Finite Group Databasebeta

← #224617

Subgroups — #224617

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36590C211yesmaximal, characteristic
2#21345C211yesmaximal, characteristic, complemented
3#20520C211yesmaximal, characteristic, complemented
4#9688C212yesmaximal
5#9687C212yesmaximal
6#9686C211yesmaximal, characteristic, complemented
7#9686C211yesmaximal, characteristic, complemented
8#8580C212yesmaximal, complemented
9#8579C211yesmaximal, characteristic
10#8227C212yesmaximal, complemented
11C4×#329C211yesmaximal, characteristic
12C2×#446—21no—
13#2861C2212yes—
14#2859C2211yescharacteristic
15#2594—22no—
16#2582C411yescharacteristic
17#2538C411yescharacteristic
18#2525—22no—
19#2514C412yes—
20#2513C412yes—
21#2494—21no—
22#2421—21no—
23#2362—21no—
24#2355C411yescharacteristic
25#2354C411yescharacteristic
26#2347—21no—
27C4×#125C2212yes—
28C4×#124C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1771—22no—
34C8×C4⋊C4—21no—
35#1767C2211yescharacteristic
36#1767C2212yes—
37#1767C2211yescharacteristic
38#1767C2212yes—
39#1745C2212yes—
40C4×#97C2211yescharacteristic
41C4×#97C2211yescharacteristic
42C4×#97C2212yes—
43C2×#330C2212yes—
44C2×#330C2212yes—
45C2×#330C2212yes—
46C2×#330C2212yes—
47C2×#329C2211yescharacteristic
48C2×#329C2211yescharacteristic
49C2×#329C2212yes—
50#1295C2211yescharacteristic
51#1295C2211yescharacteristic
52#1295C2212yes—
53C2×#125C2312yes—
54C2×#124C2311yescharacteristic
55C2×C22⋊Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×#117—21no—
62#446—28no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×#97—22no—
68C2×#97—21no—
69C2×#97—21no—
70C2×#97C2311yescharacteristic
71C2×#97C2311yescharacteristic
72C2×#97C2312yes—
73C2×#97C2312yes—
74C2×#97C2311yescharacteristic
75C2×#97C2311yescharacteristic
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—22no—
79#401—22no—
80#400—22no—
81#398—22no—
82#397—22no—
83#397—22no—
84#396—22no—
85#391—21no—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2×C411yescharacteristic
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2×C411yescharacteristic
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389C2×C412yes—
98#389C2×C412yes—
99#388—22no—
100#388—21no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—21no—
106#388—22no—
107#388—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#384C2×C411yescharacteristic
113#384C2×C411yescharacteristic
114#383—22no—
115#382—21no—
116#382C2×C412yes—
117#382C2311yescharacteristic
118#380—21no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D412yes—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C43C2312yes—
134C43C2311yescharacteristic
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—21no—
141#330—24no—
142#330—24no—
143#330—42no—
144#330—42no—
145#329—22no—
146#329—22no—
147#329—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22×C411yescharacteristic
150C22×D4C22×C411yescharacteristic
151C23×C4—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158#125—22no—
159#125—22no—
160#125—22no—
161#125—22no—
162#124—21no—
163#124—21no—
164#124—21no—
165#124—21no—
166C22⋊Q8—44no—
167C4×Q8—24no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×D4—42no—
172#117—44no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2411yescharacteristic
217C4⋊C8—24no—
218C4⋊C8—28no—
219C4⋊C8—42no—
220C4⋊C8—44no—
221#97—28no—
222#97—21no—
223#97—21no—
224#97—22no—
225#97—22no—
226#97—22no—
227#97—22no—
228#97—21no—
229#97—21no—
230#97—21no—
231#97—21no—
232#97—42no—
233#97—42no—
234#97#4112yes—
235#97#4112yes—
236C4×C8—24no—
237C4×C8—42no—
238C4×C8—44no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24—21no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—81no—
325C4⋊C4—82no—
326C4⋊C4—81no—
327C4⋊C4#10414yes—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—44no—
335C22⋊C4—81no—
336C22⋊C4—82no—
337C22⋊C4—81no—
338C22⋊C4—82no—
339C42—22no—
340C42—22no—
341C42—24no—
342C42—22no—
343C42—42no—
344C42—41no—
345C42—41no—
346C42—42no—
347C42—42no—
348C42—41no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—44no—
353C42—82no—
354C42C2×#4111yescharacteristic
355C42C2×#4111yescharacteristic
356C42H(5,𝔽2)11yescharacteristic
357C422−1+411yescharacteristic
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C22×C22⋊C411yescharacteristic
366Q8—44no—
367Q8—81no—
368Q8—81no—
369D4—81no—
370D4—81no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—84no—
413C2×C4#41312yes—
414C2×C4C2×#10414yes—
415C2×C4C2×#10412yes—
416C2×C4#42112yes—
417C2×C4C4×#4111yescharacteristic
418C2×C4#52011yescharacteristic
419C2×C4#54811yescharacteristic
420C2×C4#54811yescharacteristic
421C2×C4#55011yescharacteristic
422C2×C4#55011yescharacteristic
423C8—24no—
424C8—42no—
425C8—44no—
426C8—42no—
427C8—42no—
428C8—82no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#234111yescharacteristic
437C22C2×#41311yescharacteristic
438C22#288911yescharacteristic
439C22C22×#10411yescharacteristic
440C22C2×#42111yescharacteristic
441C22#290111yescharacteristic
442C22#290111yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—24no—
450C4—41no—
451C4—42no—
452C4—41no—
453C4—42no—
454C4—42no—
455C4—41no—
456C4—81no—
457C4—82no—
458C4—81no—
459C4—82no—
460C4—81no—
461C4#152914yes—
462C2—161no—
463C2C2×#152911yescharacteristic
464C2#1765311yescharacteristic
465C2#1768311yescharacteristic
466C2#1768711yescharacteristic
467C2#2253711yescharacteristic
468C2#2255011yescharacteristic
469C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21345C2semidirect
3. #20520C2semidirect
6. #9686C2semidirect
7. #9686C2semidirect
8. #8580C2semidirect
10. #8227C2semidirect

Maximal subgroups

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

subgroupindex
#365902
#213452
#205202
#96882
#96872
#96862
#85802
#85792
#82272
C4×#3292