Finite Group Databasebeta

← #365487

Subgroups — #365487

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2C2×#1995C211yesmaximal, characteristic
3#21323C211yesmaximal, characteristic
4#21172C211yesmaximal, characteristic
5#14354C211yesmaximal, characteristic
6#11340C212yesmaximal
7#11339C212yesmaximal
8#10892C212yesmaximal
9#10891C212yesmaximal
10#10857C211yesmaximal, characteristic
11#10701C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#428C2211yescharacteristic
14C2×#428C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2360—24no—
18#2355—22no—
19Q8×C4⋊C4—22no—
20#2330—22no—
21#2317C2212yes—
22C4×C4⋊Q8—22no—
23C2×#384C2212yes—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2212yes—
27#2274C2212yes—
28C4×#117C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1995C418yescomplemented
31#1993—22no—
32#1991—22no—
33#1949—22no—
34#1949—22no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39#1736C2211yescharacteristic
40#1736C2211yescharacteristic
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45#1444—24no—
46C22×M16C2311yescharacteristic
47C22×M16C2311yescharacteristic
48C22×M16C2312yes—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52#533Q814yes—
53C2×C4⋊Q8D412yes—
54C2×#125D412yes—
55C2×C22⋊Q8—22no—
56C2×C4×Q8—24no—
57C2×C4×Q8—22no—
58C2×#117C2311yescharacteristic
59C2×#117C2312yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—24no—
62C22×C4⋊C4D414yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68#428—42no—
69#428—42no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#106—21no—
73C2×#106—21no—
74#423—22no—
75#423C2×C414yes—
76C2×C4⋊C8—22no—
77C4×M16—22no—
78C4×M16—22no—
79C2×#97—21no—
80C2×C4×C8—21no—
81#391—24no—
82#389—24no—
83#389—28no—
84#389—24no—
85#387—28no—
86#385—24no—
87#384—22no—
88#384—21no—
89#384—21no—
90#384—22no—
91#384—24no—
92#384—42no—
93#382—22no—
94#382—22no—
95#382—21no—
96#382—21no—
97#382—24no—
98#382—42no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—24no—
106#376—21no—
107#376—22no—
108#376C2×C414yes—
109C2×#95—22no—
110C2×#95—24no—
111C2×#95—22no—
112C43—21no—
113#343—42no—
114#343—42no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—24no—
119#339—24no—
120#339C2×C418yes—
121#339C2×C418yes—
122#339Q818yescomplemented
123C22×Q8—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4C2×D414yes—
128C23×C4C2×D412yes—
129C23×C4C2411yescharacteristic
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16C22×C412yes—
142C2×M16C22×C412yes—
143C2×M16C2×Q814yes—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C4⋊Q8—24no—
148C4⋊Q8C4⋊C418yescomplemented
149#125—24no—
150#125C4⋊C418yescomplemented
151C22⋊Q8—28no—
152C22⋊Q8—28no—
153C4×Q8—28no—
154C4×Q8—28no—
155C4×Q8—88no—
156#117—22no—
157#117—22no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117C22×C414yes—
162#117C2×Q814yes—
163#117C2×Q814yes—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—84no—
185C2×C4⋊C4—88no—
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—28no—
202C2×C42C22×C414yes—
203C2×C42C22×C412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×Q814yes—
208C2×C42C2×Q812yes—
209C2×C42#4114yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212#107—42no—
213#107—42no—
214#106—42no—
215#106—42no—
216C4⋊C8—28no—
217#97—42no—
218C4×C8—42no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—22no—
223#95—22no—
224#95—48no—
225#95—42no—
226#95—42no—
227C24C22×D411yescharacteristic
228C2×Q8—28no—
229C2×Q8—44no—
230C2×Q8—44no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4C4×D414yes—
253C22×C4#12012yes—
254C22×C4#12012yes—
255C22×C4C22⋊Q814yes—
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×Q811yescharacteristic
260C22×C4C22×Q812yes—
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4111yescharacteristic
264C22×C4H(5,𝔽2)12yes—
265C22×C42−1+412yes—
266M16—24no—
267M16—24no—
268M16—24no—
269M16—24no—
270M16—42no—
271M16—42no—
272M16—42no—
273M16—42no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—216no—
286C4⋊C4—28no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—48no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—84no—
303C4⋊C4—88no—
304C4⋊C4—816no—
305C4⋊C4C2×C4⋊C418yes—
306C4⋊C4C4⋊Q814yes—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—42no—
311C42—24no—
312C42—24no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—44no—
318C42—44no—
319C42—42no—
320C42—88no—
321C42C2×C4⋊C412yes—
322C42C2×C4⋊C412yes—
323C42C4×D414yes—
324C42C4×Q818yescomplemented
325C42C22⋊Q814yes—
326C42#12312yes—
327C42#12312yes—
328C23—22no—
329C23—22no—
330C23C2×C4×D411yescharacteristic
331C23C2×#12011yescharacteristic
332C23C2×C22⋊Q811yescharacteristic
333C23#54612yes—
334C23D4×Q812yes—
335Q8—416no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—28no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4#38514yes—
368C2×C4#38514yes—
369C2×C4C22×C4⋊C411yescharacteristic
370C2×C4C2×C4×D411yescharacteristic
371C2×C4C2×C4×Q812yes—
372C2×C4#52012yes—
373C2×C4C2×C22⋊Q811yescharacteristic
374C2×C4C2×#12311yescharacteristic
375C2×C4C2×C4⋊Q811yescharacteristic
376C2×C4#53211yescharacteristic
377C2×C4#53312yes—
378C2×C4#53511yescharacteristic
379C2×C4#53611yescharacteristic
380C2×C4#54212yes—
381C2×C4D4×Q812yes—
382C2×C4#55012yes—
383C2×C4#55212yes—
384C2×C4#55412yes—
385C2×C4#55712yes—
386C2×C4Q8212yes—
387C8—42no—
388C8—42no—
389C8—44no—
390C22—22no—
391C22—22no—
392C22—24no—
393C22—22no—
394C22—22no—
395C22C2×#38511yescharacteristic
396C22#236012yes—
397C22#243111yescharacteristic
398C22#243411yescharacteristic
399C22#245112yes—
400C22#301912yes—
401C22#302012yes—
402C4—24no—
403C4—24no—
404C4—48no—
405C4—48no—
406C4—44no—
407C4—81no—
408C4—84no—
409C4—81no—
410C4—161no—
411C4—161no—
412C4C4×C4⋊Q811yescharacteristic
413C4#232611yescharacteristic
414C4Q8×C4⋊C412yes—
415C4#242312yes—
416C4#256511yescharacteristic
417C4#256711yescharacteristic
418C2—22no—
419C2—22no—
420C2#1397611yescharacteristic
421C2#2086412yes—
422C2#2559311yescharacteristic
423C2#2559411yescharacteristic
424C2#2583212yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #1995C4semidirect
122. #339Q8semidirect
148. C4⋊Q8C4⋊C4semidirect
150. #125C4⋊C4semidirect
324. C42C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#342142
C2×#19952
#213232
#211722
#143542
#113402
#113392
#108922
#108912
#108572
#107012