Finite Group Databasebeta

← #97536

Subgroups — #97536

Order 512.

1173 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic
2#11207C212yesmaximal, complemented
3#11049C211yesmaximal, characteristic
4C2×#1271C212yesmaximal, complemented
5C2×#1270C211yesmaximal, characteristic
6C22×C22⋊Q8—22no—
7C22×#102—21no—
8#2683—22no—
9#2683—24no—
10#2683C414yescomplemented
11C2×#396—22no—
12C2×#395—21no—
13C2×#390C2211yescharacteristic
14C2×#389C2212yes—
15C2×#386—21no—
16C2×#384C2211yescharacteristic
17C2×#340—21no—
18C2×#340—22no—
19C2×#340—22no—
20C2×#336C2211yescharacteristic
21C2×#336C2212yes—
22#1271—24no—
23#1270—22no—
24C23×Q8—21no—
25C23×D4D411yescharacteristic
26C24×C4—21no—
27C23×C8—21no—
28C23×C8—22no—
29C2×C22⋊Q8—28no—
30C2×C22⋊Q8—28no—
31C2×C22⋊Q8—44no—
32C2×C22⋊Q8—44no—
33C2×C22⋊Q8—44no—
34C2×C22⋊Q8—44no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D412yes—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C22⋊C4—21no—
45C22×C42C2311yescharacteristic
46C2×#102—42no—
47C2×#102—42no—
48C2×#102—42no—
49#396—28no—
50#396—28no—
51#395—24no—
52#395—24no—
53#390—22no—
54#390C2×C412yes—
55#390D412yes—
56#389—22no—
57#389—22no—
58#389C2×C414yes—
59#389D414yes—
60#386—24no—
61#386—24no—
62#384—42no—
63C2×#95—21no—
64C2×#95—22no—
65#340—44no—
66#340—42no—
67#340—44no—
68#336—22no—
69#336—22no—
70#336—22no—
71C25—21no—
72C22×Q8—24no—
73C22×Q8—44no—
74C22×Q8—44no—
75C22×Q8—81no—
76C22×Q8—81no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4—41no—
84C22×D4C22⋊C412yes—
85C22×D4SD1612yes—
86C22×D4Q1612yes—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4—44no—
92C23×C4—44no—
93C23×C4—41no—
94C23×C4—81no—
95C23×C4—81no—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C22×C8—24no—
99C22×C8—22no—
100C22×C8—42no—
101C22×C8—41no—
102C22×C8—42no—
103C22×C8—41no—
104C22×C8—42no—
105C22×C8—41no—
106C22×C8—42no—
107C22×C8—41no—
108C22×C8—42no—
109C22×C8—41no—
110C22×C8—42no—
111C22×C8—41no—
112C22⋊Q8—416no—
113C22⋊Q8—416no—
114C22⋊Q8—416no—
115C22⋊Q8—416no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4C22⋊C414yes—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4D814yes—
150C2×C4⋊C4SD1614yes—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—81no—
162C2×C22⋊C4—81no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D412yes—
168#102—48no—
169#95—24no—
170#95—24no—
171#95—24no—
172C24—22no—
173C24—22no—
174C24—22no—
175C24—42no—
176C24—42no—
177C24—41no—
178C24—42no—
179C24—41no—
180C24—42no—
181C24—41no—
182C24—41no—
183C24#12011yescharacteristic
184C2×Q8—416no—
185C2×Q8—84no—
186C2×Q8—84no—
187C2×Q8—84no—
188C2×Q8—88no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—81no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C2×D4#10314yes—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—416no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—88no—
238C22×C4—82no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—82no—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4#12012yes—
245C22×C4#12011yescharacteristic
246C22×C4C2×D812yes—
247C22×C4C2×SD1612yes—
248C22×C4C2×SD1611yescharacteristic
249C22×C4C2×Q1611yescharacteristic
250C22×C4AGL(1,ℤ/8)12yes—
251C22×C4AGL(1,ℤ/8)12yes—
252C22×C4#13711yescharacteristic
253C22×C4#13711yescharacteristic
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—44no—
259C2×C8—42no—
260C2×C8—82no—
261C2×C8—81no—
262C2×C8—81no—
263C2×C8—82no—
264C2×C8—82no—
265C2×C8—81no—
266C2×C8—82no—
267C2×C8—81no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—416no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—81no—
283C4⋊C4—82no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4#10218yescomplemented
289C22⋊C4—416no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—84no—
295C42—41no—
296C42—42no—
297C42—41no—
298C23—24no—
299C23—42no—
300C23—44no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—44no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—44no—
310C23—42no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—81no—
316C23—81no—
317C23—82no—
318C23—81no—
319C23—81no—
320C23—82no—
321C23#37911yescharacteristic
322C23#44712yes—
323C23#44811yescharacteristic
324C23#45012yes—
325C23#45111yescharacteristic
326C23#45311yescharacteristic
327C23#45412yes—
328C23#45511yescharacteristic
329C23#45612yes—
330C23B(4,𝔽2)11yescharacteristic
331C23#45811yescharacteristic
332Q8—816no—
333D4—44no—
334D4—82no—
335D4—82no—
336D4—82no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—28no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—816no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—84no—
383C2×C4—161no—
384C2×C4—162no—
385C2×C4—161no—
386C2×C4C2×#10212yes—
387C2×C4C2×#10311yescharacteristic
388C2×C4#41812yes—
389C2×C4#41911yescharacteristic
390C2×C4#44712yes—
391C2×C4#44811yescharacteristic
392C2×C4#45012yes—
393C2×C4#45111yescharacteristic
394C8—84no—
395C8—82no—
396C22—44no—
397C22—48no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—84no—
402C22—84no—
403C22—82no—
404C22—82no—
405C22—84no—
406C22—82no—
407C22#159611yescharacteristic
408C22#159812yes—
409C22#160211yescharacteristic
410C22#160412yes—
411C22#160811yescharacteristic
412C22#161012yes—
413C22#161411yescharacteristic
414C22#161612yes—
415C22#161911yescharacteristic
416C22#162712yes—
417C22#162911yescharacteristic
418C22#163712yes—
419C22#163912yes—
420C22#164912yes—
421C22#165112yes—
422C22#165711yescharacteristic
423C22#166011yescharacteristic
424C22#166412yes—
425C22#178612yes—
426C22#178711yescharacteristic
427C22#179011yescharacteristic
428C22#179211yescharacteristic
429C22#179312yes—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—48no—
439C4—84no—
440C4—82no—
441C4—162no—
442C2—84no—
443C2#773811yescharacteristic
444C2#775411yescharacteristic
445C2#776412yes—
446C2#779112yes—
447C2#779711yescharacteristic
448C2#782611yescharacteristic
449C2#840412yes—
450C2#840511yescharacteristic
451C2#844311yescharacteristic
452C2#844712yes—
453C2#850711yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11207C2semidirect
4. C2×#1271C2semidirect
10. #2683C4semidirect
288. C4⋊C4#102semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
#112072
#110492
C2×#12712
C2×#12702