Finite Group Databasebeta

← #164507

Subgroups — #164507

Order 512.

727 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36497C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic
3#20572C211yesmaximal, characteristic
4#7697C212yesmaximal
5#7683C211yesmaximal, characteristic
6#7680C211yesmaximal, characteristic
7#7657C212yesmaximal
8#7636C211yesmaximal, characteristic
9#7625C212yesmaximal
10#7591C212yesmaximal
11#7568C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2864C2212yes—
14#2859C2211yescharacteristic
15#2570C412yes—
16#2552C411yescharacteristic
17#2551C411yescharacteristic
18#2550C412yes—
19#2520—22no—
20#2462—22no—
21#2453—21no—
22#2451—21no—
23#2370—21no—
24#2365—21no—
25#2355C411yescharacteristic
26#2355C411yescharacteristic
27C4×C4⋊Q8C2212yes—
28C4×#125C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1839C2212yes—
34#1831C2212yes—
35#1770—21no—
36#1770—22no—
37#1767C2211yescharacteristic
38#1767C2211yescharacteristic
39#1767C2212yes—
40#1744C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125C2×C411yescharacteristic
58C2×C4×Q8—22no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C8×Q8—28no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×#97—21no—
68C2×#97—22no—
69C2×#97—21no—
70C2×C4×C8—21no—
71C2×C4×C8—22no—
72C2×C4×C8—21no—
73#400—22no—
74#398—22no—
75#398—22no—
76#398—22no—
77#395—22no—
78#395—22no—
79#391—21no—
80#391—21no—
81#391—22no—
82#391—22no—
83#389—21no—
84#389—21no—
85#389—22no—
86#389—21no—
87#389—21no—
88#389—22no—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2311yescharacteristic
93#389C2311yescharacteristic
94#389C2312yes—
95#387—22no—
96#387—22no—
97#387—21no—
98#387—21no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#384C2×C412yes—
106#384C2×C411yescharacteristic
107#384C2×C411yescharacteristic
108#382—21no—
109#382C2×C411yescharacteristic
110#382C2×C412yes—
111#382C2×C412yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C43C2311yescharacteristic
128C43C2312yes—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—22no—
134#336—21no—
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141C22×Q8—22no—
142C22×Q8C22⋊C411yescharacteristic
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C4⋊Q8—24no—
150C4⋊Q8—42no—
151C4⋊Q8—44no—
152#125—22no—
153#125—41no—
154#125—44no—
155#125—81no—
156#125—81no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—42no—
160C4×Q8—82no—
161C4×Q8—82no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C22×C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2411yescharacteristic
206C4⋊C8—28no—
207C4⋊C8—24no—
208C4⋊C8—42no—
209C4⋊C8—82no—
210#97—82no—
211#97—81no—
212#97—81no—
213C4×C8—28no—
214C4×C8—24no—
215C4×C8—42no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—22no—
231#95—21no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—82no—
240C2×Q8#10114yes—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C2×C8—22no—
259C2×C8—24no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C4⋊C4—21no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—82no—
304C4⋊C4—81no—
305C4⋊C4—82no—
306C4⋊C4—81no—
307C4⋊C4#10414yes—
308C4⋊C4#13112yes—
309C42—21no—
310C42—22no—
311C42—21no—
312C42—22no—
313C42—22no—
314C42—21no—
315C42—21no—
316C42—22no—
317C42—22no—
318C42—21no—
319C42—22no—
320C42—21no—
321C42—24no—
322C42—22no—
323C42—24no—
324C42—22no—
325C42—22no—
326C42—42no—
327C42—41no—
328C42—42no—
329C42—42no—
330C42—82no—
331C42—82no—
332C23C22×C22⋊C411yescharacteristic
333Q8—28no—
334Q8—88no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4C2×#10114yes—
379C2×C4C2×#10112yes—
380C2×C4C2×#10414yes—
381C2×C4C2×#10412yes—
382C2×C4#42112yes—
383C2×C4C2×#13111yescharacteristic
384C2×C4#56811yescharacteristic
385C2×C4#57611yescharacteristic
386C2×C4#57711yescharacteristic
387C2×C4#57711yescharacteristic
388C2×C4#57811yescharacteristic
389C8—42no—
390C8—44no—
391C8—42no—
392C8—81no—
393C8—82no—
394C8—81no—
395C22#288011yescharacteristic
396C22#288411yescharacteristic
397C22C22×#10111yescharacteristic
398C22#289711yescharacteristic
399C22#289811yescharacteristic
400C22C22×#10411yescharacteristic
401C22C2×#42111yescharacteristic
402C4—21no—
403C4—21no—
404C4—24no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—24no—
410C4—41no—
411C4—42no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—81no—
417C4—82no—
418C4—81no—
419C4—82no—
420C4—81no—
421C4#153114yes—
422C2#1743011yescharacteristic
423C2#1751911yescharacteristic
424C2#1752211yescharacteristic
425C2#1755011yescharacteristic
426C2#1760111yescharacteristic
427C2C2×#153111yescharacteristic
428C2#1777711yescharacteristic
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
#364972
#220222
#205722
#76972
#76832
#76802
#76572
#76362
#76252
#75912
#75682