Finite Group Databasebeta

← #481627

Subgroups — #481627

Order 512.

1049 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#35700C211yesmaximal, characteristic
2#20399C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#11782C214yesmaximal, complemented
5#11778C214yesmaximal, complemented
6#11700C212yesmaximal, complemented
7#11697C212yesmaximal
8C2×D4×Q8—21no—
9C2×D4×Q8C411yescharacteristic
10C2×#519C2211yescharacteristic
11#2983—21no—
12C22×#101—21no—
13#2873—21no—
14#2621C412yes—
15#2621C2211yescharacteristic
16#2546C411yescharacteristic
17#2546C2212yes—
18#2536C411yescharacteristic
19#2536C2212yes—
20#2429—22no—
21#2429—21no—
22#2415—22no—
23#2415—21no—
24#2392C412yes—
25#2392C2211yescharacteristic
26#2351C411yescharacteristic
27Q8×C22⋊C4—21no—
28C4×#101—22no—
29C4×#101C2214yes—
30#1742C2212yes—
31#1467C2214yes—
32#1464—22no—
33#1461—21no—
34#1458C2212yes—
35#1346C2218yes—
36#1346C2214yes—
37#1345C2214yes—
38C23×Q8—21no—
39C23×Q8D411yescharacteristic
40C22×M16—22no—
41D4×Q8—44no—
42D4×Q8—48no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—21no—
53#519—44no—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8C2312yes—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×#117C2×C412yes—
66C2×#117D412yes—
67C2×#117D411yescharacteristic
68C2×#117C2311yescharacteristic
69#446—44no—
70#432—24no—
71#431—22no—
72C2×#101—28no—
73C2×#101—24no—
74#405—44no—
75C4×M16—24no—
76#398—22no—
77#398—21no—
78#398C2×C412yes—
79#398C2×C412yes—
80#398C2×C412yes—
81#398C2312yes—
82#398C2311yescharacteristic
83#397—22no—
84#397—21no—
85#397C2×C411yescharacteristic
86#397C2×C412yes—
87#396—22no—
88#396—21no—
89#394—22no—
90#394—21no—
91#394C2×C412yes—
92#394C2311yescharacteristic
93#392C2×C411yescharacteristic
94#391—21no—
95#391—22no—
96#391—22no—
97#391—21no—
98#389—22no—
99#389—21no—
100#389—22no—
101#389—22no—
102#389—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#382C2×C412yes—
107#382C2×C411yescharacteristic
108#380—21no—
109#380—22no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C411yescharacteristic
112C4×C22⋊C4—22no—
113C4×C22⋊C4—21no—
114#343—22no—
115#343C2314yes—
116#324—24no—
117#324C2314yes—
118#323—22no—
119C22×Q8—24no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—24no—
123C22×Q8—42no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8—44no—
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22⋊C411yescharacteristic
129C22×Q8C2×D412yes—
130C22×D4—21no—
131C22×D4C22×C411yescharacteristic
132C23×C4—22no—
133C23×C4—21no—
134C23×C4C22⋊C412yes—
135C23×C4C2×D411yescharacteristic
136C2×M16—28no—
137C2×M16—24no—
138C2×M16—42no—
139C2×M16—44no—
140C22×C8—42no—
141C4⋊Q8—42no—
142C4⋊Q8—44no—
143C4⋊Q8—48no—
144C4⋊Q8—44no—
145C22⋊Q8—48no—
146C22⋊Q8—48no—
147C22⋊Q8—44no—
148C22⋊Q8—48no—
149C22⋊Q8—44no—
150C22⋊Q8—44no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—44no—
154C4×Q8—44no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×Q8—44no—
158C4×D4—42no—
159C4×D4—44no—
160C4×D4—48no—
161C4×D4—44no—
162#117—24no—
163#117—42no—
164#117—44no—
165#117—44no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2411yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22×C411yescharacteristic
198C2×C22⋊C4C2×D411yescharacteristic
199C2×C42—22no—
200C2×C42—21no—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C4⋊C8—42no—
209C4⋊C8—44no—
210#101—216no—
211C22⋊C8—24no—
212#97—44no—
213#97—44no—
214C4×C8—44no—
215C4×C8—42no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—21no—
224#95—22no—
225#95C22×C411yescharacteristic
226#95C22×C412yes—
227#95C22×C411yescharacteristic
228C24—21no—
229C24C2×C22⋊C411yescharacteristic
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—22no—
237C2×Q8—42no—
238C2×Q8—48no—
239C2×Q8—48no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—816no—
247C2×Q8#10414yes—
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4AGL(1,ℤ/8)14yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—28no—
281M16—48no—
282M16—44no—
283C2×C8—44no—
284C2×C8—48no—
285C2×C8—44no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4#9914yes—
307C22⋊C4—24no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—42no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C42—24no—
320C42—22no—
321C42—24no—
322C42—24no—
323C42—22no—
324C42—42no—
325C42—42no—
326C42—44no—
327C42—42no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23#41812yes—
340C23C22×C22⋊C411yescharacteristic
341Q8—28no—
342Q8—48no—
343Q8—44no—
344Q8—88no—
345Q8—84no—
346D4—42no—
347D4—48no—
348D4—42no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—48no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—88no—
386C2×C4—84no—
387C2×C4C2×#9914yes—
388C2×C4C2×#9912yes—
389C2×C4#41812yes—
390C2×C4C2×#10414yes—
391C2×C4C2×#10412yes—
392C2×C4C2×AGL(1,ℤ/8)12yes—
393C8—48no—
394C8—84no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—41no—
402C22—41no—
403C22—42no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22C22×#9911yescharacteristic
410C22C2×#41811yescharacteristic
411C22C22×#10411yescharacteristic
412C4—24no—
413C4—22no—
414C4—24no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—42no—
422C4—82no—
423C4—81no—
424C4—82no—
425C4—81no—
426C4#150714yes—
427C2—21no—
428C2—21no—
429C2—42no—
430C2—82no—
431C2C2×#150711yescharacteristic
432C2#3334911yescharacteristic
433C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#392C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20399C2semidirect
3. #17371C2semidirect
4. #11782C2semidirect
5. #11778C2semidirect
6. #11700C2semidirect

Maximal subgroups

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

subgroupindex
#357002
#203992
#173712
#117822
#117782
#117002
#116972