Finite Group Databasebeta

← #197898

Subgroups — #197898

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2C2×#2991C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic, complemented
4#20737C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic, complemented
6#9227C214yesmaximal, complemented
7#9220C214yesmaximal, complemented
8#8041C211yesmaximal, characteristic, complemented
9#7880C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3079—24no—
13#3079C418yescomplemented
14C2×#477—21no—
15C2×#475C2212yes—
16C2×#476—21no—
17C2×#474C2212yes—
18#2991—24no—
19#2991—24no—
20C2×C8×Q8C2212yes—
21C2×#436—22no—
22C2×#436C2212yes—
23C2×#431—21no—
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#428—21no—
27C2×#423—21no—
28C2×#423C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#417—21no—
31#2285C2211yescharacteristic
32#1981—21no—
33#1981—21no—
34#1956C2214yes—
35#1825—21no—
36#1803C2212yes—
37#1802C2212yes—
38#1788—21no—
39#1787—22no—
40C4×#103—21no—
41C4×#102—21no—
42#1546—24no—
43C2×#332C2214yes—
44#1296C2214yes—
45#1277C2214yes—
46C22×#41D411yescharacteristic
47C23×Q8—21no—
48C22×M16—21no—
49C22×M16—21no—
50C22×M16—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yes—
58C2×C4⋊Q8C2312yes—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41C2×C414yes—
62C4×#41D414yes—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#477—28no—
77#476—28no—
78#475—24no—
79#475C2×C418yes—
80#474—24no—
81#474C2×C418yes—
82C8×Q8—24no—
83C8×Q8—24no—
84#436—216no—
85#436—24no—
86#436—24no—
87#431—28no—
88#431—22no—
89#431—22no—
90#431—22no—
91#431—22no—
92#428—28no—
93C2×#107—22no—
94C2×#106—22no—
95#423—28no—
96#423—22no—
97#423—22no—
98#423C2×C414yes—
99#423D414yes—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2312yes—
108#417—28no—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#102—22no—
114C2×#102—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8D412yes—
118C2×C4×C8—22no—
119C2×C4×C8—21no—
120C2×C4×C8C2312yes—
121#384C2312yes—
122#382—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127#340—22no—
128#340—22no—
129#332—28no—
130C2×#41—22no—
131C2×#41—24no—
132C2×#41—42no—
133C2×#41C22⋊C414yes—
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8—41no—
138C22×Q8—82no—
139C22×Q8—82no—
140C22×Q8C2×D411yescharacteristic
141C22×D4—21no—
142C22×D4C2×D411yescharacteristic
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—41no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4#4111yescharacteristic
150C2×M16—24no—
151C2×M16—28no—
152C2×M16—24no—
153C2×M16—44no—
154C2×M16—84no—
155C2×M16—82no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8—24no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8C22×C414yes—
171C4×Q8—22no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8C22⋊C414yes—
175C4×Q8C22×C412yes—
176C4×Q8C2×D412yes—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—22no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4C22⋊C414yes—
185C4×D4C22×C412yes—
186C4×D4C2×D412yes—
187#117—22no—
188#117—24no—
189#117—24no—
190#117—82no—
191#117C22⋊C414yes—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—41no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—41no—
217C2×C42C22×C412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42#4111yescharacteristic
223C2×C42#4112yes—
224C2×C42C2411yescharacteristic
225#107—28no—
226#106—28no—
227C4⋊C8—28no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8C22×C414yes—
237C4⋊C8C2×D414yes—
238#103—28no—
239#103—44no—
240#103—84no—
241#103—84no—
242#102—28no—
243#102—44no—
244C22⋊C8—28no—
245C22⋊C8—24no—
246C22⋊C8—42no—
247C4×C8—28no—
248C4×C8—22no—
249C4×C8—22no—
250C4×C8—42no—
251#95—41no—
252#95—41no—
253C24—21no—
254C24#12111yescharacteristic
255#41—28no—
256#41—44no—
257#41—48no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—42no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—88no—
266C2×Q8—84no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8C2×C22⋊C412yes—
270C2×Q8#13714yes—
271C2×D4—24no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—44no—
276C2×D4C2×C22⋊C412yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—84no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12012yes—
303C22×C4#12112yes—
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12312yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4111yescharacteristic
311M16—48no—
312M16—84no—
313M16—88no—
314C2×C8—24no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—22no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—82no—
341C4⋊C4C2×C22⋊C412yes—
342C4⋊C4C2×C22⋊C412yes—
343C22⋊C4—24no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—24no—
347C22⋊C4—82no—
348C42—22no—
349C42—22no—
350C42—24no—
351C42—24no—
352C42—82no—
353C42C2×C22⋊C412yes—
354C42C4×D412yes—
355C42C4×D412yes—
356C42#12012yes—
357C42#12112yes—
358C42#12312yes—
359C42C23×C411yescharacteristic
360C42C22×D411yescharacteristic
361C42C22×D411yescharacteristic
362C42C2×#4111yescharacteristic
363C23—22no—
364C23—22no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23#38612yes—
373C23C2×#12111yescharacteristic
374Q8—22no—
375Q8—24no—
376Q8—48no—
377Q8—88no—
378Q8—82no—
379Q8—82no—
380Q8#41914yes—
381D4—22no—
382D4—22no—
383D4—48no—
384D4—42no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—48no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—88no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#38612yes—
416C2×C4#38612yes—
417C2×C4#38612yes—
418C2×C4#47112yes—
419C2×C4#47312yes—
420C2×C4C22×C22⋊C411yescharacteristic
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×C4×D411yescharacteristic
423C2×C4C2×#12011yescharacteristic
424C2×C4C2×#12111yescharacteristic
425C2×C4C2×#12311yescharacteristic
426C2×C4C2×#13712yes—
427C2×C4#57512yes—
428C2×C4#57612yes—
429C2×C4#57712yes—
430C8—44no—
431C8—42no—
432C8—44no—
433C8—42no—
434C8—82no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—82no—
444C22—82no—
445C22C2×#38611yescharacteristic
446C22#304512yes—
447C22#305612yes—
448C22C2×#47111yescharacteristic
449C22C2×#47311yescharacteristic
450C4—21no—
451C4—21no—
452C4—22no—
453C4—22no—
454C4—24no—
455C4—42no—
456C4—42no—
457C4—84no—
458C4—162no—
459C4#190814yes—
460C4C2×#41911yescharacteristic
461C4#289611yescharacteristic
462C4#294612yes—
463C4#300811yescharacteristic
464C4#301011yescharacteristic
465C4#305612yes—
466C2—42no—
467C2—82no—
468C2#1949412yes—
469C2#1949712yes—
470C2#2093211yescharacteristic
471C2#2097211yescharacteristic
472C2C2×#190811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21804C2semidirect
5. C2×#1546C2semidirect
6. #9227C2semidirect
7. #9220C2semidirect
8. #8041C2semidirect
9. #7880C2semidirect
13. #3079C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
C2×#29912
#218042
#207372
C2×#15462
#92272
#92202
#80412
#78802