Finite Group Databasebeta

← #216198

Subgroups — #216198

Order 512.

1016 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
1C2×#2332C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic, complemented
3#21912C211yesmaximal, characteristic, complemented
4#9878C212yesmaximal, complemented
5#9651C212yesmaximal
6#9082C212yesmaximal, complemented
7#9080C212yesmaximal
8#8560C212yesmaximal, complemented
9#8557C212yesmaximal
10C22×C22⋊Q8C411yescharacteristic
11C2×#431C2212yes—
12#2332—24no—
13#2332—24no—
14C2×#391C411yescharacteristic
15C2×#391C2211yescharacteristic
16C2×#388C411yescharacteristic
17C2×#388C411yescharacteristic
18C2×#388C2211yescharacteristic
19C2×#388C2211yescharacteristic
20C2×#387C411yescharacteristic
21C2×#387C411yescharacteristic
22C2×#387C411yescharacteristic
23C2×#387C2211yescharacteristic
24C2×#381C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#1939—24no—
29#1916—24no—
30#1817C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×Q8C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8C818yescomplemented
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2311yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63#431—44no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70C2×C4×C8—22no—
71C2×C4×C8—22no—
72#391—22no—
73#391—22no—
74#391—24no—
75#388—24no—
76#388—22no—
77#388—22no—
78#388—22no—
79#388—22no—
80#388—24no—
81#387—24no—
82#387—24no—
83#387—22no—
84#387—22no—
85#387—24no—
86#381—24no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D414yes—
90C4×C4⋊C4D414yes—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C411yescharacteristic
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107#336—22no—
108#336—22no—
109#336—22no—
110#336—24no—
111#336—22no—
112#324—44no—
113#323—44no—
114C25C22×C411yescharacteristic
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—81no—
118C22×Q8—81no—
119C22×Q8C2×C814yes—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—81no—
123C23×C4—81no—
124C23×C4C2×C814yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—24no—
140C22×C8—24no—
141C22⋊Q8—232no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—81no—
154C2×C4⋊C4—81no—
155C2×C4⋊C4C22⋊C414yes—
156C2×C4⋊C4C22⋊C414yes—
157C2×C4⋊C4C2×C814yes—
158C2×C4⋊C4C2×C814yes—
159C2×C4⋊C4C2×C814yes—
160C2×C4⋊C4M1614yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—81no—
174C2×C22⋊C4—81no—
175C2×C22⋊C4C2×C814yes—
176C2×C22⋊C4C2×C814yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—42no—
183C2×C42—81no—
184C2×C42—81no—
185C2×C42—81no—
186C2×C42—81no—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C4⋊C8—44no—
194C4⋊C8—82no—
195C4⋊C8—82no—
196C4⋊C8—82no—
197C22⋊C8—24no—
198C22⋊C8—24no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C4×C8—44no—
202C4×C8—82no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95C22⋊C414yes—
218#95C22⋊C414yes—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—22no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24C22×C812yes—
228C24C23×C411yescharacteristic
229C2×Q8—28no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—84no—
233C2×Q8—84no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—28no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C22×C812yes—
291C22×C4C22×C812yes—
292C22×C4C22×C812yes—
293C22×C4C22×C812yes—
294C22×C4C22×C812yes—
295C22×C4C22×C812yes—
296C22×C4C2×M1612yes—
297C22×C4C2×M1612yes—
298C22×C4C2×M1612yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4H(5,𝔽2)12yes—
302C22×C42−1+412yes—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—44no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4C22⋊C8116yes—
318C22⋊C4—28no—
319C22⋊C4—28no—
320C22⋊C4—44no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—42no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C42—22no—
327C42—22no—
328C42—24no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42—82no—
335C42—82no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—24no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—82no—
354C23—82no—
355C23—82no—
356C23—82no—
357C23C22×C22⋊C411yescharacteristic
358C23#52012yes—
359C23C23×C811yescharacteristic
360C23C22×M1611yescharacteristic
361C23#56812yes—
362Q8—88no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—88no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4C2×C22⋊C814yes—
426C2×C4C2×C22⋊C814yes—
427C2×C4C2×C22⋊C814yes—
428C2×C4#53511yescharacteristic
429C2×C4#53611yescharacteristic
430C2×C4#53712yes—
431C8—84no—
432C8—84no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—84no—
445C22—84no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22#231111yescharacteristic
451C22C22×C22⋊C811yescharacteristic
452C22#287911yescharacteristic
453C22#288311yescharacteristic
454C22#288411yescharacteristic
455C22#288811yescharacteristic
456C22#288911yescharacteristic
457C22#298812yes—
458C22#298912yes—
459C4—24no—
460C4—21no—
461C4—22no—
462C4—21no—
463C4—24no—
464C4—24no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—81no—
475C2—22no—
476C2—22no—
477C2—82no—
478C2#1735011yescharacteristic
479C2#1735211yescharacteristic
480C2#1740611yescharacteristic
481C2#1740911yescharacteristic
482C2#2184711yescharacteristic
483C2#2187711yescharacteristic
484C2#2188111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21951C2semidirect
3. #21912C2semidirect
4. #9878C2semidirect
6. #9082C2semidirect
8. #8560C2semidirect
43. C2×C22⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23322
#219512
#219122
#98782
#96512
#90822
#90802
#85602
#85572