Finite Group Databasebeta

← #214873

Subgroups — #214873

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35711C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic
3#17371C211yesmaximal, characteristic
4#9488C212yesmaximal, complemented
5#9404C212yesmaximal, complemented
6#9193C212yesmaximal
7#9033C212yesmaximal
8#8766C212yesmaximal, complemented
9#8396C212yesmaximal, complemented
10C2×Q82C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2983C2212yes—
13#2905C2212yes—
14C22×#101—21no—
15#2628C412yes—
16#2628C2211yescharacteristic
17#2581C412yes—
18#2581C2211yescharacteristic
19#2568C412yes—
20#2568C2211yescharacteristic
21#2451—21no—
22#2451—22no—
23#2439—21no—
24#2439—22no—
25#2396C412yes—
26#2396C2211yescharacteristic
27#2351—21no—
28#2351C2211yescharacteristic
29Q8×C22⋊C4—21no—
30#1801—21no—
31#1801—21no—
32#1801C2212yes—
33#1801C2212yes—
34#1742—21no—
35#1549C2214yes—
36#1487C2214yes—
37#1467—22no—
38#1464—22no—
39#1461C2212yes—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C23×Q8D411yescharacteristic
44C22×M16—22no—
45Q82—44no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2311yescharacteristic
50#519—44no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×#117C2×C412yes—
62C2×#117D411yescharacteristic
63C2×#117D412yes—
64C2×#117C2311yescharacteristic
65#446—44no—
66#432—24no—
67#431C2314yes—
68#424—44no—
69#423—22no—
70#423C2314yes—
71C2×#101—22no—
72C2×#101—22no—
73C2×#101—44no—
74#400C2×C411yescharacteristic
75#400C2×C412yes—
76#400C2×C412yes—
77#400C2×C412yes—
78#400C2×C412yes—
79#399C2×C411yescharacteristic
80#399C2×C412yes—
81#398—22no—
82#398—21no—
83#396—22no—
84#396—21no—
85#395—22no—
86#395—21no—
87#395C2×C411yescharacteristic
88#395C2×C412yes—
89#393—22no—
90#393—21no—
91#393C2×C411yescharacteristic
92#391—22no—
93#391—22no—
94#391—21no—
95#391—21no—
96#391—22no—
97#391—22no—
98#391—22no—
99#389—22no—
100#389—21no—
101#389—21no—
102#389—22no—
103#389—22no—
104#389—21no—
105#387—22no—
106#387—22no—
107#387—22no—
108#382—22no—
109#382—21no—
110#382C2×C412yes—
111#382C2311yescharacteristic
112#380—22no—
113#380—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2311yescharacteristic
118C4×C22⋊C4—22no—
119C4×C22⋊C4—21no—
120#343—22no—
121#343—22no—
122#332—44no—
123Q8⋊C8—44no—
124#324—22no—
125#324—24no—
126#323—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—42no—
133C22×Q8C22⋊C412yes—
134C22×Q8C22×C411yescharacteristic
135C22×Q8C2×D411yescharacteristic
136C22×Q8C2×D411yescharacteristic
137C23×C4C22⋊C412yes—
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—42no—
141C2×M16—44no—
142C2×M16—44no—
143C22×C8—42no—
144C4⋊Q8—24no—
145C4⋊Q8—44no—
146C4⋊Q8—48no—
147C4⋊Q8—44no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—44no—
152C4×Q8—44no—
153C4×Q8—44no—
154C4×Q8—42no—
155C4×Q8—42no—
156C4×Q8—44no—
157C4×Q8—42no—
158C4×Q8—42no—
159#117—24no—
160#117—44no—
161#117—44no—
162#117—42no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22×C411yescharacteristic
190C2×C22⋊C4C2×D411yescharacteristic
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22×C412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2411yescharacteristic
202C4⋊C8—24no—
203C4⋊C8—24no—
204C4⋊C8—24no—
205C4⋊C8—44no—
206C4⋊C8—44no—
207#101—44no—
208#101—44no—
209C22⋊C8—24no—
210#97—44no—
211C4×C8—24no—
212#95—22no—
213#95—21no—
214#95—22no—
215#95—21no—
216#95—21no—
217#95—22no—
218#95—21no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95C22×C411yescharacteristic
228#95C22×C412yes—
229#95C22×C411yescharacteristic
230C24C2×C22⋊C411yescharacteristic
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8AGL(1,ℤ/8)12yes—
250C2×Q8#13712yes—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4#13714yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274M16—44no—
275M16—44no—
276M16—84no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—84no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—44no—
299C4⋊C4—81no—
300C4⋊C4—82no—
301C4⋊C4—81no—
302C4⋊C4—82no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—82no—
308C22⋊C4—81no—
309C42—24no—
310C42—24no—
311C42—22no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—44no—
316C42—42no—
317C42—41no—
318C42—41no—
319C42—44no—
320C42H(5,𝔽2)14yes—
321C23—21no—
322C23—21no—
323C23—41no—
324C23—41no—
325C23#41912yes—
326C23C22×C22⋊C411yescharacteristic
327Q8—24no—
328Q8—24no—
329Q8—24no—
330Q8—48no—
331Q8—42no—
332Q8—42no—
333Q8—42no—
334Q8—162no—
335Q8—164no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—48no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4#41714yes—
375C2×C4#41912yes—
376C2×C4#51812yes—
377C2×C4#53412yes—
378C2×C4#53412yes—
379C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
380C2×C4C2×#13712yes—
381C2×C4C2×#13711yescharacteristic
382C8—84no—
383C8—84no—
384C22—21no—
385C22—21no—
386C22—22no—
387C22—42no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22#230811yescharacteristic
392C22C2×#41911yescharacteristic
393C22C2×#41711yescharacteristic
394C4—22no—
395C4—24no—
396C4—22no—
397C4—22no—
398C4—42no—
399C4—41no—
400C4—42no—
401C4—42no—
402C4—41no—
403C4—42no—
404C4—81no—
405C4—81no—
406C4—82no—
407C4—82no—
408C4—81no—
409C4#319312yes—
410C4#319512yes—
411C2—21no—
412C2—21no—
413C2—42no—
414C2#2176511yescharacteristic
415C2#2178011yescharacteristic
416C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9488C2semidirect
5. #9404C2semidirect
8. #8766C2semidirect
9. #8396C2semidirect

Maximal subgroups

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

subgroupindex
#357112
#206392
#173712
#94882
#94042
#91932
#90332
#87662
#83962