Finite Group Databasebeta

← #127103

Subgroups — #127103

Order 512.

734 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21740C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#7203C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6#2920—21no—
7C2×#407C411yescharacteristic
8C22×C22⋊C8—21no—
9C22×C22⋊C8C411yescharacteristic
10#1870—21no—
11#1870—21no—
12#1869C412yes—
13#1869C2212yes—
14#1821—21no—
15#1817—21no—
16#1800—21no—
17#1800C412yes—
18#1783C412yes—
19#1779—21no—
20C4×#100—21no—
21C4×#100C2212yes—
22#1459—21no—
23#1459—21no—
24#1459C412yes—
25#1459C2212yes—
26C2×#324—21no—
27C2×#323C412yes—
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C23×C8—41no—
33C23×C8C2×C412yes—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4—41no—
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4C2×C411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4D411yescharacteristic
44C2×C4×D4Q811yescharacteristic
45C2×C4×D4C2311yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4Q811yescharacteristic
49C22×C42D411yescharacteristic
50#432—41no—
51#432—41no—
52#431—41no—
53#431—41no—
54#423—22no—
55C2×#100—22no—
56C2×#100C2×C414yes—
57#407—22no—
58#407—41no—
59#407—41no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8—41no—
70C2×C22⋊C8—41no—
71C2×C22⋊C8—41no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8C2×C412yes—
74C4×M16—22no—
75#343—22no—
76#343—22no—
77#336—22no—
78#336—22no—
79#324—22no—
80#324—22no—
81#323—22no—
82#323C2×C414yes—
83C25C4211yescharacteristic
84C25C4⋊C411yescharacteristic
85C22×D4—24no—
86C22×D4—41no—
87C22×D4—41no—
88C22×D4—42no—
89C22×D4C4211yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C22×D4C22×C411yescharacteristic
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—82no—
102C23×C4—82no—
103C23×C4C4211yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C4⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×M16—24no—
117C2×M16—42no—
118C2×M16—82no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—42no—
122C22×C8—42no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8—82no—
126C22×C8—84no—
127C4×D4—41no—
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×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C4⋊C411yescharacteristic
148C2×C4⋊C4C2×D411yescharacteristic
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4C4211yescharacteristic
154C2×C22⋊C4C22⋊C411yescharacteristic
155C2×C22⋊C4C4⋊C411yescharacteristic
156C2×C22⋊C4C4⋊C411yescharacteristic
157C2×C22⋊C4C4⋊C411yescharacteristic
158C2×C22⋊C4C4⋊C411yescharacteristic
159C2×C22⋊C4C22×C411yescharacteristic
160C2×C22⋊C4C2×Q811yescharacteristic
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C4⋊C411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C4⋊C8—42no—
170C4⋊C8—42no—
171#100—28no—
172#100—44no—
173C22⋊C8—24no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—42no—
179C22⋊C8—42no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—42no—
185C22⋊C8—82no—
186C22⋊C8—82no—
187#97—42no—
188C4×C8—42no—
189C24—21no—
190C24—21no—
191C24—22no—
192C24—22no—
193C24—21no—
194C24—21no—
195C24—22no—
196C24—41no—
197C24—42no—
198C24—42no—
199C24—81no—
200C24—81no—
201C24#9511yescharacteristic
202C24#9511yescharacteristic
203C24#9511yescharacteristic
204C24C2×C4211yescharacteristic
205C24C2×C4⋊C411yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—48no—
218C2×D4—48no—
219C2×D4#13114yes—
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—21no—
230C22×C4—22no—
231C22×C4—21no—
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—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4#9511yescharacteristic
267C22×C4#9511yescharacteristic
268C22×C4#9511yescharacteristic
269C22×C4#9511yescharacteristic
270C22×C4#9511yescharacteristic
271C22×C4#9911yescharacteristic
272C22×C4#9911yescharacteristic
273C22×C4#9911yescharacteristic
274C22×C4#9911yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C4⋊C411yescharacteristic
279C22×C4C2×C4⋊C411yescharacteristic
280M16—44no—
281M16—44no—
282M16—84no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—84no—
292C2×C8—84no—
293C2×C8—84no—
294C2×C8—84no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C23—24no—
316C23—24no—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—41no—
330C23—42no—
331C23—42no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—42no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—84no—
349C23—84no—
350C23—84no—
351C23—84no—
352C23#34211yescharacteristic
353C23#34211yescharacteristic
354C23#34412yes—
355C23C2×#9511yescharacteristic
356C23#40512yes—
357C23#42412yes—
358D4—44no—
359D4—44no—
360D4—44no—
361D4—44no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4#34211yescharacteristic
410C2×C4#34211yescharacteristic
411C2×C4#40812yes—
412C2×C4C2×#9911yescharacteristic
413C2×C4C2×#9911yescharacteristic
414C8—82no—
415C8—84no—
416C8—84no—
417C8—82no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—48no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—81no—
445C22—81no—
446C22—84no—
447C22—81no—
448C22—81no—
449C22—84no—
450C22—84no—
451C22#139212yes—
452C22#172911yescharacteristic
453C22C2×#34211yescharacteristic
454C22C2×#34411yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—22no—
459C4—41no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4#212112yes—
467C2—22no—
468C2—21no—
469C2—21no—
470C2—41no—
471C2—42no—
472C2—42no—
473C2—42no—
474C2—81no—
475C2#792511yescharacteristic
476C2#1257411yescharacteristic
477C2C2×#139211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7203C2semidirect

Maximal subgroups

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

subgroupindex
#217402
#207262
C2×#14592
#72032