Finite Group Databasebeta

← #101396

Subgroups — #101396

Order 512.

614 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24301C211yesmaximal, characteristic
2#22295C211yesmaximal, characteristic
3#11010C211yesmaximal, characteristic
4#8572C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7020C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#475—21no—
10C22×#103—21no—
11C22×#103—21no—
12C4×C22⋊Q8C2211yescharacteristic
13#2058C412yescomplemented
14#2024C412yescomplemented
15#1988—21no—
16#1985—21no—
17#1940—21no—
18#1917—21no—
19#1917—21no—
20#1872—21no—
21#1853—21no—
22#1850—21no—
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25#1797C2211yescharacteristic
26#1752—21no—
27C2×#336C2211yescharacteristic
28C2×#332—21no—
29C2×#332—21no—
30C2×#328—21no—
31C4⋊#95C2211yescharacteristic
32#1294—21no—
33C23×Q8—21no—
34C23×C8D411yescharacteristic
35C23×C8D411yescharacteristic
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—42no—
40C2×C4⋊Q8—42no—
41C2×C4⋊Q8C2×C412yes—
42C2×C22⋊Q8D411yescharacteristic
43C2×C4×Q8D411yescharacteristic
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C42C2311yescharacteristic
48#475—42no—
49#475—42no—
50C2×#106—22no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8D411yescharacteristic
56C2×C4⋊C8D411yescharacteristic
57C2×#103—22no—
58C2×#103—42no—
59C2×#103—42no—
60C2×#103—42no—
61C2×#103—42no—
62C2×#103—42no—
63C2×#103—42no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×#97—21no—
68C2×#97—21no—
69#391C2×C411yescharacteristic
70#389C2×C411yescharacteristic
71#387—21no—
72#385—21no—
73#380D411yescharacteristic
74C4×C4⋊C4—21no—
75C4×C4⋊C4D411yescharacteristic
76C4×C22⋊C4—21no—
77C2×#95—21no—
78C2×#95—21no—
79#340—22no—
80#340—22no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336C2×C412yes—
90#332—42no—
91#332—42no—
92#328—42no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8—41no—
105C22×Q8—41no—
106C22×Q8—41no—
107C22×Q8C22⋊C411yescharacteristic
108C23×C4—41no—
109C23×C4—41no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4#4111yescharacteristic
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8SD1612yes—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137C22⋊Q8—21no—
138C22⋊Q8—21no—
139C22⋊Q8—21no—
140C22⋊Q8—21no—
141C4×Q8—41no—
142C4×Q8—41no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42C22×C411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42#4111yescharacteristic
171#106—44no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—81no—
179C4⋊C8—81no—
180#103—44no—
181#103—48no—
182#103—48no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189#97—81no—
190#97—81no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—41no—
197#95—41no—
198#95—81no—
199C24#12111yescharacteristic
200C2×Q8—24no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—42no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—81no—
221C2×Q8—81no—
222C2×Q8—81no—
223C2×Q8—81no—
224C2×Q8—81no—
225C2×Q8—81no—
226C2×Q8—81no—
227C2×Q8—81no—
228C2×Q8—81no—
229C2×Q8—81no—
230C2×Q8—81no—
231C2×Q8—81no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4C2×SD1611yescharacteristic
267C22×C4#13511yescharacteristic
268C22×C4AGL(1,ℤ/8)11yescharacteristic
269C22×C4AGL(1,ℤ/8)11yescharacteristic
270C22×C4#13711yescharacteristic
271C22×C4#13711yescharacteristic
272C22×C4#13711yescharacteristic
273C22×C4#13711yescharacteristic
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—81no—
292C2×C8—81no—
293C2×C8—81no—
294C2×C8—81no—
295C2×C8—81no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4—81no—
315C4⋊C4—81no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#38611yescharacteristic
335C23#46011yescharacteristic
336C23#46411yescharacteristic
337C23#46811yescharacteristic
338C23#47011yescharacteristic
339C23#47111yescharacteristic
340C23#47311yescharacteristic
341Q8—42no—
342Q8—44no—
343Q8—44no—
344Q8—81no—
345Q8—82no—
346Q8—82no—
347Q8—82no—
348Q8—82no—
349Q8—82no—
350Q8—82no—
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—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—161no—
400C2×C4—161no—
401C2×C4—161no—
402C2×C4—161no—
403C2×C4#41711yescharacteristic
404C2×C4#41911yescharacteristic
405C2×C4C4×SD1611yescharacteristic
406C2×C4#44011yescharacteristic
407C2×C4#44811yescharacteristic
408C2×C4#44911yescharacteristic
409C2×C4#45311yescharacteristic
410C2×C4#45411yescharacteristic
411C2×C4B(4,𝔽2)11yescharacteristic
412C2×C4B(4,𝔽2)11yescharacteristic
413C2×C4#46811yescharacteristic
414C2×C4#47011yescharacteristic
415C8—42no—
416C8—82no—
417C8—81no—
418C8—81no—
419C8—81no—
420C8—161no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22#168611yescharacteristic
434C22#169011yescharacteristic
435C22#170011yescharacteristic
436C22#170411yescharacteristic
437C22#188011yescharacteristic
438C22#188311yescharacteristic
439C22#188911yescharacteristic
440C22#189411yescharacteristic
441C22#190211yescharacteristic
442C22#190811yescharacteristic
443C22#190911yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—41no—
451C4—41no—
452C4—81no—
453C4—84no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—162no—
458C4#159611yescharacteristic
459C4#159711yescharacteristic
460C4#159911yescharacteristic
461C4#160011yescharacteristic
462C2—21no—
463C2—21no—
464C2—21no—
465C2—21no—
466C2#904211yescharacteristic
467C2#922811yescharacteristic
468C2#929211yescharacteristic
469C2#932111yescharacteristic
470C2#937711yescharacteristic
471C2#947111yescharacteristic
472C2#951911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2058C4semidirect
14. #2024C4semidirect

Maximal subgroups

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

subgroupindex
#243012
#222952
#110102
#85722
#83402
#70392
#70202