Finite Group Databasebeta

← #100440

Subgroups — #100440

Order 512.

557 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33106C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic, complemented
3#10806C211yesmaximal, characteristic, complemented
4#9680C211yesmaximal, characteristic
5#7664C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic, complemented
7#7029C211yesmaximal, characteristic, complemented
8C2×#480—21no—
9C2×#431—21no—
10#2856C2211yescharacteristic
11C2×C4×C4⋊C4C2211yescharacteristic
12#1982C411yescharacteristic, complemented
13#1982C411yescharacteristic, complemented
14#1947C411yescharacteristic, complemented
15#1947C411yescharacteristic, complemented
16#1939—21no—
17#1925—21no—
18#1925—21no—
19#1916—22no—
20#1874—21no—
21#1817—21no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24C8×C4⋊C4—21no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—21no—
27C2×#340—21no—
28C2×#336C2211yescharacteristic
29C2×#327—21no—
30C2×#327—21no—
31C4⋊#95C2211yescharacteristic
32#1270—22no—
33C23×C8—21no—
34C2×#127C2×C411yescharacteristic
35C2×#121—21no—
36C2×#121—21no—
37C2×#121—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C42—21no—
42C22×C42C2311yescharacteristic
43#480—42no—
44#480—42no—
45#431—42no—
46#431—42no—
47C2×#107—21no—
48C2×#107—21no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8C2×C411yescharacteristic
52C2×C4⋊C8C2×C411yescharacteristic
53C2×#102—21no—
54C2×#102—21no—
55C2×#102—21no—
56C2×#102—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8D411yescharacteristic
60C2×C22⋊C8D411yescharacteristic
61C2×C4×C8—22no—
62C2×C4×C8—21no—
63#390—21no—
64#390—21no—
65#390—21no—
66#384C2×C411yescharacteristic
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4C2×C411yescharacteristic
70C4×C4⋊C4C2×C411yescharacteristic
71C4×C4⋊C4D411yescharacteristic
72C4×C4⋊C4D411yescharacteristic
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76#340—24no—
77#340—21no—
78#340—21no—
79#340—42no—
80#336—22no—
81#336—22no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336D412yes—
87#327—42no—
88#327—42no—
89C22×D4—21no—
90C22×D4—21no—
91C22×D4—21no—
92C22×D4—41no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—41no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4#4111yescharacteristic
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111#127—41no—
112#127—41no—
113#121—41no—
114#121—41no—
115#121—41no—
116#121—41no—
117#121—41no—
118#121—41no—
119#121—41no—
120#121—41no—
121#121—81no—
122#121—81no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4C22⋊C411yescharacteristic
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C22⋊C411yescharacteristic
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C4⋊C4D812yes—
143C2×C4⋊C4SD1612yes—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4111yescharacteristic
161#107—42no—
162#107—42no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—44no—
168#102—42no—
169#102—42no—
170#102—42no—
171#102—42no—
172C22⋊C8—21no—
173C22⋊C8—21no—
174C22⋊C8—21no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8D812yes—
180C4×C8—41no—
181C4×C8—41no—
182C4×C8—44no—
183#95—21no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—41no—
188#95—41no—
189#95—81no—
190C24—41no—
191C24#12111yescharacteristic
192C2×D4—42no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—81no—
207C2×D4—81no—
208C2×D4—81no—
209C2×D4—81no—
210C2×D4—81no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×SD1611yescharacteristic
252C22×C4#13511yescharacteristic
253C22×C4#13511yescharacteristic
254C22×C4#13511yescharacteristic
255C22×C4AGL(1,ℤ/8)11yescharacteristic
256C22×C4AGL(1,ℤ/8)11yescharacteristic
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—81no—
276C2×C8—81no—
277C2×C8—81no—
278C2×C8—81no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4#10212yes—
297C4⋊C4#10212yes—
298C4⋊C4#10412yes—
299C22⋊C4—82no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23#38611yescharacteristic
330C23#45911yescharacteristic
331C23#46311yescharacteristic
332C23#46511yescharacteristic
333C23#46611yescharacteristic
334C23#47111yescharacteristic
335C23#47211yescharacteristic
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
340D4—161no—
341D4—161no—
342D4—162no—
343D4—161no—
344D4—161no—
345D4—161no—
346D4—161no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
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—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—81no—
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—161no—
398C2×C4—161no—
399C2×C4—161no—
400C2×C4—161no—
401C2×C4#41011yescharacteristic
402C2×C4#41011yescharacteristic
403C2×C4C2×#10211yescharacteristic
404C2×C4#41611yescharacteristic
405C2×C4C2×#10411yescharacteristic
406C2×C4#42111yescharacteristic
407C2×C4C4×D811yescharacteristic
408C2×C4#44211yescharacteristic
409C2×C4#44711yescharacteristic
410C2×C4#44911yescharacteristic
411C2×C4#46511yescharacteristic
412C2×C4#46611yescharacteristic
413C8—42no—
414C8—82no—
415C8—81no—
416C8—82no—
417C8—81no—
418C8—81no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22#166511yescharacteristic
439C22#166711yescharacteristic
440C22#167711yescharacteristic
441C22#168111yescharacteristic
442C22#187111yescharacteristic
443C22#187911yescharacteristic
444C22#188111yescharacteristic
445C22#189311yescharacteristic
446C22#189911yescharacteristic
447C22#190511yescharacteristic
448C22#190711yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—41no—
459C4—41no—
460C4—81no—
461C4—162no—
462C4—161no—
463C4#150011yescharacteristic
464C4#150211yescharacteristic
465C4#151711yescharacteristic
466C4#152111yescharacteristic
467C2—21no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2—321no—
472C2#899111yescharacteristic
473C2#911411yescharacteristic
474C2#916411yescharacteristic
475C2#922111yescharacteristic
476C2#926011yescharacteristic
477C2#930611yescharacteristic
478C2#934811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
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
2. #22599C2semidirect
3. #10806C2semidirect
6. #7039C2semidirect
7. #7029C2semidirect
12. #1982C4semidirect
13. #1982C4semidirect
14. #1947C4semidirect
15. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#331062
#225992
#108062
#96802
#76642
#70392
#70292