Finite Group Databasebeta

← #206051

Subgroups — #206051

Order 512.

1243 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3186C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#20528C211yesmaximal, characteristic, complemented
4#17577C212yesmaximal, complemented
5C2×#1471C212yesmaximal, complemented
6#9818C214yesmaximal, complemented
7#9815C214yesmaximal, complemented
8C22×#121C2212yes—
9#3186—24no—
10#3186—22no—
11#3186—21no—
12#3186—21no—
13C2×#480C414yescomplemented
14C2×#466C2214yes—
15C2×#447C414yescomplemented
16C2×#407C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2280C2211yescharacteristic
20#1855—24no—
21#1810C2214yescomplemented
22#1752—21no—
23#1752C2212yescomplemented
24C8×C22⋊C4—21no—
25C8×C22⋊C4C2212yescomplemented
26#1750—22no—
27#1519—24no—
28#1510—24no—
29#1507C2218yescomplemented
30#1471—24no—
31#1471C2214yescomplemented
32#1471C2214yescomplemented
33C2×#323—24no—
34C23×D4—22no—
35C23×D4C2×C412yes—
36C24×C4C2311yescharacteristic
37C22×D8C2×C414yes—
38C22×M16—22no—
39C23×C8—21no—
40C23×C8C2312yes—
41C2×#121—22no—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121—22no—
46C2×#121—22no—
47C2×#121—22no—
48C2×#121—22no—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121C2×C414yes—
52C2×#121C2314yes—
53C2×#117—21no—
54C22×C4⋊C4C2×C412yes—
55C22×C22⋊C4—22no—
56#480—28no—
57#480—24no—
58#480—24no—
59#466—24no—
60#466—24no—
61#466—24no—
62#466—24no—
63#447—28no—
64#447—24no—
65#447—24no—
66C2×#108—24no—
67C2×#107C2×C414yes—
68C2×#102C2×C418yes—
69#407—22no—
70#407—22no—
71#407—22no—
72#407C2314yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8C2×C414yes—
78C2×C22⋊C8D414yes—
79C2×C22⋊C8D414yes—
80C2×C22⋊C8D412yes—
81C2×C22⋊C8D411yescharacteristic
82C2×C22⋊C8D411yescharacteristic
83C2×C22⋊C8C2311yescharacteristic
84C2×C22⋊C8C2311yescharacteristic
85C2×#97—22no—
86C2×C4×C8—22no—
87#381D412yes—
88#380—21no—
89#380D412yes—
90#379—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4C2312yes—
93#341—24no—
94#336—22no—
95#336—22no—
96#327—216no—
97#323—28no—
98#323—28no—
99C25—22no—
100C25#4111yescharacteristic
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—42no—
109C22×D4—42no—
110C22×D4—42no—
111C22×D4—42no—
112C22×D4—84no—
113C22×D4—88no—
114C22×D4C22×C414yes—
115C23×C4—22no—
116C23×C4—82no—
117C23×C4C22×C412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4#4111yescharacteristic
125C23×C4#4112yes—
126C23×C4C2411yescharacteristic
127C2×D8—24no—
128C2×D8—24no—
129C2×D8—84no—
130C2×M16—24no—
131C2×M16—42no—
132C2×M16—84no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—24no—
136C22×C8—42no—
137C22×C8—82no—
138C22×C8—82no—
139C22×C8C22×C414yes—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8C2×D414yes—
143#121—24no—
144#121—24no—
145#121—24no—
146#121—24no—
147#121—24no—
148#121—24no—
149#121—24no—
150#121—24no—
151#121—42no—
152#121—42no—
153#121—42no—
154#121—42no—
155#121—42no—
156#121—42no—
157#121—42no—
158#121—42no—
159#117—82no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22×C414yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C42—22no—
177#108—84no—
178#107—24no—
179#107—24no—
180#102—28no—
181#102—28no—
182C22⋊C8—24no—
183C22⋊C8—21no—
184C22⋊C8—21no—
185C22⋊C8—24no—
186C22⋊C8—24no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189C22⋊C8C22⋊C418yes—
190C22⋊C8C22⋊C414yes—
191C22⋊C8C22⋊C414yes—
192C22⋊C8C2×D412yes—
193C22⋊C8C2×D412yes—
194#97—82no—
195C4×C8—82no—
196#95—22no—
197#95C2×D414yes—
198C24—24no—
199C24—24no—
200C24—42no—
201C24—48no—
202C24—42no—
203C24—42no—
204C24—84no—
205C24—81no—
206C24—84no—
207C24C4×D412yes—
208C24#12111yescharacteristic
209C24#12111yescharacteristic
210C24#12412yes—
211C24C2×#4111yescharacteristic
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—48no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—84no—
222C2×D4—84no—
223C2×D4—84no—
224C2×D4—84no—
225C2×D4—84no—
226C2×D4—88no—
227C2×D4—88no—
228C2×D4—88no—
229C2×D4—88no—
230C2×D4—88no—
231C2×D4—88no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—81no—
242C22×C4—82no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—82no—
246C22×C4—84no—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C4×D412yes—
250C22×C4C4×D412yes—
251C22×C4C4×D412yes—
252C22×C4#12112yes—
253C22×C4#12112yes—
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12412yes—
257C22×C4#12712yes—
258C22×C4#12712yes—
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4111yescharacteristic
265D8—84no—
266D8—84no—
267M16—48no—
268M16—84no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—44no—
278C2×C8—84no—
279C2×C8—84no—
280C2×C8C2×C22⋊C414yes—
281C2×C8C2×C22⋊C414yes—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—82no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—82no—
294C42—42no—
295C42—42no—
296C23—22no—
297C23—24no—
298C23—24no—
299C23—22no—
300C23—24no—
301C23—22no—
302C23—42no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—42no—
308C23—41no—
309C23—42no—
310C23—44no—
311C23—48no—
312C23—84no—
313C23—84no—
314C23—84no—
315C23—84no—
316C23—84no—
317C23—84no—
318C23—84no—
319C23—88no—
320C23—84no—
321C23—1616no—
322C23#39012yes—
323C23#39012yes—
324C23#47214yes—
325C23C2×C4×D411yescharacteristic
326C23C2×#12111yescharacteristic
327C23C2×#12411yescharacteristic
328D4—48no—
329D4—84no—
330D4—84no—
331D4—84no—
332D4—88no—
333D4—88no—
334D4—88no—
335D4—88no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4#39012yes—
362C2×C4#39012yes—
363C2×C4C22×C22⋊C411yescharacteristic
364C2×C4C2×C4×D411yescharacteristic
365C2×C4C2×#12111yescharacteristic
366C2×C4C2×#12711yescharacteristic
367C8—24no—
368C8—24no—
369C8—42no—
370C8—42no—
371C8—84no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—21no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—22no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—44no—
385C22—44no—
386C22—48no—
387C22—82no—
388C22—84no—
389C22—84no—
390C22—84no—
391C22—84no—
392C22—168no—
393C22—168no—
394C22—168no—
395C22#197814yes—
396C22C2×#39011yescharacteristic
397C22#295512yes—
398C22C2×#47212yes—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—22no—
404C4—84no—
405C4—84no—
406C4#288312yes—
407C2—22no—
408C2—22no—
409C2—24no—
410C2—41no—
411C2—42no—
412C2—164no—
413C2#2121311yescharacteristic
414C2#2124211yescharacteristic
415C2C2×#197811yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3186C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #20528C2semidirect
4. #17577C2semidirect
5. C2×#1471C2semidirect
6. #9818C2semidirect
7. #9815C2semidirect
13. C2×#480C4semidirect
15. C2×#447C4semidirect
21. #1810C22semidirect
23. #1752C22semidirect
25. C8×C22⋊C4C22semidirect
29. #1507C22semidirect
31. #1471C22semidirect
32. #1471C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31862
#219482
#205282
#175772
C2×#14712
#98182
#98152