Finite Group Databasebeta

← #109694

Subgroups — #109694

Order 512.

938 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14275C211yesmaximal, characteristic
2#13778C211yesmaximal, characteristic
3#8023C212yesmaximal
4C2×#1289C212yesmaximal
5C2×#1280C211yesmaximal, characteristic
6C22×C22⋊C8C2212yes—
7C2×#385—21no—
8C2×#380C2211yescharacteristic
9C2×#380C2212yes—
10C2×#380C2211yescharacteristic
11C2×C4×C22⋊C4—21no—
12C22×#95C2211yescharacteristic
13#1444C412yescomplemented
14#1442C412yescomplemented
15#1441—22no—
16#1436—24no—
17C22⋊#95—21no—
18C22⋊#95—21no—
19#1294—24no—
20#1289C418yescomplemented
21#1280—22no—
22C24×C4D411yescharacteristic
23C24×C4D411yescharacteristic
24C24×C4C2311yescharacteristic
25C23×C8—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—22no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C42—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8C2×C414yes—
41#385—44no—
42#380—22no—
43#380—22no—
44#380—22no—
45#380—22no—
46#380—22no—
47#380C2×C414yes—
48#380C2×C414yes—
49#380Q814yes—
50C4×C22⋊C4—42no—
51C4×C22⋊C4—42no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95C2×C412yes—
62C2×#95C2×C412yes—
63C2×#95D412yes—
64C2×#95D414yes—
65C25C2×D411yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—24no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—81no—
79C23×C4—81no—
80C23×C4—81no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4—81no—
84C23×C4C22⋊C412yes—
85C23×C4C22⋊C412yes—
86C23×C4SD1612yes—
87C23×C4Q1612yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C22×C412yes—
90C23×C4C2×D412yes—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×Q812yes—
93C23×C4#4112yes—
94C23×C4#4112yes—
95C23×C4#4112yes—
96C22×C8—24no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—82no—
100C22×C8—82no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—82no—
108C2×C4⋊C4—81no—
109C2×C4⋊C4—82no—
110C2×C4⋊C4—81no—
111C2×C4⋊C4—82no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4—81no—
114C2×C4⋊C4—81no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C42—44no—
130C2×C42—82no—
131C2×C42—82no—
132C22⋊C8—24no—
133C22⋊C8—24no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136#95—22no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—44no—
141#95—44no—
142#95—42no—
143#95—82no—
144#95—82no—
145#95—82no—
146#95—82no—
147#95C4214yescomplemented
148#95C4⋊C418yescomplemented
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24C2×C22⋊C411yescharacteristic
154C24C4×D412yes—
155C24#12011yescharacteristic
156C24C22⋊Q812yes—
157C24#12311yescharacteristic
158C24C2×SD1611yescharacteristic
159C24C2×Q1611yescharacteristic
160C24#13512yes—
161C24AGL(1,ℤ/8)11yescharacteristic
162C24AGL(1,ℤ/8)12yes—
163C24#13711yescharacteristic
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—28no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—84no—
228C22×C4—84no—
229C22×C4#10314yes—
230C22×C4C2×C4211yescharacteristic
231C22×C4C2×C4⋊C412yes—
232C22×C4#11712yes—
233C22×C4#11711yescharacteristic
234C22×C4C4×D412yes—
235C22×C4C4×D412yes—
236C22×C4C4×D412yes—
237C22×C4C4×Q812yes—
238C22×C4C4×Q812yes—
239C22×C4#12112yes—
240C22×C4C22⋊Q812yes—
241C22×C4#12312yes—
242C22×C4#12411yescharacteristic
243C22×C4#12512yes—
244C22×C4#12612yes—
245C22×C4C4⋊Q811yescharacteristic
246C2×C8—44no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—84no—
250C2×C8—84no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4—88no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C42—88no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—42no—
279C23—42no—
280C23—41no—
281C23—41no—
282C23—42no—
283C23—41no—
284C23—42no—
285C23—41no—
286C23—42no—
287C23—42no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—41no—
292C23—42no—
293C23—41no—
294C23C4×C22⋊C411yescharacteristic
295C23#38512yes—
296C23#38611yescharacteristic
297C23#38712yes—
298C23#39111yescharacteristic
299C23C2×#10311yescharacteristic
300C23#41611yescharacteristic
301C23#41711yescharacteristic
302C23#41811yescharacteristic
303C23C4×SD1612yes—
304C23C4×Q1612yes—
305C23#44012yes—
306C23#44212yes—
307C23#44912yes—
308C23#45011yescharacteristic
309C23#45111yescharacteristic
310C23#47612yes—
311C23#47712yes—
312C23#47812yes—
313C23#47912yes—
314C23#48212yes—
315C23#48311yescharacteristic
316C23#48411yescharacteristic
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—84no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4C4×C4⋊C412yes—
376C2×C4#38712yes—
377C2×C4#38812yes—
378C2×C4#38912yes—
379C8—84no—
380C22—21no—
381C22—22no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—21no—
405C22—21no—
406C22—22no—
407C22—22no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22#143611yescharacteristic
417C22C4×#10311yescharacteristic
418C22#176211yescharacteristic
419C22#186312yes—
420C22#186412yes—
421C22#187211yescharacteristic
422C22#187511yescharacteristic
423C22#191111yescharacteristic
424C22#191211yescharacteristic
425C22#191311yescharacteristic
426C22#191311yescharacteristic
427C22#192012yes—
428C22#192112yes—
429C22#198711yescharacteristic
430C22#198911yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—44no—
436C4—88no—
437C4—84no—
438C4—82no—
439C4—161no—
440C4—161no—
441C2—22no—
442C2—21no—
443C2—22no—
444C2—22no—
445C2—22no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2#919911yescharacteristic
450C2#920011yescharacteristic
451C2#920511yescharacteristic
452C2#920611yescharacteristic
453C2#1088511yescharacteristic
454C2#1088811yescharacteristic
455C2#1091511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1444C4semidirect
14. #1442C4semidirect
20. #1289C4semidirect
147. #95C42semidirect
148. #95C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#142752
#137782
#80232
C2×#12892
C2×#12802