Finite Group Databasebeta

← #207039

Subgroups — #207039

Order 512.

938 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C4×#423C211yesmaximal, characteristic
3C2×#1745C211yesmaximal, characteristic
4#9895C212yesmaximal, complemented
5#9894C212yesmaximal, complemented
6#9893C214yesmaximal, complemented
7#8029C214yesmaximal
8C2×#423—21no—
9C2×C4×M16C2212yes—
10C22×#97C2212yes—
11#2864C2212yes—
12#2863—21no—
13#2863—21no—
14#2862C2211yescharacteristic
15#2859C2211yescharacteristic
16#2857C411yescharacteristic
17#2857C411yescharacteristic
18#2855—21no—
19#2855—21no—
20#2851C412yes—
21C4×C4⋊Q8C411yescharacteristic
22C4×C4⋊Q8C411yescharacteristic
23C4×#125C412yes—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26#2295—21no—
27#2295—21no—
28#2283C2212yes—
29C2×C43C2211yescharacteristic
30#1767C2212yes—
31#1767C2212yes—
32C4×C4⋊C8—21no—
33C4×C4⋊C8—21no—
34#1745—21no—
35#1745—21no—
36#1745C2212yes—
37#1745C2212yes—
38#1728—21no—
39C2×#336—21no—
40C2×#336—21no—
41#1295C2218yes—
42#1295C2218yes—
43C22×M16—22no—
44C23×C8—22no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125C2×C412yes—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59#423—28no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C4×M16—28no—
63C2×#97—22no—
64C2×#97—24no—
65C2×#97Q818yescomplemented
66C2×#97C2314yes—
67C2×#97C2314yes—
68C2×C4×C8—22no—
69#391—21no—
70#391—21no—
71#391—21no—
72#391—21no—
73#391—21no—
74#391—21no—
75#391—21no—
76#391—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#385—22no—
98#385—22no—
99#384C2×C411yescharacteristic
100#384C2×C411yescharacteristic
101#383—22no—
102#382C2×C412yes—
103#382C2312yes—
104#382C2312yes—
105#381—22no—
106#380—21no—
107#380—21no—
108#380C2×C412yes—
109#380C2×C412yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C22⋊C4—22no—
117#376—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43C2311yescharacteristic
125C43C2311yescharacteristic
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—28no—
142C2×M16—44no—
143C2×M16—44no—
144C22×C8—28no—
145C22×C8—24no—
146C22×C8—24no—
147C22×C8—44no—
148C22×C8—44no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151#125—44no—
152C22⋊Q8—44no—
153C22⋊Q8—44no—
154C4×Q8—44no—
155C4×Q8—44no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q814yes—
210C2×C42#4114yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—28no—
213C4⋊C8—28no—
214#97—28no—
215#97—24no—
216#97—28no—
217#97C2×Q818yes—
218C4×C8—44no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233C24C2×C22⋊C411yescharacteristic
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4#10414yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C4×Q814yes—
274C22×C4C22⋊Q814yes—
275C22×C4#12412yes—
276C22×C4C4⋊Q812yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—416no—
281C2×C8—28no—
282C2×C8—28no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—416no—
286C2×C8—44no—
287C2×C8—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—41no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—44no—
331C42—44no—
332C42C4×Q814yes—
333C42C22⋊Q814yes—
334C42#12412yes—
335C42C4⋊Q812yes—
336C42C22×Q812yes—
337C42C2×#4112yes—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23#39112yes—
343C23C2×#10412yes—
344C23#42112yes—
345C23C22×C22⋊C411yescharacteristic
346Q8—84no—
347Q8—84no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—28no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4#39112yes—
399C2×C4#39112yes—
400C2×C4#39112yes—
401C2×C4#41312yes—
402C2×C4#41312yes—
403C2×C4C2×#10412yes—
404C2×C4C2×#10412yes—
405C2×C4#42112yes—
406C2×C4#42112yes—
407C2×C4C2×C4×Q812yes—
408C2×C4C2×C22⋊Q812yes—
409C2×C4C2×#12411yescharacteristic
410C2×C4C2×C4⋊Q811yescharacteristic
411C8—48no—
412C8—48no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22#198614yes—
420C22C2×#39111yescharacteristic
421C22C2×#41311yescharacteristic
422C22C2×#41311yescharacteristic
423C22C22×#10411yescharacteristic
424C22C2×#42111yescharacteristic
425C22C2×#42111yescharacteristic
426C22C2×#42111yescharacteristic
427C4—21no—
428C4—21no—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4#199514yes—
446C2—24no—
447C2#1765011yescharacteristic
448C2#1765111yescharacteristic
449C2#1765911yescharacteristic
450C2#1766211yescharacteristic
451C2C2×#198611yescharacteristic
452C2C2×#199511yescharacteristic
453C2#2133711yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9895C2semidirect
5. #9894C2semidirect
6. #9893C2semidirect
65. C2×#97Q8semidirect

Maximal subgroups

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

subgroupindex
#366352
C4×#4232
C2×#17452
#98952
#98942
#98932
#80292