Finite Group Databasebeta

← #164359

Subgroups — #164359

Order 512.

792 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#36312C211yesmaximal, characteristic
2#20573C211yesmaximal, characteristic, complemented
3#20572C211yesmaximal, characteristic, complemented
4#7692C212yesmaximal
5#7686C212yesmaximal
6#7683C211yesmaximal, characteristic, complemented
7#7680C211yesmaximal, characteristic, complemented
8#7635C211yesmaximal, characteristic
9#7630C212yesmaximal, complemented
10#7567C211yesmaximal, characteristic
11#7562C212yesmaximal, complemented
12C2×#446—21no—
13C2×C8×Q8—21no—
14#2860C2212yes—
15#2859C2211yescharacteristic
16#2582C411yescharacteristic
17#2538C411yescharacteristic
18#2529—22no—
19#2519—22no—
20#2508C412yes—
21#2496—21no—
22#2448C412yes—
23#2423—21no—
24#2362—21no—
25#2354C411yescharacteristic
26Q8×C4⋊C4C411yescharacteristic
27#2346—21no—
28C4×#124C2211yescharacteristic
29#2295C2212yes—
30C42×Q8C2211yescharacteristic
31#1771—22no—
32#1770—21no—
33#1770—22no—
34C8×C4⋊C4—21no—
35#1767C2211yescharacteristic
36#1767C2211yescharacteristic
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39#1745C2212yes—
40#1744C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C2×#126—22no—
54C2×#125—21no—
55C2×#124C2×C411yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×#117—21no—
64#446—28no—
65C8×Q8—28no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×#97—22no—
71C2×#97—21no—
72C2×#97—22no—
73C2×#97—21no—
74C2×C4×C8—21no—
75C2×C4×C8—22no—
76C2×C4×C8—21no—
77C2×C4×C8—22no—
78#400—22no—
79#399—22no—
80#398—22no—
81#398—22no—
82#397—22no—
83#396—22no—
84#391—21no—
85#391C2×C412yes—
86#391C2311yescharacteristic
87#390C2×C412yes—
88#390C2×C412yes—
89#390C2311yescharacteristic
90#389—21no—
91#389—21no—
92#389—21no—
93#389—22no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2312yes—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—21no—
104#388—21no—
105#388—22no—
106#388—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#384—21no—
111#384C2×C411yescharacteristic
112#384C2×C411yescharacteristic
113#382—21no—
114#382—22no—
115#382C2×C411yescharacteristic
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129#376—22no—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—21no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2312yes—
142C22×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×D4C22×C411yescharacteristic
145C23×C4—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152#126—82no—
153#126—82no—
154#125—81no—
155#125—81no—
156#124—22no—
157#124—41no—
158C22⋊Q8—44no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×D4—42no—
166#117—44no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—28no—
214C4⋊C8—24no—
215C4⋊C8—24no—
216C4⋊C8—28no—
217#97—28no—
218#97—81no—
219#97—82no—
220#97—81no—
221C4×C8—28no—
222C4×C8—24no—
223C4×C8—24no—
224C4×C8—82no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C24—21no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8#10414yes—
247C2×Q8#13112yes—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—24no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—24no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4#10414yes—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C42—22no—
324C42—21no—
325C42—22no—
326C42—21no—
327C42—21no—
328C42—22no—
329C42—22no—
330C42—21no—
331C42—21no—
332C42—22no—
333C42—22no—
334C42—21no—
335C42—22no—
336C42—24no—
337C42—22no—
338C42—22no—
339C42—24no—
340C42—42no—
341C42—41no—
342C42—44no—
343C42—42no—
344C42—44no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23C22×C22⋊C411yescharacteristic
353Q8—28no—
354Q8—42no—
355Q8—42no—
356D4—81no—
357D4—81no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—84no—
399C2×C4#41312yes—
400C2×C4C2×#10414yes—
401C2×C4C2×#10414yes—
402C2×C4C2×#10412yes—
403C2×C4C2×#10412yes—
404C2×C4C2×#13111yescharacteristic
405C2×C4#56811yescharacteristic
406C2×C4#57611yescharacteristic
407C2×C4#57711yescharacteristic
408C2×C4#57711yescharacteristic
409C2×C4#57811yescharacteristic
410C8—42no—
411C8—44no—
412C8—42no—
413C8—44no—
414C8—42no—
415C8—42no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22#288011yescharacteristic
424C22#288411yescharacteristic
425C22C2×#41311yescharacteristic
426C22#289711yescharacteristic
427C22#289811yescharacteristic
428C22C22×#10411yescharacteristic
429C22C22×#10411yescharacteristic
430C4—21no—
431C4—21no—
432C4—24no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—24no—
438C4—42no—
439C4—41no—
440C4—41no—
441C4—42no—
442C4—41no—
443C4—42no—
444C4—82no—
445C4—81no—
446C4—82no—
447C4—81no—
448C4#152414yes—
449C2—161no—
450C2#1742311yescharacteristic
451C2#1751911yescharacteristic
452C2#1752011yescharacteristic
453C2#1754911yescharacteristic
454C2#1755011yescharacteristic
455C2C2×#152411yescharacteristic
456C2#1778111yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20573C2semidirect
3. #20572C2semidirect
6. #7683C2semidirect
7. #7680C2semidirect
9. #7630C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#363122
#205732
#205722
#76922
#76862
#76832
#76802
#76352
#76302
#75672
#75622