Finite Group Databasebeta

← #115152

Subgroups — #115152

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic
2C2×#1823C212yesmaximal
3C22×#324C211yesmaximal, characteristic
4#14339C211yesmaximal, characteristic
5C2×#1294C212yesmaximal
6C23×C4⋊C4—21no—
7C23×C4⋊C4C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10C2×#398—24no—
11C2×#398—24no—
12C2×#398—22no—
13C2×#398C414yescomplemented
14C2×#381—21no—
15C2×#380C2212yes—
16C22×#95—21no—
17C22×#95C2211yescharacteristic
18#1823—24no—
19#1823—24no—
20C2×#340—24no—
21C2×#324—24no—
22C2×#324—22no—
23#1441—24no—
24C22⋊#95—21no—
25#1294—24no—
26C24×C4—21no—
27C24×C4—21no—
28C24×C4D411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4C2311yescharacteristic
31C23×C8—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—28no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—44no—
47C22×C4⋊C4—42no—
48C22×C4⋊C4—42no—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4D414yes—
53C22×C4⋊C4D414yes—
54C22×C22⋊C4—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8D414yes—
61#398—216no—
62#398—28no—
63#398—28no—
64#381—42no—
65#381—42no—
66#380—22no—
67#380—22no—
68#380—22no—
69#380—22no—
70C2×#95—28no—
71C2×#95—24no—
72C2×#95—22no—
73C2×#95—21no—
74C2×#95—24no—
75C2×#95—22no—
76C2×#95—24no—
77C2×#95C2×C412yes—
78C2×#95D414yes—
79#340—216no—
80#324—28no—
81C25C2×D411yescharacteristic
82C23×C4—28no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—24no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—24no—
100C23×C4—44no—
101C23×C4—44no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—42no—
105C23×C4—44no—
106C23×C4—41no—
107C23×C4—42no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—41no—
114C23×C4—41no—
115C23×C4C22⋊C412yes—
116C23×C4C22⋊C412yes—
117C23×C4D812yes—
118C23×C4SD1612yes—
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4#4112yes—
126C23×C4#4112yes—
127C22×C8—24no—
128C22×C8—48no—
129C22×C8—42no—
130C22×C8—42no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—216no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—48no—
151C2×C4⋊C4—416no—
152C2×C4⋊C4—416no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4C22⋊C414yes—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C22⋊C8—24no—
178C22⋊C8—24no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181#95—216no—
182#95—22no—
183#95—24no—
184#95—48no—
185#95—42no—
186#95—42no—
187#95—84no—
188#95C22⋊C414yes—
189C24—24no—
190C24—24no—
191C24C2×C22⋊C411yescharacteristic
192C24#12012yes—
193C24#12011yescharacteristic
194C24#12112yes—
195C24#12311yescharacteristic
196C24C2×D811yescharacteristic
197C24C2×SD1611yescharacteristic
198C24#13512yes—
199C24AGL(1,ℤ/8)11yescharacteristic
200C24AGL(1,ℤ/8)12yes—
201C24#13711yescharacteristic
202C22×C4—28no—
203C22×C4—28no—
204C22×C4—216no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—416no—
232C22×C4—48no—
233C22×C4—48no—
234C22×C4—44no—
235C22×C4—48no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—48no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—82no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—82no—
281C22×C4—84no—
282C22×C4#10214yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4#12012yes—
288C22×C4#12011yescharacteristic
289C22×C4#12012yes—
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4#12112yes—
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4#12312yes—
296C22×C4#12311yescharacteristic
297C22×C4#12412yes—
298C22×C4#12712yes—
299C2×C8—48no—
300C2×C8—44no—
301C2×C8—84no—
302C2×C8—84no—
303C4⋊C4—28no—
304C4⋊C4—432no—
305C4⋊C4—48no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—84no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C23—28no—
319C23—28no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—24no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—24no—
329C23—24no—
330C23—24no—
331C23—24no—
332C23#37911yescharacteristic
333C23#38611yescharacteristic
334C23#38611yescharacteristic
335C23#39212yes—
336C23#39412yes—
337C23C2×#10211yescharacteristic
338C23#41611yescharacteristic
339C23#41711yescharacteristic
340C23#41811yescharacteristic
341C23#44712yes—
342C23#44711yescharacteristic
343C23#44811yescharacteristic
344C23#44912yes—
345C23#44912yes—
346C23#45012yes—
347C23#45212yes—
348C23#45614yes—
349C23#45814yes—
350C23#46512yes—
351C23#46612yes—
352C23#46812yes—
353C23#46912yes—
354C23#48011yescharacteristic
355C23#48111yescharacteristic
356C23#48212yes—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—416no—
380C2×C4—416no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—82no—
408C2×C4—88no—
409C2×C4—88no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4#38612yes—
414C2×C4#39012yes—
415C2×C4#39212yes—
416C2×C4#39412yes—
417C8—84no—
418C22—22no—
419C22—22no—
420C22—24no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—24no—
426C22—24no—
427C22—24no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22#143811yescharacteristic
440C22#160918yes—
441C22#161014yes—
442C22#161114yes—
443C22#178611yescharacteristic
444C22#178811yescharacteristic
445C22#187411yescharacteristic
446C22#187511yescharacteristic
447C22#188612yes—
448C22#189012yes—
449C22#189311yescharacteristic
450C22#189511yescharacteristic
451C22#199912yes—
452C22#200012yes—
453C22#200514yes—
454C22#202214yes—
455C22#203312yes—
456C22#203412yes—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—22no—
461C4—48no—
462C4—44no—
463C4—84no—
464C4—82no—
465C4—164no—
466C2—21no—
467C2—21no—
468C2—21no—
469C2—22no—
470C2—22no—
471C2—21no—
472C2#902612yes—
473C2#902712yes—
474C2#991914yes—
475C2#992714yes—
476C2#1096911yescharacteristic
477C2#1097811yescharacteristic
478C2#1099211yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. C2×#398C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3982
C2×#18232
C22×#3242
#143392
C2×#12942