Finite Group Databasebeta

← #127387

Subgroups — #127387

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21738C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3#20722C211yesmaximal, characteristic
4#7236C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6#2920—21no—
7C22×C22⋊C8C411yescharacteristic, complemented
8C22×C22⋊C8C411yescharacteristic, complemented
9#2280—21no—
10#1869—21no—
11#1869—21no—
12#1869C412yescomplemented
13#1869C2212yes—
14#1821C412yescomplemented
15#1819C412yes—
16#1817—21no—
17#1800—21no—
18#1800C2212yes—
19#1799—21no—
20#1799C2212yes—
21#1783—21no—
22#1781—21no—
23#1779—21no—
24C4×#100—21no—
25C4×#100C412yescomplemented
26C4×#99—21no—
27C4×#99C412yes—
28C23×D4D411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C23×C8—21no—
32C23×C8C2×C412yes—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4Q811yescharacteristic
44C2×C4×D4C2311yescharacteristic
45C2×#117—41no—
46C22×C4⋊C4C2×C411yescharacteristic
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×#100—22no—
57C2×#99—22no—
58C2×#99—22no—
59#407—22no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8—41no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8C2×C412yes—
70C4×M16—22no—
71#381—41no—
72#380—22no—
73#380—41no—
74#379—22no—
75C4×C22⋊C4—22no—
76C4×C22⋊C4—41no—
77#343—22no—
78#343C2×C414yes—
79#342—22no—
80#342C2×C414yes—
81#336—22no—
82#336—22no—
83C25C22⋊C411yescharacteristic
84C25C4⋊C411yescharacteristic
85C22×D4—24no—
86C22×D4—41no—
87C22×D4—41no—
88C22×D4—42no—
89C22×D4C22⋊C411yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C22×D4C2×D411yescharacteristic
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×C4C4⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×M16—24no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—42no—
120C22×C8—42no—
121C22×C8—82no—
122C22×C8—82no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
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—
139#117—84no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4C4211yescharacteristic
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C4⋊C411yescharacteristic
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C22⋊C4—24no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4C4211yescharacteristic
155C2×C22⋊C4C22⋊C411yescharacteristic
156C2×C22⋊C4C4⋊C411yescharacteristic
157C2×C22⋊C4C4⋊C411yescharacteristic
158C2×C22⋊C4C4⋊C411yescharacteristic
159C2×C22⋊C4C4⋊C411yescharacteristic
160C2×C22⋊C4C22×C411yescharacteristic
161C2×C22⋊C4C2×Q811yescharacteristic
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—42no—
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C4⋊C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C4⋊C8—42no—
172C4⋊C8—42no—
173#100—44no—
174#100—44no—
175#99—44no—
176#99—44no—
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—
189#95—42no—
190#95—42no—
191C24—22no—
192C24—21no—
193C24—21no—
194C24—22no—
195C24—21no—
196C24—21no—
197C24—22no—
198C24—41no—
199C24—81no—
200C24—81no—
201C24—81no—
202C24—81no—
203C24#9511yescharacteristic
204C24#9511yescharacteristic
205C24#9511yescharacteristic
206C24C2×C22⋊C411yescharacteristic
207C24C2×C4⋊C411yescharacteristic
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—48no—
220C2×D4—48no—
221C2×D4#13514yes—
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—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
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—42no—
248C22×C4—41no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4#9511yescharacteristic
270C22×C4#9511yescharacteristic
271C22×C4#9511yescharacteristic
272C22×C4#9511yescharacteristic
273C22×C4#9511yescharacteristic
274C22×C4#9911yescharacteristic
275C22×C4#9911yescharacteristic
276C22×C4#9911yescharacteristic
277C22×C4#9911yescharacteristic
278C22×C4C2×C4211yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C4⋊C411yescharacteristic
282C22×C4C2×C4⋊C411yescharacteristic
283M16—44no—
284M16—44no—
285C2×C8—44no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—84no—
293C2×C8—84no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—84no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—84no—
310C22⋊C4—84no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—84no—
320C23—24no—
321C23—24no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—84no—
354C23—84no—
355C23—84no—
356C23—84no—
357C23H(3,ℤ/4)12yes—
358C23#34211yescharacteristic
359C23#34211yescharacteristic
360C23C2×#9511yescharacteristic
361C23#41612yes—
362C23#42712yes—
363D4—44no—
364D4—44no—
365D4—44no—
366D4—44no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4#34211yescharacteristic
418C2×C4#34211yescharacteristic
419C2×C4C2×#9911yescharacteristic
420C2×C4C2×#9911yescharacteristic
421C2×C4#41612yes—
422C8—82no—
423C8—84no—
424C8—82no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22—84no—
456C22—84no—
457C22—84no—
458C22—84no—
459C22C2×H(3,ℤ/4)11yescharacteristic
460C22#173511yescharacteristic
461C22C2×#34211yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—164no—
474C4#212112yes—
475C2—21no—
476C2—21no—
477C2—42no—
478C2—41no—
479C2—41no—
480C2—42no—
481C2—42no—
482C2—81no—
483C2#798211yescharacteristic
484C2#1257411yescharacteristic
485C2#1258911yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4D4
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. #7236C2semidirect
7. C22×C22⋊C8C4semidirect
8. C22×C22⋊C8C4semidirect
12. #1869C4semidirect
14. #1821C4semidirect
25. C4×#100C4semidirect

Maximal subgroups

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

subgroupindex
#217382
#207262
#207222
#72362