Finite Group Databasebeta

← #6434618

Subgroups — #6434618

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C212yesmaximal, complemented
2#30379C214yesmaximal, complemented
3#30336C214yesmaximal, complemented
4#30296C211yesmaximal, characteristic, complemented
5#30247C211yesmaximal, characteristic
6#30197C212yesmaximal, complemented
7#29782C214yesmaximal, complemented
8#23780C214yesmaximal, complemented
9#23682C214yesmaximal, complemented
10#21538C214yesmaximal, complemented
11C4×#423C211yesmaximal, characteristic, complemented
12C2×#529—22no—
13C2×#469C2214yes—
14C2×#442C2214yes—
15C2×#423C2212yes—
16C2×C4×M16C2212yes—
17#2861—22no—
18#2860—22no—
19#2860C2214yes—
20#2859C2212yes—
21#2858C2212yes—
22#2857C2212yes—
23#2856C2212yes—
24#2854—22no—
25#2854C2214yes—
26#2850—22no—
27C4×C4⋊Q8C2212yes—
28C4×#127C2212yes—
29C4×#124—22no—
30C4×#121—22no—
31C4×#121C2214yes—
32#2293—22no—
33#2293C2214yes—
34C2×C43C2211yescharacteristic
35#2100C2214yes—
36#2094C2214yes—
37#2088C2218yes—
38#2084C2214yes—
39#2071C2214yes—
40#2067C2218yes—
41#2065C2218yes—
42#2057C2214yes—
43#2054C2218yes—
44#2041C2214yes—
45#2023C2214yes—
46#2011C2214yes—
47#1970C2214yes—
48#1941C2214yes—
49#1934C2218yes—
50#1895C2218yes—
51#1809C2214yes—
52#1776C2214yes—
53#1767C2214yes—
54C4×C4⋊C8C2214yes—
55#1762C2218yes—
56#1728C2211yescharacteristic
57#1728C2211yescharacteristic
58C22×D8C2314yes—
59C22×M16C2312yes—
60C22×M16C2312yes—
61C2×C4⋊Q8C2312yes—
62C2×#127C2312yes—
63#529—416no—
64C2×#126—24no—
65C2×#125—22no—
66C2×#124—22no—
67C2×#124—22no—
68C2×#123—24no—
69C2×C22⋊Q8—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121C2314yes—
73C2×C4×Q8—22no—
74C2×C4×D4—24no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4C2314yes—
78C2×#117—24no—
79C22×C42D412yes—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#469—216no—
85#442—48no—
86C2×#107C2314yes—
87C2×#106C2314yes—
88C2×#106C2314yes—
89#423—28no—
90C2×C4⋊C8C2318yes—
91C2×#102C2318yes—
92C2×#102C2318yes—
93C2×#102C2318yes—
94C4×M16—44no—
95C2×#97C2314yes—
96C2×C4×C8C2314yes—
97#391—22no—
98#391—22no—
99#390—22no—
100#390—22no—
101#390—22no—
102#390—22no—
103#390—22no—
104#390—22no—
105#390C2314yes—
106#390C2314yes—
107#390C2314yes—
108#390C2314yes—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#389C2314yes—
117#388—24no—
118#388—24no—
119#388—24no—
120#388—24no—
121#388—24no—
122#388—24no—
123#388—24no—
124#388—24no—
125#386—24no—
126#386—24no—
127#384C2312yes—
128#384C2312yes—
129#382—22no—
130#380—22no—
131#380C2314yes—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2314yes—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—24no—
136C43D412yes—
137C43D412yes—
138C43C2312yes—
139C43C2312yes—
140#340C2318yes—
141#340C2318yes—
142#340C2318yes—
143#336C2314yes—
144#336C2314yes—
145C22×Q8—22no—
146C22×D4—24no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4C2414yes—
150C23×C4—24no—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2411yescharacteristic
156C2×D8—28no—
157C2×D8—44no—
158C2×D8—44no—
159C2×M16—24no—
160C2×M16—42no—
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—42no—
164C2×M16#4118yes—
165C22×C8C2414yes—
166C22×C8C2414yes—
167C4⋊Q8—44no—
168#127—44no—
169#126—416no—
170#125—48no—
171#124—48no—
172#124—48no—
173#123—416no—
174C22⋊Q8—48no—
175#121—216no—
176#121—48no—
177#121—48no—
178C4×Q8—48no—
179C4×D4—28no—
180C4×D4—28no—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—416no—
184#117—416no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2414yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C42—24no—
209C2×C42—28no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—44no—
214C2×C42—42no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2412yes—
228C2×C42C2411yescharacteristic
229C2×C42C2412yes—
230C2×C42C2411yescharacteristic
231#107—44no—
232#106—28no—
233#106—44no—
234C4⋊C8—216no—
235#102—216no—
236#102—48no—
237#102—48no—
238#97—28no—
239C4×C8—44no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246C24—24no—
247C24C22×D411yescharacteristic
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×D4—28no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—28no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—48no—
260C2×D4—48no—
261C2×D4—48no—
262C2×D4—44no—
263C2×D4—44no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4C22×D412yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D412yes—
286C22×C4C22×D412yes—
287C22×C4C22×D412yes—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4C2×#4114yes—
291C22×C4C2511yescharacteristic
292D8—416no—
293M16—48no—
294M16—48no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8C2×#4118yes—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—28no—
305C4⋊C4—48no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C22⋊C4—48no—
317C22⋊C4—48no—
318C22⋊C4—48no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—48no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C22⋊C4—48no—
325C42—22no—
326C42—22no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—21no—
335C42—21no—
336C42—24no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—48no—
341C42AGL(1,ℤ/8)18yes—
342C42H(5,𝔽2)12yes—
343C422−1+412yes—
344C23—21no—
345C23—21no—
346C23—22no—
347C23—22no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23#53212yes—
356C23C23×D411yescharacteristic
357Q8—88no—
358D4—48no—
359D4—48no—
360D4—88no—
361D4—88no—
362D4—816no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—816no—
394C2×C4#53212yes—
395C2×C4#53714yes—
396C2×C4C2×AGL(1,ℤ/8)14yes—
397C2×C4C2×AGL(1,ℤ/8)14yes—
398C2×C4C2×AGL(1,ℤ/8)14yes—
399C2×C4#57514yes—
400C2×C4C22×#4112yes—
401C2×C4C2×H(5,𝔽2)11yescharacteristic
402C2×C4C2×2−1+411yescharacteristic
403C2×C4#58512yes—
404C8—28no—
405C8—44no—
406C22—24no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—84no—
412C22—84no—
413C22—84no—
414C22—84no—
415C22—84no—
416C22—84no—
417C22—84no—
418C22C2×#53211yescharacteristic
419C22C2×#53711yescharacteristic
420C22#344911yescharacteristic
421C22C22×AGL(1,ℤ/8)12yes—
422C22C2×#57512yes—
423C4—24no—
424C4—22no—
425C4—24no—
426C4—24no—
427C4—24no—
428C4—24no—
429C4—42no—
430C4—84no—
431C4—84no—
432C4—84no—
433C4#345912yes—
434C4#346112yes—
435C2—42no—
436C2—164no—
437C2#3459411yescharacteristic
438C2#6070012yes—
439C2#6091612yes—
440C2#6096411yescharacteristic
441C2#6096811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#30247C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ 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. #30379C2semidirect
3. #30336C2semidirect
4. #30296C2semidirect
6. #30197C2semidirect
7. #29782C2semidirect
8. #23780C2semidirect
9. #23682C2semidirect
10. #21538C2semidirect
11. C4×#423C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#303792
#303362
#302962
#302472
#301972
#297822
#237802
#236822
#215382
C4×#4232