Finite Group Databasebeta

← #109774

Subgroups — #109774

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21982C211yesmaximal, characteristic
2#13779C211yesmaximal, characteristic
3#10840C211yesmaximal, characteristic
4#8043C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7C2×#1274C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11#2285C2211yescharacteristic
12C2×#384C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4—21no—
15C2×#376C2211yescharacteristic
16#1947—22no—
17#1852—22no—
18#1845C412yescomplemented
19#1834C412yescomplemented
20C2×#339—21no—
21C2×#339—21no—
22#1436—22no—
23#1436—22no—
24C4⋊#95—21no—
25#1296—22no—
26#1294—22no—
27#1286C414yescomplemented
28#1274C414yescomplemented
29C22×M16—21no—
30C2×#117—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4D412yes—
35C22×C42—22no—
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42C2311yescharacteristic
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431C2×C412yes—
45#431C2×C412yes—
46#431C2×C412yes—
47C2×#107—24no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423C2×C412yes—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8C2×C412yes—
60#384—21no—
61#384—21no—
62#384C2×C411yescharacteristic
63#384C2×C411yescharacteristic
64#384C2×C412yes—
65#384D412yes—
66#382—21no—
67#382—21no—
68#382—21no—
69#382C2×C412yes—
70#382Q812yes—
71#380—21no—
72#380—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—42no—
76C4×C4⋊C4—42no—
77#376—21no—
78#376—21no—
79#376C2×C412yes—
80#376Q812yes—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D412yes—
88#340—24no—
89#339—42no—
90#339—42no—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C2×M16—22no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C22×C8—22no—
105C22×C8—41no—
106C22×C8—41no—
107#117—41no—
108#117—41no—
109#117—41no—
110#117—41no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—82no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4C22⋊C414yes—
135C2×C22⋊C4—41no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—44no—
145C2×C42—41no—
146C2×C42—82no—
147C2×C42—82no—
148C2×C42C4⋊C412yes—
149C2×C42C4⋊C412yes—
150C2×C42D812yes—
151C2×C42Q1612yes—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C411yescharacteristic
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×Q811yescharacteristic
157C2×C42C2×Q811yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164#107—84no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—41no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—41no—
180C4×C8C4214yescomplemented
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—44no—
189#95—41no—
190#95—41no—
191#95—82no—
192#95—82no—
193#95C4⋊C414yes—
194C24#12011yescharacteristic
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C4⋊C411yescharacteristic
231C22×C4C2×C4⋊C411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×Q1611yescharacteristic
249C22×C4#13511yescharacteristic
250C22×C4#13511yescharacteristic
251C22×C4AGL(1,ℤ/8)11yescharacteristic
252C22×C4AGL(1,ℤ/8)11yescharacteristic
253C22×C4#13711yescharacteristic
254C22×C4#13711yescharacteristic
255M16—42no—
256M16—42no—
257M16—81no—
258M16—81no—
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—81no—
265C2×C8—81no—
266C2×C8—81no—
267C2×C8—81no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C22⋊C4—81no—
289C22⋊C4—81no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—41no—
296C42—41no—
297C42—88no—
298C42—81no—
299C42—81no—
300C42#10714yescomplemented
301C42C2×C4211yescharacteristic
302C42#11711yescharacteristic
303C42#11711yescharacteristic
304C42#11711yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×Q811yescharacteristic
308C42C4×Q811yescharacteristic
309C42C4×Q811yescharacteristic
310C42C4×Q811yescharacteristic
311C42#12411yescharacteristic
312C42#12511yescharacteristic
313C42#12511yescharacteristic
314C42#12611yescharacteristic
315C42#12611yescharacteristic
316C42C4⋊Q811yescharacteristic
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23#38511yescharacteristic
322C23#38511yescharacteristic
323C23#38611yescharacteristic
324C23#44711yescharacteristic
325C23#44911yescharacteristic
326C23#45111yescharacteristic
327C23#45211yescharacteristic
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4C4×C22⋊C411yescharacteristic
379C2×C4C4×C4⋊C411yescharacteristic
380C2×C4C4×C4⋊C411yescharacteristic
381C2×C4#38711yescharacteristic
382C2×C4#38711yescharacteristic
383C2×C4#38711yescharacteristic
384C2×C4#38711yescharacteristic
385C2×C4#38811yescharacteristic
386C2×C4#38811yescharacteristic
387C2×C4#38911yescharacteristic
388C2×C4#38911yescharacteristic
389C2×C4#39111yescharacteristic
390C2×C4C2×#10711yescharacteristic
391C2×C4#42711yescharacteristic
392C2×C4#42811yescharacteristic
393C2×C4#42811yescharacteristic
394C2×C4C4×D811yescharacteristic
395C2×C4C4×D811yescharacteristic
396C2×C4C4×Q1611yescharacteristic
397C2×C4C4×Q1611yescharacteristic
398C2×C4#44011yescharacteristic
399C2×C4#44011yescharacteristic
400C2×C4#44111yescharacteristic
401C2×C4#44211yescharacteristic
402C2×C4#45911yescharacteristic
403C2×C4#46211yescharacteristic
404C2×C4#46311yescharacteristic
405C2×C4#46411yescharacteristic
406C2×C4#47411yescharacteristic
407C2×C4#47711yescharacteristic
408C2×C4#47811yescharacteristic
409C2×C4#47911yescharacteristic
410C2×C4#48011yescharacteristic
411C2×C4#48211yescharacteristic
412C2×C4#48411yescharacteristic
413C2×C4#48511yescharacteristic
414C8—44no—
415C8—82no—
416C8—82no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22#143611yescharacteristic
430C22#185211yescharacteristic
431C22#185411yescharacteristic
432C22#186211yescharacteristic
433C22#186311yescharacteristic
434C22#187811yescharacteristic
435C22#187911yescharacteristic
436C22#190411yescharacteristic
437C22#190511yescharacteristic
438C22#191111yescharacteristic
439C22#191211yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—44no—
449C4—84no—
450C4—82no—
451C4—84no—
452C4—161no—
453C4—161no—
454C4C4×#10711yescharacteristic
455C4#177611yescharacteristic
456C4#191811yescharacteristic
457C4#192111yescharacteristic
458C4#192511yescharacteristic
459C4#192811yescharacteristic
460C4#194711yescharacteristic
461C4#194911yescharacteristic
462C2—21no—
463C2—21no—
464C2—21no—
465C2—21no—
466C2#929711yescharacteristic
467C2#929811yescharacteristic
468C2#929911yescharacteristic
469C2#930011yescharacteristic
470C2#1088811yescharacteristic
471C2#1090311yescharacteristic
472C2#1090711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1845C4semidirect
19. #1834C4semidirect
27. #1286C4semidirect
28. #1274C4semidirect
180. C4×C8C42semidirect
300. C42#107semidirect

Maximal subgroups

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

subgroupindex
#219822
#137792
#108402
#80432
#80262
C2×#12862
C2×#12742