Finite Group Databasebeta

← #215174

Subgroups — #215174

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C4×#432C211yesmaximal, characteristic, complemented
3#20463C211yesmaximal, characteristic, complemented
4#9687C214yesmaximal, complemented
5#9686C212yesmaximal, complemented
6#9685C212yesmaximal, complemented
7#8336C214yesmaximal, complemented
8C2×#529C2212yes—
9C2×#432—21no—
10C2×#431—21no—
11#2861C2212yes—
12#2860—21no—
13#2860—21no—
14#2859C2211yescharacteristic
15#2858C2211yescharacteristic
16#2857C411yescharacteristic
17#2856C411yescharacteristic
18#2854—21no—
19#2854—21no—
20#2850C412yes—
21C4×C4⋊Q8C411yescharacteristic
22C4×#127C411yescharacteristic
23C4×#124C412yes—
24C4×#121—21no—
25C4×#121—21no—
26#2293—21no—
27#2293—21no—
28C2×C43C2211yescharacteristic
29#1767—21no—
30#1767C2212yes—
31#1767C2212yes—
32C4×C4⋊C8—21no—
33#1749—21no—
34#1749C2212yes—
35#1749C2212yes—
36C4×C22⋊C8—21no—
37#1745—21no—
38#1744—21no—
39C4×#97C2212yes—
40C4×#97C2212yes—
41C2×#330C2218yes—
42C2×#329C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45#529—48no—
46C2×#126—22no—
47C2×#125C2312yes—
48C2×#124C2×C412yes—
49C2×#124C2312yes—
50C2×#123—22no—
51C2×C22⋊Q8C2×C412yes—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121C2×C412yes—
55C2×C4×Q8C2×C412yes—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yes—
60C2×#117—22no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#432—28no—
69#431—28no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×#97—22no—
75C2×#97—22no—
76C2×#97C2314yes—
77C2×#97C2314yes—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80#391C2×C412yes—
81#391C2×C412yes—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390C2×C412yes—
91#390C2×C412yes—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#386—22no—
109#386—22no—
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#382C2×C412yes—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43C2311yescharacteristic
126C43C2311yescharacteristic
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#330—48no—
132#329—48no—
133C22×Q8C22×C412yes—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C22×C412yes—
138C23×C4—22no—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C22×C8—24no—
147C22×C8—24no—
148C4⋊Q8—42no—
149#127—42no—
150#126—48no—
151#125—28no—
152#124—28no—
153#124—44no—
154#123—48no—
155C22⋊Q8—44no—
156#121—44no—
157#121—44no—
158#121—44no—
159C4×Q8—44no—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—48no—
163C4×D4—44no—
164#117—48no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—28no—
225C4⋊C8—28no—
226C22⋊C8—28no—
227C22⋊C8—28no—
228#97—28no—
229#97—24no—
230#97—28no—
231#97—24no—
232#97—44no—
233C4×C8—28no—
234C4×C8—44no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241C24—22no—
242C24C2×C22⋊C411yescharacteristic
243C2×Q8—24no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×D4—24no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4#10414yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C2×C8—28no—
300C2×C8—28no—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—44no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—44no—
355C42#10414yes—
356C42H(5,𝔽2)14yes—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23C2×#10412yes—
371C23C22×C22⋊C411yescharacteristic
372Q8—84no—
373D4—84no—
374D4—84no—
375D4—88no—
376D4—84no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—28no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—88no—
424C2×C4#41312yes—
425C2×C4#41312yes—
426C2×C4C2×#10412yes—
427C2×C4C2×#10412yes—
428C2×C4C2×#10412yes—
429C2×C4C2×#10412yes—
430C2×C4C2×#10412yes—
431C2×C4#42112yes—
432C2×C4#42112yes—
433C2×C4#51812yes—
434C2×C4#53512yes—
435C2×C4#53512yes—
436C8—48no—
437C8—48no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22#230911yescharacteristic
453C22C2×#41311yescharacteristic
454C22C2×#41311yescharacteristic
455C22C22×#10411yescharacteristic
456C22C22×#10411yescharacteristic
457C22C2×#42111yescharacteristic
458C22C2×#42111yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4#152414yes—
476C2—42no—
477C2—162no—
478C2C2×#152411yescharacteristic
479C2#1764911yescharacteristic
480C2#1765111yescharacteristic
481C2#1766811yescharacteristic
482C2#2180011yescharacteristic
483C2#2181511yescharacteristic
484C2#2181511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#432C2semidirect
3. #20463C2semidirect
4. #9687C2semidirect
5. #9686C2semidirect
6. #9685C2semidirect
7. #8336C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C4×#4322
#204632
#96872
#96862
#96852
#83362