Finite Group Databasebeta

← #110599

Subgroups — #110599

Order 512.

1252 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×#1824C212yesmaximal
2C2×#1445C211yesmaximal, characteristic
3#13778C211yesmaximal, characteristic
4C2×#1294C212yesmaximal
5C2×#1280C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#380C2212yes—
9C2×C4×C22⋊C4—21no—
10C22×#95—21no—
11C22×#95—21no—
12C22×#95C2211yescharacteristic
13C22×#95C2211yescharacteristic
14#1824—24no—
15#1824C418yescomplemented
16C2×#340—24no—
17#1445—22no—
18#1445C414yescomplemented
19#1436—24no—
20C22⋊#95—21no—
21#1294—24no—
22#1280—22no—
23C24×C4—21no—
24C24×C4—21no—
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C24×C4C2311yescharacteristic
28C23×C8—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4C2×C414yes—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4D414yes—
38C22×C22⋊C4—22no—
39C22×C42—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8C2×C414yes—
46#380—22no—
47#380—22no—
48#380C2×C414yes—
49#380D414yes—
50C4×C22⋊C4—42no—
51C4×C22⋊C4—42no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—24no—
59C2×#95—24no—
60C2×#95—21no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—44no—
64C2×#95—44no—
65C2×#95C2×C412yes—
66C2×#95C2×C412yes—
67C2×#95D412yes—
68#340—216no—
69C25C2×D411yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—24no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—24no—
78C23×C4—24no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—41no—
97C23×C4—42no—
98C23×C4—42no—
99C23×C4—41no—
100C23×C4—42no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4C22⋊C412yes—
104C23×C4C22⋊C412yes—
105C23×C4SD1612yes—
106C23×C4Q1612yes—
107C23×C4C22×C412yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4#4112yes—
113C23×C4#4112yes—
114C23×C4#4112yes—
115C22×C8—24no—
116C22×C8—44no—
117C22×C8—44no—
118C22×C8—42no—
119C22×C8—42no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4C4214yescomplemented
139C2×C4⋊C4C22⋊C418yes—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C42—44no—
147C2×C42—82no—
148C2×C42—82no—
149C22⋊C8—24no—
150C22⋊C8—24no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—48no—
159#95—48no—
160#95—44no—
161#95—41no—
162#95—41no—
163#95—82no—
164#95—82no—
165C24—22no—
166C24—22no—
167C24—22no—
168C24—22no—
169C24C2×C22⋊C411yescharacteristic
170C24C4×D412yes—
171C24#12011yescharacteristic
172C24#12112yes—
173C24#12311yescharacteristic
174C24C2×SD1611yescharacteristic
175C24C2×Q1611yescharacteristic
176C24#13512yes—
177C24AGL(1,ℤ/8)11yescharacteristic
178C24#13711yescharacteristic
179C24#13712yes—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—28no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—44no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—48no—
217C22×C4—48no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—48no—
222C22×C4—48no—
223C22×C4—48no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—84no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4—84no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—84no—
262C22×C4#10314yes—
263C22×C4C2×C4211yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4#11712yes—
266C22×C4#11711yescharacteristic
267C22×C4C4×D412yes—
268C22×C4C4×D412yes—
269C22×C4C4×D412yes—
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4#12112yes—
273C22×C4#12112yes—
274C22×C4#12312yes—
275C22×C4#12411yescharacteristic
276C22×C4#12412yes—
277C22×C4#12612yes—
278C22×C4#12711yescharacteristic
279C2×C8—48no—
280C2×C8—44no—
281C2×C8—82no—
282C2×C8—82no—
283C2×C8—84no—
284C4⋊C4—28no—
285C4⋊C4—48no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—84no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C42—88no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—24no—
306C23—24no—
307C23—24no—
308C23—24no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—24no—
316C23—21no—
317C23—21no—
318C23—24no—
319C23C4×C22⋊C411yescharacteristic
320C23#38611yescharacteristic
321C23#38612yes—
322C23#38812yes—
323C23#39011yescharacteristic
324C23C2×#10311yescharacteristic
325C23#41611yescharacteristic
326C23#41711yescharacteristic
327C23#41911yescharacteristic
328C23C4×SD1612yes—
329C23C4×Q1612yes—
330C23#44012yes—
331C23#44112yes—
332C23#45011yescharacteristic
333C23#45111yescharacteristic
334C23#45212yes—
335C23#45312yes—
336C23#45412yes—
337C23#45812yes—
338C23#45812yes—
339C23#46112yes—
340C23#46212yes—
341C23#46312yes—
342C23#46412yes—
343C23#48311yescharacteristic
344C23#48411yescharacteristic
345C23#48512yes—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—42no—
380C2×C4—48no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—88no—
389C2×C4—88no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—88no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—88no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4C4×C22⋊C412yes—
406C2×C4#38812yes—
407C2×C4#38812yes—
408C2×C4#39012yes—
409C8—84no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—24no—
425C22—24no—
426C22—24no—
427C22—24no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—24no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22#143711yescharacteristic
439C22#161312yes—
440C22#161412yes—
441C22#161514yes—
442C22#161612yes—
443C22#161712yes—
444C22#161814yes—
445C22C4×#10311yescharacteristic
446C22#176311yescharacteristic
447C22#187211yescharacteristic
448C22#187311yescharacteristic
449C22#187712yes—
450C22#187812yes—
451C22#188314yes—
452C22#188711yescharacteristic
453C22#188811yescharacteristic
454C22#189011yescharacteristic
455C22#189011yescharacteristic
456C22#189014yes—
457C22#192712yes—
458C22#192812yes—
459C22#198111yescharacteristic
460C22#198311yescharacteristic
461C4—22no—
462C4—22no—
463C4—22no—
464C4—22no—
465C4—44no—
466C4—84no—
467C4—82no—
468C4—88no—
469C4—84no—
470C2—21no—
471C2—22no—
472C2—22no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2#902111yescharacteristic
477C2#902211yescharacteristic
478C2#902311yescharacteristic
479C2#902411yescharacteristic
480C2#904514yes—
481C2#904614yes—
482C2#1093211yescharacteristic
483C2#1093611yescharacteristic
484C2#1095811yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×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
15. #1824C4semidirect
18. #1445C4semidirect
138. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#18242
C2×#14452
#137782
C2×#12942
C2×#12802