Finite Group Databasebeta

← #110601

Subgroups — #110601

Order 512.

1252 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1823C212yesmaximal
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#1823—24no—
15#1823C418yescomplemented
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—42no—
65C2×#95—42no—
66C2×#95C2×C412yes—
67C2×#95C2×C412yes—
68C2×#95D412yes—
69#340—216no—
70C25C2×D411yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—24no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—24no—
79C23×C4—24no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—22no—
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—41no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—42no—
102C23×C4—41no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—41no—
106C23×C4C22⋊C412yes—
107C23×C4C22⋊C412yes—
108C23×C4D812yes—
109C23×C4SD1612yes—
110C23×C4C22×C412yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4#4112yes—
116C23×C4#4112yes—
117C23×C4#4112yes—
118C22×C8—24no—
119C22×C8—44no—
120C22×C8—44no—
121C22×C8—42no—
122C22×C8—42no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4C4214yescomplemented
142C2×C4⋊C4C22⋊C418yes—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C42—44no—
150C2×C42—82no—
151C2×C42—82no—
152C22⋊C8—24no—
153C22⋊C8—24no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—48no—
162#95—44no—
163#95—48no—
164#95—41no—
165#95—41no—
166#95—82no—
167#95—82no—
168C24—22no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24C2×C22⋊C411yescharacteristic
173C24C4×D412yes—
174C24#12011yescharacteristic
175C24#12112yes—
176C24#12311yescharacteristic
177C24C2×D811yescharacteristic
178C24C2×SD1611yescharacteristic
179C24#13512yes—
180C24AGL(1,ℤ/8)11yescharacteristic
181C24AGL(1,ℤ/8)12yes—
182C24#13711yescharacteristic
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—28no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—44no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—44no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—48no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—48no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—48no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—84no—
257C22×C4—84no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—82no—
270C22×C4—84no—
271C22×C4#10214yes—
272C22×C4C2×C4211yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4#11712yes—
275C22×C4#11711yescharacteristic
276C22×C4C4×D412yes—
277C22×C4C4×D412yes—
278C22×C4C4×D412yes—
279C22×C4C4×D412yes—
280C22×C4C4×D412yes—
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12312yes—
284C22×C4#12411yescharacteristic
285C22×C4#12412yes—
286C22×C4#12612yes—
287C22×C4#12711yescharacteristic
288C2×C8—48no—
289C2×C8—44no—
290C2×C8—82no—
291C2×C8—82no—
292C2×C8—84no—
293C4⋊C4—28no—
294C4⋊C4—48no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—84no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C42—88no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—24no—
309C23—24no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—24no—
325C23—21no—
326C23—21no—
327C23—24no—
328C23C4×C22⋊C411yescharacteristic
329C23#38611yescharacteristic
330C23#38612yes—
331C23#38812yes—
332C23#39011yescharacteristic
333C23C2×#10211yescharacteristic
334C23#41611yescharacteristic
335C23#41711yescharacteristic
336C23#41811yescharacteristic
337C23C4×D812yes—
338C23C4×SD1612yes—
339C23#44012yes—
340C23#44212yes—
341C23#44711yescharacteristic
342C23#44811yescharacteristic
343C23#44912yes—
344C23#45512yes—
345C23#45612yes—
346C23#45812yes—
347C23#45812yes—
348C23#45912yes—
349C23#46012yes—
350C23#46312yes—
351C23#46412yes—
352C23#48011yescharacteristic
353C23#48111yescharacteristic
354C23#48212yes—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—42no—
387C2×C4—48no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—88no—
398C2×C4—88no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4C4×C22⋊C412yes—
417C2×C4#38812yes—
418C2×C4#38812yes—
419C2×C4#39012yes—
420C8—84no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—22no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—24no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22#143711yescharacteristic
450C22#160712yes—
451C22#160812yes—
452C22#160914yes—
453C22#161012yes—
454C22#161112yes—
455C22#161214yes—
456C22C4×#10211yescharacteristic
457C22#176211yescharacteristic
458C22#187411yescharacteristic
459C22#187511yescharacteristic
460C22#187912yes—
461C22#188012yes—
462C22#188214yes—
463C22#188511yescharacteristic
464C22#188611yescharacteristic
465C22#189011yescharacteristic
466C22#189011yescharacteristic
467C22#189014yes—
468C22#192512yes—
469C22#192612yes—
470C22#198211yescharacteristic
471C22#198311yescharacteristic
472C4—22no—
473C4—22no—
474C4—22no—
475C4—22no—
476C4—44no—
477C4—84no—
478C4—82no—
479C4—88no—
480C4—84no—
481C2—21no—
482C2—22no—
483C2—22no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2#902511yescharacteristic
488C2#902611yescharacteristic
489C2#902711yescharacteristic
490C2#902811yescharacteristic
491C2#904314yes—
492C2#904414yes—
493C2#1093111yescharacteristic
494C2#1093611yescharacteristic
495C2#1095711yescharacteristic
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. #1823C4semidirect
18. #1445C4semidirect
141. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#18232
C2×#14452
#137782
C2×#12942
C2×#12802