Finite Group Databasebeta

← #7542899

Subgroups — #7542899

Order 512.

1925 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#32258C214yesmaximal, complemented
4#29208C212yesmaximal, complemented
5#26296C214yesmaximal, complemented
6#25338C214yesmaximal, complemented
7#25010C214yesmaximal, complemented
8#24088C214yesmaximal, complemented
9#23173C211yesmaximal, characteristic, complemented
10D4×#107C214yesmaximal, complemented
11C4×#474C212yesmaximal, complemented
12C2×#559—21no—
13C2×#547C2212yes—
14C2×#532—21no—
15C2×#480C2214yes—
16C2×#474C2212yes—
17C2×C8×D4C2214yes—
18C22×#107C2214yes—
19#2868C2211yescharacteristic
20#2857—21no—
21#2724—22no—
22#2582—21no—
23#2547C2212yes—
24#2547C2214yes—
25#2537C2214yes—
26#2486C2214yes—
27#2447—24no—
28#2432—22no—
29#2422—24no—
30D4×C4⋊C4C2214yes—
31D4×C4⋊C4C2214yes—
32C4×C4⋊Q8C2212yes—
33C4×#127C2211yescharacteristic
34C42×D4C2211yescharacteristic
35#2075C2218yes—
36#2075C2218yes—
37#2057C2218yes—
38#2033C2218yes—
39#1990C2214yes—
40#1988C2212yes—
41#1954C2214yes—
42#1947C2214yes—
43#1947C2214yes—
44#1947C2212yes—
45#1918C2214yes—
46#1893C2214yes—
47#1869C2214yes—
48#1862C2214yes—
49#1852C2218yes—
50#1852C2218yes—
51#1845C2214yes—
52#1823C2214yes—
53#1808C2218yes—
54#1802C2212yes—
55C4×#107C2214yes—
56C4×C4⋊C8C2212yes—
57C4×#102C2214yes—
58C22×#41—22no—
59C23×C8C2314yes—
60#559—48no—
61#547—44no—
62#547—44no—
63#532—48no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2312yes—
68C2×#127D411yescharacteristic
69C2×#124—22no—
70C2×#124—24no—
71C2×#123—24no—
72C2×#123—24no—
73C2×C22⋊Q8—22no—
74C2×#121—24no—
75C2×#121C2314yes—
76C2×C4×Q8—22no—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4C2312yes—
80C2×C4×D4C2311yescharacteristic
81C2×C4×D4C2314yes—
82C2×C4×D4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C42—21no—
86#480—48no—
87#474—28no—
88C8×D4—216no—
89C2×#107—48no—
90C2×#107C2314yes—
91C2×#107C2314yes—
92C2×#107C2318yes—
93C2×#107C2318yes—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×#102C2314yes—
97C2×#102C2314yes—
98C2×C22⋊C8C2318yes—
99C2×C22⋊C8C2314yes—
100C2×C4×C8C2314yes—
101#398—24no—
102#397—24no—
103#397—24no—
104#397—28no—
105#394—28no—
106#391—24no—
107#391—22no—
108#391—22no—
109#390D414yes—
110#390D412yes—
111#389—24no—
112#389—24no—
113#389—22no—
114#389C2314yes—
115#385—28no—
116#385—28no—
117#384—22no—
118#384D411yescharacteristic
119#384C2312yes—
120#384C2314yes—
121#381—24no—
122#380—24no—
123#380C2318yes—
124#380C2318yes—
125#380C2314yes—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2314yes—
128C4×C4⋊C4C2312yes—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—22no—
131C4×C22⋊C4—21no—
132C43C2311yescharacteristic
133#340C2318yes—
134#340C2318yes—
135#340C2314yes—
136#340C2314yes—
137#340C2314yes—
138#340C2314yes—
139#336C2314yes—
140C2×#41—48no—
141C2×#41—48no—
142C2×#41—84no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×D4—24no—
146C22×D4C2×D414yes—
147C22×D4C2×D412yes—
148C22×D4C2411yescharacteristic
149C23×C4—21no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4C2414yes—
154C22×C8—216no—
155C22×C8—44no—
156C22×C8C2414yes—
157C22×C8C2414yes—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—44no—
163C4⋊Q8—82no—
164#127D818yes—
165#124—48no—
166#124—416no—
167#123—416no—
168#123—416no—
169C22⋊Q8—48no—
170#121—28no—
171#121—44no—
172#121—416no—
173C4×Q8—48no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—28no—
177C4×D4—24no—
178C4×D4—28no—
179C4×D4—24no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—416no—
183C4×D4—416no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2412yes—
202C2×C4⋊C4C2414yes—
203C2×C4⋊C4C2411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4C2×D418yes—
211C2×C22⋊C4C2×D414yes—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—21no—
215C2×C42—44no—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220C2×C42C2412yes—
221#107—28no—
222#107—28no—
223#107—48no—
224#107—48no—
225C4⋊C8—24no—
226C4⋊C8#4118yescomplemented
227#102—28no—
228#102—28no—
229C22⋊C8—216no—
230C22⋊C8—44no—
231C4×C8—28no—
232#95—22no—
233#95—21no—
234#95—24no—
235#95—24no—
236#95—28no—
237#95—28no—
238#95—24no—
239C24—21no—
240C24C22×D414yes—
241#41—832no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×D4—28no—
249C2×D4—28no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—22no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—48no—
256C2×D4—48no—
257C2×D4—84no—
258C2×D4C2×D8116yes—
259C2×D4C2×D818yes—
260C2×D4H(5,𝔽2)12yes—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—42no—
272C22×C4—48no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—84no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D414yes—
284C22×C4C22×D412yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D412yes—
287C22×C4C2511yescharacteristic
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—416no—
294C2×C8C2×#4118yes—
295C4⋊C4—28no—
296C4⋊C4—21no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—42no—
306C4⋊C4—48no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—82no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4H(5,𝔽2)12yes—
316C22⋊C4—24no—
317C22⋊C4—216no—
318C22⋊C4—44no—
319C22⋊C4—48no—
320C22⋊C4—416no—
321C22⋊C4—416no—
322C22⋊C4—48no—
323C22⋊C4—44no—
324C42—24no—
325C42—21no—
326C42—21no—
327C42—41no—
328C42—44no—
329C42—42no—
330C42—48no—
331C42C2×D814yes—
332C42C2×#4114yes—
333C422−1+414yes—
334C23—24no—
335C23—28no—
336C23—24no—
337C23—28no—
338C23—22no—
339C23—41no—
340C23—42no—
341C23—41no—
342C23—42no—
343C23C22×D818yes—
344C23C23×D411yescharacteristic
345Q8—88no—
346Q8—88no—
347D4—232no—
348D4—24no—
349D4—48no—
350D4—41no—
351D4—41no—
352D4—816no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—28no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—48no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—816no—
381C2×C4—88no—
382C2×C4—168no—
383C2×C4#54714yes—
384C2×C4#54714yes—
385C2×C4C22×D812yes—
386C2×C4C22×D814yes—
387C2×C4#57814yes—
388C2×C4C22×#4112yes—
389C2×C4C2×H(5,𝔽2)11yescharacteristic
390C2×C4C2×H(5,𝔽2)11yescharacteristic
391C2×C4C2×2−1+412yes—
392C8—28no—
393C8—44no—
394C22—216no—
395C22—28no—
396C22—48no—
397C22—44no—
398C22—44no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—82no—
403C22—82no—
404C22C2×#54712yes—
405C22#348411yescharacteristic
406C22C23×D811yescharacteristic
407C22C2×#57812yes—
408C22#358511yescharacteristic
409C4—24no—
410C4—22no—
411C4—42no—
412C4—44no—
413C4—41no—
414C4—82no—
415C4—84no—
416C4—84no—
417C4—84no—
418C4#333418yes—
419C4#348111yescharacteristic
420C4#352212yes—
421C4#355011yescharacteristic
422C2—48no—
423C2—82no—
424C2#5907011yescharacteristic
425C2C2×#333412yes—
426C2#6128212yes—
427C2#6155911yescharacteristic
428C2#6166511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59827C2semidirect
3. #32258C2semidirect
4. #29208C2semidirect
5. #26296C2semidirect
6. #25338C2semidirect
7. #25010C2semidirect
8. #24088C2semidirect
9. #23173C2semidirect
10. D4×#107C2semidirect
11. C4×#474C2semidirect
226. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#598272
#361272
#322582
#292082
#262962
#253382
#250102
#240882
#231732
D4×#1072
C4×#4742