Finite Group Databasebeta

← #6499273

Subgroups — #6499273

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic
3#33069C212yesmaximal, complemented
4#33067C212yesmaximal, complemented
5#29988C214yesmaximal, complemented
6#29274C214yesmaximal, complemented
7#29257C214yesmaximal, complemented
8#29164C218yesmaximal
9#21964C212yesmaximal
10#21894C212yesmaximal
11#20752C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#423C2212yes—
14#2864—22no—
15#2859C2211yescharacteristic
16#2859C2211yescharacteristic
17#2859C2212yes—
18#2851—21no—
19#2851—21no—
20#2851C2214yes—
21C4×C4⋊Q8C2212yes—
22C4×C4⋊Q8C2212yes—
23C4×#125—21no—
24C4×#125—21no—
25C4×#125—22no—
26C4×#125C2214yes—
27C4×C22⋊Q8—22no—
28C4×C22⋊Q8C2214yes—
29#2296—22no—
30#2296—21no—
31#2295—21no—
32#2295C2212yes—
33#2285C2212yes—
34#2285C2211yescharacteristic
35C42×Q8—21no—
36C42×Q8C2212yes—
37C4×#117—21no—
38C2×C43C2211yescharacteristic
39#2101C2218yes—
40#2097C2218yes—
41#2059C2218yes—
42#2026C2218yes—
43#1988C2214yes—
44#1981C2214yes—
45#1948C2214yes—
46#1948C2214yes—
47#1947C2214yes—
48#1947C2214yes—
49#1921C2218yes—
50#1919C2218yes—
51#1894C2218yes—
52#1865C2218yes—
53#1863C2218yes—
54#1831C2212yes—
55#1831C2212yes—
56#1831C2214yes—
57#1809C2218yes—
58#1803C2214yes—
59#1803C2214yes—
60#1728C2214yes—
61C22×M16C2314yes—
62#533—24no—
63#533Q818yes—
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×#125—21no—
67C2×#125—21no—
68C2×#125—21no—
69C2×#125C2314yes—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8C2312yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8C2312yes—
75C2×C4×Q8C2314yes—
76C2×#117—21no—
77C2×#117—21no—
78C2×#117—22no—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D412yes—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84C2×#107C2318yes—
85C2×#106C2318yes—
86#423—28no—
87C2×C4⋊C8C2314yes—
88C2×C4⋊C8C2314yes—
89C2×#103C2318yes—
90C2×#103C2318yes—
91#391—22no—
92#391C2314yes—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389C2314yes—
103#389C2314yes—
104#389C2314yes—
105#387—24no—
106#387—24no—
107#387—24no—
108#387—24no—
109#387—24no—
110#387—24no—
111#385—24no—
112#385—24no—
113#385—24no—
114#384C2312yes—
115#384C2312yes—
116#384C2312yes—
117#382—21no—
118#382—21no—
119#382—21no—
120#382—22no—
121#382C2314yes—
122#380—22no—
123#380—21no—
124#380C2312yes—
125#380C2314yes—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2312yes—
140C4×C22⋊C4—24no—
141#376—22no—
142C43—22no—
143C43D412yes—
144C43D411yescharacteristic
145C43D411yescharacteristic
146C43C2312yes—
147C43C2311yescharacteristic
148C43C2311yescharacteristic
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2318yes—
154#336C2318yes—
155C22×Q8—21no—
156C22×Q8C2412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2411yescharacteristic
162C2×M16—28no—
163C2×M16—44no—
164C2×M16—44no—
165C22×C8C2418yes—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8C2×Q814yes—
169C4⋊Q8C2×Q814yes—
170#125—28no—
171#125—44no—
172#125—44no—
173#125—44no—
174#125—44no—
175C22⋊Q8—28no—
176C22⋊Q8—24no—
177C22⋊Q8C2×Q818yes—
178C4×Q8—28no—
179C4×Q8—24no—
180C4×Q8—44no—
181C4×Q8—44no—
182C4×Q8—48no—
183C4×Q8C2×Q818yes—
184#117—28no—
185#117—44no—
186#117—48no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—44no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×Q814yes—
236C2×C42#4114yes—
237C2×C42C2412yes—
238C2×C42C2412yes—
239C2×C42C2411yescharacteristic
240C2×C42C2411yescharacteristic
241#107—48no—
242#106—48no—
243C4⋊C8—28no—
244C4⋊C8—28no—
245#103—48no—
246#103—48no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—24no—
251#95—22no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259C24C22×D411yescharacteristic
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—44no—
265C2×Q8C22×Q814yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4C22⋊Q814yes—
283C22×C4C22⋊Q814yes—
284C22×C4C22×D412yes—
285C22×C4C22×D412yes—
286C22×C4C22×D412yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D412yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×Q812yes—
294C22×C4C2×#4112yes—
295C22×C4C2511yescharacteristic
296M16—88no—
297C2×C8—28no—
298C2×C8—28no—
299C2×C8—28no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4C22×Q814yes—
325C22⋊C4—28no—
326C22⋊C4—44no—
327C22⋊C4—48no—
328C42—28no—
329C42—24no—
330C42—22no—
331C42—22no—
332C42—24no—
333C42—24no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—42no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—48no—
345C42—48no—
346C42—44no—
347C42—48no—
348C42C22⋊Q814yes—
349C42C22⋊Q814yes—
350C42#13714yes—
351C42C22×Q812yes—
352C42C22×Q812yes—
353C42C2×#4112yes—
354C42C2×#4112yes—
355C23—22no—
356C23—21no—
357C23—21no—
358C23—22no—
359C23C2×C22⋊Q812yes—
360C23C23×D411yescharacteristic
361Q8—24no—
362Q8—24no—
363Q8—48no—
364Q8—44no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4C2×C22⋊Q812yes—
401C2×C4C2×C22⋊Q812yes—
402C2×C4C2×C22⋊Q812yes—
403C2×C4C2×C22⋊Q812yes—
404C2×C4C2×C22⋊Q812yes—
405C2×C4C2×#13714yes—
406C2×C4C2×#13712yes—
407C2×C4#57512yes—
408C2×C4#57514yes—
409C2×C4C23×Q811yescharacteristic
410C2×C4C22×#4111yescharacteristic
411C2×C4#58512yes—
412C2×C4#58512yes—
413C8—48no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—42no—
418C22—42no—
419C22C22×C22⋊Q811yescharacteristic
420C22#344912yes—
421C22C22×#13711yescharacteristic
422C22C2×#57512yes—
423C22C2×#57511yescharacteristic
424C4—22no—
425C4—24no—
426C4—22no—
427C4—22no—
428C4—44no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—44no—
433C4—84no—
434C4—84no—
435C4—84no—
436C4#307814yes—
437C4#308014yes—
438C4#350411yescharacteristic
439C4#350411yescharacteristic
440C4#351012yes—
441C2—42no—
442C2#3533411yescharacteristic
443C2C2×#307811yescharacteristic
444C2C2×#308011yescharacteristic
445C2#6095212yes—
446C2#6096712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #33069C2semidirect
4. #33067C2semidirect
5. #29988C2semidirect
6. #29274C2semidirect
7. #29257C2semidirect

Maximal subgroups

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

subgroupindex
#366402
C4×#5332
#330692
#330672
#299882
#292742
#292572
#291642
#219642
#218942
#207522