Finite Group Databasebeta

← #207055

Subgroups — #207055

Order 512.

990 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic
3C2×C4×#97C211yesmaximal, characteristic
4#9892C214yesmaximal, complemented
5#9891C212yesmaximal, complemented
6#9890C212yesmaximal, complemented
7#8036C214yesmaximal
8C2×#432C2212yes—
9C22×#97—21no—
10C22×#97C2212yes—
11C22×C4×C8—21no—
12#2864C2212yes—
13#2863—21no—
14#2863—21no—
15#2862C2211yescharacteristic
16#2859C2211yescharacteristic
17#2857C411yescharacteristic
18#2857C411yescharacteristic
19#2855—21no—
20#2855—21no—
21#2851C412yes—
22C4×C4⋊Q8C411yescharacteristic
23C4×C4⋊Q8C411yescharacteristic
24C4×#125C412yes—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27#2295—21no—
28#2295—21no—
29#2283C2212yes—
30C2×C43C2211yescharacteristic
31#1842—21no—
32#1831—21no—
33#1797—21no—
34#1797—21no—
35#1767C2212yes—
36#1767C2212yes—
37C4×#97—21no—
38C4×#97—21no—
39C4×#97C2212yes—
40C4×#97C2212yes—
41#1295C2218yes—
42#1295C2218yes—
43C23×C8—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58#432—28no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×#97—28no—
64C2×#97—22no—
65C2×#97—22no—
66C2×#97—22no—
67C2×#97—24no—
68C2×#97Q818yescomplemented
69C2×#97C2314yes—
70C2×#97C2314yes—
71C2×C4×C8—28no—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#384C2×C411yescharacteristic
105#384C2×C411yescharacteristic
106#383—22no—
107#382C2×C412yes—
108#382C2312yes—
109#382C2312yes—
110#381—22no—
111#380—21no—
112#380—21no—
113#380C2×C412yes—
114#380C2×C412yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C22⋊C4—22no—
122#376—22no—
123C43C2×C411yescharacteristic
124C43C2×C411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43D411yescharacteristic
129C43C2311yescharacteristic
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C22×C8—28no—
143C22×C8—28no—
144C22×C8—28no—
145C22×C8—24no—
146C22×C8—24no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149#125—44no—
150C22⋊Q8—44no—
151C22⋊Q8—44no—
152C4×Q8—44no—
153C4×Q8—44no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
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×Q814yes—
208C2×C42#4114yes—
209C2×C42C2411yescharacteristic
210C4⋊C8—44no—
211C4⋊C8—44no—
212C4⋊C8—44no—
213C22⋊C8—44no—
214#97—28no—
215#97—28no—
216#97—28no—
217#97—24no—
218#97—28no—
219#97C2×Q818yes—
220C4×C8—28no—
221C4×C8—28no—
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—
233#95—22no—
234#95—22no—
235#95—22no—
236C24C2×C22⋊C411yescharacteristic
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
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—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4#10414yes—
264C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C4×Q814yes—
277C22×C4C22⋊Q814yes—
278C22×C4#12412yes—
279C22×C4C4⋊Q812yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C2×C8—28no—
284C2×C8—28no—
285C2×C8—232no—
286C2×C8—28no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—44no—
333C42—44no—
334C42C4×Q814yes—
335C42C22⋊Q814yes—
336C42#12412yes—
337C42C4⋊Q812yes—
338C42C22×Q812yes—
339C42C2×#4112yes—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23#39112yes—
345C23C2×#10412yes—
346C23C2×#10412yes—
347C23C22×C22⋊C411yescharacteristic
348Q8—84no—
349Q8—84no—
350C2×C4—28no—
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—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—28no—
374C2×C4—28no—
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—42no—
384C2×C4—42no—
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×C4C2×#10412yes—
406C2×C4C2×#10412yes—
407C2×C4#42112yes—
408C2×C4#42112yes—
409C2×C4C2×C4×Q812yes—
410C2×C4C2×C22⋊Q812yes—
411C2×C4C2×#12411yescharacteristic
412C2×C4C2×C4⋊Q811yescharacteristic
413C8—216no—
414C8—84no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22#199414yes—
422C22C2×#39111yescharacteristic
423C22C2×#41311yescharacteristic
424C22C2×#41311yescharacteristic
425C22C22×#10411yescharacteristic
426C22C22×#10411yescharacteristic
427C22C2×#42111yescharacteristic
428C22C2×#42111yescharacteristic
429C4—21no—
430C4—21no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—41no—
442C4—41no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4#153214yes—
448C2—24no—
449C2C2×#153211yescharacteristic
450C2#1764611yescharacteristic
451C2#1764911yescharacteristic
452C2#1765011yescharacteristic
453C2#2130611yescharacteristic
454C2C2×#199411yescharacteristic
455C2#2133711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9892C2semidirect
5. #9891C2semidirect
6. #9890C2semidirect
68. C2×#97Q8semidirect

Maximal subgroups

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

subgroupindex
#366352
#208102
C2×C4×#972
#98922
#98912
#98902
#80362