Finite Group Databasebeta

← #185962

Subgroups — #185962

Order 512.

2526 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2#21385C212yesmaximal, complemented
3C2×#1518C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic
5#13941C211yesmaximal, characteristic
6#8405C218yesmaximal, complemented
7#3501—28no—
8#3501—24no—
9#3501C418yescomplemented
10C2×#554—22no—
11C2×#554—24no—
12C2×D4×Q8—22no—
13C2×#544—21no—
14C2×#543C2214yes—
15C2×#533C2212yes—
16C22×C22⋊Q8—22no—
17C2×#518—21no—
18C22×C4×D4C2211yescharacteristic
19C2×#431C2212yes—
20C2×#390—21no—
21C2×#382C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1987—22no—
24#1981—22no—
25#1872C2218yes—
26C4×#103—24no—
27#1518—28no—
28C2×#328C2218yes—
29#1460—24no—
30C2×#323C2214yes—
31#1439C2214yes—
32C4⋊#95—21no—
33C22⋊#95—22no—
34C23×Q8—22no—
35C23×D4D411yescharacteristic
36C24×C4D412yes—
37#554—232no—
38#554—216no—
39#554—216no—
40#554—232no—
41D4×Q8—216no—
42D4×Q8—216no—
43#544—28no—
44#544—28no—
45#543—216no—
46#543—28no—
47#543C2×C4116yes—
48#533—24no—
49#533—24no—
50#533C2×C418yes—
51#533D418yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8D412yes—
54C2×C4⋊Q8D412yes—
55C2×#125—24no—
56C2×#125—22no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—216no—
60C2×C22⋊Q8—28no—
61C2×C22⋊Q8—216no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8C2314yes—
66#518—28no—
67#518—28no—
68C2×C4×Q8—22no—
69C2×C4×Q8D414yes—
70C2×C4×D4—24no—
71C2×C4×D4—24no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—24no—
75C2×C4×D4C2×C414yes—
76C2×C4×D4D414yes—
77C2×#117—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4C2312yes—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#431—24no—
89#431—24no—
90C2×C4⋊C8—22no—
91C2×#103—28no—
92C2×C22⋊C8C2314yes—
93C2×C4×C8—22no—
94#390—44no—
95#382—24no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—24no—
101C2×#95C2312yes—
102#340—28no—
103#328—216no—
104#323—28no—
105C25C2×D412yes—
106C22×Q8—24no—
107C22×Q8—216no—
108C22×Q8—22no—
109C22×Q8—88no—
110C22×Q8C2×D414yes—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4C22⋊C414yes—
116C22×D4SD1614yes—
117C22×D4Q1614yes—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—84no—
126C23×C4C22⋊C418yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—24no—
132C4⋊Q8—232no—
133C4⋊Q8—24no—
134C4⋊Q8—24no—
135C4⋊Q8C22⋊C418yescomplemented
136C4⋊Q8C22⋊C418yescomplemented
137#125—232no—
138#125—28no—
139#125—28no—
140C22⋊Q8—216no—
141C22⋊Q8—232no—
142C22⋊Q8—232no—
143C22⋊Q8—232no—
144C22⋊Q8—232no—
145C22⋊Q8—232no—
146C22⋊Q8—28no—
147C22⋊Q8—216no—
148C22⋊Q8—28no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8—232no—
152C22⋊Q8C22×C418yes—
153C22⋊Q8C2×D418yes—
154C4×Q8—28no—
155C4×Q8—28no—
156C4×Q8—28no—
157C4×Q8C22⋊C4116yescomplemented
158C4×D4—232no—
159C4×D4—232no—
160C4×D4—28no—
161C4×D4—28no—
162C4×D4—28no—
163C4×D4—28no—
164#117—28no—
165#117—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—216no—
169C2×C4⋊C4—232no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C22⋊C414yes—
183C2×C4⋊C4C2×D414yes—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—216no—
186C2×C22⋊C4—216no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—48no—
192C2×C22⋊C4—84no—
193C2×C22⋊C4C22×C414yes—
194C2×C22⋊C4C2×D414yes—
195C2×C42—24no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42C22×C412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D414yes—
203C2×C42C2×D412yes—
204C4⋊C8—44no—
205#103—416no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C4×C8—44no—
209#95—24no—
210#95—41no—
211#95—41no—
212#95—42no—
213#95—84no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—44no—
219C24—44no—
220C24C2×C22⋊C414yes—
221C24C2×SD1614yes—
222C24C2×Q1614yes—
223C24C22×D411yescharacteristic
224C2×Q8—216no—
225C2×Q8—24no—
226C2×Q8—28no—
227C2×Q8—48no—
228C2×Q8—48no—
229C2×Q8—88no—
230C2×Q8—832no—
231C2×Q8C2×C22⋊C418yes—
232C2×D4—28no—
233C2×D4—28no—
234C2×D4—28no—
235C2×D4—28no—
236C2×D4—28no—
237C2×D4—28no—
238C2×D4#103116yescomplemented
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—216no—
243C22×C4—216no—
244C22×C4—22no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—48no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—84no—
264C22×C4—88no—
265C22×C4—88no—
266C22×C4—88no—
267C22×C4—88no—
268C22×C4C2×C22⋊C414yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12012yes—
274C22×C4#12012yes—
275C22×C4#12012yes—
276C22×C4C2×SD1612yes—
277C22×C4C2×Q1612yes—
278C22×C4#13712yes—
279C22×C4#13712yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D412yes—
282C22×C4C22×D412yes—
283C22×C4C22×D411yescharacteristic
284C2×C8—48no—
285C2×C8—44no—
286C4⋊C4—216no—
287C4⋊C4—232no—
288C4⋊C4—28no—
289C4⋊C4—28no—
290C4⋊C4—24no—
291C4⋊C4—28no—
292C4⋊C4—216no—
293C4⋊C4—28no—
294C4⋊C4—28no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—44no—
298C4⋊C4—84no—
299C4⋊C4C2×C22⋊C418yes—
300C22⋊C4—232no—
301C22⋊C4—28no—
302C22⋊C4—28no—
303C22⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—816no—
306C42—216no—
307C42—84no—
308C42C2×C22⋊C414yes—
309C42C2×C22⋊C414yes—
310C42#12014yes—
311C42#12014yes—
312C23—22no—
313C23—22no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23—24no—
320C23—48no—
321C23—44no—
322C23—44no—
323C23—48no—
324C23—44no—
325C23—44no—
326C23—82no—
327C23—84no—
328C23—84no—
329C23—82no—
330C23C2×#10318yes—
331C23C22×C22⋊C411yescharacteristic
332C23C2×#12011yescharacteristic
333C23C2×#12011yescharacteristic
334C23C22×SD1611yescharacteristic
335C23C22×Q1611yescharacteristic
336C23C2×#13711yescharacteristic
337C23C2×#13711yescharacteristic
338Q8—48no—
339Q8—416no—
340Q8—84no—
341Q8—816no—
342D4—232no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—416no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—88no—
363C2×C4—82no—
364C2×C4—816no—
365C2×C4—816no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4#37912yes—
376C2×C4#37912yes—
377C2×C4#37914yes—
378C2×C4C2×#10314yes—
379C2×C4#41914yes—
380C2×C4#44814yes—
381C2×C4#44814yes—
382C2×C4#45114yes—
383C2×C4#45114yes—
384C2×C4C22×C22⋊C412yes—
385C2×C4C2×#12012yes—
386C8—84no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—28no—
396C22—48no—
397C22—48no—
398C22—48no—
399C22—44no—
400C22—44no—
401C22—84no—
402C22—84no—
403C22—84no—
404C22C2×#37911yescharacteristic
405C22C22×#10311yescharacteristic
406C22C2×#41911yescharacteristic
407C22C2×#44811yescharacteristic
408C22C2×#44811yescharacteristic
409C22C2×#45111yescharacteristic
410C22C2×#45111yescharacteristic
411C22#301912yes—
412C22#302512yes—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—48no—
418C4—44no—
419C4—48no—
420C4—84no—
421C4—162no—
422C4#178718yes—
423C2—21no—
424C2—21no—
425C2—21no—
426C2—21no—
427C2—48no—
428C2#1821212yes—
429C2#1821312yes—
430C2C2×#178711yescharacteristic
431C2#2060811yescharacteristic
432C2#2061611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21385C2semidirect
3. C2×#1518C2semidirect
6. #8405C2semidirect
9. #3501C4semidirect
135. C4⋊Q8C22⋊C4semidirect
136. C4⋊Q8C22⋊C4semidirect
157. C4×Q8C22⋊C4semidirect
238. C2×D4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
#213852
C2×#15182
C2×#14602
#139412
#84052