Finite Group Databasebeta

← #109594

Subgroups — #109594

Order 512.

880 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#97C211yesmaximal, characteristic
2#14707C211yesmaximal, characteristic
3#10693C211yesmaximal, characteristic
4C2×#1295C212yesmaximal
5C2×#1295C212yesmaximal
6C22×#97—21no—
7C22×#97C2212yes—
8C22×#97C2212yes—
9C22×C4×C8—21no—
10C2×#389C411yescharacteristic
11C2×#389C411yescharacteristic
12C2×#382C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×C43C2211yescharacteristic
15C2×#336—21no—
16C2×#336—21no—
17C4×#97—24no—
18#1436—21no—
19#1436—21no—
20#1436—21no—
21#1436—21no—
22C4⋊#95C411yescharacteristic
23C4⋊#95C411yescharacteristic
24#1295—24no—
25#1295C418yescomplemented
26C23×C8—22no—
27C23×C8—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42C2311yescharacteristic
41C2×#97—24no—
42C2×#97—24no—
43C2×#97—24no—
44C2×#97—22no—
45C2×#97—22no—
46C2×#97—22no—
47C2×#97—22no—
48C2×#97—22no—
49C2×#97—22no—
50C2×#97—22no—
51C2×#97—22no—
52C2×#97C2×C414yes—
53C2×#97C2×C414yes—
54C2×#97D414yes—
55C2×#97Q814yes—
56C2×C4×C8—24no—
57C2×C4×C8—24no—
58C2×C4×C8—24no—
59#389—42no—
60#389—42no—
61#382—22no—
62#382—22no—
63#382—22no—
64#382C2×C414yes—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C43—22no—
76C43—22no—
77#336—24no—
78#336—24no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C22×C8—24no—
91C22×C8—24no—
92C22×C8—24no—
93C22×C8—24no—
94C22×C8—24no—
95C22×C8—24no—
96C22×C8—24no—
97C22×C8—24no—
98C22×C8—24no—
99C22×C8—24no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8—42no—
103C22×C8—42no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4C4212yes—
123C2×C4⋊C4C4212yes—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42—41no—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42—41no—
144C2×C42C22×C412yes—
145C2×C42C2×D412yes—
146C2×C42C2×Q812yes—
147C2×C42#4112yes—
148C2×C42#4112yes—
149C2×C42#4112yes—
150#97—216no—
151#97—24no—
152#97—24no—
153#97—24no—
154#97—24no—
155#97—24no—
156#97—42no—
157#97—42no—
158#97C4⋊C418yescomplemented
159C4×C8—216no—
160#95—24no—
161#95—24no—
162#95—42no—
163#95—42no—
164#95—42no—
165#95—42no—
166#95—42no—
167#95—42no—
168#95—41no—
169#95—42no—
170#95—41no—
171C24C2×C22⋊C411yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4#10012yes—
233C22×C4#10112yes—
234C22×C4#10412yes—
235C22×C4#10412yes—
236C22×C4#10412yes—
237C22×C4#10412yes—
238C22×C4#10412yes—
239C22×C4#10412yes—
240C22×C4C2×C4211yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4C4×D412yes—
243C22×C4C4×D412yes—
244C22×C4C4×D412yes—
245C22×C4C4×Q812yes—
246C22×C4#12011yescharacteristic
247C22×C4#12112yes—
248C22×C4C22⋊Q812yes—
249C22×C4#12311yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4C4⋊Q811yescharacteristic
252C2×C8—28no—
253C2×C8—28no—
254C2×C8—28no—
255C2×C8—28no—
256C2×C8—28no—
257C2×C8—28no—
258C2×C8—28no—
259C2×C8—28no—
260C2×C8—44no—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—44no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—84no—
273C4⋊C4—84no—
274C42—24no—
275C42—24no—
276C42—24no—
277C42—24no—
278C42—44no—
279C42—44no—
280C42—44no—
281C42—44no—
282C42C2×C4⋊C412yes—
283C42#11712yes—
284C42C4×D412yes—
285C42C4×Q812yes—
286C42C22⋊Q812yes—
287C42#12312yes—
288C42#12512yes—
289C42#12612yes—
290C23C4×C22⋊C411yescharacteristic
291C23#38611yescharacteristic
292C23#39111yescharacteristic
293C23C2×#10011yescharacteristic
294C23C2×#10111yescharacteristic
295C23#41311yescharacteristic
296C23#41311yescharacteristic
297C23C2×#10411yescharacteristic
298C23C2×#10411yescharacteristic
299C23C2×#10411yescharacteristic
300C23C2×#10411yescharacteristic
301C23C2×#10411yescharacteristic
302C23C2×#10411yescharacteristic
303C23#42111yescharacteristic
304C23#42111yescharacteristic
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4#32914yes—
384C2×C4#33014yes—
385C2×C4C4×C4⋊C412yes—
386C2×C4#38512yes—
387C2×C4#38712yes—
388C2×C4#38712yes—
389C2×C4#38812yes—
390C2×C4#38912yes—
391C8—216no—
392C8—48no—
393C22#143611yescharacteristic
394C22C2×#32911yescharacteristic
395C22C2×#33011yescharacteristic
396C22#152411yescharacteristic
397C22#152411yescharacteristic
398C22#152411yescharacteristic
399C22#152411yescharacteristic
400C22#152511yescharacteristic
401C22#152611yescharacteristic
402C22#152711yescharacteristic
403C22#152811yescharacteristic
404C22#152911yescharacteristic
405C22#152911yescharacteristic
406C22#153011yescharacteristic
407C22#153111yescharacteristic
408C22#153211yescharacteristic
409C22#153211yescharacteristic
410C22C4×#10011yescharacteristic
411C22C4×#10111yescharacteristic
412C22C4×#10411yescharacteristic
413C22C4×#10411yescharacteristic
414C22C4×#10411yescharacteristic
415C22#175911yescharacteristic
416C22#187111yescharacteristic
417C22#187111yescharacteristic
418C22#189611yescharacteristic
419C22#189611yescharacteristic
420C22#190011yescharacteristic
421C22#190111yescharacteristic
422C22#198611yescharacteristic
423C22#198611yescharacteristic
424C22#199411yescharacteristic
425C22#199411yescharacteristic
426C22#199711yescharacteristic
427C22#199811yescharacteristic
428C4—22no—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—24no—
433C4—24no—
434C4—22no—
435C4—22no—
436C4—42no—
437C4—42no—
438C4—82no—
439C4—82no—
440C4—161no—
441C4—161no—
442C2C4×#32911yescharacteristic
443C2C4×#33011yescharacteristic
444C2#822611yescharacteristic
445C2#822711yescharacteristic
446C2#911911yescharacteristic
447C2#912011yescharacteristic
448C2#912111yescharacteristic
449C2#912211yescharacteristic
450C2#989011yescharacteristic
451C2#989111yescharacteristic
452C2#989211yescharacteristic
453C2#989211yescharacteristic
454C2#1087811yescharacteristic
455C2#1088611yescharacteristic
456C2#1091311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1295C4semidirect
158. #97C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#972
#147072
#106932
C2×#12952