Finite Group Databasebeta

← #164425

Subgroups — #164425

Order 512.

709 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36313C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic
3#20572C211yesmaximal, characteristic
4#7713C212yesmaximal
5#7682C211yesmaximal, characteristic
6#7682C211yesmaximal, characteristic
7#7669C212yesmaximal
8#7636C211yesmaximal, characteristic
9#7615C212yesmaximal
10#7581C212yesmaximal
11#7568C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2863C2212yes—
14#2862C2211yescharacteristic
15#2568C411yescharacteristic
16#2546C411yescharacteristic
17#2528—22no—
18#2524—22no—
19#2514—21no—
20#2514C412yes—
21#2453C412yes—
22#2446—21no—
23#2365—21no—
24#2365C411yescharacteristic
25#2365C411yescharacteristic
26#2356—21no—
27C4×#125C2211yescharacteristic
28#2295C2212yes—
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#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C411yescharacteristic
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
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#401—22no—
74#400—22no—
75#400—22no—
76#400—22no—
77#398—22no—
78#398—22no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2312yes—
89#389—22no—
90#389—21no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2311yescharacteristic
95#389C2311yescharacteristic
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#387—22no—
106#387—22no—
107#383—22no—
108#383—22no—
109#382—21no—
110#382—21no—
111#382—21no—
112#382C2×C411yescharacteristic
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×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D412yes—
126#376—22no—
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—21no—
142C22×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C412yes—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—44no—
153#125—22no—
154#125—41no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—44no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2411yescharacteristic
200C4⋊C8—28no—
201C4⋊C8—24no—
202C4⋊C8—42no—
203C4⋊C8—82no—
204#97—82no—
205#97—81no—
206#97—81no—
207C4×C8—28no—
208C4×C8—24no—
209C4×C8—42no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—22no—
215#95—22no—
216#95—21no—
217#95—21no—
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—21no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—42no—
238C2×Q8—42no—
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⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C2×C8—22no—
261C2×C8—24no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—41no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4#13112yes—
304C42—22no—
305C42—21no—
306C42—22no—
307C42—21no—
308C42—22no—
309C42—21no—
310C42—21no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—22no—
316C42—22no—
317C42—24no—
318C42—22no—
319C42—22no—
320C42—42no—
321C42—42no—
322C42—41no—
323C42—44no—
324C42—44no—
325C42—42no—
326C42—42no—
327C23C22×C22⋊C411yescharacteristic
328Q8—28no—
329Q8—82no—
330Q8—82no—
331Q8—84no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4#41312yes—
377C2×C4C2×#10414yes—
378C2×C4C2×#10412yes—
379C2×C4#42112yes—
380C2×C4C2×#13111yescharacteristic
381C2×C4#56811yescharacteristic
382C2×C4#57711yescharacteristic
383C2×C4#57711yescharacteristic
384C2×C4#57811yescharacteristic
385C2×C4#57811yescharacteristic
386C8—44no—
387C8—42no—
388C8—42no—
389C8—81no—
390C8—82no—
391C8—81no—
392C22#288011yescharacteristic
393C22#288411yescharacteristic
394C22C2×#41311yescharacteristic
395C22#289811yescharacteristic
396C22#289811yescharacteristic
397C22C22×#10411yescharacteristic
398C22C2×#42111yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—24no—
406C4—42no—
407C4—41no—
408C4—42no—
409C4—41no—
410C4—42no—
411C4—42no—
412C4—41no—
413C4—82no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—82no—
418C4#153214yes—
419C2#1742811yescharacteristic
420C2#1751811yescharacteristic
421C2#1752511yescharacteristic
422C2#1754911yescharacteristic
423C2#1760111yescharacteristic
424C2C2×#153211yescharacteristic
425C2#1777811yescharacteristic
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
#363132
#220222
#205722
#77132
#76822
#76692
#76362
#76152
#75812
#75682