Finite Group Databasebeta

← #99526

Subgroups — #99526

Order 512.

614 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22650C211yesmaximal, characteristic, complemented
2C4×#459C211yesmaximal, characteristic
3#9674C211yesmaximal, characteristic, complemented
4#9638C211yesmaximal, characteristic, complemented
5C4×D4⋊C8C211yesmaximal, characteristic, complemented
6#7031C211yesmaximal, characteristic, complemented
7#7015C211yesmaximal, characteristic, complemented
8C2×#459C411yescharacteristic
9C2×C4×D8—21no—
10C2×#436—21no—
11C2×#423—21no—
12C4×#127C2211yescharacteristic
13C42×D4C2211yescharacteristic
14#1982C411yescharacteristic
15#1947C411yescharacteristic
16#1939—21no—
17#1925—21no—
18#1879—21no—
19#1851—22no—
20#1842C2211yescharacteristic
21#1802C411yescharacteristic
22#1797—21no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×#102—21no—
27C42×C8C2211yescharacteristic
28C2×D4⋊C8—22no—
29C2×D4⋊C8—21no—
30C2×D4⋊C8—21no—
31C2×D4⋊C8—21no—
32#1279—21no—
33C22×D8—21no—
34C22×M16—22no—
35C2×#127C2×C411yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C22×C42—21no—
43#459—22no—
44#459—22no—
45C4×D8—42no—
46C4×D8—42no—
47#436—42no—
48#436—42no—
49C2×#107—21no—
50#423—42no—
51#423—42no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8C2×C411yescharacteristic
57C2×C4⋊C8C2×C411yescharacteristic
58C2×C4⋊C8D411yescharacteristic
59C2×C4⋊C8D411yescharacteristic
60C2×C4⋊C8D412yes—
61C2×#102—21no—
62C2×#102—21no—
63C2×#102—21no—
64C2×#102—21no—
65C2×C22⋊C8—22no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72#390—21no—
73#390—21no—
74#390—21no—
75#384C2×C411yescharacteristic
76C4×C4⋊C4C2×C411yescharacteristic
77C4×C4⋊C4D411yescharacteristic
78C4×C4⋊C4D411yescharacteristic
79C4×C22⋊C4—21no—
80C4×C22⋊C4—21no—
81C4×C22⋊C4—21no—
82C43C2311yescharacteristic
83#340—21no—
84#340—21no—
85#336—22no—
86#336—21no—
87#336—21no—
88#336—21no—
89D4⋊C8—24no—
90D4⋊C8—44no—
91D4⋊C8—42no—
92D4⋊C8—42no—
93C22×D4—21no—
94C22×D4—21no—
95C22×D4—21no—
96C22×D4C22⋊C411yescharacteristic
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—41no—
101C2×D8—24no—
102C2×D8—42no—
103C2×D8—42no—
104C2×M16—44no—
105C2×M16—44no—
106C2×M16—82no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—41no—
113C22×C8—41no—
114#127—21no—
115#127—21no—
116#127—21no—
117#127M1612yescomplemented
118C4×D4—22no—
119C4×D4—21no—
120C4×D4—21no—
121C4×D4—21no—
122C4×D4—21no—
123C4×D4—21no—
124C4×D4—21no—
125C4×D4—41no—
126C4×D4—42no—
127C4×D4—42no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4D812yes—
137C4×D4SD1612yes—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164#107—42no—
165C4⋊C8—22no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—42no—
173C4⋊C8—41no—
174C4⋊C8—44no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8M1612yescomplemented
180C4⋊C8D812yescomplemented
181#102—24no—
182#102—42no—
183#102—42no—
184#102—42no—
185C22⋊C8—44no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—41no—
190C4×C8—41no—
191C4×C8—41no—
192C4×C8—41no—
193C4×C8—41no—
194C4×C8—41no—
195C4×C8—41no—
196C4×C8—41no—
197C4×C8—41no—
198#95—21no—
199#95—21no—
200#95—21no—
201C24—21no—
202C24—41no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#10212yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12311yescharacteristic
258D8—48no—
259M16—88no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—81no—
280C2×C8—81no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4#10212yes—
291C4⋊C4#10412yes—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—82no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42C2×M1611yescharacteristic
316C42C2×M1611yescharacteristic
317C42#13111yescharacteristic
318C42#13111yescharacteristic
319C42C2×D811yescharacteristic
320C42C2×D811yescharacteristic
321C42C2×SD1611yescharacteristic
322C42#13511yescharacteristic
323C42AGL(1,ℤ/8)11yescharacteristic
324C42AGL(1,ℤ/8)11yescharacteristic
325C42AGL(1,ℤ/8)11yescharacteristic
326C42#13711yescharacteristic
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23#38611yescharacteristic
342D4—22no—
343D4—22no—
344D4—44no—
345D4—42no—
346D4—42no—
347D4—42no—
348D4—41no—
349D4—42no—
350D4—41no—
351D4—82no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—161no—
388C2×C4—161no—
389C2×C4—161no—
390C2×C4—161no—
391C2×C4#40711yescharacteristic
392C2×C4#40811yescharacteristic
393C2×C4C2×#10211yescharacteristic
394C2×C4#41711yescharacteristic
395C2×C4C2×#10411yescharacteristic
396C2×C4#42111yescharacteristic
397C2×C4#43511yescharacteristic
398C2×C4#43511yescharacteristic
399C2×C4#43611yescharacteristic
400C2×C4#43611yescharacteristic
401C2×C4C4×D811yescharacteristic
402C2×C4#44211yescharacteristic
403C2×C4#44311yescharacteristic
404C2×C4#44411yescharacteristic
405C2×C4#44711yescharacteristic
406C2×C4#44711yescharacteristic
407C2×C4#44811yescharacteristic
408C2×C4#44911yescharacteristic
409C2×C4#45311yescharacteristic
410C2×C4#45411yescharacteristic
411C2×C4#45911yescharacteristic
412C2×C4#46311yescharacteristic
413C2×C4#48011yescharacteristic
414C2×C4#48111yescharacteristic
415C8—44no—
416C8—42no—
417C8—42no—
418C8—82no—
419C8—82no—
420C8—81no—
421C8—81no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22#187011yescharacteristic
437C22#187111yescharacteristic
438C22#187411yescharacteristic
439C22#187611yescharacteristic
440C22#187911yescharacteristic
441C22#188111yescharacteristic
442C22#188311yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—41no—
450C4—42no—
451C4—42no—
452C4—81no—
453C4—161no—
454C4#148611yescharacteristic
455C4#148911yescharacteristic
456C4#149211yescharacteristic
457C4#149511yescharacteristic
458C4#158111yescharacteristic
459C4#158611yescharacteristic
460C4#158911yescharacteristic
461C4#159411yescharacteristic
462C4#161911yescharacteristic
463C4#162211yescharacteristic
464C4#162511yescharacteristic
465C4#162711yescharacteristic
466C2—82no—
467C2—162no—
468C2#894411yescharacteristic
469C2#895111yescharacteristic
470C2#895611yescharacteristic
471C2#897011yescharacteristic
472C2#897711yescharacteristic
473C2#899111yescharacteristic
474C2#899511yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC23#386
prime radicalC2×#459C4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22650C2semidirect
3. #9674C2semidirect
4. #9638C2semidirect
5. C4×D4⋊C8C2semidirect
6. #7031C2semidirect
7. #7015C2semidirect
117. #127M16semidirect
179. C4⋊C8M16semidirect
180. C4⋊C8D8semidirect

Maximal subgroups

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

subgroupindex
#226502
C4×#4592
#96742
#96382
C4×D4⋊C82
#70312
#70152