Finite Group Databasebeta

← #6651609

Subgroups — #6651609

Order 512.

1616 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
1#35179C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3#33099C212yesmaximal
4#32694C214yesmaximal, complemented
5#32644C214yesmaximal, complemented
6#32591C218yesmaximal, complemented
7#29641C212yesmaximal
8#27610C212yesmaximal
9#24609C214yesmaximal
10#24063C212yesmaximal
11#21822C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8—21no—
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#483C2214yes—
17C2×#432C2212yes—
18#2855C2212yes—
19#2855C2212yes—
20#2627—21no—
21#2627—21no—
22#2582C2214yes—
23#2581—22no—
24#2568C2212yes—
25#2567C2212yes—
26#2565—22no—
27#2546—22no—
28#2546—22no—
29#2546C2214yes—
30#2546C2214yes—
31C2×#398—21no—
32C2×#398—21no—
33#2317—21no—
34#2310—21no—
35C2×#391—21no—
36C2×#389C2212yes—
37C2×#389C2212yes—
38#2285C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#2099C2214yes—
41#2088C2218yes—
42#2083C2214yes—
43#2068C2218yes—
44#2058C2218yes—
45#2058C2214yes—
46#2043C2214yes—
47#2024C2218yes—
48#1985C2214yes—
49#1985C2214yes—
50#1943C2218yes—
51#1935C2218yes—
52#1928C2218yes—
53#1921C2218yes—
54#1918C2218yes—
55#1872C2214yes—
56#1865C2218yes—
57#1852C2214yes—
58#1823C2214yes—
59#1817C2212yes—
60#1817C2212yes—
61C23×Q8—21no—
62C23×Q8D411yescharacteristic
63#533—44no—
64C2×C4⋊Q8—24no—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—44no—
68C2×C4⋊Q8—48no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2314yes—
72C2×#125—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8C2312yes—
75C2×C4×Q8—21no—
76C2×C4×Q8—44no—
77C2×C4×Q8D412yes—
78C2×C4×Q8D412yes—
79C2×C4×Q8C2312yes—
80C2×C4×Q8C2312yes—
81C2×C4×Q8C2312yes—
82C2×#117—21no—
83C2×#117—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#483—48no—
97#432—44no—
98C2×#107C2318yes—
99C2×#106C2318yes—
100C2×C4⋊C8C2314yes—
101C2×#103C2318yes—
102C2×#103C2318yes—
103C2×C22⋊C8C2314yes—
104C2×#97C2314yes—
105#400—22no—
106#400—22no—
107#398—24no—
108#398—24no—
109#398—24no—
110#398—24no—
111#398—24no—
112#398—24no—
113#395—22no—
114#395—22no—
115#391—22no—
116#391—22no—
117#391—24no—
118#391—24no—
119#391—24no—
120#391C2314yes—
121#391C2314yes—
122#389—24no—
123#389—22no—
124#389—22no—
125#389—24no—
126#389C2314yes—
127#389C2314yes—
128#389C2314yes—
129#389C2318yes—
130#389C2314yes—
131#387—24no—
132#387—24no—
133#385—22no—
134#385—22no—
135#384—22no—
136#384C2312yes—
137#382—22no—
138#382—21no—
139#381—22no—
140#381—22no—
141#380—21no—
142#380C2312yes—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4C2312yes—
146C4×C4⋊C4C2314yes—
147C2×#95—21no—
148C2×#95—22no—
149C2×#95—22no—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2318yes—
156#336C2314yes—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—21no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—44no—
165C22×Q8C2×D412yes—
166C22×Q8C2×D412yes—
167C22×Q8C2412yes—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C22×C8C2418yes—
178C4⋊Q8—44no—
179C4⋊Q8—44no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—48no—
183C4⋊Q8—416no—
184C4⋊Q8—416no—
185#125—44no—
186C22⋊Q8—28no—
187C22⋊Q8—44no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×Q8—42no—
194C4×Q8—44no—
195C4×Q8—44no—
196#117—44no—
197#117—44no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2412yes—
229C2×C4⋊C4C2414yes—
230C2×C4⋊C4C2414yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—44no—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2411yescharacteristic
245C2×C42C2411yescharacteristic
246C2×C42C2412yes—
247C2×C42C2412yes—
248#107—216no—
249#106—48no—
250C4⋊C8—44no—
251#103—216no—
252#103—48no—
253C22⋊C8—28no—
254#97—44no—
255#95—24no—
256#95—28no—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—22no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267C24C22×D411yescharacteristic
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—28no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—416no—
277C2×Q8—42no—
278C2×Q8—48no—
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—82no—
282C2×Q8—816no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4AGL(1,ℤ/8)14yes—
309C22×C4#13714yes—
310C22×C4C22×D412yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D412yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D412yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D412yes—
319C22×C4H(5,𝔽2)14yes—
320C22×C4C2511yescharacteristic
321C2×C8—28no—
322C2×C8—28no—
323C2×C8—28no—
324C4⋊C4—28no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—28no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—48no—
336C4⋊C4—416no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—44no—
346C4⋊C4—82no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C42−1+418yes—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C42—24no—
353C42—22no—
354C42—22no—
355C42—24no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—48no—
363C42—44no—
364C42—44no—
365C23—22no—
366C23—22no—
367C23#53412yes—
368C23C2×AGL(1,ℤ/8)12yes—
369C23C2×#13712yes—
370C23C23×D411yescharacteristic
371Q8—48no—
372Q8—48no—
373Q8—48no—
374Q8—816no—
375Q8—84no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—28no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—28no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—44no—
399C2×C4—416no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—88no—
416C2×C4#53412yes—
417C2×C4C2×AGL(1,ℤ/8)12yes—
418C2×C4C2×AGL(1,ℤ/8)12yes—
419C2×C4C2×#13714yes—
420C2×C4C2×H(5,𝔽2)12yes—
421C2×C4C2×2−1+414yes—
422C8—48no—
423C22—22no—
424C22—24no—
425C22—24no—
426C22—22no—
427C22#319314yes—
428C22C2×#53411yescharacteristic
429C22#348612yes—
430C22C22×AGL(1,ℤ/8)11yescharacteristic
431C22C22×#13711yescharacteristic
432C22#358612yes—
433C4—22no—
434C4—22no—
435C4—24no—
436C4—22no—
437C4—42no—
438C4—42no—
439C4—41no—
440C4—42no—
441C4—41no—
442C4—48no—
443C4—84no—
444C4—82no—
445C4—84no—
446C4—82no—
447C4#354414yes—
448C2—24no—
449C2#3725811yescharacteristic
450C2C2×#319311yescharacteristic
451C2#6075411yescharacteristic
452C2#6182712yes—
453C2#6182912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32694C2semidirect
5. #32644C2semidirect
6. #32591C2semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×#25462
#330992
#326942
#326442
#325912
#296412
#276102
#246092
#240632
#218222