Finite Group Databasebeta

← #298636

Subgroups — #298636

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic
2#33207C211yesmaximal, characteristic
3#25902C211yesmaximal, characteristic
4#25856C211yesmaximal, characteristic
5#25813C211yesmaximal, characteristic
6#25722C211yesmaximal, characteristic
7C2×#1707C212yesmaximal, complemented
8#10318C212yesmaximal, complemented
9#10314C212yesmaximal, complemented
10#9819C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#500C2212yes—
15C2×#498—28no—
16C2×#498—22no—
17C2×#498C2212yes—
18C2×#498C2211yescharacteristic
19C2×#498C2212yes—
20C2×#494C2212yes—
21C2×#487C2212yes—
22C2×#487C2212yes—
23C2×#451—22no—
24C22×#106—22no—
25C22×C4×C8C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2057—22no—
29#2043—22no—
30#2024—22no—
31#2016—22no—
32#1990C2212yes—
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2212yes—
36#1956C2212yes—
37#1956C2212yes—
38#1941—22no—
39#1872—22no—
40#1852—22no—
41#1844C2211yescharacteristic
42#1807—22no—
43#1787—22no—
44C4×C22⋊C8C2211yescharacteristic
45C4×C22⋊C8C2211yescharacteristic
46#1707—28no—
47C2×C82C2212yes—
48#1273C2212yes—
49C23×Q8—21no—
50C23×Q8—21no—
51C22×Q16—24no—
52C23×C8D412yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8—42no—
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#500—24no—
75#500—42no—
76#498—216no—
77#498—24no—
78#498—24no—
79#498—24no—
80#498—24no—
81#498D418yes—
82#494—24no—
83#494—42no—
84#487—24no—
85#487—24no—
86#487—42no—
87#487—42no—
88#451—48no—
89C2×#107—24no—
90C2×#106—216no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×C22⋊C8D412yes—
100C2×C22⋊C8D412yes—
101C2×C4×C8—24no—
102C2×C4×C8D412yes—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2312yes—
105C2×C4×C8C2312yes—
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#391—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—24no—
123#336D412yes—
124#336D412yes—
125C82—24no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—44no—
131C22×Q8—44no—
132C22×Q8—41no—
133C22×Q8—41no—
134C22×Q8—44no—
135C22×Q8—44no—
136C22×Q8—82no—
137C22×Q8—82no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C2×Q16—48no—
144C2×Q16—48no—
145C2×Q16—84no—
146C22×C8—24no—
147C22×C8—24no—
148C22×C8Q1618yescomplemented
149C22×C8C2×D414yes—
150C22×C8C2×D414yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8C2×D414yes—
172C4⋊Q8C2×D414yes—
173C22⋊Q8—44no—
174C22⋊Q8—44no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209#107—48no—
210#106—216no—
211#106—216no—
212#106—44no—
213#106—44no—
214#103—48no—
215#103—48no—
216#103—48no—
217#103—84no—
218C22⋊C8—24no—
219C22⋊C8SD1618yes—
220C4×C8—24no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8C2×D414yes—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231C24#12711yescharacteristic
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—42no—
242C2×Q8—44no—
243C2×Q8—42no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—816no—
249C2×Q8—816no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×Q8—82no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4#12711yescharacteristic
274C22×C4#12711yescharacteristic
275C22×C4#12711yescharacteristic
276C22×C4C2×SD1614yes—
277C22×C4C2×Q1614yes—
278C22×C4#13514yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282Q16—816no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—216no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8#12014yes—
292C2×C8C2×SD1618yes—
293C2×C8C2×Q1614yes—
294C2×C8C2×Q1614yes—
295C2×C8#13714yes—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C42—21no—
325C42—21no—
326C42—22no—
327C42—22no—
328C42C22×D412yes—
329C42H(5,𝔽2)12yes—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#49212yes—
335C23#49412yes—
336C23#49512yes—
337C23C2×#12711yescharacteristic
338Q8—48no—
339Q8—48no—
340Q8—88no—
341Q8—88no—
342Q8—84no—
343Q8—84no—
344Q8—84no—
345Q8—84no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—88no—
359C2×C4—88no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4#49212yes—
377C2×C4#49412yes—
378C2×C4#49512yes—
379C2×C4C2×#12011yescharacteristic
380C2×C4#53511yescharacteristic
381C2×C4D4212yes—
382C2×C4D4212yes—
383C2×C4C22×SD1612yes—
384C2×C4C22×Q1612yes—
385C2×C4C2×#13512yes—
386C2×C4C2×#13712yes—
387C2×C4#57712yes—
388C2×C4#57712yes—
389C8—24no—
390C8—28no—
391C8—44no—
392C8#45118yescomplemented
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22#170714yescomplemented
400C22#243311yescharacteristic
401C22C2×#49211yescharacteristic
402C22C2×#49411yescharacteristic
403C22C2×#49511yescharacteristic
404C22#315211yescharacteristic
405C22#315411yescharacteristic
406C22#315411yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—84no—
411C4—84no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4C2×#45112yes—
417C4#320912yes—
418C4D4×SD1614yes—
419C4#328214yes—
420C2—24no—
421C2C2×#170711yescharacteristic
422C2#1962211yescharacteristic
423C2#1962611yescharacteristic
424C2#1966311yescharacteristic
425C2#2572211yescharacteristic
426C2#2577411yescharacteristic
427C2#2577611yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
7. C2×#1707C2semidirect
8. #10318C2semidirect
9. #10314C2semidirect
10. #9819C2semidirect
148. C22×C8Q16semidirect
392. C8#451semidirect
399. C22#1707semidirect

Maximal subgroups

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

subgroupindex
C22×#4982
#332072
#259022
#258562
#258132
#257222
C2×#17072
#103182
#103142
#98192
C8×C22⋊C82