Finite Group Databasebeta

← #240989

Subgroups — #240989

Order 512.

887 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#21370C211yesmaximal, characteristic
3#17749C211yesmaximal, characteristic
4#17746C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6#17464C211yesmaximal, characteristic
7#10556C218yesmaximal, complemented
8#7938C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3165—22no—
12#3165—22no—
13#3165—22no—
14#3165C2214yes—
15C2×#501—22no—
16C2×#500—21no—
17C2×#498—21no—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#428—22no—
22C2×#423C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#1950C2214yes—
25#1855—22no—
26C4×#108—22no—
27#1559—22no—
28#1558—22no—
29#1550C2214yes—
30#1548—22no—
31#1548—21no—
32#1548C2214yes—
33#1546—22no—
34#1546—22no—
35#1546—21no—
36C2×#332C2214yes—
37#1485—22no—
38#1481C2214yes—
39#1479—22no—
40#1283C2214yes—
41C23×Q8—21no—
42C22×M16—21no—
43C22×M16D412yes—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533D412yes—
51#533C2312yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8C2312yes—
66C2×#125—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#501—28no—
76#501—28no—
77#500—24no—
78#500—24no—
79#498—24no—
80#498—24no—
81#432—21no—
82#432—21no—
83#432—21no—
84#432—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431D412yes—
91#431D412yes—
92#431C2312yes—
93C2×#108—24no—
94#428—28no—
95#428—28no—
96C2×#107—24no—
97C2×#106—24no—
98#423—21no—
99#423—21no—
100#423—21no—
101#423—21no—
102#423—21no—
103#423C2312yes—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107#407—21no—
108#407—21no—
109C2×C22⋊C8—21no—
110C4×M16—21no—
111C4×M16—21no—
112C4×M16—21no—
113C4×M16—21no—
114C4×M16D412yes—
115C4×M16C2312yes—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119#341—24no—
120#333—28no—
121#333—24no—
122#332—24no—
123#332—28no—
124Q8⋊C8—28no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—41no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×Q8—82no—
134C22×Q8—82no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4#4111yescharacteristic
138C23×C4#4111yescharacteristic
139C2×M16—22no—
140C2×M16—28no—
141C2×M16—22no—
142C2×M16—41no—
143C2×M16—41no—
144C2×M16—41no—
145C2×M16—82no—
146C2×M16C2×D414yes—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8C2×D414yes—
172C4⋊Q8C2×D412yes—
173C4⋊Q8C2×D412yes—
174#125—22no—
175#125—22no—
176#125—41no—
177#125—41no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194#117—22no—
195#117—22no—
196#117—41no—
197#117—41no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C42—41no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230#108—48no—
231#107—216no—
232#106—216no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241C4⋊C8—41no—
242C4⋊C8C2×D414yes—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—41no—
247C22⋊C8—41no—
248C22⋊C8—41no—
249#97—22no—
250#97—22no—
251#97—41no—
252#97—41no—
253C4×C8—22no—
254C4×C8—22no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—41no—
259C24#12411yescharacteristic
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—42no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—82no—
271C2×Q8—84no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×Q8—84no—
275C2×Q8—82no—
276C2×Q8—82no—
277C2×Q8—84no—
278C2×Q8—82no—
279C2×Q8—82no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—84no—
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12311yescharacteristic
299C22×C4#12311yescharacteristic
300C22×C4#12311yescharacteristic
301C22×C4#12311yescharacteristic
302C22×C4#12411yescharacteristic
303C22×C4#12411yescharacteristic
304C22×C4#12411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308M16—28no—
309M16—24no—
310M16—44no—
311M16—82no—
312M16—82no—
313M16—82no—
314M16#13718yes—
315C2×C8—28no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—44no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—82no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C42—42no—
346C42—42no—
347C42#12011yescharacteristic
348C42#12011yescharacteristic
349C42#12111yescharacteristic
350C42#12111yescharacteristic
351C42#12111yescharacteristic
352C42#12111yescharacteristic
353C42#12111yescharacteristic
354C42#12111yescharacteristic
355C42#12311yescharacteristic
356C42#12311yescharacteristic
357C42#12311yescharacteristic
358C42#12311yescharacteristic
359C42C22×D411yescharacteristic
360C42C22×D411yescharacteristic
361C42C22×D411yescharacteristic
362C42C2×#4111yescharacteristic
363C23—22no—
364C23—22no—
365C23—41no—
366C23#39411yescharacteristic
367C23#39411yescharacteristic
368C23C2×#12411yescharacteristic
369Q8—42no—
370Q8—42no—
371Q8—82no—
372Q8—84no—
373Q8—84no—
374Q8—82no—
375Q8—82no—
376Q8—84no—
377Q8—82no—
378Q8—82no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—82no—
403C2×C4—84no—
404C2×C4—82no—
405C2×C4#39411yescharacteristic
406C2×C4#39411yescharacteristic
407C2×C4#39411yescharacteristic
408C2×C4#39411yescharacteristic
409C2×C4#39411yescharacteristic
410C2×C4#39411yescharacteristic
411C2×C4#47112yes—
412C2×C4#47212yes—
413C2×C4C2×#12011yescharacteristic
414C2×C4C2×#12111yescharacteristic
415C2×C4C2×#12111yescharacteristic
416C2×C4C2×#12111yescharacteristic
417C2×C4C2×#12311yescharacteristic
418C2×C4C2×#12311yescharacteristic
419C2×C4C2×#13714yes—
420C8—28no—
421C8—44no—
422C8—82no—
423C8—82no—
424C8—82no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—44no—
433C22#205112yes—
434C22C2×#39411yescharacteristic
435C22#313912yes—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—82no—
444C4—84no—
445C4—82no—
446C4#204912yes—
447C4#205112yes—
448C4#300412yes—
449C4#303912yes—
450C4C2×#47111yescharacteristic
451C4C2×#47211yescharacteristic
452C2—22no—
453C2—22no—
454C2—41no—
455C2#2350711yescharacteristic
456C2C2×#204911yescharacteristic
457C2C2×#205111yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10556C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#213702
#177492
#177462
C2×#15462
#174642
#105562
#79382