Finite Group Databasebeta

← #238132

Subgroups — #238132

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3#21385C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5#10353C214yesmaximal, complemented
6#10352C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2571—22no—
15#2571—22no—
16#2571—22no—
17#2571C2214yes—
18C2×#400—22no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#384C2211yescharacteristic
22C2×#382—21no—
23C2×#382—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1990C2212yes—
27#1989—22no—
28#1988C2212yes—
29#1987C2212yes—
30#1987C2212yes—
31#1981C2212yes—
32#1956C2212yes—
33#1872—22no—
34#1872—21no—
35#1872—21no—
36#1803—22no—
37C4×#103—22no—
38#1736—21no—
39#1736—21no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×M16—21no—
44#533—22no—
45#533—22no—
46#533—22no—
47#533Q814yes—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×#117—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64C22×C42—21no—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#431—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71#423—22no—
72#423—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—24no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8C2312yes—
84#400—28no—
85#400—28no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389—24no—
90#384—21no—
91#384—21no—
92#384D412yes—
93#384C2312yes—
94#382—24no—
95#382—24no—
96#382—24no—
97#382—24no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4C2312yes—
105C4×C4⋊C4C2312yes—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111#340—24no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118C22×Q8—21no—
119C22×Q8—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×M16—42no—
129C2×M16—42no—
130C2×M16—42no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8C2×Q812yes—
141C4⋊Q8C2×Q812yes—
142#125—24no—
143#125—42no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—42no—
151C4×Q8—42no—
152#117—24no—
153#117—42no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—84no—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D414yes—
186C2×C22⋊C4—41no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—44no—
192C2×C42—41no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×Q812yes—
199C2×C42#4112yes—
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—42no—
208C4⋊C8—42no—
209#103—44no—
210#103—44no—
211#103—84no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—42no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224C24#12011yescharacteristic
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4#12011yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4C22⋊Q812yes—
263C22×C4C22⋊Q812yes—
264C22×C4C22⋊Q812yes—
265C22×C4#12412yes—
266C22×C4#12412yes—
267C22×C4#12412yes—
268C22×C4#13512yes—
269C22×C4#13512yes—
270C22×C4#13712yes—
271C22×C4#13712yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275M16—84no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—82no—
314C4⋊C4—88no—
315C22⋊C4—44no—
316C42—44no—
317C42—42no—
318C42—42no—
319C42—44no—
320C42—44no—
321C42—44no—
322C42C22⋊Q812yes—
323C42C22⋊Q812yes—
324C42C22⋊Q812yes—
325C42#12412yes—
326C42#12412yes—
327C42#12412yes—
328C42C22×Q811yescharacteristic
329C42C2×#4111yescharacteristic
330C42C2×#4111yescharacteristic
331C42C2×#4111yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23#39312yes—
337C23#45212yes—
338C23#45212yes—
339C23C2×#12011yescharacteristic
340Q8—44no—
341Q8—44no—
342Q8—82no—
343Q8—82no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—88no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4#39312yes—
381C2×C4#39312yes—
382C2×C4#39312yes—
383C2×C4#45212yes—
384C2×C4#45212yes—
385C2×C4#47914yes—
386C2×C4#47914yes—
387C2×C4#48814yes—
388C2×C4#49114yes—
389C2×C4C2×C22⋊Q811yescharacteristic
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4C2×C22⋊Q811yescharacteristic
392C2×C4C2×#12411yescharacteristic
393C2×C4C2×#12411yescharacteristic
394C2×C4C2×#12411yescharacteristic
395C2×C4C2×#13511yescharacteristic
396C2×C4C2×#13511yescharacteristic
397C2×C4C2×#13711yescharacteristic
398C2×C4C2×#13711yescharacteristic
399C2×C4#57512yes—
400C2×C4#57512yes—
401C8—82no—
402C8—82no—
403C8—82no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22#202614yes—
411C22C2×#39311yescharacteristic
412C22C2×#45211yescharacteristic
413C22C2×#45211yescharacteristic
414C22#300511yescharacteristic
415C22#300511yescharacteristic
416C22#301911yescharacteristic
417C22#302011yescharacteristic
418C4—22no—
419C4—21no—
420C4—21no—
421C4—22no—
422C4—84no—
423C4—82no—
424C4—82no—
425C4—82no—
426C4—82no—
427C4—82no—
428C4—82no—
429C4#202614yes—
430C4C2×#47911yescharacteristic
431C4C2×#47911yescharacteristic
432C4#308012yes—
433C4C2×#48811yescharacteristic
434C4C2×#49111yescharacteristic
435C4#314212yes—
436C2—24no—
437C2#1875111yescharacteristic
438C2#1876311yescharacteristic
439C2#1877311yescharacteristic
440C2#1878111yescharacteristic
441C2C2×#202611yescharacteristic
442C2#2345311yescharacteristic
443C2#2346211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10353C2semidirect
6. #10352C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
#218602
#213852
#213122
#103532
#103522
#80432
#79012
#78922