Finite Group Databasebeta

← #161409

Subgroups — #161409

Order 512.

892 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20463C211yesmaximal, characteristic
3C4×#431C211yesmaximal, characteristic
4#7682C212yesmaximal, complemented
5#7680C212yesmaximal, complemented
6#7675C214yesmaximal
7#7562C214yesmaximal, complemented
8C2×#431—21no—
9C2×#431—21no—
10#2864—21no—
11#2863—21no—
12#2863C412yes—
13#2862C2211yescharacteristic
14#2859C2211yescharacteristic
15#2857C411yescharacteristic
16#2857C411yescharacteristic
17#2855—21no—
18#2855C2212yes—
19#2851—21no—
20C4×C4⋊Q8C411yescharacteristic
21C4×C4⋊Q8C411yescharacteristic
22C4×#125—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8C412yes—
25#2295—21no—
26#2295C2212yes—
27#2283—21no—
28C2×C43C2211yescharacteristic
29#1767—21no—
30#1767C2212yes—
31C4×C4⋊C8—21no—
32C4×C4⋊C8C2212yes—
33#1749—21no—
34#1749C2212yes—
35C4×C22⋊C8—21no—
36C4×C22⋊C8C2212yes—
37#1744C2212yes—
38#1744C2212yes—
39C4×#97—21no—
40C42×C8—21no—
41#1272C2218yes—
42#1271C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42C2311yescharacteristic
57#431—28no—
58#431—28no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×#97—22no—
64C2×#97—22no—
65C2×C4×C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8—22no—
69#391—21no—
70#391—21no—
71#391—21no—
72#391—21no—
73#391C2×C412yes—
74#391C2×C412yes—
75#391C2×C412yes—
76#391C2312yes—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2312yes—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#385—22no—
98#385—22no—
99#384C2×C411yescharacteristic
100#384C2×C411yescharacteristic
101#383—22no—
102#382—21no—
103#382—21no—
104#382—21no—
105#381—22no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380C2×C412yes—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C412yes—
116C4×C22⋊C4—22no—
117#376—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43C2311yescharacteristic
125C43C2311yescharacteristic
126#336—22no—
127#336—22no—
128#336C2314yes—
129#336C2314yes—
130C22×Q8—21no—
131C22×Q8C22×C412yes—
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C22×C8—24no—
140C22×C8—24no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143#125—44no—
144C22⋊Q8—44no—
145C22⋊Q8—44no—
146C4×Q8—28no—
147C4×Q8—44no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2411yescharacteristic
203C4⋊C8—28no—
204C4⋊C8—28no—
205C22⋊C8—28no—
206C22⋊C8—28no—
207#97—44no—
208#97—44no—
209C4×C8—28no—
210C4×C8—28no—
211C4×C8—44no—
212C4×C8—44no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227C24C2×C22⋊C411yescharacteristic
228C2×Q8—24no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4#10414yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—28no—
274C2×C8—28no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—44no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42—41no—
321C42—44no—
322C42—44no—
323C42#10414yes—
324C42#13114yes—
325C42#13514yes—
326C42#13714yes—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23C2×#10412yes—
334C23C22×C22⋊C411yescharacteristic
335Q8—84no—
336Q8—84no—
337C2×C4—28no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—28no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4#40812yes—
383C2×C4#40812yes—
384C2×C4#41012yes—
385C2×C4#41312yes—
386C2×C4#41612yes—
387C2×C4#41612yes—
388C2×C4#41912yes—
389C2×C4#41912yes—
390C2×C4C2×#10412yes—
391C2×C4C2×#10412yes—
392C2×C4C2×#10412yes—
393C2×C4C2×#10412yes—
394C2×C4C2×#10412yes—
395C2×C4C2×#13112yes—
396C2×C4C2×#13512yes—
397C2×C4C2×#13712yes—
398C8—48no—
399C8—48no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22C2×#40811yescharacteristic
408C22C2×#41011yescharacteristic
409C22C2×#41311yescharacteristic
410C22C2×#41611yescharacteristic
411C22C2×#41911yescharacteristic
412C22C22×#10411yescharacteristic
413C22C22×#10411yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—82no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4#148514yes—
431C4#152414yes—
432C2—42no—
433C2#1737611yescharacteristic
434C2C2×#148511yescharacteristic
435C2#1747311yescharacteristic
436C2#1753311yescharacteristic
437C2#1754511yescharacteristic
438C2C2×#152411yescharacteristic
439C2#1771811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7682C2semidirect
5. #7680C2semidirect
7. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#204632
C4×#4312
#76822
#76802
#76752
#75622