Finite Group Databasebeta

← #164356

Subgroups — #164356

Order 512.

771 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#36497C211yesmaximal, characteristic
2#20573C211yesmaximal, characteristic
3#20573C211yesmaximal, characteristic
4#7683C211yesmaximal, characteristic
5#7680C211yesmaximal, characteristic
6#7677C212yesmaximal
7#7677C212yesmaximal
8#7638C212yesmaximal
9#7636C211yesmaximal, characteristic
10#7570C212yesmaximal
11#7568C211yesmaximal, characteristic
12C2×#446—21no—
13C2×#446—21no—
14#2864C2212yes—
15#2859C2211yescharacteristic
16#2570C412yes—
17#2552C411yescharacteristic
18#2551C411yescharacteristic
19#2550C412yes—
20#2520—22no—
21#2462—22no—
22#2453—21no—
23#2451—21no—
24#2370—21no—
25#2365—21no—
26#2355C411yescharacteristic
27#2355C411yescharacteristic
28C4×C4⋊Q8C2212yes—
29C4×#125C2211yescharacteristic
30C42×Q8C2211yescharacteristic
31#1771—22no—
32#1771—22no—
33C8×C4⋊C4—21no—
34C8×C4⋊C4—21no—
35#1767C2211yescharacteristic
36#1767C2211yescharacteristic
37C4×C4⋊C8C2212yes—
38C4×C4⋊C8C2212yes—
39#1745C2212yes—
40#1745C2212yes—
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2211yescharacteristic
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2212yes—
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
62#446—28no—
63#446—28no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×#97—22no—
69C2×#97—22no—
70C2×#97—21no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76#400—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#395—22no—
81#395—22no—
82#391—21no—
83#391—21no—
84#391—22no—
85#391—22no—
86#389—21no—
87#389—21no—
88#389—22no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2311yescharacteristic
96#389C2311yescharacteristic
97#389C2312yes—
98#387—22no—
99#387—22no—
100#387—21no—
101#387—21no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#384C2×C412yes—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#382—21no—
112#382C2×C411yescharacteristic
113#382C2×C412yes—
114#382C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D412yes—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2312yes—
142C22×Q8—22no—
143C22×Q8C22⋊C411yescharacteristic
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—24no—
151C4⋊Q8—42no—
152C4⋊Q8—44no—
153#125—22no—
154#125—41no—
155#125—44no—
156#125—81no—
157#125—81no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—82no—
162C4×Q8—82no—
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⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C412yes—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—28no—
210C4⋊C8—28no—
211#97—28no—
212#97—28no—
213#97—81no—
214#97—81no—
215C4×C8—24no—
216C4×C8—24no—
217C4×C8—82no—
218C4×C8—82no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—21no—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—82no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—41no—
279C4⋊C4—21no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
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—44no—
303C4⋊C4—42no—
304C4⋊C4—82no—
305C4⋊C4—81no—
306C4⋊C4—82no—
307C4⋊C4—81no—
308C4⋊C4#10114yes—
309C4⋊C4#10414yes—
310C4⋊C4#10414yes—
311C4⋊C4#13112yes—
312C42—21no—
313C42—22no—
314C42—21no—
315C42—22no—
316C42—22no—
317C42—21no—
318C42—22no—
319C42—21no—
320C42—22no—
321C42—21no—
322C42—22no—
323C42—21no—
324C42—22no—
325C42—24no—
326C42—22no—
327C42—22no—
328C42—24no—
329C42—24no—
330C42—41no—
331C42—42no—
332C42—42no—
333C42—82no—
334C42—82no—
335C23C22×C22⋊C411yescharacteristic
336Q8—44no—
337Q8—88no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4C2×#10114yes—
381C2×C4C2×#10112yes—
382C2×C4C2×#10414yes—
383C2×C4C2×#10414yes—
384C2×C4C2×#10412yes—
385C2×C4C2×#10412yes—
386C2×C4C2×#13111yescharacteristic
387C2×C4#56811yescharacteristic
388C2×C4#57611yescharacteristic
389C2×C4#57711yescharacteristic
390C2×C4#57711yescharacteristic
391C2×C4#57811yescharacteristic
392C8—42no—
393C8—44no—
394C8—42no—
395C8—42no—
396C8—44no—
397C8—42no—
398C22#288011yescharacteristic
399C22#288411yescharacteristic
400C22C22×#10111yescharacteristic
401C22#289711yescharacteristic
402C22#289811yescharacteristic
403C22C22×#10411yescharacteristic
404C22C22×#10411yescharacteristic
405C4—21no—
406C4—21no—
407C4—24no—
408C4—24no—
409C4—24no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—41no—
418C4—42no—
419C4—82no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—82no—
424C4#152614yes—
425C2#1743011yescharacteristic
426C2#1751711yescharacteristic
427C2#1751911yescharacteristic
428C2#1754911yescharacteristic
429C2#1755011yescharacteristic
430C2C2×#152611yescharacteristic
431C2#1777411yescharacteristic
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
#205732
#76832
#76802
#76772
#76382
#76362
#75702
#75682