Finite Group Databasebeta

← #115154

Subgroups — #115154

Order 512.

1362 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#400C211yesmaximal, characteristic
2C2×#1824C212yesmaximal
3C22×#324C211yesmaximal, characteristic
4#14339C211yesmaximal, characteristic
5C2×#1294C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8C2212yes—
9C2×#400—24no—
10C2×#400—22no—
11C2×#400—24no—
12C2×#400C414yescomplemented
13C2×#381—21no—
14C2×#380C2212yes—
15C22×#95—21no—
16C22×#95—21no—
17C22×#95C2211yescharacteristic
18#1824—24no—
19#1824—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—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4D414yes—
48C22×C4⋊C4D414yes—
49C22×C22⋊C4—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8D414yes—
56#400—216no—
57#400—28no—
58#400—28no—
59#381—42no—
60#381—42no—
61#380—22no—
62#380—22no—
63#380—22no—
64#380—22no—
65C2×#95—28no—
66C2×#95—28no—
67C2×#95—24no—
68C2×#95—24no—
69C2×#95—22no—
70C2×#95—21no—
71C2×#95—24no—
72C2×#95—22no—
73C2×#95—24no—
74C2×#95C2×C412yes—
75C2×#95D414yes—
76#340—216no—
77#324—28no—
78C25C2×D411yescharacteristic
79C23×C4—22no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—22no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4—24no—
96C23×C4—44no—
97C23×C4—44no—
98C23×C4—44no—
99C23×C4—44no—
100C23×C4—48no—
101C23×C4—41no—
102C23×C4—42no—
103C23×C4—41no—
104C23×C4—42no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4C22⋊C412yes—
108C23×C4C22⋊C412yes—
109C23×C4SD1612yes—
110C23×C4Q1612yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4#4112yes—
118C23×C4#4112yes—
119C22×C8—24no—
120C22×C8—48no—
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—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C22⋊C8—24no—
165C22⋊C8—24no—
166C22⋊C8—42no—
167C22⋊C8—42no—
168#95—216no—
169#95—216no—
170#95—22no—
171#95—24no—
172#95—48no—
173#95—42no—
174#95—42no—
175#95—84no—
176#95C22⋊C414yes—
177C24—24no—
178C24—24no—
179C24C2×C22⋊C411yescharacteristic
180C24#12012yes—
181C24#12011yescharacteristic
182C24#12112yes—
183C24#12311yescharacteristic
184C24C2×SD1611yescharacteristic
185C24C2×Q1611yescharacteristic
186C24#13512yes—
187C24AGL(1,ℤ/8)11yescharacteristic
188C24#13711yescharacteristic
189C24#13712yes—
190C22×C4—28no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—48no—
229C22×C4—48no—
230C22×C4—48no—
231C22×C4—416no—
232C22×C4—48no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—82no—
250C22×C4—88no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4—88no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4#10314yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C4×D412yes—
261C22×C4C4×D412yes—
262C22×C4#12012yes—
263C22×C4#12011yescharacteristic
264C22×C4#12012yes—
265C22×C4#12112yes—
266C22×C4#12112yes—
267C22×C4#12112yes—
268C22×C4#12112yes—
269C22×C4#12112yes—
270C22×C4#12312yes—
271C22×C4#12311yescharacteristic
272C22×C4#12412yes—
273C22×C4#12712yes—
274C2×C8—48no—
275C2×C8—44no—
276C2×C8—84no—
277C2×C8—84no—
278C4⋊C4—28no—
279C4⋊C4—48no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—84no—
288C22⋊C4—82no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C22⋊C4—82no—
292C23—28no—
293C23—28no—
294C23—24no—
295C23—24no—
296C23—24no—
297C23—24no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—24no—
303C23—24no—
304C23—24no—
305C23—24no—
306C23#37911yescharacteristic
307C23#38611yescharacteristic
308C23#38611yescharacteristic
309C23#39212yes—
310C23#39412yes—
311C23C2×#10311yescharacteristic
312C23#41611yescharacteristic
313C23#41711yescharacteristic
314C23#41911yescharacteristic
315C23#44812yes—
316C23#44912yes—
317C23#45011yescharacteristic
318C23#45112yes—
319C23#45111yescharacteristic
320C23#45212yes—
321C23#45212yes—
322C23#45414yes—
323C23#45814yes—
324C23#46512yes—
325C23#46712yes—
326C23#46812yes—
327C23#47012yes—
328C23#48311yescharacteristic
329C23#48411yescharacteristic
330C23#48512yes—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—88no—
369C2×C4—88no—
370C2×C4—84no—
371C2×C4—88no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—88no—
375C2×C4—84no—
376C2×C4—82no—
377C2×C4—88no—
378C2×C4—88no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4#38612yes—
383C2×C4#39012yes—
384C2×C4#39212yes—
385C2×C4#39412yes—
386C8—84no—
387C22—22no—
388C22—22no—
389C22—24no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—24no—
395C22—24no—
396C22—24no—
397C22—24no—
398C22—24no—
399C22—24no—
400C22—24no—
401C22—24no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22#143811yescharacteristic
409C22#161518yes—
410C22#161614yes—
411C22#161714yes—
412C22#178711yescharacteristic
413C22#178811yescharacteristic
414C22#187211yescharacteristic
415C22#187311yescharacteristic
416C22#188812yes—
417C22#189012yes—
418C22#189211yescharacteristic
419C22#189411yescharacteristic
420C22#200012yes—
421C22#200112yes—
422C22#200314yes—
423C22#202214yes—
424C22#203512yes—
425C22#203612yes—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—44no—
431C4—84no—
432C4—84no—
433C4—82no—
434C4—164no—
435C2—21no—
436C2—21no—
437C2—21no—
438C2—22no—
439C2—22no—
440C2—21no—
441C2#902212yes—
442C2#902312yes—
443C2#992114yes—
444C2#992914yes—
445C2#1097011yescharacteristic
446C2#1097611yescharacteristic
447C2#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
12. C2×#400C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4002
C2×#18242
C22×#3242
#143392
C2×#12942