Finite Group Databasebeta

← #417015

Subgroups — #417015

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35339C211yesmaximal, characteristic
2#21324C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5#11296C214yesmaximal
6#11276C214yesmaximal
7#10858C211yesmaximal, characteristic
8#10840C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2864—21no—
14#2864—21no—
15#2647—22no—
16#2570—22no—
17#2570—22no—
18#2569—22no—
19#2569—22no—
20#2317C2212yes—
21#2316C2212yes—
22#2285C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1993—21no—
27#1993—21no—
28#1990—22no—
29#1988—22no—
30#1987—22no—
31C2×#343C2214yes—
32#1736C2212yes—
33#1736C2214yes—
34#1736C2212yes—
35C2×#339C2212yes—
36C2×#339C2214yes—
37C2×#339C2212yes—
38#1446—22no—
39#1445—22no—
40C4⋊#95C2214yes—
41C23×Q8—21no—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8Q814yes—
52C2×#117C2311yescharacteristic
53C2×#117C2312yes—
54C2×#117C2311yescharacteristic
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60#423—22no—
61#423—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C4×M16—22no—
65C4×M16—22no—
66C2×#97—21no—
67C2×C4×C8—21no—
68#400—24no—
69#400—24no—
70#398—24no—
71#398—24no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#387—24no—
79#387—24no—
80#387—24no—
81#387—24no—
82#385—24no—
83#384—22no—
84#384D412yes—
85#382—22no—
86#382—21no—
87#382—21no—
88#382—21no—
89#382—21no—
90#382D411yescharacteristic
91#382D411yescharacteristic
92#380—24no—
93#380—21no—
94#380—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4D412yes—
98C4×C4⋊C4D412yes—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101#376—21no—
102#376—22no—
103#376—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—28no—
107C2×#95D412yes—
108C2×#95D412yes—
109C43—21no—
110#343—28no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#339—24no—
116#339—24no—
117#339—28no—
118C22×Q8—24no—
119C22×Q8—41no—
120C22×Q8—41no—
121C22×Q8—42no—
122C22×Q8—84no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C2×D414yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16#4114yes—
139C22×C8—22no—
140C22×C8—22no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149#117—24no—
150#117—22no—
151#117—22no—
152#117—24no—
153#117—22no—
154#117—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×Q814yes—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×Q812yes—
197C2×C42#4114yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200C2×C42#4112yes—
201C4⋊C8—42no—
202C4⋊C8—42no—
203#97—42no—
204C4×C8—42no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—42no—
220#95—42no—
221#95—88no—
222C24C22×D411yescharacteristic
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—84no—
226C2×Q8—84no—
227C2×Q8—88no—
228C2×Q8—88no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4C22⋊Q812yes—
253C22×C4C22⋊Q812yes—
254C22×C4#12412yes—
255C22×C4#12412yes—
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×Q811yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4112yes—
264C22×C42−1+412yes—
265C22×C42−1+412yes—
266M16—24no—
267M16—24no—
268M16—42no—
269M16—42no—
270M16—44no—
271M16—44no—
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4—84no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—44no—
302C42—24no—
303C42—24no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—82no—
311C42—82no—
312C42C22⋊Q812yes—
313C42C22⋊Q812yes—
314C42C22⋊Q812yes—
315C42C22⋊Q812yes—
316C42#12412yes—
317C42#12412yes—
318C42#12412yes—
319C42#12412yes—
320C23—22no—
321C23—22no—
322C23C2×#12011yescharacteristic
323C23C2×C22⋊Q811yescharacteristic
324C23C2×#12411yescharacteristic
325C23#54712yes—
326C23#54812yes—
327Q8—88no—
328Q8—88no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—84no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—88no—
362C2×C4#39312yes—
363C2×C4#39312yes—
364C2×C4#39312yes—
365C2×C4#39312yes—
366C2×C4C2×C22⋊Q811yescharacteristic
367C2×C4C2×C22⋊Q811yescharacteristic
368C2×C4C2×#12411yescharacteristic
369C2×C4C2×#12411yescharacteristic
370C2×C4#52912yes—
371C2×C4#52912yes—
372C2×C4#53211yescharacteristic
373C2×C4#53211yescharacteristic
374C2×C4#53611yescharacteristic
375C2×C4#53611yescharacteristic
376C2×C4#54112yes—
377C2×C4#54112yes—
378C2×C4#54712yes—
379C2×C4#54812yes—
380C2×C4#55212yes—
381C2×C4#55212yes—
382C2×C4#55712yes—
383C2×C4#55712yes—
384C8—44no—
385C8—82no—
386C22—24no—
387C22—24no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22C2×#39311yescharacteristic
393C22#243211yescharacteristic
394C22#243211yescharacteristic
395C22#245312yes—
396C22#248912yes—
397C22#302112yes—
398C22#302212yes—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—84no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4—84no—
409C4—84no—
410C4#245312yes—
411C4#248912yes—
412C4#256511yescharacteristic
413C4#256511yescharacteristic
414C4#264211yescharacteristic
415C4#286111yescharacteristic
416C2—22no—
417C2—22no—
418C2#1523811yescharacteristic
419C2#2346311yescharacteristic
420C2#2346411yescharacteristic
421C2#2567012yes—
422C2#2569212yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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
#353392
#213242
#213132
#142822
#112962
#112762
#108582
#108402
#107012