Finite Group Databasebeta

← #162467

Subgroups — #162467

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35187C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic
3#20570C211yesmaximal, characteristic
4#7709C212yesmaximal
5#7697C212yesmaximal
6#7689C212yesmaximal
7#7677C212yesmaximal
8#7588C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12C2×#446—21no—
13C2×C8×Q8—21no—
14#2864C2212yes—
15#2863C2212yes—
16#2655—22no—
17#2582C411yescharacteristic
18#2571C411yescharacteristic
19#2568C411yescharacteristic
20#2546C411yescharacteristic
21#2524—22no—
22#2514—21no—
23#2514C412yes—
24#2508C412yes—
25#2446—21no—
26#2327—21no—
27#2320—21no—
28C4×C4⋊Q8C2211yescharacteristic
29#2295C2211yescharacteristic
30C42×Q8C2211yescharacteristic
31#1939—21no—
32#1939—22no—
33#1831C2212yes—
34#1831C2211yescharacteristic
35#1771—22no—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2211yescharacteristic
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60#446—28no—
61C8×Q8—44no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×#97—22no—
69C2×#97—22no—
70C2×C4×C8—21no—
71C2×C4×C8—22no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74#401—22no—
75#400—22no—
76#400—22no—
77#400—22no—
78#400—22no—
79#398—22no—
80#398—22no—
81#398—22no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—21no—
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2×C411yescharacteristic
90#391C2311yescharacteristic
91#389—21no—
92#389—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C411yescharacteristic
99#389C2312yes—
100#389C2311yescharacteristic
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—21no—
111#387—21no—
112#384C2×C411yescharacteristic
113#383—22no—
114#383—22no—
115#382—21no—
116#382—22no—
117#382C2×C412yes—
118#382C2×C412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D412yes—
129#376—21no—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—22no—
133#336—21no—
134#336—21no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8C22⋊C411yescharacteristic
144C22×Q8C22×C411yescharacteristic
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C4⋊Q8—24no—
152C4⋊Q8—44no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—24no—
158C4×Q8—42no—
159C4×Q8—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42C22×C412yes—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2411yescharacteristic
198C4⋊C8—24no—
199C4⋊C8—28no—
200C4⋊C8—44no—
201C4⋊C8—42no—
202C4⋊C8—44no—
203C4⋊C8—42no—
204#97—28no—
205#97—82no—
206C4×C8—24no—
207C4×C8—44no—
208C4×C8—42no—
209C4×C8—81no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
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—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8#10414yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—41no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C42—24no—
303C42—22no—
304C42—22no—
305C42—21no—
306C42—22no—
307C42—21no—
308C42—22no—
309C42—22no—
310C42—24no—
311C42—41no—
312C42—42no—
313C42—41no—
314C42—42no—
315C42—41no—
316C42—42no—
317C42—44no—
318C42—42no—
319C42#13114yes—
320C42AGL(1,ℤ/8)12yes—
321C42#13712yes—
322C23C22×C22⋊C411yescharacteristic
323Q8—28no—
324Q8—82no—
325Q8—84no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4#40812yes—
369C2×C4#40812yes—
370C2×C4#41712yes—
371C2×C4#41712yes—
372C2×C4C2×#10414yes—
373C2×C4C2×#10412yes—
374C2×C4C2×#13112yes—
375C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
376C2×C4C2×#13711yescharacteristic
377C2×C4#57711yescharacteristic
378C2×C4#57811yescharacteristic
379C8—42no—
380C8—44no—
381C8—42no—
382C8—42no—
383C8—82no—
384C8—81no—
385C22C2×#40811yescharacteristic
386C22#288411yescharacteristic
387C22#288911yescharacteristic
388C22C2×#41711yescharacteristic
389C22#289811yescharacteristic
390C22C22×#10411yescharacteristic
391C22#290111yescharacteristic
392C4—24no—
393C4—22no—
394C4—22no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—42no—
401C4—42no—
402C4—41no—
403C4—42no—
404C4—82no—
405C4—81no—
406C4—81no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4#148314yes—
411C2#1739111yescharacteristic
412C2C2×#148311yescharacteristic
413C2#1748411yescharacteristic
414C2#1754911yescharacteristic
415C2#1762811yescharacteristic
416C2#1765411yescharacteristic
417C2#1776311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#351872
#227342
#205702
#77092
#76972
#76892
#76772
#75882
#75772
#75702
#75622