Finite Group Databasebeta

← #238144

Subgroups — #238144

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5#10357C214yesmaximal, complemented
6#10350C214yesmaximal, complemented
7#8045C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C2×#533—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103—22no—
16#2552C2212yes—
17#2551C2212yes—
18#2550—22no—
19#2549—22no—
20#2440—22no—
21#2399—22no—
22C4×C4⋊Q8—22no—
23#2296—21no—
24#2285C2211yescharacteristic
25C4×#117C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990C2214yes—
28#1988C2214yes—
29#1987—22no—
30#1987—22no—
31#1981—22no—
32#1981C2212yes—
33#1981C2212yes—
34#1824—22no—
35#1736—21no—
36#1736—21no—
37#1296C2214yes—
38#1279C2212yes—
39#1279C2214yes—
40#1279C2212yes—
41C23×Q8—21no—
42C22×M16—21no—
43C23×C8—21no—
44#533—44no—
45#533—44no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8Q814yescomplemented
55C2×C4⋊Q8C2312yes—
56C2×#125—22no—
57C2×C22⋊Q8—24no—
58C2×C4×Q8—24no—
59C2×#117—21no—
60C2×#117D411yescharacteristic
61C2×#117D411yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D412yes—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#431—22no—
68#431—22no—
69#423—22no—
70#423—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—48no—
81C2×C22⋊C8—21no—
82C2×C4×C8C2312yes—
83#398—24no—
84#398—24no—
85#395—24no—
86#395—24no—
87#391—24no—
88#389—24no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#385—24no—
94#384—22no—
95#384C2312yes—
96#382—21no—
97#380—21no—
98#380D412yes—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4C2311yescharacteristic
107C4×C4⋊C4C2311yescharacteristic
108C4×C4⋊C4C2312yes—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111#376—21no—
112C2×#95—21no—
113C43—21no—
114#340—28no—
115#340—24no—
116#340—24no—
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8—41no—
123C22×Q8—42no—
124C23×C4—22no—
125C23×C4—41no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—44no—
136C22×C8—42no—
137C4⋊Q8—24no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—44no—
144C4⋊Q8—44no—
145C4⋊Q8—44no—
146C4⋊Q8—44no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8C2×Q814yes—
151#125—44no—
152#125—44no—
153C22⋊Q8—416no—
154C4×Q8—416no—
155#117—22no—
156#117—22no—
157#117—42no—
158#117—42no—
159#117SD1614yes—
160#117Q1614yes—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D414yes—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×Q812yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203C2×C42#4112yes—
204C2×C42C2411yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—42no—
210C4⋊C8—42no—
211C4⋊C8—42no—
212#103—48no—
213#103—48no—
214C22⋊C8—42no—
215C4×C8—22no—
216C4×C8—22no—
217#95—24no—
218#95—24no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—22no—
226C24#12011yescharacteristic
227C2×Q8—24no—
228C2×Q8—48no—
229C2×Q8—48no—
230C2×Q8—416no—
231C2×Q8—42no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—84no—
236C2×Q8—84no—
237C2×Q8—84no—
238C2×Q8—84no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—82no—
252C22×C4—84no—
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4C22⋊Q812yes—
257C22×C4C22⋊Q812yes—
258C22×C4C22⋊Q812yes—
259C22×C4#12412yes—
260C22×C4#12412yes—
261C22×C4#12412yes—
262C22×C4C2×SD1612yes—
263C22×C4C2×Q1612yes—
264C22×C4AGL(1,ℤ/8)12yes—
265C22×C4#13712yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269M16—84no—
270C2×C8—44no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—84no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—48no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—84no—
300C4⋊C4C2×SD1614yes—
301C4⋊C4C2×Q1614yes—
302C4⋊C4AGL(1,ℤ/8)14yes—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—44no—
306C42—24no—
307C42—24no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42—42no—
313C42—42no—
314C42—44no—
315C42—44no—
316C42—82no—
317C42C22⋊Q812yes—
318C42C22⋊Q812yes—
319C42C22⋊Q812yes—
320C42#12412yes—
321C42#12412yes—
322C42#12412yes—
323C42C22×Q811yescharacteristic
324C42C2×#4111yescharacteristic
325C42C2×#4111yescharacteristic
326C42C2×#4111yescharacteristic
327C23—22no—
328C23—22no—
329C23#39312yes—
330C23#45012yes—
331C23#45112yes—
332C23C2×#12011yescharacteristic
333Q8—48no—
334Q8—88no—
335Q8—816no—
336Q8—82no—
337Q8—82no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—88no—
363C2×C4—88no—
364C2×C4—84no—
365C2×C4#39312yes—
366C2×C4#39312yes—
367C2×C4#39312yes—
368C2×C4#45012yes—
369C2×C4#45112yes—
370C2×C4#47514yes—
371C2×C4#47714yes—
372C2×C4#48714yes—
373C2×C4#48914yes—
374C2×C4C2×C22⋊Q811yescharacteristic
375C2×C4C2×C22⋊Q811yescharacteristic
376C2×C4C2×C22⋊Q811yescharacteristic
377C2×C4C2×#12411yescharacteristic
378C2×C4C2×#12411yescharacteristic
379C2×C4C2×#12411yescharacteristic
380C2×C4C22×SD1611yescharacteristic
381C2×C4C22×Q1611yescharacteristic
382C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C2×AGL(1,ℤ/8)12yes—
384C2×C4C2×#13711yescharacteristic
385C2×C4#57512yes—
386C8—82no—
387C8—82no—
388C8—82no—
389C22—24no—
390C22—24no—
391C22—22no—
392C22—22no—
393C22#202414yes—
394C22C2×#39311yescharacteristic
395C22C2×#45011yescharacteristic
396C22C2×#45111yescharacteristic
397C22#300612yes—
398C22#302212yes—
399C4—22no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—44no—
404C4—44no—
405C4—44no—
406C4—82no—
407C4—84no—
408C4—84no—
409C4#202414yes—
410C4C2×#47511yescharacteristic
411C4C2×#47711yescharacteristic
412C4#307712yes—
413C4C2×#48711yescharacteristic
414C4C2×#48911yescharacteristic
415C4#313811yescharacteristic
416C4#314011yescharacteristic
417C2—24no—
418C2#1879912yes—
419C2#1880112yes—
420C2C2×#202411yescharacteristic
421C2#2345411yescharacteristic
422C2#2346411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10357C2semidirect
6. #10350C2semidirect
54. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#349382
#218342
#213132
C2×#19812
#103572
#103502
#80452
#79012
#78912