Finite Group Databasebeta

← #182699

Subgroups — #182699

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#529C211yesmaximal, characteristic
2#20364C211yesmaximal, characteristic, complemented
3C2×C4×#97C211yesmaximal, characteristic, complemented
4C2×#1524C212yesmaximal
5#8227C214yesmaximal, complemented
6#8226C214yesmaximal, complemented
7#8029C212yesmaximal, complemented
8C2×#529C2211yescharacteristic
9C2×C4×M16C2212yes—
10C22×#97—21no—
11C22×#97C2212yes—
12C22×C4×C8—21no—
13C4×#125C2212yes—
14C4×#126—22no—
15C4×#124C2212yes—
16C4×C22⋊Q8C412yes—
17C4×#123—22no—
18C4×#121C412yes—
19#2298—21no—
20#2295C411yescharacteristic
21#2293C411yescharacteristic
22#2290—21no—
23#2283C2211yescharacteristic
24C42×Q8C411yescharacteristic
25C42×D4—21no—
26C42×D4C411yescharacteristic
27C4×#117—21no—
28C2×C43C2211yescharacteristic
29#1745C2214yes—
30#1744—21no—
31#1744—21no—
32#1728—22no—
33C4×#97—22no—
34C4×#97C2214yes—
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—28no—
66C2×#97—24no—
67C2×#97—22no—
68C2×#97—24no—
69C2×#97C2×C418yescomplemented
70C2×#97C2314yes—
71C2×#97C2314yes—
72C2×C4×C8—28no—
73C2×C4×C8—24no—
74C2×C4×C8—22no—
75#391C2×C412yes—
76#390C2×C412yes—
77#389C2×C412yes—
78#389C2×C412yes—
79#388—22no—
80#388—22no—
81#388—22no—
82#388—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#386—22no—
88#385—22no—
89#383—21no—
90#382C2312yes—
91#381—21no—
92#380C2×C412yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C411yescharacteristic
102C4×C4⋊C4C2×C411yescharacteristic
103C4×C22⋊C4—22no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111#376—21no—
112#376—21no—
113C43—21no—
114C43C2×C411yescharacteristic
115C43C2×C411yescharacteristic
116C43D412yes—
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43C2312yes—
120#336—24no—
121#336—24no—
122#330C2×C4116yescomplemented
123#329C2×C4116yescomplemented
124C22×Q8C22×C411yescharacteristic
125C22×D4—21no—
126C22×D4C22×C411yescharacteristic
127C23×C4—21no—
128C23×C4—22no—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C2×M16—28no—
135C2×M16—48no—
136C22×C8—28no—
137C22×C8—28no—
138C22×C8—28no—
139C22×C8—24no—
140C22×C8—24no—
141#126—28no—
142#125—22no—
143#125C22×C414yes—
144#124—22no—
145#124C22×C414yes—
146#123—28no—
147C22⋊Q8—22no—
148C22⋊Q8C4214yes—
149#121—22no—
150#121C4214yes—
151C4×Q8—22no—
152C4×Q8—44no—
153C4×Q8C4214yes—
154C4×D4—28no—
155C4×D4—22no—
156C4×D4—44no—
157C4×D4—48no—
158C4×D4C4214yes—
159#117—28no—
160#117—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C42—21no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—44no—
190C2×C42—44no—
191C2×C42—44no—
192C2×C42C22⋊C412yes—
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C414yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4114yes—
211C2×C42C2411yescharacteristic
212#97—216no—
213#97—28no—
214#97—24no—
215#97—24no—
216#97C22×C418yes—
217#97C22×C418yes—
218C4×C8—216no—
219C4×C8—28no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95—22no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—22no—
231C24—21no—
232C24C2×C22⋊C411yescharacteristic
233C2×Q8—21no—
234C2×Q8—42no—
235C2×Q8C2×C4212yes—
236C2×D4—24no—
237C2×D4—21no—
238C2×D4—42no—
239C2×D4—44no—
240C2×D4C2×C4212yes—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4C2×C4212yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4#11712yes—
271C22×C4C4×D414yes—
272C22×C4C4×D414yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276M16—416no—
277C2×C8—28no—
278C2×C8—28no—
279C2×C8—232no—
280C2×C8—28no—
281C2×C8—28no—
282C2×C8—48no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4C2×C4212yes—
295C4⋊C4C2×C4212yes—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C42—24no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—24no—
310C42—22no—
311C42—22no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—44no—
319C42—44no—
320C42—44no—
321C42—42no—
322C42—42no—
323C42#10414yes—
324C42#10414yes—
325C42C2×C4212yes—
326C42#11712yes—
327C42C4×D414yes—
328C42C4×D414yes—
329C42C23×C412yes—
330C42C2×#4112yes—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—22no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23C4×C22⋊C412yes—
340C23#42114yes—
341C23C22×C22⋊C411yescharacteristic
342Q8—42no—
343Q8—42no—
344D4—42no—
345D4—42no—
346D4—48no—
347C2×C4—216no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—48no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—88no—
383C2×C4C4×C22⋊C412yes—
384C2×C4C4×C22⋊C412yes—
385C2×C4C4×C22⋊C412yes—
386C2×C4#41314yes—
387C2×C4C2×#10414yes—
388C2×C4C2×#10414yes—
389C2×C4C2×#10412yes—
390C2×C4C2×#10412yes—
391C2×C4C22×C4211yescharacteristic
392C2×C4C2×#11711yescharacteristic
393C2×C4C2×C4×D412yes—
394C2×C4C2×C4×D412yes—
395C8—216no—
396C8—48no—
397C22—22no—
398C22—22no—
399C22—24no—
400C22—24no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—82no—
405C22—82no—
406C22#175914yes—
407C22C2×C4×C22⋊C411yescharacteristic
408C22C2×#41312yes—
409C22C22×#10411yescharacteristic
410C22C22×#10411yescharacteristic
411C22C2×#42112yes—
412C4—21no—
413C4—22no—
414C4—21no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—24no—
419C4—24no—
420C4—24no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—82no—
426C4—82no—
427C4C4×#10414yes—
428C2—24no—
429C2—82no—
430C2#1764212yes—
431C2#1764612yes—
432C2C4×#41311yescharacteristic
433C2C2×C4×#10411yescharacteristic
434C2C2×#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×C4×#97C2semidirect
5. #8227C2semidirect
6. #8226C2semidirect
7. #8029C2semidirect
35. #1524C4semidirect
64. C4×M16C2×C4semidirect
69. C2×#97C2×C4semidirect
122. #330C2×C4semidirect
123. #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×C4×#972
C2×#15242
#82272
#82262
#80292