Finite Group Databasebeta

← #239129

Subgroups — #239129

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#21385C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4#21146C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6C2×#1548C211yesmaximal, characteristic
7#10544C214yesmaximal, complemented
8#10543C214yesmaximal, complemented
9#7873C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—24no—
13#3165—24no—
14C2×#501—22no—
15C2×#500C2212yes—
16C2×#498C2212yes—
17C2×#431C2211yescharacteristic
18C2×#428—21no—
19C2×#428—21no—
20C2×#423—21no—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1987—21no—
26#1987—21no—
27#1987C2212yes—
28#1987C2214yes—
29#1981C2212yes—
30#1940—22no—
31#1939C2214yes—
32#1872—22no—
33C4×#103—21no—
34C4×#103—21no—
35C2×#340—22no—
36#1728—21no—
37#1550—24no—
38#1548—24no—
39C2×#333C2214yes—
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1277C2214yes—
43C23×Q8—21no—
44C22×M16—21no—
45C22×M16D411yescharacteristic
46C22×M16D411yescharacteristic
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533D414yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8D414yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2312yes—
61C2×C4⋊Q8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#501—216no—
74#500—24no—
75#500—24no—
76#498—24no—
77#498—24no—
78#431—22no—
79#431—22no—
80#428—28no—
81#428—28no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#106—22no—
85C2×#106—22no—
86#423—22no—
87#423—22no—
88#423—44no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×#103—22no—
98C2×#103—22no—
99C2×C22⋊C8—21no—
100C4×M16—22no—
101C4×M16—22no—
102C2×#97D412yes—
103C2×C4×C8—21no—
104C2×C4×C8C2312yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D411yescharacteristic
109C4×C4⋊C4C2312yes—
110C2×#95—21no—
111#340—22no—
112#340—22no—
113#340—24no—
114#340—24no—
115#336—24no—
116#333—28no—
117#333—24no—
118#332—24no—
119C22×Q8—24no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—41no—
124C22×Q8—42no—
125C22×Q8—84no—
126C23×C4—22no—
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4#4111yescharacteristic
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—42no—
136C2×M16—42no—
137C2×M16—42no—
138C2×M16SD1614yes—
139C2×M16Q1614yes—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—44no—
144C22×C8—42no—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—22no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—22no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—22no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8C2×D414yes—
163C4⋊Q8C2×D412yes—
164C4⋊Q8C2×D412yes—
165#125—24no—
166#125—42no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—42no—
174C4×Q8—42no—
175#117—24no—
176#117—42no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—41no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4111yescharacteristic
210C2×C42C2411yescharacteristic
211#107—28no—
212#107—28no—
213#106—28no—
214#106—28no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—48no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224#103—44no—
225#103—44no—
226C22⋊C8—42no—
227#97—24no—
228C4×C8—22no—
229C4×C8—22no—
230C4×C8—42no—
231#95—21no—
232#95—21no—
233#95—22no—
234C24#12111yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—24no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—84no—
245C2×Q8—82no—
246C2×Q8—84no—
247C2×Q8—88no—
248C2×Q8—88no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—82no—
261C22×C4—84no—
262C22×C4#12012yes—
263C22×C4#12112yes—
264C22×C4#12112yes—
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12312yes—
269C22×C4#12312yes—
270C22×C4#12412yes—
271C22×C4C2×SD1612yes—
272C22×C4C2×Q1612yes—
273C22×C4#13712yes—
274C22×C4#13712yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4111yescharacteristic
278M16—28no—
279M16—28no—
280M16—84no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—44no—
293C2×C8—84no—
294C2×C8C2×SD1614yes—
295C2×C8C2×Q1614yes—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4#13714yes—
317C22⋊C4—44no—
318C42—42no—
319C42—42no—
320C42—44no—
321C42—42no—
322C42—42no—
323C42#12012yes—
324C42#12112yes—
325C42#12112yes—
326C42#12312yes—
327C42#12312yes—
328C42#12412yes—
329C42C22×D411yescharacteristic
330C42C22×D411yescharacteristic
331C42C2×#4111yescharacteristic
332C42C2×#4111yescharacteristic
333C23—22no—
334C23—22no—
335C23#39412yes—
336C23#46112yes—
337C23#46212yes—
338C23C2×#12111yescharacteristic
339Q8—42no—
340Q8—42no—
341Q8—82no—
342Q8—84no—
343Q8—82no—
344Q8—88no—
345Q8—84no—
346Q8—82no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—88no—
365C2×C4—88no—
366C2×C4—82no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4#39412yes—
370C2×C4#39412yes—
371C2×C4#39412yes—
372C2×C4#44814yes—
373C2×C4#45114yes—
374C2×C4#46112yes—
375C2×C4#46212yes—
376C2×C4#48714yes—
377C2×C4#49114yes—
378C2×C4C2×#12011yescharacteristic
379C2×C4C2×#12111yescharacteristic
380C2×C4C2×#12111yescharacteristic
381C2×C4C2×#12311yescharacteristic
382C2×C4C2×#12311yescharacteristic
383C2×C4C2×#12411yescharacteristic
384C2×C4C22×SD1611yescharacteristic
385C2×C4C22×Q1611yescharacteristic
386C2×C4C2×#13711yescharacteristic
387C2×C4C2×#13711yescharacteristic
388C2×C4C2×#13712yes—
389C2×C4#57512yes—
390C8—24no—
391C8—24no—
392C8—82no—
393C8—82no—
394C8—82no—
395C22—24no—
396C22—22no—
397C22—24no—
398C22—22no—
399C22#204314yes—
400C22C2×#39411yescharacteristic
401C22C2×#46111yescharacteristic
402C22C2×#46211yescharacteristic
403C22#305612yes—
404C22#306812yes—
405C4—21no—
406C4—21no—
407C4—22no—
408C4—22no—
409C4—44no—
410C4—82no—
411C4—84no—
412C4—82no—
413C4—82no—
414C4#204314yes—
415C4C2×#44811yescharacteristic
416C4C2×#45111yescharacteristic
417C4#303712yes—
418C4#303911yescharacteristic
419C4#304011yescharacteristic
420C4C2×#48711yescharacteristic
421C4C2×#49111yescharacteristic
422C2—24no—
423C2#1916512yes—
424C2#1917312yes—
425C2C2×#204311yescharacteristic
426C2#2352111yescharacteristic
427C2#2354711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10544C2semidirect
8. #10543C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#213852
C2×#19872
#211462
C2×#15502
C2×#15482
#105442
#105432
#78732