Finite Group Databasebeta

← #164851

Subgroups — #164851

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36454C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic
3#22012C211yesmaximal, characteristic
4#7721C212yesmaximal, complemented
5#7715C212yesmaximal, complemented
6#7682C211yesmaximal, characteristic, complemented
7#7682C211yesmaximal, characteristic, complemented
8#7648C212yesmaximal, complemented
9#7636C211yesmaximal, characteristic, complemented
10#7600C212yesmaximal, complemented
11#7568C211yesmaximal, characteristic, complemented
12#2862C2211yescharacteristic
13#2855C2212yes—
14#2575—22no—
15#2547C411yescharacteristic
16#2542C411yescharacteristic
17#2517—22no—
18#2511—21no—
19#2489C412yes—
20#2447—21no—
21#2421C412yes—
22#2367—21no—
23#2365C411yescharacteristic
24#2365C411yescharacteristic
25#2355—21no—
26C4×#125C2211yescharacteristic
27C4×C22⋊Q8C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1939—21no—
33#1924—22no—
34#1924—22no—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1767C2211yescharacteristic
40#1767C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125C2×C411yescharacteristic
56C2×#124—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C412yes—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C22×C42C2312yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#400—22no—
78#400—22no—
79#399—22no—
80#397—22no—
81#397—22no—
82#397—22no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—21no—
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2312yes—
90#390—21no—
91#389—21no—
92#389C2×C412yes—
93#389C2311yescharacteristic
94#389C2311yescharacteristic
95#388—22no—
96#388—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#385—22no—
106#385—22no—
107#384—21no—
108#382—21no—
109#382—21no—
110#382—22no—
111#382C2×C411yescharacteristic
112#381—22no—
113#380C2×C412yes—
114#380C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—22no—
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129C43C2311yescharacteristic
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8—21no—
141C22×Q8C22×C412yes—
142C22×D4C22⋊C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154#125—22no—
155#125—41no—
156#124—44no—
157C22⋊Q8—24no—
158C22⋊Q8—42no—
159C22⋊Q8—82no—
160C4×Q8—44no—
161C4×Q8—82no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
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⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—82no—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2411yescharacteristic
206C4⋊C8—42no—
207C4⋊C8—42no—
208C4⋊C8—82no—
209C4⋊C8—82no—
210C22⋊C8—82no—
211C22⋊C8—82no—
212C22⋊C8—82no—
213C22⋊C8—82no—
214#97—81no—
215#97—81no—
216C4×C8—42no—
217C4×C8—42no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—21no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237C24C2×C22⋊C412yes—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4#13112yes—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—82no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—42no—
334C42—42no—
335C42—41no—
336C42—42no—
337C42—44no—
338C42—42no—
339C42—42no—
340C42—82no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23C22×C22⋊C411yescharacteristic
349Q8—82no—
350Q8—82no—
351Q8—84no—
352D4—44no—
353C2×C4—21no—
354C2×C4—21no—
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—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4#41312yes—
402C2×C4#42112yes—
403C2×C4#42112yes—
404C2×C4C2×#13111yescharacteristic
405C2×C4#56811yescharacteristic
406C2×C4#57711yescharacteristic
407C2×C4#57711yescharacteristic
408C2×C4#57811yescharacteristic
409C2×C4#57811yescharacteristic
410C8—81no—
411C8—81no—
412C8—82no—
413C8—82no—
414C8—81no—
415C8—81no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22#288011yescharacteristic
424C22#288311yescharacteristic
425C22C2×#41311yescharacteristic
426C22#289811yescharacteristic
427C22#289811yescharacteristic
428C22C2×#42111yescharacteristic
429C22C2×#42111yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—42no—
440C4—41no—
441C4—42no—
442C4—42no—
443C4—82no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—82no—
448C4#153514yes—
449C2—82no—
450C2#1742911yescharacteristic
451C2#1752311yescharacteristic
452C2#1752511yescharacteristic
453C2#1760011yescharacteristic
454C2#1760011yescharacteristic
455C2C2×#153511yescharacteristic
456C2#1777811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7721C2semidirect
5. #7715C2semidirect
6. #7682C2semidirect
7. #7682C2semidirect
8. #7648C2semidirect
9. #7636C2semidirect
10. #7600C2semidirect
11. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#364542
#220122
#77212
#77152
#76822
#76482
#76362
#76002
#75682