Finite Group Databasebeta

← #127403

Subgroups — #127403

Order 512.

627 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21380C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#20637C211yesmaximal, characteristic
4#7238C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6#2920—21no—
7C2×#432C411yescharacteristic
8C2×#423—21no—
9C2×#413—21no—
10C2×C4×M16C411yescharacteristic
11#1984C2212yes—
12#1954C412yes—
13#1870—21no—
14#1870—21no—
15#1822C412yes—
16#1801—21no—
17#1801C2212yes—
18#1801C2212yes—
19#1800—21no—
20#1800—21no—
21#1800C412yes—
22#1785—21no—
23C4×#101—21no—
24C4×#100C412yes—
25#1749—21no—
26#1749—21no—
27#1742C412yes—
28C22×#41D411yescharacteristic
29C22×M16—21no—
30C22×M16—22no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C4×#41—41no—
35C2×C4×Q8D411yescharacteristic
36C2×C4×Q8C2311yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×#117C2×C411yescharacteristic
44C2×#117C2×C411yescharacteristic
45C2×#117Q811yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49#432—41no—
50#432—41no—
51#432—41no—
52#432—41no—
53#431—41no—
54#431—41no—
55#423—22no—
56#423—22no—
57#423—22no—
58#423—41no—
59#423—41no—
60C2×C4⋊C8—41no—
61C2×C4⋊C8—41no—
62C2×C4⋊C8C2×C412yes—
63#413—44no—
64C2×#101—22no—
65C2×#101—22no—
66C2×#100—22no—
67C2×#100—22no—
68#407—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C4×M16—22no—
73C4×M16—41no—
74C4×M16—41no—
75C2×#97—21no—
76C2×#97—21no—
77C2×C4×C8C2×C412yes—
78#343—22no—
79#343—22no—
80#343C2×C414yes—
81#343C2×C414yes—
82#336—22no—
83#336—22no—
84C2×#41—22no—
85C2×#41—22no—
86C2×#41—41no—
87C2×#41—41no—
88C2×#41—42no—
89C22×Q8C2×D411yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C22⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C23×C4C4211yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C23×C4C4⋊C411yescharacteristic
102C2×M16—24no—
103C2×M16—24no—
104C2×M16—44no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—44no—
108C2×M16—44no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—42no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121C4×D4—41no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128#117—41no—
129#117—41no—
130#117—41no—
131#117—41no—
132#117—41no—
133#117—41no—
134C2×C4⋊C4C22⋊C411yescharacteristic
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C4⋊C411yescharacteristic
137C2×C4⋊C4C22×C411yescharacteristic
138C2×C4⋊C4C22×C411yescharacteristic
139C2×C4⋊C4C2×Q811yescharacteristic
140C2×C22⋊C4C4211yescharacteristic
141C2×C22⋊C4C4211yescharacteristic
142C2×C22⋊C4C4⋊C411yescharacteristic
143C2×C22⋊C4C4⋊C411yescharacteristic
144C2×C22⋊C4C4⋊C411yescharacteristic
145C2×C22⋊C4C4⋊C411yescharacteristic
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42C4211yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C4⋊C411yescharacteristic
163C2×C42C4⋊C411yescharacteristic
164C2×C42C4⋊C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C4⋊C8—42no—
169C4⋊C8—42no—
170C4⋊C8—42no—
171C4⋊C8—42no—
172C4⋊C8—42no—
173C4⋊C8—42no—
174C4⋊C8—42no—
175#101—42no—
176#101—44no—
177#101—42no—
178#100—44no—
179#100—44no—
180C22⋊C8—44no—
181C22⋊C8—44no—
182C22⋊C8—42no—
183C22⋊C8—42no—
184C22⋊C8—42no—
185#97—42no—
186#97—41no—
187#97—41no—
188#97—41no—
189#97—41no—
190C4×C8—42no—
191C4×C8—42no—
192C24#9511yescharacteristic
193C24#9511yescharacteristic
194C24#9511yescharacteristic
195#41—44no—
196#41—44no—
197#41—44no—
198#41—44no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8AGL(1,ℤ/8)12yes—
202C2×Q8#13712yes—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
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—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4#9511yescharacteristic
245C22×C4#9511yescharacteristic
246C22×C4#9511yescharacteristic
247C22×C4#9511yescharacteristic
248C22×C4#9511yescharacteristic
249C22×C4C2×C4211yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4C2×C4⋊C411yescharacteristic
256M16—48no—
257M16—44no—
258M16—44no—
259M16—44no—
260M16—44no—
261M16—84no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—44no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—44no—
269C2×C8—42no—
270C2×C8—84no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42#9911yescharacteristic
302C42#9911yescharacteristic
303C42#9911yescharacteristic
304C42#9911yescharacteristic
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23H(3,ℤ/4)12yes—
320C23C2×#9511yescharacteristic
321Q8—42no—
322Q8—42no—
323D4—44no—
324D4—44no—
325D4—42no—
326D4—42no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4#34211yescharacteristic
370C2×C4#34211yescharacteristic
371C2×C4#34211yescharacteristic
372C2×C4#34211yescharacteristic
373C2×C4C2×#9911yescharacteristic
374C2×C4C2×#9911yescharacteristic
375C2×C4#41712yes—
376C2×C4#41811yescharacteristic
377C2×C4#41911yescharacteristic
378C2×C4#42812yes—
379C8—44no—
380C8—84no—
381C8—84no—
382C8—82no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—42no—
387C22—42no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C2×H(3,ℤ/4)11yescharacteristic
401C22#173611yescharacteristic
402C22C2×#34211yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—22no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4#213212yes—
419C2—21no—
420C2—21no—
421C2—42no—
422C2—81no—
423C2—81no—
424C2#798311yescharacteristic
425C2#1257911yescharacteristic
426C2#1258511yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7238C2semidirect

Maximal subgroups

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

subgroupindex
#213802
#207282
#206372
#72382