Finite Group Databasebeta

← #403157

Subgroups — #403157

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1948C211yesmaximal, characteristic
2C2×#1919C211yesmaximal, characteristic
3#15153C211yesmaximal, characteristic
4#13818C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11255C211yesmaximal, characteristic
7#11227C211yesmaximal, characteristic
8#11220C212yesmaximal
9#11220C212yesmaximal
10#10908C212yesmaximal
11#10807C212yesmaximal
12C22×#106C2212yes—
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#400—21no—
16C2×#400—21no—
17C2×#389C2211yescharacteristic
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22#1948—24no—
23#1919—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1445—22no—
36#1436—22no—
37C4⋊#95—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4×#95—21no—
42C4×#95C2211yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42—21no—
57C22×C42—22no—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×#106—22no—
61C2×#106—22no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—24no—
65C2×#106Q814yes—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—22no—
68#400—44no—
69#400—44no—
70#389—24no—
71#389—24no—
72#389—22no—
73#389—22no—
74#384—22no—
75#384—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—42no—
79C4×C4⋊C4D414yes—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95D411yescharacteristic
91C2×#95D412yes—
92C2×#95D411yescharacteristic
93#340—24no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—42no—
101#336—24no—
102#336—22no—
103#336—42no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8C2×Q812yes—
131C22×C8#4114yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×Q812yes—
171C2×C4⋊C4C2×Q812yes—
172C2×C4⋊C4#4114yes—
173C2×C4⋊C4#4112yes—
174C2×C4⋊C4#4114yes—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—24no—
184C2×C42—44no—
185C2×C42—44no—
186C2×C42—82no—
187C2×C42—82no—
188C2×C42C2×D412yes—
189C2×C42#4112yes—
190#106—28no—
191#106—48no—
192C4⋊C8—28no—
193#95—28no—
194#95—24no—
195#95—22no—
196#95—42no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—42no—
201#95—44no—
202#95—44no—
203#95—42no—
204#95—81no—
205#95SD1614yes—
206C24C22×D411yescharacteristic
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4#12112yes—
263C22×C4#12112yes—
264C22×C4C2×SD1614yes—
265C22×C4C2×SD1612yes—
266C22×C4AGL(1,ℤ/8)12yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C4C2×#4112yes—
273C22×C4C2×#4111yescharacteristic
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C42−1+412yes—
276C22×C42−1+411yescharacteristic
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—84no—
315C4⋊C4—88no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4C22⋊Q814yes—
319C4⋊C4#12414yes—
320C42—22no—
321C42—22no—
322C42—24no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—84no—
328C42—88no—
329C23C2×#12111yescharacteristic
330C23#52512yes—
331C23#54611yescharacteristic
332C23#54712yes—
333C23D4×Q811yescharacteristic
334C23C22×SD1611yescharacteristic
335C23C2×AGL(1,ℤ/8)11yescharacteristic
336C23#57512yes—
337C23#57711yescharacteristic
338C23#57712yes—
339C23#57811yescharacteristic
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4#46014yes—
400C2×C4C2×C22⋊Q811yescharacteristic
401C2×C4#52511yescharacteristic
402C2×C4C2×#12411yescharacteristic
403C2×C4#52912yes—
404C2×C4#52912yes—
405C2×C4#53211yescharacteristic
406C2×C4#53312yes—
407C2×C4#53611yescharacteristic
408C2×C4#53711yescharacteristic
409C2×C4#53912yes—
410C2×C4#54112yes—
411C2×C4#55212yes—
412C2×C4#55212yes—
413C2×C4#55211yescharacteristic
414C2×C4#55312yes—
415C2×C4#55411yescharacteristic
416C2×C4#55612yes—
417C2×C4#55712yes—
418C8—44no—
419C8—42no—
420C8—81no—
421C22#241311yescharacteristic
422C22#241711yescharacteristic
423C22#246212yes—
424C22#246612yes—
425C22#285011yescharacteristic
426C22C2×#46011yescharacteristic
427C22#304011yescharacteristic
428C22#304211yescharacteristic
429C22#304611yescharacteristic
430C22#310312yes—
431C22#311512yes—
432C22#312012yes—
433C22#312312yes—
434C22#329811yescharacteristic
435C22#331112yes—
436C22#333211yescharacteristic
437C22#333512yes—
438C22#334512yes—
439C22#334812yes—
440C22#335212yes—
441C22Q8×SD1611yescharacteristic
442C22#338611yescharacteristic
443C22#338812yes—
444C4—24no—
445C4—44no—
446C4—44no—
447C4—42no—
448C4—44no—
449C4—82no—
450C4—84no—
451C4—82no—
452C4—162no—
453C4—161no—
454C4#239611yescharacteristic
455C4#253312yes—
456C4#256211yescharacteristic
457C4#256811yescharacteristic
458C4#259012yes—
459C4#266411yescharacteristic
460C2#1486911yescharacteristic
461C2#2457811yescharacteristic
462C2#2459511yescharacteristic
463C2#2481311yescharacteristic
464C2#2483411yescharacteristic
465C2#2686012yes—
466C2#2688012yes—
467C2#2703412yes—
468C2#2705412yes—
469C2#3302511yescharacteristic
470C2#3304011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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×#19482
C2×#19192
#151532
#138182
#113572
#112552
#112272
#112202
#109082
#108072