Finite Group Databasebeta

← #416770

Subgroups — #416770

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1847C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic
3#14348C211yesmaximal, characteristic
4#14277C211yesmaximal, characteristic
5#11257C212yesmaximal
6#11256C212yesmaximal
7#11220C212yesmaximal
8#11219C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1847—24no—
24#1835—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1444—22no—
34#1442—22no—
35#1442—22no—
36#1442—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
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×#97—22no—
63C2×#97—22no—
64C2×#97Q814yes—
65C2×C4×C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8Q814yes—
68#389—42no—
69#389—42no—
70#384—22no—
71#384—22no—
72#384—22no—
73#384—22no—
74#382—22no—
75#382—22no—
76#382—22no—
77#382D414yes—
78#376—22no—
79#376—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D412yes—
91C2×#95D412yes—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#336—24no—
99#336—24no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2411yescharacteristic
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×C8—22no—
120C22×C8—22no—
121C22×C8C2×Q814yes—
122C22×C8C2×Q814yes—
123C22×C8#4114yes—
124C22×C8#4114yes—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
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—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42C2×D412yes—
172C2×C42C2×Q812yes—
173C2×C42C2×Q812yes—
174C2×C42#4112yes—
175C2×C42#4112yes—
176C2×C42#4112yes—
177#107—48no—
178#106—48no—
179#97—28no—
180C4×C8—28no—
181#95—28no—
182#95—24no—
183#95—24no—
184#95—42no—
185#95—42no—
186#95—44no—
187#95—82no—
188#95—82no—
189#95—82no—
190#95—82no—
191#95—82no—
192#95—82no—
193C24C22×D411yescharacteristic
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4#12114yes—
250C22×C4#12314yes—
251C22×C4C4⋊Q814yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×Q811yescharacteristic
254C22×C4C22×Q811yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4112yes—
259C22×C4C2×#4112yes—
260C22×C4H(5,𝔽2)12yes—
261C22×C42−1+412yes—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8#12518yes—
276C4⋊C4—24no—
277C4⋊C4—24no—
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—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—88no—
293C4⋊C4—88no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C42—42no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42C22⋊Q814yes—
301C42C22⋊Q814yes—
302C42#12514yes—
303C42#12514yes—
304C23C2×#12111yescharacteristic
305C23C2×#12311yescharacteristic
306C23C2×C4⋊Q811yescharacteristic
307C23#53211yescharacteristic
308C23#53211yescharacteristic
309C23#53511yescharacteristic
310C23#53611yescharacteristic
311C23#57612yes—
312C23#57712yes—
313C23#57712yes—
314C23#57812yes—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
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—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4#39818yes—
373C2×C4C2×C22⋊Q811yescharacteristic
374C2×C4C2×C22⋊Q811yescharacteristic
375C2×C4C2×#12511yescharacteristic
376C2×C4C2×#12511yescharacteristic
377C2×C4C2×#12512yes—
378C2×C4#53312yes—
379C2×C4#54412yes—
380C2×C4#54412yes—
381C2×C4#54612yes—
382C2×C4#54812yes—
383C2×C4#55212yes—
384C2×C4#55312yes—
385C2×C4#55512yes—
386C2×C4#55612yes—
387C2×C4#55612yes—
388C2×C4#55912yes—
389C8—28no—
390C8—44no—
391C22C2×#39811yescharacteristic
392C22#249911yescharacteristic
393C22#250011yescharacteristic
394C22#254411yescharacteristic
395C22#254511yescharacteristic
396C22#266011yescharacteristic
397C22#285911yescharacteristic
398C22#304512yes—
399C22#304612yes—
400C22#309612yes—
401C22#309712yes—
402C22#316312yes—
403C22#316412yes—
404C22#317812yes—
405C22#317812yes—
406C22#317912yes—
407C22#317912yes—
408C22#322011yescharacteristic
409C22#322111yescharacteristic
410C22#322211yescharacteristic
411C22#322311yescharacteristic
412C22#324312yes—
413C22#324412yes—
414C4—28no—
415C4—82no—
416C4—82no—
417C4—82no—
418C4—81no—
419C4—81no—
420C4—162no—
421C4—162no—
422C4—161no—
423C4—161no—
424C4#243912yes—
425C4#244212yes—
426C4#257012yes—
427C4#257112yes—
428C2#1523111yescharacteristic
429C2#2366212yes—
430C2#2788712yes—
431C2#2790211yescharacteristic
432C2#2790311yescharacteristic
433C2#2862911yescharacteristic
434C2#2863011yescharacteristic
435C2#2866812yes—
436C2#3031311yescharacteristic
437C2#3031411yescharacteristic
438C2#3326212yes—
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×#18472
C2×#18352
#143482
#142772
#112572
#112562
#112202
#112192
#108432
#108412
#106942