Finite Group Databasebeta

← #162425

Subgroups — #162425

Order 512.

788 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35176C211yesmaximal, characteristic
2#22696C211yesmaximal, characteristic
3#22008C211yesmaximal, characteristic
4#7727C212yesmaximal, complemented
5#7715C212yesmaximal, complemented
6#7667C212yesmaximal, complemented
7#7655C212yesmaximal, complemented
8#7590C211yesmaximal, characteristic, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7571C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12C2×C8×D4—21no—
13#2857C2212yes—
14#2855C2212yes—
15#2667—22no—
16#2552C411yescharacteristic
17#2546C411yescharacteristic
18#2542C411yescharacteristic
19#2538C411yescharacteristic
20#2522—22no—
21#2507—21no—
22#2454—21no—
23#2429C412yes—
24#2423C412yes—
25#2330—21no—
26#2319—21no—
27C4×C4⋊Q8C2211yescharacteristic
28#2295C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1939—21no—
33#1923—22no—
34#1850—22no—
35#1831C2211yescharacteristic
36#1817C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1744C2211yescharacteristic
42C2×#336C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1272C2212yes—
47#1272C2212yes—
48#1272C2211yescharacteristic
49#1272C2212yes—
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2211yescharacteristic
53C23×C8—22no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C412yes—
56C2×#125—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C412yes—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C22×C42C2312yes—
66C8×D4—44no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#399—22no—
79#399—22no—
80#398—22no—
81#398—22no—
82#398—22no—
83#398—22no—
84#396—22no—
85#396—22no—
86#391C2×C412yes—
87#391C2×C411yescharacteristic
88#391C2312yes—
89#391C2311yescharacteristic
90#390—21no—
91#390—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389C2×C412yes—
102#389C2×C411yescharacteristic
103#389C2311yescharacteristic
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#387—21no—
109#387—21no—
110#387—22no—
111#385—22no—
112#384—21no—
113#384C2×C411yescharacteristic
114#384C2×C412yes—
115#381—22no—
116#380—22no—
117#380—22no—
118#380C2×C412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129#376—21no—
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—22no—
134#336—21no—
135#336C2311yescharacteristic
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141C22×Q8C22×C411yescharacteristic
142C22×Q8C22×C412yes—
143C22×D4C22⋊C411yescharacteristic
144C23×C4C22×C412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—44no—
154C22×C8—44no—
155C22×C8—82no—
156C4⋊Q8—24no—
157C4⋊Q8—82no—
158#125—44no—
159C22⋊Q8—82no—
160C22⋊Q8—82no—
161C22⋊Q8—82no—
162C4×Q8—24no—
163C4×Q8—82no—
164C4×D4—24no—
165C4×D4—42no—
166C4×D4SD1614yes—
167C4×D4Q1614yes—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—82no—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2411yescharacteristic
212C4⋊C8—42no—
213C4⋊C8—42no—
214C4⋊C8—42no—
215C4⋊C8—82no—
216C4⋊C8—82no—
217C22⋊C8—44no—
218C22⋊C8—82no—
219C22⋊C8—82no—
220C4×C8—42no—
221C4×C8—42no—
222C4×C8—81no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239C24C2×C22⋊C412yes—
240C2×Q8—22no—
241C2×Q8—24no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—41no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4#10314yes—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×SD1614yes—
272C22×C4C2×Q1614yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—88no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—81no—
310C4⋊C4—82no—
311C4⋊C4—81no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4#10314yes—
319C22⋊C4—24no—
320C22⋊C4—42no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4#10318yes—
326C42—21no—
327C42—21no—
328C42—22no—
329C42—41no—
330C42—42no—
331C42—42no—
332C42—41no—
333C42—42no—
334C42—41no—
335C42—42no—
336C42—42no—
337C42—82no—
338C42#13114yes—
339C42C2×SD1612yes—
340C42C2×Q1612yes—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23C2×#10314yes—
348C23C22×C22⋊C411yescharacteristic
349Q8—82no—
350Q8—84no—
351D4—28no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—84no—
396C2×C4#40812yes—
397C2×C4#40812yes—
398C2×C4C2×#10312yes—
399C2×C4C2×#10314yes—
400C2×C4C2×#10312yes—
401C2×C4#42112yes—
402C2×C4C2×#13112yes—
403C2×C4C22×SD1611yescharacteristic
404C2×C4C22×Q1611yescharacteristic
405C2×C4#57711yescharacteristic
406C2×C4#57811yescharacteristic
407C8—42no—
408C8—81no—
409C8—82no—
410C8—81no—
411C8—81no—
412C8—82no—
413C22—24no—
414C22—24no—
415C22—24no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22C2×#40811yescharacteristic
421C22#288311yescharacteristic
422C22#288811yescharacteristic
423C22C22×#10311yescharacteristic
424C22#289811yescharacteristic
425C22C2×#42111yescharacteristic
426C22#290111yescharacteristic
427C4—22no—
428C4—22no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—24no—
434C4—22no—
435C4—41no—
436C4—42no—
437C4—42no—
438C4—82no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—82no—
444C4#148114yes—
445C2—44no—
446C2#1738911yescharacteristic
447C2C2×#148111yescharacteristic
448C2#1748411yescharacteristic
449C2#1760011yescharacteristic
450C2#1761411yescharacteristic
451C2#1768411yescharacteristic
452C2#1772211yescharacteristic
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. #7727C2semidirect
5. #7715C2semidirect
6. #7667C2semidirect
7. #7655C2semidirect
8. #7590C2semidirect
9. #7577C2semidirect
10. #7571C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#351762
#226962
#220082
#77272
#77152
#76672
#76552
#75902
#75772
#75712
#75622