Finite Group Databasebeta

← #493976

Subgroups — #493976

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2#20727C212yesmaximal
3#12860C212yesmaximal
4#12802C214yesmaximal, complemented
5#12700C212yesmaximal
6C4×#356C214yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
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×#384C411yescharacteristic
21C2×C4×C4⋊C4—22no—
22C2×C43C411yescharacteristic
23C2×#356C2218yes—
24#1821—22no—
25#1801—22no—
26#1801C2214yes—
27#1784C2214yes—
28C4×#101—22no—
29C4×#101C2214yes—
30#1394C2218yes—
31C23×Q8C2×C411yescharacteristic
32C23×Q8C2311yescharacteristic
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—28no—
35C2×C4⋊Q8C2×C412yes—
36C2×C4⋊Q8C2312yes—
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⋊C4C2×C411yescharacteristic
60C22×C4⋊C4D412yes—
61C22×C42—21no—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D412yes—
68#432—42no—
69#432—42no—
70#431—42no—
71#431—42no—
72#423—24no—
73C2×#101—24no—
74C2×#101C2318yes—
75#407—24no—
76C4×M16—24no—
77#391—22no—
78#391—22no—
79#391C2×C412yes—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2312yes—
83#389—24no—
84#389—24no—
85#389—24no—
86#389—24no—
87#384—22no—
88#384C2×C412yes—
89#384C2×C412yes—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C2×#95—22no—
95C2×#95—22no—
96C43—21no—
97C43—21no—
98C43C2×C412yes—
99C43C2×C412yes—
100#356—216no—
101#343—24no—
102#343—24no—
103C22×Q8—24no—
104C22×Q8—24no—
105C22×Q8—21no—
106C22×Q8—41no—
107C22×Q8—41no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8C22×C412yes—
113C22×Q8C22×C411yescharacteristic
114C22×Q8C2×D412yes—
115C22×Q8C2411yescharacteristic
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C2×D412yes—
125C2×M16—28no—
126C4⋊Q8—216no—
127C4⋊Q8—24no—
128C4⋊Q8—24no—
129C4⋊Q8—24no—
130C4⋊Q8#4118yes—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—42no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×Q8—42no—
151C2×C4⋊C4—216no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C42—28no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—44no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—42no—
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C412yes—
203C2×C42C22⋊C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C4⋊C8—44no—
211C4⋊C8—44no—
212#101—216no—
213#101—48no—
214C22⋊C8—44no—
215C22⋊C8—44no—
216#97—44no—
217C4×C8—44no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222C24C2×C22⋊C411yescharacteristic
223C2×Q8—24no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—48no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8#11714yes—
242C2×Q8C2×#4114yes—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4#9911yescharacteristic
276C22×C4#9911yescharacteristic
277C22×C4#9911yescharacteristic
278C22×C4#9911yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D412yes—
289M16—48no—
290M16—48no—
291C2×C8—48no—
292C4⋊C4—216no—
293C4⋊C4—216no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C42—28no—
308C42—28no—
309C42—24no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—22no—
324C42—22no—
325C42—21no—
326C42—21no—
327C42—42no—
328C42—44no—
329C42—44no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42#11714yes—
336C23—21no—
337C23—22no—
338C23—22no—
339C23—21no—
340C23C2×#9911yescharacteristic
341C23C2×#9911yescharacteristic
342C23C22×C22⋊C411yescharacteristic
343Q8—44no—
344Q8—48no—
345Q8—44no—
346Q8—44no—
347Q8—44no—
348Q8—48no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—216no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—48no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4#35614yescomplemented
397C2×C4C2×#9911yescharacteristic
398C2×C4C2×#9911yescharacteristic
399C2×C4C2×#9911yescharacteristic
400C2×C4C2×#9911yescharacteristic
401C2×C4#41012yes—
402C2×C4C2×#11712yes—
403C2×C4#52514yes—
404C8—88no—
405C22—24no—
406C22—24no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22C2×#35612yes—
414C22#213212yes—
415C22#228011yescharacteristic
416C22C22×#9911yescharacteristic
417C22C2×#41011yescharacteristic
418C4—28no—
419C4—21no—
420C4—21no—
421C4—24no—
422C4—22no—
423C4—22no—
424C4—44no—
425C4—41no—
426C4—41no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—44no—
431C4—82no—
432C4C2×#35612yes—
433C4C2×#35612yes—
434C2—21no—
435C2—21no—
436C2—22no—
437C2—22no—
438C2#2072211yescharacteristic
439C2C22×#35611yescharacteristic
440C2C2×#213211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
socleC22C22×#99
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12802C2semidirect
6. C4×#356C2semidirect
396. C2×C4#356semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
#207272
#128602
#128022
#127002
C4×#3562