Finite Group Databasebeta

← #194874

Subgroups — #194874

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2#34057C211yesmaximal, characteristic
3#21841C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic
5#9024C214yesmaximal, complemented
6#9021C214yesmaximal, complemented
7#8021C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C22⋊Q8—21no—
10C22×C22⋊Q8—21no—
11C22×C4×D4C2211yescharacteristic
12C2×#432C2212yes—
13#2542—24no—
14#2542C418yescomplemented
15C2×#399C2212yes—
16C2×#399C2212yes—
17#2318—24no—
18#2315C2212yes—
19#2313—21no—
20#2312C2212yes—
21#2308—21no—
22C2×#390—21no—
23#2290—24no—
24C2×#384—21no—
25#2283C2212yes—
26#2280—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4—21no—
29#1993—24no—
30#1988—24no—
31#1824C2218yes—
32#1466—24no—
33C2×#323C2212yes—
34C2×#323C2212yes—
35#1294C2218yes—
36C23×Q8—21no—
37C23×D4D411yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8C2×C414yescomplemented
45C2×C22⋊Q8—28no—
46C2×C22⋊Q8—28no—
47C2×C22⋊Q8—48no—
48C2×C22⋊Q8—48no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—24no—
52C2×C4×D4C2×C414yes—
53C2×C4×D4D414yes—
54C2×#117—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4C2311yescharacteristic
65C22×C22⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—22no—
68C22×C42C2311yescharacteristic
69#432—24no—
70#432—24no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8C2314yes—
73C2×#97—22no—
74#399—24no—
75#399—24no—
76#399C2×C418yes—
77#399C2×C418yescomplemented
78#390—28no—
79#388—28no—
80#387—28no—
81#386—28no—
82#385—28no—
83#384—28no—
84#383—22no—
85#382D414yes—
86#381—22no—
87#381—22no—
88#380—22no—
89#380—22no—
90#380C2314yes—
91#379—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C414yescomplemented
94C4×C4⋊C4D414yes—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100#376—22no—
101C2×#95—22no—
102C2×#95C2312yes—
103#340—216no—
104#323—24no—
105#323—24no—
106C25C2×D411yescharacteristic
107C25C2×D411yescharacteristic
108C22×Q8—24no—
109C22×Q8—44no—
110C22×Q8—44no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—42no—
114C22×D4—24no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—42no—
118C22×D4C22⋊C414yes—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4C22⋊C414yes—
131C23×C4C22⋊C414yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—24no—
138C4⋊Q8—28no—
139C4⋊Q8—28no—
140C4⋊Q8—44no—
141C4⋊Q8—44no—
142C22⋊Q8—432no—
143C22⋊Q8—432no—
144C4×D4—28no—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—28no—
148#117—84no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4C22⋊C414yes—
165C2×C4⋊C4C22×C414yes—
166C2×C4⋊C4C2×D418yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4C22×C412yes—
182C2×C22⋊C4C22×C412yes—
183C2×C22⋊C4C2×D412yes—
184C2×C22⋊C4C2×D412yes—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—44no—
189C2×C42—84no—
190C2×C42C22×C412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42#4114yes—
195C4⋊C8—44no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198#97—44no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—22no—
205#95—24no—
206#95C22×C414yes—
207C24—21no—
208C24—21no—
209C24—22no—
210C24—21no—
211C24—22no—
212C24—21no—
213C24—22no—
214C24—42no—
215C24—42no—
216C24—42no—
217C24—42no—
218C24C2×C22⋊C412yes—
219C24C2×C22⋊C412yes—
220C24#13714yes—
221C24C22×D411yescharacteristic
222C2×Q8—48no—
223C2×Q8—48no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—84no—
227C2×Q8—88no—
228C2×Q8—84no—
229C2×D4—28no—
230C2×D4—28no—
231C2×D4—44no—
232C2×D4—48no—
233C2×D4—48no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—28no—
238C22×C4—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—42no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—48no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4#12012yes—
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12312yes—
277C22×C4#12312yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D412yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4112yes—
282C2×C8—48no—
283C2×C8—44no—
284C4⋊C4—216no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—48no—
288C4⋊C4—416no—
289C4⋊C4—416no—
290C4⋊C4—44no—
291C4⋊C4—48no—
292C4⋊C4—84no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—416no—
298C22⋊C4—416no—
299C22⋊C4—48no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—84no—
303C22⋊C4—84no—
304C42—48no—
305C42—48no—
306C42—84no—
307C42C4×D414yes—
308C42C4×D414yes—
309C42#12114yes—
310C42#12114yes—
311C23—22no—
312C23—22no—
313C23—24no—
314C23—22no—
315C23—24no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—48no—
325C23—48no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—81no—
334C23—84no—
335C23—81no—
336C23—81no—
337C23—84no—
338C23#41914yes—
339C23C22×C22⋊C411yescharacteristic
340C23C2×#12011yescharacteristic
341C23C2×#12311yescharacteristic
342C23C2×#13712yes—
343C23#57512yes—
344Q8—88no—
345Q8—88no—
346D4—416no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—416no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—84no—
381C2×C4—816no—
382C2×C4—84no—
383C2×C4#38614yes—
384C2×C4#38612yes—
385C2×C4#38612yes—
386C2×C4#45314yes—
387C2×C4C2×C4×D412yes—
388C2×C4C2×#12112yes—
389C8—84no—
390C22—24no—
391C22—24no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—24no—
397C22—44no—
398C22—44no—
399C22—48no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—84no—
407C22—82no—
408C22—84no—
409C22—84no—
410C22C2×#38611yescharacteristic
411C22C2×#41911yescharacteristic
412C22#289611yescharacteristic
413C22#300712yes—
414C22C2×#45312yes—
415C22#302812yes—
416C22#309512yes—
417C4—24no—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—48no—
422C4—42no—
423C4—42no—
424C4—88no—
425C4—84no—
426C4#188714yes—
427C2—21no—
428C2—21no—
429C2—22no—
430C2—44no—
431C2—82no—
432C2#1834314yes—
433C2#2091111yescharacteristic
434C2C2×#188711yescharacteristic
435C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21841C2semidirect
5. #9024C2semidirect
6. #9021C2semidirect
7. #8021C2semidirect
14. #2542C4semidirect
44. C2×C4⋊Q8C2×C4semidirect
77. #399C2×C4semidirect
93. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#340572
#218412
C2×#14662
#90242
#90212
#80212