Finite Group Databasebeta

← #6437317

Subgroups — #6437317

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C212yesmaximal, complemented
2#30384C214yesmaximal, complemented
3#30365C214yesmaximal, complemented
4#30342C214yesmaximal, complemented
5#30324C214yesmaximal, complemented
6#23837C211yesmaximal, characteristic, complemented
7#23815C212yesmaximal, complemented
8#23798C211yesmaximal, characteristic, complemented
9#23697C214yesmaximal, complemented
10#21277C214yesmaximal, complemented
11#20463C211yesmaximal, characteristic, complemented
12C2×#529—22no—
13C2×C4×D8C2214yes—
14C2×#431C2212yes—
15#2861—22no—
16#2860—22no—
17#2860C2214yes—
18#2859C2212yes—
19#2858C2212yes—
20#2857C2212yes—
21#2856C2212yes—
22#2854—22no—
23#2854C2214yes—
24#2850—22no—
25C4×C4⋊Q8C2212yes—
26C4×#127C2212yes—
27C4×#124—22no—
28C4×#121—22no—
29C4×#121C2214yes—
30#2293—22no—
31#2293C2214yes—
32C2×C43C2211yescharacteristic
33#2099C2214yes—
34#2093C2214yes—
35#2087C2214yes—
36#2084C2214yes—
37#2076C2214yes—
38#2071C2214yes—
39#2065C2218yes—
40#2058C2214yes—
41#2054C2214yes—
42#2041C2214yes—
43#2034C2218yes—
44#2023C2214yes—
45#2011C2214yes—
46#1999C2214yes—
47#1972C2218yes—
48#1943C2218yes—
49#1936C2218yes—
50#1936C2218yes—
51C4×#106C2214yes—
52#1767C2214yes—
53C4×#102C2218yes—
54#1749C2214yes—
55#1744C2211yescharacteristic
56#1744C2211yescharacteristic
57C4×#97C2212yes—
58C22×D8C2314yes—
59C2×C4⋊Q8C2312yes—
60C2×#127C2312yes—
61#529—416no—
62C2×#126—24no—
63C2×#125—22no—
64C2×#124—22no—
65C2×#124—22no—
66C2×#123—24no—
67C2×C22⋊Q8—22no—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121C2314yes—
71C2×C4×Q8—22no—
72C2×C4×D4—24no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4C2314yes—
76C2×#117—24no—
77C22×C42D412yes—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2312yes—
82C4×D8—216no—
83#431—28no—
84C2×#107C2314yes—
85C2×#106C2318yes—
86C2×C4⋊C8C2314yes—
87C2×#102C2318yes—
88C2×#102C2318yes—
89C2×#102C2318yes—
90C2×C22⋊C8C2314yes—
91C2×#97C2318yes—
92C2×C4×C8C2312yes—
93C2×C4×C8C2312yes—
94#391—22no—
95#391—22no—
96#390—22no—
97#390—22no—
98#390—22no—
99#390—22no—
100#390—22no—
101#390—22no—
102#390C2314yes—
103#390C2314yes—
104#390C2314yes—
105#390C2314yes—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#388—24no—
115#388—24no—
116#388—24no—
117#388—24no—
118#388—24no—
119#388—24no—
120#388—24no—
121#388—24no—
122#386—24no—
123#386—24no—
124#384C2312yes—
125#384C2312yes—
126#382—22no—
127#380—22no—
128#380C2314yes—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2314yes—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—24no—
133C43D411yescharacteristic
134C43D411yescharacteristic
135C43D412yes—
136C43C2311yescharacteristic
137C43C2312yes—
138C43C2311yescharacteristic
139#340C2314yes—
140#340C2318yes—
141#340C2318yes—
142#340C2314yes—
143#336C2314yes—
144#336C2314yes—
145C22×Q8—22no—
146C22×D4—24no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4C2414yes—
150C23×C4—24no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2411yescharacteristic
156C2×D8—28no—
157C2×D8—48no—
158C22×C8C2414yes—
159C22×C8C2414yes—
160C4⋊Q8—44no—
161#127—44no—
162#126—416no—
163#125—48no—
164#124—48no—
165#124—48no—
166#123—416no—
167C22⋊Q8—48no—
168#121—48no—
169#121—48no—
170#121—48no—
171C4×Q8—48no—
172C4×D4—216no—
173C4×D4—416no—
174C4×D4—48no—
175C4×D4—48no—
176#117—416no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C2414yes—
187C2×C4⋊C4C2414yes—
188C2×C4⋊C4C2414yes—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C42—28no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—42no—
204C2×C42—44no—
205C2×C42—42no—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2412yes—
221C2×C42C2412yes—
222C2×C42C2411yescharacteristic
223C2×C42C2411yescharacteristic
224#107—28no—
225#106—48no—
226C4⋊C8—28no—
227#102—216no—
228#102—48no—
229#102—48no—
230C22⋊C8—28no—
231#97—48no—
232C4×C8—24no—
233C4×C8#4118yes—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240C24—24no—
241C24C22×D411yescharacteristic
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×D4—28no—
246C2×D4—24no—
247C2×D4—28no—
248C2×D4—24no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—48no—
254C2×D4—48no—
255C2×D4—48no—
256C2×D4—44no—
257C2×D4—44no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D412yes—
279C22×C4C22×D412yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D412yes—
283C22×C4C22×D412yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C2511yescharacteristic
287D8—416no—
288C2×C8—28no—
289C2×C8—24no—
290C2×C8—28no—
291C2×C8C2×#4118yes—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—28no—
295C4⋊C4—24no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—48no—
305C22⋊C4—48no—
306C22⋊C4—48no—
307C22⋊C4—48no—
308C22⋊C4—48no—
309C22⋊C4—48no—
310C22⋊C4—48no—
311C22⋊C4—48no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C42—22no—
317C42—22no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—24no—
322C42—22no—
323C42—24no—
324C42—24no—
325C42—42no—
326C42—42no—
327C42—41no—
328C42—41no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42—48no—
333C42AGL(1,ℤ/8)18yes—
334C42C2×#4114yes—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—21no—
339C23—44no—
340C23—44no—
341C23—44no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23#57512yes—
347C23C23×D411yescharacteristic
348Q8—88no—
349D4—48no—
350D4—48no—
351D4—88no—
352D4—88no—
353D4—816no—
354C2×C4—28no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—816no—
387C2×C4#53212yes—
388C2×C4#53212yes—
389C2×C4C2×AGL(1,ℤ/8)14yes—
390C2×C4C2×AGL(1,ℤ/8)14yes—
391C2×C4C2×AGL(1,ℤ/8)14yes—
392C2×C4#57512yes—
393C2×C4C22×#4112yes—
394C2×C4#58514yes—
395C8—28no—
396C8—44no—
397C22—22no—
398C22—24no—
399C22—41no—
400C22—42no—
401C22—41no—
402C22—84no—
403C22—84no—
404C22—84no—
405C22—84no—
406C22—84no—
407C22—84no—
408C22—84no—
409C22C2×#53211yescharacteristic
410C22#344912yes—
411C22C22×AGL(1,ℤ/8)12yes—
412C22C2×#57511yescharacteristic
413C22C2×#57511yescharacteristic
414C4—22no—
415C4—22no—
416C4—24no—
417C4—24no—
418C4—41no—
419C4—42no—
420C4—42no—
421C4—41no—
422C4—84no—
423C4—84no—
424C4—84no—
425C4#318414yes—
426C4#345514yes—
427C2—42no—
428C2—164no—
429C2#3464311yescharacteristic
430C2#6068811yescharacteristic
431C2C2×#318411yescharacteristic
432C2#6094912yes—
433C2#6095112yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36631C2semidirect
2. #30384C2semidirect
3. #30365C2semidirect
4. #30342C2semidirect
5. #30324C2semidirect
6. #23837C2semidirect
7. #23815C2semidirect
8. #23798C2semidirect
9. #23697C2semidirect
10. #21277C2semidirect
11. #20463C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#303842
#303652
#303422
#303242
#238372
#238152
#237982
#236972
#212772
#204632