Finite Group Databasebeta

← #480391

Subgroups — #480391

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2#17365C212yesmaximal
3#11532C214yesmaximal, complemented
4#11512C212yesmaximal
5#11506C212yesmaximal
6#11478C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic
8C22×C4×Q8C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—22no—
11C4×C4⋊Q8—22no—
12C4×C4⋊Q8—22no—
13C4×C4⋊Q8C412yes—
14C4×C4⋊Q8C412yes—
15C4×C4⋊Q8C2212yes—
16C4×C4⋊Q8C2212yes—
17C2×#391C411yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#389—22no—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4—22no—
22C2×C43C411yescharacteristic
23#1467—22no—
24#1467C2214yes—
25#1463—22no—
26#1463C2214yes—
27#1461—22no—
28#1461C2214yes—
29#1325C2218yes—
30#1319C2218yes—
31C23×Q8C2×C411yescharacteristic
32C23×Q8C2×C411yescharacteristic
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—28no—
35C2×C4⋊Q8C2×C412yes—
36C2×C4⋊Q8C2×C412yes—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—41no—
40C2×C4×Q8—41no—
41C2×C4×Q8—41no—
42C2×C4×Q8—41no—
43C2×C4×Q8—41no—
44C2×C4×Q8—41no—
45C2×C4×Q8—41no—
46C2×C4×Q8—41no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D412yes—
52C2×C4×Q8D412yes—
53C2×C4×Q8C2311yescharacteristic
54C2×C4×Q8C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D412yes—
68#432—24no—
69#432—42no—
70#431—24no—
71#431—42no—
72#423—42no—
73#407—24no—
74C4×M16—42no—
75#391—22no—
76#391—22no—
77#391C2×C412yes—
78#391C2×C412yes—
79#391C2×C412yes—
80#391C2312yes—
81#389—24no—
82#389—24no—
83#389—24no—
84#389—24no—
85#384—22no—
86#384C2×C412yes—
87#384C2312yes—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92C2×#95—22no—
93C2×#95—22no—
94C43—21no—
95C43—21no—
96C43C2×C412yes—
97C43C2×C412yes—
98#324—24no—
99#324—24no—
100#324—24no—
101#324C2318yes—
102C22×Q8—24no—
103C22×Q8—24no—
104C22×Q8—21no—
105C22×Q8—41no—
106C22×Q8—41no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—41no—
110C22×Q8—41no—
111C22×Q8C22⋊C412yes—
112C22×Q8C22×C412yes—
113C22×Q8C22×C411yescharacteristic
114C22×Q8C22×C411yescharacteristic
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D412yes—
124C2×M16—44no—
125C4⋊Q8—216no—
126C4⋊Q8—24no—
127C4⋊Q8—24no—
128C4⋊Q8—24no—
129C4⋊Q8M1618yescomplemented
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×Q8—42no—
150C2×C4⋊C4—216no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—48no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2411yescharacteristic
175C2×C42—28no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—24no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—44no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—42no—
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C4⋊C8—44no—
210C4⋊C8—44no—
211C22⋊C8—28no—
212C22⋊C8—44no—
213#97—44no—
214C4×C8—44no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219C24C2×C22⋊C411yescharacteristic
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—42no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—48no—
232C2×Q8—48no—
233C2×Q8—48no—
234C2×Q8—48no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8C2×M1614yes—
239C2×Q8C2×M1614yes—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4#9911yescharacteristic
273C22×C4#9911yescharacteristic
274C22×C4#9911yescharacteristic
275C22×C4#9911yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D412yes—
286M16—88no—
287C2×C8—48no—
288C4⋊C4—216no—
289C4⋊C4—216no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—48no—
299C4⋊C4—48no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C42—28no—
304C42—28no—
305C42—24no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—22no—
320C42—22no—
321C42—21no—
322C42—21no—
323C42—42no—
324C42—44no—
325C42—44no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42C2×M1614yes—
332C23—21no—
333C23—21no—
334C23—22no—
335C23—22no—
336C23C2×#9911yescharacteristic
337C23C2×#9911yescharacteristic
338C23C22×C22⋊C411yescharacteristic
339Q8—44no—
340Q8—48no—
341Q8—44no—
342Q8—44no—
343Q8—44no—
344Q8—48no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—216no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—24no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4#35614yes—
393C2×C4#40714yes—
394C2×C4C2×#9911yescharacteristic
395C2×C4C2×#9911yescharacteristic
396C2×C4C2×#9911yescharacteristic
397C2×C4C2×#9911yescharacteristic
398C2×C4#41312yes—
399C2×C4C22×M1612yes—
400C8—88no—
401C22—24no—
402C22—24no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22#212612yes—
410C22C2×#35612yes—
411C22C2×#40711yescharacteristic
412C22C22×#9911yescharacteristic
413C22C2×#41311yescharacteristic
414C4—28no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—22no—
419C4—24no—
420C4—44no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—42no—
425C4—42no—
426C4—44no—
427C4—82no—
428C4C2×#35612yes—
429C4C2×#35612yes—
430C2—21no—
431C2—21no—
432C2—22no—
433C2—22no—
434C2#1736211yescharacteristic
435C2C2×#212611yescharacteristic
436C2C22×#35611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11532C2semidirect
6. #11478C2semidirect
129. C4⋊Q8M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
#173652
#115322
#115122
#115062
#114782