Finite Group Databasebeta

← #374607

Subgroups — #374607

Order 512.

1639 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#21306C212yesmaximal, complemented
4C2×C4×#100C211yesmaximal, characteristic
5#11224C214yesmaximal, complemented
6#10837C212yesmaximal, complemented
7#10740C214yesmaximal
8C2×#533C2211yescharacteristic
9C2×#518C2212yes—
10C22×C4×D4C2211yescharacteristic
11C22×#100C2212yes—
12C2×C4×M16C2212yes—
13#2848—22no—
14#2626—22no—
15#2580—22no—
16#2566C414yescomplemented
17#2563C412yescomplemented
18#2542C412yescomplemented
19#2538—22no—
20#2349—24no—
21D4×C4⋊C4—24no—
22#2317C2212yes—
23#2316C2212yes—
24#2315—22no—
25#2309C2212yes—
26#2285C2211yescharacteristic
27#2285C2212yes—
28C2×#380—22no—
29#1986—24no—
30#1986—24no—
31C4×#100—22no—
32C4×#100—22no—
33C2×#343C2212yes—
34#1733C2218yes—
35C2×H(3,ℤ/4)C2218yes—
36C23×D4C2311yescharacteristic
37C24×C4D412yes—
38C22×M16C2314yes—
39#533—22no—
40#533—22no—
41#533—22no—
42#533—22no—
43C2×C4⋊Q8C2×C411yescharacteristic, complemented
44C2×C4⋊Q8C2×C411yescharacteristic, complemented
45C2×#125C2×C412yescomplemented
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—24no—
49#518—28no—
50#518—44no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—28no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C412yes—
60C2×C4×D4D412yes—
61C2×C4×D4D412yes—
62C2×#117C2312yes—
63C2×#117C2312yes—
64C2×#117C2314yes—
65C22×C4⋊C4—24no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—24no—
68C22×C22⋊C4D412yes—
69C22×C42C2311yescharacteristic
70C2×#100—24no—
71C2×#100—24no—
72C2×#100Q818yescomplemented
73C4×M16—22no—
74C4×M16—22no—
75C4×M16—22no—
76C4×M16—22no—
77C2×#97—22no—
78C2×C4×C8—22no—
79#399—24no—
80#396—28no—
81#393—24no—
82#390—24no—
83#390C2×C414yes—
84#390C2×C414yes—
85#389—24no—
86#389—28no—
87#385—216no—
88#385—28no—
89#385—24no—
90#384—22no—
91#384—21no—
92#384C2×C414yescomplemented
93#382—21no—
94#382—22no—
95#382C2×C414yescomplemented
96#380—24no—
97#380—24no—
98#380—24no—
99#380—22no—
100#380—24no—
101#380—21no—
102#380—21no—
103#380—22no—
104#379—24no—
105C4×C4⋊C4C2×C412yescomplemented
106C4×C4⋊C4C2×C412yescomplemented
107C4×C4⋊C4D414yes—
108C4×C4⋊C4D414yes—
109C2×#95—24no—
110#343—24no—
111#339—216no—
112H(3,ℤ/4)—48no—
113C25C2×D412yes—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4C22×C412yes—
121C22×D4C2×D412yes—
122C22×D4C2×Q814yes—
123C22×D4#4114yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—28no—
129C23×C4—84no—
130C23×C4C22⋊C414yes—
131C23×C4C22⋊C414yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—24no—
141C2×M16C2×Q818yes—
142C22×C8—24no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147#125—24no—
148#125—24no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—42no—
152C22⋊Q8—41no—
153C22⋊Q8—42no—
154C22⋊Q8—41no—
155C22⋊Q8—88no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—432no—
169#117—28no—
170#117—28no—
171#117—24no—
172#117—24no—
173#117—24no—
174#117—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—88no—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—48no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4C22⋊C414yes—
205C2×C22⋊C4C22⋊C414yes—
206C2×C42—22no—
207C2×C42C22×C414yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D418yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4112yes—
216C2×C42#4112yes—
217#100—216no—
218#97—42no—
219#97—42no—
220C4×C8—42no—
221C4×C8—42no—
222#95—28no—
223#95—22no—
224#95—22no—
225#95—28no—
226#95—24no—
227#95—24no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—82no—
233C24—82no—
234C24C2×C22⋊C414yes—
235C24#12414yes—
236C24C4⋊Q814yes—
237C24C22×D411yescharacteristic
238C2×Q8—44no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—82no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—28no—
248C2×D4—28no—
249C2×D4—28no—
250C2×D4—28no—
251C2×D4C4×Q818yes—
252C2×D4C22⋊Q818yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—28no—
263C22×C4—42no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—416no—
267C22×C4—84no—
268C22×C4—81no—
269C22×C4—84no—
270C22×C4—82no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4—88no—
274C22×C4—88no—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C414yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12012yes—
283C22×C4#12311yescharacteristic
284C22×C4#12312yes—
285C22×C4#12311yescharacteristic
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D412yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×Q812yes—
290C22×C4C2×#4112yes—
291C22×C4C2×#4112yes—
292C22×C4H(5,𝔽2)12yes—
293C22×C42−1+412yes—
294M16—28no—
295M16—28no—
296M16—44no—
297M16—44no—
298C2×C8—28no—
299C2×C8—44no—
300C2×C8—44no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—48no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—48no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C4⋊C4—88no—
318C4⋊C4—88no—
319C22⋊C4—28no—
320C22⋊C4—28no—
321C22⋊C4—416no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—88no—
329C22⋊C4—88no—
330C42—28no—
331C42—216no—
332C42—44no—
333C42C4×D412yes—
334C42C4×D412yes—
335C42C4×D412yes—
336C42C4×D412yes—
337C42#12112yes—
338C42#12112yes—
339C42#12112yes—
340C42#12112yes—
341C23—24no—
342C23—24no—
343C23—24no—
344C23—24no—
345C23—24no—
346C23—24no—
347C23—41no—
348C23—41no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—82no—
354C23—82no—
355C23—84no—
356C23—84no—
357C23—84no—
358C23—84no—
359C23#39118yes—
360C23C22×C22⋊C411yescharacteristic
361C23C2×#12011yescharacteristic
362C23C2×#12311yescharacteristic
363C23C2×#12411yescharacteristic
364C23C2×C4⋊Q811yescharacteristic
365C23#53511yescharacteristic
366C23#53611yescharacteristic
367Q8—82no—
368Q8—84no—
369Q8—82no—
370D4—216no—
371D4—216no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—48no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—44no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4—82no—
400C2×C4—1616no—
401C2×C4#38612yes—
402C2×C4#38612yes—
403C2×C4#38611yescharacteristic
404C2×C4#38611yescharacteristic
405C2×C4#38611yescharacteristic
406C2×C4#38611yescharacteristic
407C2×C4C2×C4×D412yes—
408C2×C4C2×C4×Q812yes—
409C2×C4#52012yes—
410C2×C4C2×#12112yes—
411C2×C4C2×C22⋊Q812yes—
412C2×C4#53712yes—
413C2×C4#54614yes—
414C2×C4D4×Q814yes—
415C2×C4#55214yes—
416C2×C4#55414yes—
417C8—48no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—28no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—48no—
430C22—48no—
431C22—48no—
432C22—44no—
433C22—44no—
434C22—84no—
435C22—84no—
436C22—84no—
437C22C2×#38611yescharacteristic
438C22C2×#39111yescharacteristic
439C22#231111yescharacteristic
440C22#243411yescharacteristic
441C22#243511yescharacteristic
442C22#250211yescharacteristic
443C22#250311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—22no—
448C4—22no—
449C4—44no—
450C4—84no—
451C4—88no—
452C4—82no—
453C4—82no—
454C4#191112yes—
455C4#236014yes—
456C4#241714yes—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—48no—
464C2#1413911yescharacteristic
465C2C2×#191111yescharacteristic
466C2#2101211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21306C2semidirect
5. #11224C2semidirect
6. #10837C2semidirect
16. #2566C4semidirect
17. #2563C4semidirect
18. #2542C4semidirect
43. C2×C4⋊Q8C2×C4semidirect
44. C2×C4⋊Q8C2×C4semidirect
45. C2×#125C2×C4semidirect
72. C2×#100Q8semidirect
92. #384C2×C4semidirect
95. #382C2×C4semidirect
105. C4×C4⋊C4C2×C4semidirect
106. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#351722
#341812
#213062
C2×C4×#1002
#112242
#108372
#107402