Finite Group Databasebeta

← #127148

Subgroups — #127148

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21849C211yesmaximal, characteristic
2C4×#413C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#7206C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6#2920—21no—
7C2×#431C411yescharacteristic
8C2×#423C411yescharacteristic
9C2×#413C412yes—
10C2×C4×M16—21no—
11#1984C2212yes—
12#1954C412yes—
13#1870—21no—
14#1870—21no—
15#1797—21no—
16#1797—21no—
17C4×#101C412yes—
18C4×#101C2212yes—
19C4×#100—21no—
20C4×#100—21no—
21#1742—21no—
22#1465—21no—
23#1465C412yes—
24#1464—21no—
25#1464C412yes—
26#1461C2212yes—
27#1460—21no—
28C22×#41C2×C411yescharacteristic
29C22×M16—21no—
30C22×M16—22no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C4×#41—41no—
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8C2311yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×#117C2×C411yescharacteristic
44C2×#117D411yescharacteristic
45C2×#117Q811yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49#432—22no—
50#432—41no—
51#431—22no—
52#431—41no—
53#431—41no—
54#431—41no—
55#423—41no—
56#423—41no—
57#423—41no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2×C412yes—
61#413—28no—
62C2×#101C2×C414yes—
63C2×#100—22no—
64#407—22no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C4×M16—22no—
69C4×M16—22no—
70C4×M16—41no—
71C4×M16—41no—
72C4×M16—41no—
73C2×#97—41no—
74C2×C4×C8—41no—
75C2×C4×C8C2×C412yes—
76#343—22no—
77#343—22no—
78#336—22no—
79#336—22no—
80#324—22no—
81#324C2×C414yes—
82#323—22no—
83#323—22no—
84C2×#41—24no—
85C2×#41—41no—
86C2×#41—41no—
87C2×#41—42no—
88C22×Q8C22×C411yescharacteristic
89C22×D4C4211yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C23×C4C4211yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C2×M16—28no—
102C2×M16—24no—
103C2×M16—44no—
104C2×M16—42no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—82no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—42no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×D4—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—
127#117—41no—
128#117—41no—
129#117—41no—
130#117—41no—
131#117—41no—
132#117—41no—
133C2×C4⋊C4C4211yescharacteristic
134C2×C4⋊C4C22⋊C411yescharacteristic
135C2×C4⋊C4C4⋊C411yescharacteristic
136C2×C4⋊C4C22×C411yescharacteristic
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×Q811yescharacteristic
139C2×C22⋊C4C22⋊C411yescharacteristic
140C2×C22⋊C4C22⋊C411yescharacteristic
141C2×C22⋊C4C4⋊C411yescharacteristic
142C2×C22⋊C4C4⋊C411yescharacteristic
143C2×C22⋊C4C4⋊C411yescharacteristic
144C2×C22⋊C4C4⋊C411yescharacteristic
145C2×C42—21no—
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×C42C4211yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C4⋊C411yescharacteristic
162C2×C42C4⋊C411yescharacteristic
163C2×C42C4⋊C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C4⋊C8—42no—
168C4⋊C8—42no—
169C4⋊C8—42no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174#101—24no—
175#101M1618yescomplemented
176#100—28no—
177C22⋊C8—24no—
178C22⋊C8—42no—
179C22⋊C8—42no—
180C22⋊C8—82no—
181C22⋊C8—82no—
182#97—42no—
183#97—42no—
184#97—42no—
185C4×C8—42no—
186C4×C8—42no—
187C4×C8—42no—
188C4×C8—42no—
189C24#9511yescharacteristic
190C24#9511yescharacteristic
191C24#9511yescharacteristic
192#41—48no—
193#41—48no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8C2×M1614yes—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4#9511yescharacteristic
240C22×C4#9511yescharacteristic
241C22×C4#9511yescharacteristic
242C22×C4#9511yescharacteristic
243C22×C4#9511yescharacteristic
244C22×C4C2×C4211yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C4⋊C411yescharacteristic
249C22×C4C2×C4⋊C411yescharacteristic
250C22×C4C2×C4⋊C411yescharacteristic
251M16—28no—
252M16—48no—
253M16—44no—
254M16—84no—
255M16—84no—
256C2×C8—44no—
257C2×C8—48no—
258C2×C8—44no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
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#9911yescharacteristic
296C42#9911yescharacteristic
297C42#9911yescharacteristic
298C42#9911yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23#34412yes—
314C23C2×#9511yescharacteristic
315Q8—44no—
316D4—44no—
317D4—44no—
318D4—44no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4#34211yescharacteristic
364C2×C4#34211yescharacteristic
365C2×C4#34211yescharacteristic
366C2×C4#34211yescharacteristic
367C2×C4C4×M1612yes—
368C2×C4#40712yes—
369C2×C4C2×#9911yescharacteristic
370C2×C4C2×#9911yescharacteristic
371C2×C4#42312yes—
372C8—48no—
373C8—82no—
374C8—84no—
375C8—82no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22#172811yescharacteristic
394C22C2×#34211yescharacteristic
395C22C2×#34411yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—22no—
402C4—22no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4#139512yes—
410C4#213411yescharacteristic
411C4#213511yescharacteristic
412C2—21no—
413C2—21no—
414C2—42no—
415C2—81no—
416C2—81no—
417C2#792911yescharacteristic
418C2#1258711yescharacteristic
419C2C2×#139511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C2×#342
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #7206C2semidirect
175. #101M16semidirect

Maximal subgroups

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

subgroupindex
#218492
C4×#4132
#173662
#72062