Finite Group Databasebeta

← #415451

Subgroups — #415451

Order 512.

988 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C212yesmaximal
2#14277C211yesmaximal, characteristic
3#13779C211yesmaximal, characteristic
4#11358C211yesmaximal, characteristic
5#11358C211yesmaximal, characteristic
6#11324C211yesmaximal, characteristic
7#11323C211yesmaximal, characteristic
8#11220C212yesmaximal
9#11219C212yesmaximal
10#10849C212yesmaximal
11#10833C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C2×#389C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×#376C2211yescharacteristic
23#1949—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1444—22no—
34#1442—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C42—22no—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#107—24no—
57C2×#107—24no—
58C2×#107—42no—
59C2×#106—24no—
60C2×#106—24no—
61C2×#106—42no—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8Q818yes—
65#389—42no—
66#389—42no—
67#384—24no—
68#384—24no—
69#382—22no—
70#382—22no—
71#382D414yes—
72#382Q814yes—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—24no—
75#376—24no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95D412yes—
86C2×#95D412yes—
87#340—24no—
88#340—24no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#336—22no—
94#336—22no—
95#336—42no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2411yescharacteristic
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8C2×Q814yes—
120C22×C8C2×Q814yes—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
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—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×Q814yes—
158C2×C4⋊C4#4114yes—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—44no—
166C2×C42—82no—
167C2×C42—82no—
168C2×C42C2×D412yes—
169C2×C42C2×Q812yes—
170C2×C42C2×Q814yes—
171C2×C42#4112yes—
172C2×C42#4114yes—
173C2×C42#4112yes—
174#107—48no—
175#106—48no—
176C4⋊C8—216no—
177#95—28no—
178#95—28no—
179#95—24no—
180#95—24no—
181#95—42no—
182#95—42no—
183#95—44no—
184#95—82no—
185#95—82no—
186#95—82no—
187#95—82no—
188C24C22×D411yescharacteristic
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
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—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4#12114yes—
244C22×C4C22⋊Q814yes—
245C22×C4#12314yes—
246C22×C4C4⋊Q814yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×Q812yes—
249C22×C4C22×Q811yescharacteristic
250C22×C4C2×#4112yes—
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4112yes—
254C22×C4H(5,𝔽2)11yescharacteristic
255C22×C4H(5,𝔽2)12yes—
256C22×C42−1+411yescharacteristic
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
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—42no—
286C4⋊C4—42no—
287C4⋊C4—88no—
288C4⋊C4—88no—
289C4⋊C4—84no—
290C4⋊C4—84no—
291C42—24no—
292C42—24no—
293C42—42no—
294C42—42no—
295C42—88no—
296C42C22⋊Q818yes—
297C42#12518yes—
298C23C2×#12111yescharacteristic
299C23C2×C22⋊Q811yescharacteristic
300C23C2×#12311yescharacteristic
301C23#52511yescharacteristic
302C23C2×C4⋊Q811yescharacteristic
303C23#53511yescharacteristic
304C23#53711yescharacteristic
305C23#57612yes—
306C23#57712yes—
307C23#57712yes—
308C23#57812yes—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
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—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4#39818yes—
367C2×C4C2×C22⋊Q812yes—
368C2×C4C2×#12512yes—
369C2×C4#52912yes—
370C2×C4#53312yes—
371C2×C4#54012yes—
372C2×C4#54212yes—
373C2×C4#54612yes—
374C2×C4D4×Q811yescharacteristic
375C2×C4#55011yescharacteristic
376C2×C4#55312yes—
377C2×C4#55412yes—
378C2×C4#55412yes—
379C2×C4#55411yescharacteristic
380C2×C4#55611yescharacteristic
381C2×C4#55712yes—
382C2×C4Q8212yes—
383C8—44no—
384C8—44no—
385C22C2×#39811yescharacteristic
386C22#253511yescharacteristic
387C22#253811yescharacteristic
388C22#256711yescharacteristic
389C22#258411yescharacteristic
390C22#262411yescharacteristic
391C22#285511yescharacteristic
392C22#304512yes—
393C22#304612yes—
394C22#308312yes—
395C22#308412yes—
396C22#309612yes—
397C22#309712yes—
398C22#311812yes—
399C22#312112yes—
400C22#316312yes—
401C22#316412yes—
402C22#322011yescharacteristic
403C22#322111yescharacteristic
404C22#322211yescharacteristic
405C22#322311yescharacteristic
406C22#325712yes—
407C22#325812yes—
408C4—28no—
409C4—44no—
410C4—84no—
411C4—82no—
412C4—82no—
413C4—84no—
414C4—81no—
415C4—81no—
416C4—161no—
417C4—161no—
418C4#245112yes—
419C4#245412yes—
420C4#250712yes—
421C4#250912yes—
422C2#1519411yescharacteristic
423C2#2366212yes—
424C2#2817111yescharacteristic
425C2#2817311yescharacteristic
426C2#2829312yes—
427C2#2921511yescharacteristic
428C2#2921611yescharacteristic
429C2#2932512yes—
430C2#2993311yescharacteristic
431C2#2993411yescharacteristic
432C2#3315512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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×#19492
#142772
#137792
#113582
#113242
#113232
#112202
#112192
#108492
#108332