Finite Group Databasebeta

← #182705

Subgroups — #182705

Order 512.

1148 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#529C211yesmaximal, characteristic
2#20364C211yesmaximal, characteristic, complemented
3C2×#1745C211yesmaximal, characteristic, complemented
4C2×#1524C212yesmaximal
5#8225C214yesmaximal, complemented
6#8224C214yesmaximal, complemented
7#8029C212yesmaximal, complemented
8C2×#529C2211yescharacteristic
9C2×C4×M16C2212yes—
10C22×#97C2212yes—
11C4×#125C2212yes—
12C4×#126—22no—
13C4×#124C2212yes—
14C4×C22⋊Q8C412yes—
15C4×#123—22no—
16C4×#121C412yes—
17#2298—21no—
18#2295C411yescharacteristic
19#2293C411yescharacteristic
20#2290—21no—
21#2283C2211yescharacteristic
22C42×Q8C411yescharacteristic
23C42×D4—21no—
24C42×D4C411yescharacteristic
25C4×#117—21no—
26C2×C43C2211yescharacteristic
27#1745—22no—
28#1745C2214yes—
29#1745C2214yes—
30#1744—21no—
31#1744—21no—
32#1728—22no—
33C2×#336—21no—
34C2×#336—21no—
35#1524C4116yescomplemented
36C2×#330C2214yes—
37C2×#329C2214yes—
38#1295C2218yes—
39C22×M16—22no—
40C23×C8—22no—
41#529—24no—
42#529C2×C418yes—
43C2×#126—21no—
44C2×#125C2311yescharacteristic
45C2×#124C2311yescharacteristic
46C2×#123—21no—
47C2×C22⋊Q8C2×C411yescharacteristic
48C2×#121C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C412yes—
51C2×C4×D4—21no—
52C2×C4×D4—22no—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C412yes—
55C2×#117—21no—
56C2×#117—21no—
57C22×C42—21no—
58C22×C42C2×C412yes—
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D412yes—
62C22×C42C2311yescharacteristic
63C4×M16—24no—
64C4×M16C2×C418yescomplemented
65C2×#97—22no—
66C2×#97—24no—
67C2×#97C2×C418yescomplemented
68C2×#97C2314yes—
69C2×#97C2314yes—
70C2×C4×C8—22no—
71#391C2×C412yes—
72#390C2×C412yes—
73#389C2×C412yes—
74#389C2×C412yes—
75#388—22no—
76#388—22no—
77#388—22no—
78#388—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#386—22no—
84#385—22no—
85#383—21no—
86#382C2312yes—
87#381—21no—
88#380C2×C412yes—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C411yescharacteristic
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C22⋊C4—22no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107#376—21no—
108#376—21no—
109C43—21no—
110C43C2×C411yescharacteristic
111C43C2×C411yescharacteristic
112C43D412yes—
113C43D411yescharacteristic
114C43D411yescharacteristic
115C43C2312yes—
116#336—24no—
117#336—24no—
118#336—24no—
119#336—24no—
120#330C2×C4116yescomplemented
121#329C2×C4116yescomplemented
122C22×Q8C22×C411yescharacteristic
123C22×D4—21no—
124C22×D4C22×C411yescharacteristic
125C23×C4—21no—
126C23×C4—22no—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—28no—
133C2×M16—48no—
134C22×C8—28no—
135C22×C8—24no—
136C22×C8—24no—
137C22×C8—44no—
138C22×C8—44no—
139#126—28no—
140#125—22no—
141#125C22×C414yes—
142#124—22no—
143#124C22×C414yes—
144#123—28no—
145C22⋊Q8—22no—
146C22⋊Q8C4214yes—
147#121—22no—
148#121C4214yes—
149C4×Q8—22no—
150C4×Q8—44no—
151C4×Q8C4214yes—
152C4×D4—28no—
153C4×D4—22no—
154C4×D4—44no—
155C4×D4—48no—
156C4×D4C4214yes—
157#117—28no—
158#117—44no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—44no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—44no—
189C2×C42—44no—
190C2×C42C22⋊C412yes—
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22×C412yes—
199C2×C42C22×C412yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C414yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4114yes—
209C2×C42C2411yescharacteristic
210#97—28no—
211#97—24no—
212#97—24no—
213#97C22×C418yes—
214#97C22×C418yes—
215C4×C8—28no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—22no—
227C24—21no—
228C24C2×C22⋊C411yescharacteristic
229C2×Q8—21no—
230C2×Q8—42no—
231C2×Q8C2×C4212yes—
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—42no—
235C2×D4—44no—
236C2×D4C2×C4212yes—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4C2×C4212yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4#11712yes—
267C22×C4C4×D414yes—
268C22×C4C4×D414yes—
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272M16—416no—
273C2×C8—28no—
274C2×C8—28no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—416no—
278C2×C8—48no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4C2×C4212yes—
291C4⋊C4C2×C4212yes—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—44no—
317C42—44no—
318C42—44no—
319C42—42no—
320C42—42no—
321C42C2×C4212yes—
322C42#11712yes—
323C42C4×D414yes—
324C42C4×D414yes—
325C42C23×C412yes—
326C42C2×#4112yes—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—42no—
332C23—41no—
333C23—42no—
334C23—41no—
335C23C4×C22⋊C412yes—
336C23#42114yes—
337C23C22×C22⋊C411yescharacteristic
338Q8—42no—
339Q8—42no—
340D4—42no—
341D4—42no—
342D4—48no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—48no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—88no—
381C2×C4C4×C22⋊C412yes—
382C2×C4C4×C22⋊C412yes—
383C2×C4C4×C22⋊C412yes—
384C2×C4#41314yes—
385C2×C4#42112yes—
386C2×C4#42112yes—
387C2×C4C22×C4211yescharacteristic
388C2×C4C2×#11711yescharacteristic
389C2×C4C2×C4×D412yes—
390C2×C4C2×C4×D412yes—
391C8—48no—
392C8—48no—
393C22—22no—
394C22—22no—
395C22—24no—
396C22—24no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—82no—
401C22—82no—
402C22#175914yes—
403C22C2×C4×C22⋊C411yescharacteristic
404C22C2×#41312yes—
405C22C2×#42111yescharacteristic
406C22C2×#42112yes—
407C22C2×#42111yescharacteristic
408C4—21no—
409C4—22no—
410C4—21no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—24no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—82no—
422C4—82no—
423C4#175914yes—
424C2—24no—
425C2—82no—
426C2#1765912yes—
427C2#1766212yes—
428C2C4×#41311yescharacteristic
429C2C2×#175911yescharacteristic
430C2C2×#175911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20364C2semidirect
3. C2×#1745C2semidirect
5. #8225C2semidirect
6. #8224C2semidirect
7. #8029C2semidirect
35. #1524C4semidirect
64. C4×M16C2×C4semidirect
67. C2×#97C2×C4semidirect
120. #330C2×C4semidirect
121. #329C2×C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5292
#203642
C2×#17452
C2×#15242
#82252
#82242
#80292