Finite Group Databasebeta

← #414925

Subgroups — #414925

Order 512.

1072 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#21322C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic
5#21300C211yesmaximal, characteristic
6C2×C4×#108C211yesmaximal, characteristic
7#11290C214yesmaximal
8#11249C214yesmaximal
9#10857C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—24no—
13#3165—24no—
14C2×#501—22no—
15C2×#500—21no—
16C2×#498—21no—
17C22×#108C2212yes—
18C2×#428C2214yes—
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#1997—24no—
25#1992—22no—
26#1990—22no—
27#1990—22no—
28#1990—21no—
29#1990—21no—
30#1984—22no—
31#1984—22no—
32C4×#108—24no—
33C2×#343C2214yes—
34C2×#343C2214yes—
35C2×#341C2212yes—
36#1736C2214yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39#1729C2212yes—
40#1729C2212yes—
41C23×Q8—21no—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C22×M16C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D414yes—
61C2×#125D412yes—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117C2312yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C42C2311yescharacteristic
73#501—216no—
74#500—28no—
75#498—28no—
76C2×#108—24no—
77C2×#108—24no—
78C2×#108Q818yescomplemented
79#428—28no—
80#428—28no—
81C2×#107—24no—
82C2×#106—24no—
83#423—22no—
84#423—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C4×M16—22no—
88C4×M16—22no—
89C4×M16—22no—
90C4×M16D414yes—
91C2×#97—21no—
92C2×#97—21no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97C2×C4×C8D412yes—
98C2×C4×C8D412yes—
99#384—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#343—28no—
104#343—28no—
105#341—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#336—24no—
111#336—24no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—24no—
115C22×Q8—41no—
116C22×Q8—41no—
117C22×Q8—42no—
118C22×Q8—82no—
119C22×Q8—82no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4#4111yescharacteristic
123C23×C4#4111yescharacteristic
124C23×C4C2411yescharacteristic
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—24no—
135C2×M16—22no—
136C2×M16—24no—
137C2×M16C2×D414yes—
138C2×M16C2×Q814yes—
139C2×M16C2×Q814yes—
140C2×M16#4114yes—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×Q814yes—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166#125—24no—
167#125—24no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—42no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C22⋊C4—22no—
207C2×C42—22no—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4112yes—
215C2×C42#4112yes—
216#108—216no—
217#107—216no—
218#106—216no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221#97—24no—
222#97—42no—
223C4×C8—24no—
224C4×C8—24no—
225C4×C8—24no—
226C4×C8—42no—
227C24C2×#4111yescharacteristic
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—88no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—88no—
240C2×Q8—82no—
241C2×Q8—82no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12312yes—
262C22×C4#12312yes—
263C22×C4#12412yes—
264C22×C4#12412yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×Q812yes—
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4112yes—
273C22×C42−1+412yes—
274C22×C42−1+412yes—
275M16—24no—
276M16—24no—
277M16—24no—
278M16—24no—
279M16—24no—
280M16—42no—
281M16—42no—
282M16—44no—
283M16—44no—
284M16C22⋊Q818yes—
285C2×C8—24no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8C4⋊Q814yes—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C42—42no—
328C42—42no—
329C42#12012yes—
330C42#12012yes—
331C42#12112yes—
332C42#12112yes—
333C42#12112yes—
334C42#12112yes—
335C42#12312yes—
336C42#12312yes—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23C2×#12111yescharacteristic
342C23C2×#12311yescharacteristic
343C23C2×#12411yescharacteristic
344C23#55612yes—
345C23#55712yes—
346Q8—88no—
347Q8—84no—
348Q8—88no—
349Q8—84no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—84no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—164no—
373C2×C4#39412yes—
374C2×C4#39412yes—
375C2×C4#39412yes—
376C2×C4#39412yes—
377C2×C4#47112yes—
378C2×C4#47112yes—
379C2×C4#47212yes—
380C2×C4#47312yes—
381C2×C4C2×#12011yescharacteristic
382C2×C4C2×#12111yescharacteristic
383C2×C4C2×#12111yescharacteristic
384C2×C4C2×C22⋊Q812yes—
385C2×C4C2×#12311yescharacteristic
386C2×C4C2×C4⋊Q811yescharacteristic
387C2×C4#53211yescharacteristic
388C2×C4#53312yes—
389C2×C4#53612yes—
390C2×C4#53611yescharacteristic
391C2×C4#53611yescharacteristic
392C2×C4#54112yes—
393C2×C4#54812yes—
394C2×C4#54812yes—
395C2×C4D4×Q812yes—
396C2×C4D4×Q812yes—
397C2×C4#55612yes—
398C2×C4#55712yes—
399C8—24no—
400C8—28no—
401C8—42no—
402C8—44no—
403C8—82no—
404C22—22no—
405C22—22no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22C2×#39411yescharacteristic
413C22#244612yes—
414C22#254511yescharacteristic
415C22#254611yescharacteristic
416C22#256812yes—
417C22C2×#47111yescharacteristic
418C22C2×#47111yescharacteristic
419C22C2×#47211yescharacteristic
420C22C2×#47311yescharacteristic
421C4—22no—
422C4—22no—
423C4—82no—
424C4—84no—
425C4—82no—
426C4—162no—
427C4#205114yes—
428C4#205214yes—
429C4#240912yes—
430C4#244612yes—
431C4#254511yescharacteristic
432C4#254611yescharacteristic
433C4#262111yescharacteristic
434C4#285711yescharacteristic
435C2—22no—
436C2—22no—
437C2#1518011yescharacteristic
438C2C2×#205111yescharacteristic
439C2C2×#205211yescharacteristic
440C2#2866911yescharacteristic
441C2#2867011yescharacteristic
442C2#2869911yescharacteristic
443C2#2870011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
78. C2×#108Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#31652
C2×#19972
#213222
C2×#19902
#213002
C2×C4×#1082
#112902
#112492
#108572