Finite Group Databasebeta

← #471739

Subgroups — #471739

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic
2C2×#1916C211yesmaximal, characteristic
3#15151C211yesmaximal, characteristic
4#15128C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11292C212yesmaximal
8#11227C212yesmaximal
9#11219C212yesmaximal
10#11219C212yesmaximal
11#10862C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#398—21no—
17C2×#398—21no—
18C2×#395—21no—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1990—24no—
24#1916—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1446—21no—
37#1445—21no—
38#1445—21no—
39#1444—21no—
40#1441—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8—21no—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—42no—
69C2×#107—42no—
70C2×#107Q818yescomplemented
71C2×C4⋊C8—22no—
72C2×C4⋊C8—42no—
73C2×C4×C8—42no—
74C2×C4×C8D414yes—
75#398—44no—
76#398—44no—
77#395—44no—
78#389—24no—
79#389—42no—
80#389—42no—
81#389—42no—
82#384—22no—
83#384—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98#340—24no—
99#340—24no—
100#340—24no—
101#340—42no—
102#340—42no—
103#340—42no—
104#340—42no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109C22×Q8—44no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—82no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C22×C8C2×Q814yes—
145C4⋊Q8—28no—
146C4⋊Q8—48no—
147C4⋊Q8—44no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4D814yes—
189C2×C4⋊C4C2×Q814yes—
190C2×C4⋊C4C2×Q814yes—
191C2×C4⋊C4#4114yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42C2×D412yes—
205C2×C42#4112yes—
206#107—48no—
207C4⋊C8—44no—
208C4×C8—44no—
209#95—22no—
210#95—22no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—42no—
215#95—44no—
216#95—42no—
217#95—44no—
218#95—42no—
219#95—42no—
220#95—82no—
221C24C22×D411yescharacteristic
222C2×Q8—48no—
223C2×Q8—88no—
224C2×Q8—88no—
225C2×Q8—84no—
226C2×Q8—84no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4#12112yes—
280C22×C4#12112yes—
281C22×C4C2×D812yes—
282C22×C4C2×D812yes—
283C22×C4C2×D812yes—
284C22×C4#13512yes—
285C22×C4AGL(1,ℤ/8)12yes—
286C22×C4#13712yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×Q812yes—
289C22×C4C2×#4112yes—
290C22×C4C2×#4111yescharacteristic
291C22×C4H(5,𝔽2)12yes—
292C22×C4H(5,𝔽2)11yescharacteristic
293C22×C4H(5,𝔽2)11yescharacteristic
294C22×C42−1+412yes—
295C22×C42−1+411yescharacteristic
296C22×C42−1+411yescharacteristic
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8C4⋊Q814yes—
317C4⋊C4—28no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—48no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—88no—
337C4⋊C4—88no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C42—22no—
341C42—22no—
342C42—24no—
343C42—41no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—42no—
350C23C2×#12111yescharacteristic
351C23#53511yescharacteristic
352C23#53611yescharacteristic
353C23#54612yes—
354C23D4×Q812yes—
355C23C22×D811yescharacteristic
356C23C2×#13511yescharacteristic
357C23C2×AGL(1,ℤ/8)11yescharacteristic
358C23C2×#13711yescharacteristic
359C23#57712yes—
360C23#57812yes—
361Q8—816no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—44no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4#46614yes—
421C2×C4#46814yes—
422C2×C4#52511yescharacteristic
423C2×C4C2×C4⋊Q811yescharacteristic
424C2×C4#53411yescharacteristic
425C2×C4#53611yescharacteristic
426C2×C4#53711yescharacteristic
427C2×C4#53711yescharacteristic
428C2×C4#54112yes—
429C2×C4#54212yes—
430C2×C4#54211yescharacteristic
431C2×C4#54211yescharacteristic
432C2×C4#54312yes—
433C2×C4#54412yes—
434C2×C4#55112yes—
435C2×C4#55412yes—
436C2×C4#55712yes—
437C2×C4#55712yes—
438C2×C4#56212yes—
439C2×C4#56312yes—
440C8—24no—
441C8—42no—
442C8—81no—
443C8—81no—
444C22#253411yescharacteristic
445C22#254611yescharacteristic
446C22#262011yescharacteristic
447C22#269212yes—
448C22#271312yes—
449C22C2×#46611yescharacteristic
450C22C2×#46811yescharacteristic
451C22#305711yescharacteristic
452C22#305911yescharacteristic
453C22#320511yescharacteristic
454C22#321011yescharacteristic
455C22#322512yes—
456C22#322611yescharacteristic
457C22#322811yescharacteristic
458C22#324212yes—
459C22#329412yes—
460C22#329612yes—
461C22#330812yes—
462C22#331212yes—
463C22Q8×D812yes—
464C22#338312yes—
465C22#338412yes—
466C22#338712yes—
467C4—24no—
468C4—24no—
469C4—42no—
470C4—42no—
471C4—84no—
472C4—84no—
473C4—81no—
474C4—84no—
475C4—81no—
476C4—162no—
477C4#258411yescharacteristic
478C4#263711yescharacteristic
479C4#265911yescharacteristic
480C4#275512yes—
481C4#279912yes—
482C4#281411yescharacteristic
483C2#1680611yescharacteristic
484C2#2809411yescharacteristic
485C2#2810511yescharacteristic
486C2#2867511yescharacteristic
487C2#2868411yescharacteristic
488C2#2984211yescharacteristic
489C2#2984811yescharacteristic
490C2#3151812yes—
491C2#3152112yes—
492C2#3213012yes—
493C2#3213312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
70. C2×#107Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19902
C2×#19162
#151512
#151282
#113842
#113732
#112922
#112272
#112192
#108622