Finite Group Databasebeta

← #161341

Subgroups — #161341

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3C4×#431C211yesmaximal, characteristic
4#7663C212yesmaximal, complemented
5#7660C212yesmaximal, complemented
6#7657C214yesmaximal, complemented
7#7564C214yesmaximal
8C2×#431—21no—
9#2864C412yes—
10#2864C2212yes—
11#2859C411yescharacteristic
12#2859C411yescharacteristic
13#2859C2211yescharacteristic
14#2859C2211yescharacteristic
15#2851—21no—
16#2851—21no—
17#2851—21no—
18#2851—21no—
19C4×#125—21no—
20C4×#125—21no—
21C4×#125—21no—
22C4×#125—21no—
23#2296C412yes—
24#2296C2212yes—
25#2285C411yescharacteristic
26#2285C411yescharacteristic
27C2×C43C2211yescharacteristic
28#1839C2212yes—
29#1831C2212yes—
30#1817—21no—
31#1797—21no—
32C4×C4⋊C8—21no—
33C4×C4⋊C8—21no—
34C4×C4⋊C8C2212yes—
35C4×C22⋊C8—21no—
36C4×C22⋊C8C2212yes—
37C4×C22⋊C8C2212yes—
38#1744—21no—
39#1744C2212yes—
40C42×C8—21no—
41#1271C2218yes—
42#1271C2218yes—
43C2×#125—21no—
44C2×#125—21no—
45C2×#125—21no—
46C2×#125—21no—
47C2×#117C2×C412yes—
48C2×#117C2×C412yes—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42C2311yescharacteristic
56#431—28no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C4×C8—22no—
62C2×C4×C8—22no—
63C2×C4×C8—22no—
64C2×C4×C8—22no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389C2×C412yes—
74#389C2×C412yes—
75#389C2×C412yes—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2312yes—
80#389C2312yes—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#385—22no—
90#385—22no—
91#385—22no—
92#385—22no—
93#384C2×C411yescharacteristic
94#384C2×C411yescharacteristic
95#384C2×C411yescharacteristic
96#384C2×C411yescharacteristic
97#382—21no—
98#382—21no—
99#382—21no—
100#382—21no—
101#382C2×C412yes—
102#382C2×C412yes—
103#380—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C43C2×C411yescharacteristic
116C43C2×C411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43C2311yescharacteristic
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336C2314yes—
128#336C2314yes—
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C22×C8—24no—
137C22×C8—24no—
138#125—44no—
139#125—44no—
140#125—44no—
141#125—44no—
142#117—28no—
143#117—44no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C22×C412yes—
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—44no—
205C22⋊C8—28no—
206C22⋊C8—44no—
207C4×C8—28no—
208C4×C8—44no—
209C4×C8—44no—
210C4×C8—44no—
211#95—22no—
212#95—22no—
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—
223C24C2×C22⋊C411yescharacteristic
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
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—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4#13114yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C2×C8—28no—
264C2×C8—44no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—44no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
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—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—44no—
316C42—44no—
317C42#10114yes—
318C42#10414yes—
319C42#13514yes—
320C42#13514yes—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23#40812yes—
328C23C22×C22⋊C411yescharacteristic
329C2×C4—28no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
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—28no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4#40812yes—
375C2×C4#41012yes—
376C2×C4C2×#10112yes—
377C2×C4C2×#10112yes—
378C2×C4C2×#10112yes—
379C2×C4#41612yes—
380C2×C4#41612yes—
381C2×C4#41612yes—
382C2×C4#41612yes—
383C2×C4C2×#10412yes—
384C2×C4C2×#10412yes—
385C2×C4C2×#10412yes—
386C2×C4#42112yes—
387C2×C4C2×#13112yes—
388C2×C4C2×#13512yes—
389C2×C4C2×#13512yes—
390C8—48no—
391C8—84no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22C2×#40811yescharacteristic
400C22C2×#41011yescharacteristic
401C22C22×#10111yescharacteristic
402C22C2×#41611yescharacteristic
403C22C2×#41611yescharacteristic
404C22C22×#10411yescharacteristic
405C22C2×#42111yescharacteristic
406C4—22no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—41no—
417C4—41no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4#147014yes—
423C4#154414yes—
424C2—42no—
425C2C2×#147011yescharacteristic
426C2#1745811yescharacteristic
427C2#1747311yescharacteristic
428C2#1753311yescharacteristic
429C2#1757311yescharacteristic
430C2#1764811yescharacteristic
431C2C2×#154411yescharacteristic
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. #7663C2semidirect
5. #7660C2semidirect
6. #7657C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#207772
C4×#4312
#76632
#76602
#76572
#75642