Finite Group Databasebeta

← #101375

Subgroups — #101375

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24299C211yesmaximal, characteristic
2#22168C211yesmaximal, characteristic
3#11384C211yesmaximal, characteristic, complemented
4#8571C211yesmaximal, characteristic
5#8342C211yesmaximal, characteristic, complemented
6#7039C211yesmaximal, characteristic, complemented
7#7020C211yesmaximal, characteristic, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C2×#476—21no—
10C2×C8×D4—21no—
11C4×#121C2211yescharacteristic
12#2058C412yescomplemented
13#2023C412yescomplemented
14#1987—21no—
15#1923—21no—
16#1916—21no—
17#1874—21no—
18#1869—21no—
19#1869—21no—
20#1853—21no—
21#1850—21no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25C2×#340—21no—
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#333—21no—
29C2×#333—21no—
30C2×#327—21no—
31C4⋊#95C2211yescharacteristic
32#1294—21no—
33C23×Q8—21no—
34C23×C8—21no—
35C23×C8D411yescharacteristic
36C23×C8D411yescharacteristic
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—42no—
41C2×C4⋊Q8—42no—
42C2×C4⋊Q8C2×C412yes—
43C2×#121D411yescharacteristic
44C2×C4×D4—21no—
45C2×C4×D4D411yescharacteristic
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C42C2311yescharacteristic
50#476—42no—
51#476—42no—
52C8×D4—42no—
53C8×D4—42no—
54C2×#106—22no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8D411yescharacteristic
59C2×C4⋊C8D411yescharacteristic
60C2×#102—22no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C4×C8—21no—
68#390C2×C411yescharacteristic
69#389C2×C411yescharacteristic
70#388—21no—
71#386—21no—
72#380D411yescharacteristic
73C4×C4⋊C4D411yescharacteristic
74C4×C22⋊C4—21no—
75C2×#95—21no—
76C2×#95—21no—
77#340—22no—
78#340—22no—
79#340—42no—
80#340—42no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336C2×C412yes—
88#333—42no—
89#333—42no—
90#327—42no—
91C22×Q8—22no—
92C22×Q8—44no—
93C22×Q8—41no—
94C22×Q8—81no—
95C22×Q8—81no—
96C22×Q8—82no—
97C22×D4—21no—
98C22×D4C22⋊C411yescharacteristic
99C23×C4—21no—
100C23×C4—41no—
101C23×C4—41no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—42no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—81no—
120C22×C8—81no—
121C22×C8SD1612yes—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8—41no—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132#121—21no—
133#121—21no—
134#121SD1612yes—
135#121Q1612yes—
136C4×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
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×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C42—21no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173#106—44no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—42no—
181#102—44no—
182C22⋊C8—42no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—81no—
190C22⋊C8—81no—
191C4×C8—42no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—41no—
196#95—41no—
197#95—81no—
198C24—21no—
199C24#12111yescharacteristic
200C2×Q8—24no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—82no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—88no—
209C2×Q8—88no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—21no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4#10312yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4C2×SD1611yescharacteristic
260C22×C4C2×SD1611yescharacteristic
261C22×C4C2×Q1611yescharacteristic
262C22×C4#13511yescharacteristic
263C22×C4#13511yescharacteristic
264C22×C4#13511yescharacteristic
265C22×C4AGL(1,ℤ/8)11yescharacteristic
266C22×C4#13711yescharacteristic
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—81no—
285C2×C8—84no—
286C2×C8—81no—
287C2×C8—81no—
288C2×C8—81no—
289C2×C8—81no—
290C4⋊C4—21no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C4⋊C4#10312yes—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23#38611yescharacteristic
336C23#46111yescharacteristic
337C23#46311yescharacteristic
338C23#46511yescharacteristic
339C23#46711yescharacteristic
340C23#47111yescharacteristic
341C23#47211yescharacteristic
342Q8—84no—
343Q8—84no—
344Q8—84no—
345Q8—84no—
346D4—42no—
347D4—42no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
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—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—161no—
402C2×C4—161no—
403C2×C4—161no—
404C2×C4—161no—
405C2×C4C2×#10311yescharacteristic
406C2×C4#41611yescharacteristic
407C2×C4C4×SD1611yescharacteristic
408C2×C4#44011yescharacteristic
409C2×C4#45011yescharacteristic
410C2×C4#45211yescharacteristic
411C2×C4#45511yescharacteristic
412C2×C4#45611yescharacteristic
413C2×C4B(4,𝔽2)11yescharacteristic
414C2×C4B(4,𝔽2)11yescharacteristic
415C2×C4#46511yescharacteristic
416C2×C4#46711yescharacteristic
417C8—42no—
418C8—82no—
419C8—81no—
420C8—81no—
421C8—81no—
422C8—161no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22#167111yescharacteristic
443C22#167511yescharacteristic
444C22#167711yescharacteristic
445C22#168311yescharacteristic
446C22#187711yescharacteristic
447C22#188211yescharacteristic
448C22#188911yescharacteristic
449C22#189211yescharacteristic
450C22#190311yescharacteristic
451C22#190711yescharacteristic
452C22#190911yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—41no—
460C4—41no—
461C4—42no—
462C4—84no—
463C4—81no—
464C4—81no—
465C4—162no—
466C4#160111yescharacteristic
467C4#160311yescharacteristic
468C4#160411yescharacteristic
469C4#160611yescharacteristic
470C2—21no—
471C2—21no—
472C2—21no—
473C2—21no—
474C2—82no—
475C2#903911yescharacteristic
476C2#921511yescharacteristic
477C2#928711yescharacteristic
478C2#931511yescharacteristic
479C2#937511yescharacteristic
480C2#946711yescharacteristic
481C2#952011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
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
3. #11384C2semidirect
5. #8342C2semidirect
6. #7039C2semidirect
7. #7020C2semidirect
12. #2058C4semidirect
13. #2023C4semidirect

Maximal subgroups

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

subgroupindex
#242992
#221682
#113842
#85712
#83422
#70392
#70202