Finite Group Databasebeta

← #127486

Subgroups — #127486

Order 512.

696 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#21740C211yesmaximal, characteristic
2#17364C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#7247C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6#2920—21no—
7C2×#407—21no—
8C2×#407C411yescharacteristic
9C22×C22⋊C8C411yescharacteristic
10#1870—21no—
11#1870—21no—
12#1869C412yes—
13#1869C2212yes—
14#1817—21no—
15#1779—21no—
16#1466—21no—
17#1466C2212yes—
18#1465—21no—
19#1465—21no—
20#1465C412yes—
21#1465C2212yes—
22#1462—21no—
23#1462C412yes—
24#1460—21no—
25#1460C412yes—
26C2×#324C412yes—
27C2×#323—21no—
28C23×D4Q811yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—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⋊C4C2×C411yescharacteristic
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42D411yescharacteristic
50#432—22no—
51#432—41no—
52#431—22no—
53#431—41no—
54#423—41no—
55#407—22no—
56#407—22no—
57#407—22no—
58#407—41no—
59#407—41no—
60#407—41no—
61#407—41no—
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⋊C8C2×C412yes—
72C4×M16—41no—
73#336—22no—
74#336—22no—
75#324—22no—
76#324C2×C414yes—
77#323—22no—
78#323—22no—
79#323—22no—
80#323—22no—
81#323—22no—
82#323C2×C414yes—
83C25C4⋊C411yescharacteristic
84C25C4⋊C411yescharacteristic
85C22×D4—42no—
86C22×D4—41no—
87C22×D4—41no—
88C22×D4—42no—
89C22×D4C4⋊C411yescharacteristic
90C22×D4C4⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C22×D4C2×Q811yescharacteristic
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×C4C4211yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—42no—
119C2×M16—42no—
120C2×M16—82no—
121C2×M16—84no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—42no—
125C22×C8—42no—
126C22×C8—82no—
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⋊C4C4211yescharacteristic
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C4⋊C411yescharacteristic
148C2×C4⋊C4C22×C411yescharacteristic
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⋊C4C22⋊C411yescharacteristic
156C2×C22⋊C4C22⋊C411yescharacteristic
157C2×C22⋊C4C4⋊C411yescharacteristic
158C2×C22⋊C4C4⋊C411yescharacteristic
159C2×C22⋊C4C22×C411yescharacteristic
160C2×C22⋊C4C2×D411yescharacteristic
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—
171C22⋊C8—24no—
172C22⋊C8—24no—
173C22⋊C8—42no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—42no—
183C22⋊C8—82no—
184C22⋊C8—82no—
185#97—42no—
186C4×C8—42no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—22no—
192C24—41no—
193C24—41no—
194C24—41no—
195C24—42no—
196C24—42no—
197C24—81no—
198C24—81no—
199C24#9511yescharacteristic
200C24#9511yescharacteristic
201C24#9511yescharacteristic
202C24C2×C4⋊C411yescharacteristic
203C24C2×C4⋊C411yescharacteristic
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—48no—
217C2×D4—48no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
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—42no—
244C22×C4—41no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4#9511yescharacteristic
265C22×C4#9511yescharacteristic
266C22×C4#9511yescharacteristic
267C22×C4#9511yescharacteristic
268C22×C4#9511yescharacteristic
269C22×C4#9911yescharacteristic
270C22×C4#9911yescharacteristic
271C22×C4#9911yescharacteristic
272C22×C4#9911yescharacteristic
273C22×C4C2×C4211yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C4⋊C411yescharacteristic
278M16—84no—
279M16—84no—
280M16—84no—
281M16—84no—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—84no—
291C2×C8—84no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C23—22no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—84no—
348C23—84no—
349C23—84no—
350C23—84no—
351C23#34211yescharacteristic
352C23#34211yescharacteristic
353C23C2×#9511yescharacteristic
354C23#43012yes—
355D4—44no—
356D4—44no—
357D4—44no—
358D4—44no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
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—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
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#33812yes—
407C2×C4#34211yescharacteristic
408C2×C4#34211yescharacteristic
409C2×C4C2×#9911yescharacteristic
410C2×C4C2×#9911yescharacteristic
411C8—82no—
412C8—84no—
413C8—84no—
414C8—82no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—84no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—84no—
446C22—84no—
447C22—84no—
448C22#139512yes—
449C22C2×#33811yescharacteristic
450C22#173811yescharacteristic
451C22C2×#34211yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4#212112yes—
464C2—21no—
465C2—21no—
466C2—22no—
467C2—41no—
468C2—42no—
469C2—81no—
470C2—81no—
471C2—81no—
472C2#800911yescharacteristic
473C2#1257411yescharacteristic
474C2C2×#139511yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #7247C2semidirect

Maximal subgroups

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

subgroupindex
#217402
#173642
C2×#14652
#72472