Finite Group Databasebeta

← #128129

Subgroups — #128129

Order 512.

745 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21740C211yesmaximal, characteristic
2C2×#1800C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#7321C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#423—21no—
7C22×#100C412yes—
8C2×#407—21no—
9C2×#407C411yescharacteristic
10C22×C22⋊C8C411yescharacteristic
11#1870C412yes—
12#1870C2212yes—
13#1869—21no—
14#1869—21no—
15#1817—21no—
16#1800—21no—
17#1800—21no—
18#1800C412yes—
19#1800C2212yes—
20#1779—21no—
21C2×#343—21no—
22#1465—21no—
23#1465C2212yes—
24#1463—21no—
25#1462C412yes—
26#1459—21no—
27#1459C412yes—
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—22no—
32C22×M16—41no—
33C22×M16C2×C412yes—
34C23×C8—21no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4—41no—
38C2×C4×D4—41no—
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4C2×C411yescharacteristic
41C2×C4×D4C2×C411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4D411yescharacteristic
44C2×C4×D4D411yescharacteristic
45C2×C4×D4Q811yescharacteristic
46C2×C4×D4C2311yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4Q811yescharacteristic
50C22×C42D411yescharacteristic
51#423—22no—
52#423—22no—
53#423—41no—
54#423—41no—
55C2×C4⋊C8—41no—
56C2×C4⋊C8—41no—
57C2×#100—22no—
58C2×#100—44no—
59C2×#100C2×C414yes—
60#407—22no—
61#407—22no—
62#407—41no—
63#407—41no—
64#407—41no—
65#407—41no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74C2×C22⋊C8—41no—
75C2×C22⋊C8C2×C412yes—
76#343—22no—
77#343—22no—
78#336—22no—
79#336—22no—
80#324—22no—
81#324—22no—
82#323—22no—
83#323C2×C414yes—
84C25C4211yescharacteristic
85C25C4⋊C411yescharacteristic
86C22×D4—22no—
87C22×D4—22no—
88C22×D4—42no—
89C22×D4—41no—
90C22×D4—41no—
91C22×D4C4211yescharacteristic
92C22×D4C22⋊C411yescharacteristic
93C22×D4C4⋊C411yescharacteristic
94C22×D4C22×C411yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4C4211yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—24no—
119C2×M16—42no—
120C2×M16—42no—
121C2×M16—42no—
122C2×M16—42no—
123C2×M16—44no—
124C2×M16—44no—
125C2×M16—82no—
126C2×M16—84no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—42no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8—82no—
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—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C4⋊C411yescharacteristic
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4C4211yescharacteristic
160C2×C22⋊C4C22⋊C411yescharacteristic
161C2×C22⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4C4⋊C411yescharacteristic
163C2×C22⋊C4C4⋊C411yescharacteristic
164C2×C22⋊C4C4⋊C411yescharacteristic
165C2×C22⋊C4C22×C411yescharacteristic
166C2×C22⋊C4C2×Q811yescharacteristic
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C4⋊C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C4⋊C8—42no—
176C4⋊C8—42no—
177C4⋊C8—42no—
178C4⋊C8—42no—
179#100—44no—
180#100—44no—
181C22⋊C8—24no—
182C22⋊C8—42no—
183C22⋊C8—42no—
184C22⋊C8—42no—
185C22⋊C8—42no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—42no—
191C22⋊C8—82no—
192C22⋊C8—82no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—22no—
198C24—22no—
199C24—41no—
200C24—41no—
201C24—42no—
202C24—42no—
203C24—81no—
204C24—81no—
205C24#9511yescharacteristic
206C24#9511yescharacteristic
207C24#9511yescharacteristic
208C24C2×C4211yescharacteristic
209C24C2×C4⋊C411yescharacteristic
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4#10012yes—
227C2×D4#10112yes—
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—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4#9511yescharacteristic
276C22×C4#9511yescharacteristic
277C22×C4#9511yescharacteristic
278C22×C4#9511yescharacteristic
279C22×C4#9511yescharacteristic
280C22×C4#9911yescharacteristic
281C22×C4#9911yescharacteristic
282C22×C4#9911yescharacteristic
283C22×C4#9911yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C4⋊C411yescharacteristic
288C22×C4C2×C4⋊C411yescharacteristic
289M16—44no—
290M16—44no—
291M16—84no—
292M16—84no—
293M16—84no—
294M16—84no—
295C2×C8—44no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—44no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—84no—
304C2×C8—84no—
305C2×C8—84no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—84no—
361C23—84no—
362C23—84no—
363C23—84no—
364C23#34211yescharacteristic
365C23#34211yescharacteristic
366C23#34312yes—
367C23#34312yes—
368C23C2×#9511yescharacteristic
369D4—42no—
370D4—42no—
371D4—44no—
372D4—44no—
373D4—42no—
374D4—42no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4#34211yescharacteristic
424C2×C4#34211yescharacteristic
425C2×C4#34412yes—
426C2×C4C2×#9911yescharacteristic
427C2×C4C2×#9911yescharacteristic
428C2×C4C2×#10011yescharacteristic
429C2×C4C2×#10111yescharacteristic
430C8—82no—
431C8—84no—
432C8—84no—
433C8—82no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—81no—
460C22—84no—
461C22—84no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—84no—
466C22—84no—
467C22#139212yes—
468C22C2×#34211yescharacteristic
469C22C2×#34311yescharacteristic
470C22C2×#34411yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—22no—
476C4—21no—
477C4—41no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4#212112yes—
484C2—21no—
485C2—21no—
486C2—22no—
487C2—41no—
488C2—42no—
489C2—42no—
490C2—81no—
491C2—81no—
492C2#810711yescharacteristic
493C2#1257411yescharacteristic
494C2C2×#139211yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #7321C2semidirect

Maximal subgroups

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

subgroupindex
#217402
C2×#18002
#173622
#73212