Finite Group Databasebeta

← #195258

Subgroups — #195258

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#34074C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#14274C211yesmaximal, characteristic
6#9044C218yesmaximal
7#8026C211yesmaximal, characteristic
8#7906C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3229—24no—
12#3229C418yescomplemented
13C2×#498—21no—
14C2×#487—21no—
15C2×#483C2214yes—
16C2×#475—22no—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2331—21no—
20#2331—21no—
21#2330—22no—
22#2329—22no—
23#2329C2214yes—
24#2296—21no—
25#2295—22no—
26#2295—22no—
27C2×#391—21no—
28C2×#391—21no—
29#2285C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C4×#117—21no—
32C2×#376C2211yescharacteristic
33#1990—22no—
34#1988—22no—
35#1823C2214yes—
36#1518—24no—
37C2×#328C2214yes—
38#1445—22no—
39#1445C2214yes—
40#1294C2214yes—
41#1280C2214yes—
42C23×Q8—21no—
43#533—22no—
44#533—22no—
45#533C2×C414yes—
46#533D414yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8D411yescharacteristic
56C2×C4⋊Q8D411yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2312yes—
60C2×C4×Q8—21no—
61C2×C4×Q8D412yes—
62C2×#117—21no—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42C2311yescharacteristic
70#498—28no—
71#487—28no—
72#483—28no—
73#483C2×C4116yes—
74#475—216no—
75#431—22no—
76#431—22no—
77#431—22no—
78#431C2×C414yes—
79C2×#106—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×#103—24no—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#391—24no—
88#391—24no—
89#391—44no—
90#391—44no—
91#389—24no—
92#389—22no—
93#389—22no—
94#389—24no—
95#387—24no—
96#387—24no—
97#387—24no—
98#387—24no—
99#385—24no—
100#384D412yes—
101#382—21no—
102#382—22no—
103#382—22no—
104#380—21no—
105#380C2312yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D412yes—
112C4×C22⋊C4—24no—
113#376—22no—
114#376—21no—
115#376—21no—
116#376—21no—
117#376—21no—
118C2×#95—21no—
119C2×#95—24no—
120C2×#95—24no—
121C2×#95D412yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C43—22no—
125#340—28no—
126#328—28no—
127C22×Q8—24no—
128C22×Q8—21no—
129C22×Q8—41no—
130C22×Q8—41no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8C2×D412yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8C22⋊C414yes—
157C4⋊Q8C22⋊C414yes—
158#125—24no—
159#125—24no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8C22×C414yes—
165C22⋊Q8C2×D414yes—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8C22⋊C418yes—
170#117—24no—
171#117—24no—
172#117—42no—
173#117—42no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4C22×C414yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D414yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42SD1614yes—
212C2×C42C22×C412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4112yes—
219C2×C42#4112yes—
220#106—216no—
221C4⋊C8—24no—
222C4⋊C8—42no—
223#103—216no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8C22×C414yes—
228C4×C8—24no—
229C4×C8—42no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—48no—
243#95—42no—
244#95—48no—
245C24C22×D411yescharacteristic
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—48no—
252C2×Q8—48no—
253C2×Q8—82no—
254C2×Q8—82no—
255C2×Q8—82no—
256C2×Q8—82no—
257C2×Q8—82no—
258C2×Q8—84no—
259C2×Q8—84no—
260C2×Q8C2×C22⋊C414yes—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
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—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4C4×D412yes—
288C22×C4C4×D412yes—
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4C2×SD1614yes—
296C22×C4C2×SD1612yes—
297C22×C4#13512yes—
298C22×C4AGL(1,ℤ/8)12yes—
299C22×C4#13712yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C2×C8—24no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—42no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—28no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—82no—
326C4⋊C4—84no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C4⋊C4C2×C22⋊C414yes—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C42—28no—
335C42—24no—
336C42—24no—
337C42—42no—
338C42—42no—
339C42—44no—
340C42—84no—
341C42—84no—
342C42C2×C22⋊C412yes—
343C42C2×C22⋊C412yes—
344C42C4×D412yes—
345C42C4×D412yes—
346C42#12112yes—
347C42#12112yes—
348C42#12312yes—
349C42#12312yes—
350C23—22no—
351C23—22no—
352C23C2×C4×D411yescharacteristic
353C23C2×#12011yescharacteristic
354C23C2×#12111yescharacteristic
355C23C22×SD1611yescharacteristic
356C23C2×#13511yescharacteristic
357C23C2×AGL(1,ℤ/8)11yescharacteristic
358C23C2×#13711yescharacteristic
359Q8—44no—
360Q8—82no—
361Q8—84no—
362Q8—88no—
363Q8—88no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4#38612yes—
404C2×C4#38612yes—
405C2×C4#38612yes—
406C2×C4#38612yes—
407C2×C4C4×SD1614yes—
408C2×C4#44014yes—
409C2×C4#44812yes—
410C2×C4#44812yes—
411C2×C4#44912yes—
412C2×C4#44912yes—
413C2×C4#46014yes—
414C2×C4#46414yes—
415C2×C4C22×C22⋊C411yescharacteristic
416C2×C4C2×C4×D411yescharacteristic
417C2×C4C2×#12111yescharacteristic
418C2×C4C2×#12311yescharacteristic
419C8—44no—
420C8—82no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22C2×#38611yescharacteristic
430C22C2×C4×SD1611yescharacteristic
431C22C2×#44011yescharacteristic
432C22C2×#44811yescharacteristic
433C22C2×#44911yescharacteristic
434C22#302212yes—
435C22#302812yes—
436C22C2×#46011yescharacteristic
437C22C2×#46411yescharacteristic
438C4—22no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—44no—
445C4—44no—
446C4—44no—
447C4—84no—
448C4—82no—
449C4—84no—
450C4—162no—
451C4#188014yes—
452C4#188014yes—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2#1831012yes—
458C2#1831212yes—
459C2C2×#188011yescharacteristic
460C2#2094411yescharacteristic
461C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #3229C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#340742
#218332
C2×#15182
#142742
#90442
#80262
#79062