Finite Group Databasebeta

← #359989

Subgroups — #359989

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#433C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3C2×#1825C212yesmaximal
4#14343C211yesmaximal, characteristic
5#11228C212yesmaximal, complemented
6#11227C212yesmaximal, complemented
7#10881C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10730C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#433—28no—
14C2×#433—24no—
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2316—24no—
20#2316C418yescomplemented
21C2×#389—22no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#380—22no—
25C2×#380C2212yes—
26#1825—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35#1445—22no—
36#1440—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#117—24no—
45C2×#117—24no—
46C2×#117—22no—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117C2×C414yes—
50C2×#117C2×C414yes—
51C2×#117D414yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C414yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D412yes—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#433—232no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8D418yes—
77C2×#97—24no—
78C2×#97—22no—
79C2×C4×C8—24no—
80C2×C4×C8—22no—
81#389—216no—
82#389—24no—
83#389—24no—
84#389C2×C418yes—
85#389C2×C418yes—
86#380—216no—
87#380—24no—
88#380—24no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104C25C2×D411yescharacteristic
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—82no—
117C23×C4—82no—
118C23×C4C22×C412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D414yes—
124C23×C4#4114yes—
125C23×C4C2411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8C2×D414yes—
139C22×C8C2×D414yes—
140#117—216no—
141#117—28no—
142#117—28no—
143#117—216no—
144#117—44no—
145#117—44no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—216no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—216no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4C22⋊C418yes—
180C2×C4⋊C4C22×C414yes—
181C2×C4⋊C4C22×C414yes—
182C2×C4⋊C4#4114yes—
183C2×C4⋊C4#4114yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4C22⋊C418yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—44no—
198C2×C42C22×C412yes—
199C2×C42C22×C412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C4⋊C8—216no—
203C22⋊C8—216no—
204#97—28no—
205C4×C8—28no—
206#95—28no—
207#95—28no—
208#95—44no—
209#95—82no—
210#95—82no—
211#95—82no—
212#95—82no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24C2×C22⋊C412yes—
221C24#12114yes—
222C24C22×D411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—216no—
255C22×C4—24no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—88no—
281C22×C4—88no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4C2×C22⋊C414yes—
285C22×C4C2×C22⋊C414yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C4×D418yes—
288C22×C4#12014yes—
289C22×C4#12114yes—
290C22×C4#12314yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D412yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C22×C4C2×#4112yes—
297C22×C42−1+412yes—
298C22×C42−1+412yes—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—216no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—216no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—88no—
331C4⋊C4—88no—
332C4⋊C4#11718yes—
333C22⋊C4—216no—
334C22⋊C4—48no—
335C22⋊C4—88no—
336C42—24no—
337C42—24no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—48no—
343C23—22no—
344C23—24no—
345C23—24no—
346C23—24no—
347C23—24no—
348C23—22no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23#38614yes—
364C23C22×C22⋊C411yescharacteristic
365C23C2×#12112yes—
366C23#52511yescharacteristic
367C23#52511yescharacteristic
368C23#53612yes—
369C23#57612yes—
370C23#57712yes—
371C23#57712yes—
372C23#57812yes—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—28no—
378C2×C4—28no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—816no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4#38614yes—
425C2×C4C2×#11712yes—
426C2×C4C2×C4×D412yes—
427C2×C4#51911yescharacteristic
428C2×C4#51911yescharacteristic
429C2×C4C2×#12011yescharacteristic
430C2×C4C2×#12311yescharacteristic
431C2×C4#52512yes—
432C2×C4#53612yes—
433C2×C4#54712yes—
434C2×C4#54712yes—
435C2×C4#54712yes—
436C2×C4#54712yes—
437C2×C4#55212yes—
438C2×C4#55212yes—
439C2×C4#55212yes—
440C2×C4#55212yes—
441C8—44no—
442C8—44no—
443C22—24no—
444C22—28no—
445C22—42no—
446C22—42no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—44no—
452C22#228011yescharacteristic
453C22C2×#38611yescharacteristic
454C22#231011yescharacteristic
455C22#241312yes—
456C22#241312yes—
457C22#289712yes—
458C22#289812yes—
459C22#306012yes—
460C22#306112yes—
461C22#306212yes—
462C22#306312yes—
463C22#311812yes—
464C22#312112yes—
465C22#312211yescharacteristic
466C22#312311yescharacteristic
467C22#312611yescharacteristic
468C22#312711yescharacteristic
469C22#324112yes—
470C22#324212yes—
471C22#324312yes—
472C22#324412yes—
473C4—24no—
474C4—48no—
475C4—44no—
476C4—44no—
477C4—84no—
478C4—162no—
479C4—162no—
480C4#234412yes—
481C4#234412yes—
482C4#240612yes—
483C4#248412yes—
484C2—44no—
485C2#1387011yescharacteristic
486C2#2074411yescharacteristic
487C2#2074511yescharacteristic
488C2#2097612yes—
489C2#2184312yes—
490C2#2461312yes—
491C2#2461512yes—
492C2#2461712yes—
493C2#2461912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11228C2semidirect
6. #11227C2semidirect
7. #10881C2semidirect
8. #10880C2semidirect
20. #2316C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4332
C2×#23162
C2×#18252
#143432
#112282
#112272
#108812
#108802
#107302