Finite Group Databasebeta

← #216349

Subgroups — #216349

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3#20808C211yesmaximal, characteristic, complemented
4#9864C212yesmaximal, complemented
5#9674C212yesmaximal, complemented
6#9074C214yesmaximal, complemented
7#8567C214yesmaximal, complemented
8C2×#529—21no—
9C2×#431—21no—
10C2×#431C2212yes—
11#2861—21no—
12#2860—21no—
13#2860C412yes—
14#2859C411yescharacteristic
15#2858C411yescharacteristic
16#2857C411yescharacteristic
17#2856C411yescharacteristic
18#2854—21no—
19#2854C412yes—
20#2850—21no—
21C4×C4⋊Q8C2211yescharacteristic
22C4×#127C2211yescharacteristic
23C4×#124—21no—
24C4×#121—21no—
25C4×#121C2212yes—
26#2293—21no—
27#2293C2212yes—
28C2×C43C2211yescharacteristic
29#1842—21no—
30#1842C2212yes—
31#1831—21no—
32#1831—21no—
33#1797—21no—
34#1797—21no—
35#1797—21no—
36#1797C2212yes—
37C4×C4⋊C8—21no—
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2212yes—
40C4×C4⋊C8C2212yes—
41C2×D4⋊C8C2218yes—
42#1279C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45#529—48no—
46C2×#126—22no—
47C2×#125—21no—
48C2×#124—21no—
49C2×#124—21no—
50C2×#123—22no—
51C2×C22⋊Q8—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121C2×C412yes—
55C2×C4×Q8—21no—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2312yes—
60C2×#117—22no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#431—44no—
69#431—44no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2314yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8C2314yes—
80#391—21no—
81#391—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390—21no—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2×C412yes—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
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#382—21no—
113#380—21no—
114#380C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2312yes—
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—
131D4⋊C8—48no—
132C22×Q8—21no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C22×C412yes—
137C23×C4—22no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C22×C8—24no—
146C22×C8—24no—
147C4⋊Q8—24no—
148#127—24no—
149#126—48no—
150#125—44no—
151#124—44no—
152#124—44no—
153#123—48no—
154C22⋊Q8—44no—
155#121—28no—
156#121—44no—
157#121—44no—
158C4×Q8—44no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—44no—
162C4×D4—48no—
163C4×D4—42no—
164C4×D4—42no—
165#117—48no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
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×C22⋊C4—22no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—42no—
198C2×C42—42no—
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×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2411yescharacteristic
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8—44no—
230C4⋊C8—44no—
231C4⋊C8—44no—
232C4⋊C8—44no—
233C4⋊C8—44no—
234C22⋊C8—44no—
235C22⋊C8—44no—
236C4×C8—28no—
237C4×C8—44no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244C24—22no—
245C24C2×C22⋊C411yescharacteristic
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
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×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4#13114yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—44no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C42—22no—
336C42—21no—
337C42—21no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—22no—
344C42—21no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—44no—
362C42#13114yes—
363C42AGL(1,ℤ/8)12yes—
364C42AGL(1,ℤ/8)14yes—
365C42#13712yes—
366C42H(5,𝔽2)12yes—
367C422−1+412yes—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23#40812yes—
382C23C22×C22⋊C411yescharacteristic
383Q8—84no—
384D4—48no—
385D4—84no—
386D4—88no—
387D4—84no—
388C2×C4—28no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4#40812yes—
435C2×C4#40812yes—
436C2×C4#40812yes—
437C2×C4#41712yes—
438C2×C4#41712yes—
439C2×C4#41812yes—
440C2×C4#41812yes—
441C2×C4#42112yes—
442C2×C4#42112yes—
443C2×C4#52012yes—
444C2×C4#53511yescharacteristic
445C2×C4#53611yescharacteristic
446C2×C4#53712yes—
447C2×C4C2×#13112yes—
448C2×C4C2×#13112yes—
449C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
450C2×C4C2×AGL(1,ℤ/8)12yes—
451C2×C4C2×#13711yescharacteristic
452C8—84no—
453C8—84no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22#231111yescharacteristic
469C22C2×#40811yescharacteristic
470C22C2×#40811yescharacteristic
471C22C2×#41811yescharacteristic
472C22C2×#41711yescharacteristic
473C22C2×#42111yescharacteristic
474C22C2×#42111yescharacteristic
475C4—22no—
476C4—22no—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4#149514yes—
491C4#299414yes—
492C4#325312yes—
493C4#325412yes—
494C2—42no—
495C2—162no—
496C2#1746311yescharacteristic
497C2#1746611yescharacteristic
498C2C2×#149511yescharacteristic
499C2#1747811yescharacteristic
500C2#2184811yescharacteristic
501C2#2189311yescharacteristic
502C2#2190011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2856C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20808C2semidirect
3. #20808C2semidirect
4. #9864C2semidirect
5. #9674C2semidirect
6. #9074C2semidirect
7. #8567C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#208082
#98642
#96742
#90742
#85672