Finite Group Databasebeta

← #387105

Subgroups — #387105

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C211yesmaximal, characteristic
2#21303C212yesmaximal
3#11383C211yesmaximal, characteristic
4#11383C211yesmaximal, characteristic
5#11204C212yesmaximal
6#11204C212yesmaximal
7#10913C212yesmaximal
8#10913C212yesmaximal
9#10802C212yesmaximal
10C22×#125C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#423C2212yes—
13C2×C4×M16C2212yes—
14#2584—22no—
15#2584—22no—
16#2584—22no—
17#2584—22no—
18C2×#398C412yes—
19C2×#395C412yes—
20C2×#389C414yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1985—22no—
25#1985—22no—
26#1984—22no—
27#1984—22no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#343C2211yescharacteristic
32C2×#343C2211yescharacteristic
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2211yescharacteristic
36C2×#343C2211yescharacteristic
37C2×#343C2212yes—
38#1728C2212yes—
39#1728C2212yes—
40#1728C2212yes—
41#1728C2212yes—
42C23×Q8C2×C411yescharacteristic
43C22×M16C2312yes—
44C22×M16C2312yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—42no—
50C2×#125—41no—
51C2×#125—41no—
52C2×C4×Q8—41no—
53C2×C4×Q8—42no—
54C2×C4×Q8—41no—
55C2×C4×Q8D412yes—
56C2×C4×Q8D412yes—
57C2×C4×Q8D412yes—
58C2×C4×Q8D412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C414yes—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C414yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C42C2×C412yes—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#423—42no—
72#423—42no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C4×M16—42no—
76C4×M16—42no—
77C2×#97—22no—
78C2×C4×C8—22no—
79#398—22no—
80#398—24no—
81#398—22no—
82#395—22no—
83#395—24no—
84#395—22no—
85#389—24no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C2×#95C2×C412yes—
100C2×#95C2×C414yes—
101C2×#95C2×C414yes—
102#343—24no—
103#343—24no—
104#343—24no—
105#343—41no—
106#343—41no—
107#343—42no—
108#343—42no—
109#343—41no—
110#343—41no—
111#343Q818yescomplemented
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120C22×Q8—41no—
121C22×Q8—41no—
122C22×Q8—44no—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22⋊C412yes—
125C22×Q8C22⋊C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C414yes—
133C23×C4C2411yescharacteristic
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16C2×Q814yes—
157C2×M16C2×Q814yes—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162#125—44no—
163#125—44no—
164#125—44no—
165#125—44no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—84no—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C2×Q814yes—
211C2×C4⋊C4#4114yes—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—82no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C4⋊C8—44no—
228C4⋊C8—44no—
229#97—44no—
230C4×C8—44no—
231#95—22no—
232#95—22no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237C24C23×C411yescharacteristic
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—41no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—48no—
246C2×Q8—48no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C4×Q818yes—
283C22×C4C22×D412yes—
284C22×C4C22×Q812yes—
285C22×C4C2×#4112yes—
286C22×C4H(5,𝔽2)12yes—
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C4H(5,𝔽2)11yescharacteristic
289C22×C42−1+412yes—
290C22×C42−1+412yes—
291M16—24no—
292M16—24no—
293M16—24no—
294M16—24no—
295M16—42no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—42no—
301M16—42no—
302M16—42no—
303M16—42no—
304M16—42no—
305M16—42no—
306M16—42no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—48no—
331C4⋊C4—48no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—48no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—84no—
354C23—22no—
355C23—21no—
356C23—21no—
357C23C22×C22⋊C411yescharacteristic
358C23C2×C4×Q812yes—
359C23#51811yescharacteristic
360C23#51811yescharacteristic
361C23#51912yes—
362Q8—44no—
363Q8—48no—
364Q8—44no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—28no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#41314yes—
406C2×C4#53412yes—
407C2×C4#53412yes—
408C2×C4#54212yes—
409C2×C4#54212yes—
410C2×C4#54412yes—
411C2×C4#54412yes—
412C2×C4#54812yes—
413C2×C4#54812yes—
414C2×C4D4×Q812yes—
415C2×C4D4×Q812yes—
416C2×C4#55412yes—
417C2×C4#55511yescharacteristic
418C2×C4Q8211yescharacteristic
419C2×C4#56311yescharacteristic
420C2×C4#56311yescharacteristic
421C2×C4#56312yes—
422C8—44no—
423C8—44no—
424C8—82no—
425C8—82no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22#230811yescharacteristic
435C22#233112yes—
436C22Q8×C22⋊C412yes—
437C22#236911yescharacteristic
438C22#238111yescharacteristic
439C22C2×#41312yes—
440C22#288811yescharacteristic
441C22#288911yescharacteristic
442C4—24no—
443C4—48no—
444C4—41no—
445C4—41no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—82no—
452C4#272912yes—
453C4#272912yes—
454C4#273112yes—
455C4#273412yes—
456C2—22no—
457C2—21no—
458C2—21no—
459C2#1445011yescharacteristic
460C2#2175411yescharacteristic
461C2#2175711yescharacteristic
462C2#2251512yes—
463C2#2251712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
111. #343Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
#213032
#113832
#112042
#109132
#108022