic
471C2#2370511yescharacteristic
472C2#2478011yescharacteristic
473C2#2478511yescharacteristic
474C2#2636812yes—
475C2#2638312yes—
476C2#2685612yes—
477C2#2687012yes—
478C2#2867511yescharacteristic
479C2#2868011yescharacteristic
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×#20862
C2×#19422
C2×#19162
C2×C4×#1072
#140462
#113792
#113282
#112322
#112192
#108842