Finite Group Databasebeta

← #127210

Subgroups — #127210

Order 512.

690 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20727C211yesmaximal, characteristic
2#17365C211yesmaximal, characteristic
3#13786C211yesmaximal, characteristic
4#7215C214yesmaximal
5C22×C4×Q8C2211yescharacteristic
6#2920—21no—
7#2920—21no—
8C2×#391—21no—
9C2×C4×C4⋊C4C411yescharacteristic
10C2×#376C411yescharacteristic
11#1821C2212yes—
12#1801—21no—
13#1801C412yescomplemented
14#1784—21no—
15C4×#101—21no—
16C4×#101C412yescomplemented
17#1467—21no—
18#1467C412yescomplemented
19#1463—21no—
20#1463C2212yes—
21#1461—21no—
22#1461C412yescomplemented
23#1436C412yes—
24#1436C2212yes—
25C4⋊#95—21no—
26#1434—21no—
27#1434—21no—
28C23×Q8D411yescharacteristic
29C2×C4×Q8—41no—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×C4×Q8—41no—
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×C4×Q8Q811yescharacteristic
40C2×C4×Q8Q811yescharacteristic
41C22×C4⋊C4—21no—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4C2311yescharacteristic
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C412yes—
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50#432—22no—
51#432—41no—
52#432—41no—
53#432—41no—
54#431—22no—
55#431—41no—
56#431—41no—
57#431—41no—
58#423—22no—
59#423—41no—
60C2×#101—22no—
61C2×#101—22no—
62#407—22no—
63#407—22no—
64C4×M16—22no—
65C4×M16—41no—
66#391—28no—
67C4×C4⋊C4—41no—
68C4×C4⋊C4—41no—
69#376—41no—
70#376—41no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95C2×C412yes—
78#343—22no—
79#343C2×C414yes—
80#324—22no—
81#324—22no—
82#324—22no—
83#324C2×C414yes—
84C22×Q8—24no—
85C22×Q8—41no—
86C22×Q8—41no—
87C22×Q8—42no—
88C22×Q8C22⋊C411yescharacteristic
89C22×Q8C22⋊C411yescharacteristic
90C22×Q8C4⋊C411yescharacteristic
91C22×Q8C4⋊C411yescharacteristic
92C23×C4—22no—
93C23×C4—22no—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—24no—
102C2×M16—42no—
103C4×Q8—41no—
104C4×Q8—41no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8—41no—
118C4×Q8—41no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C4211yescharacteristic
129C2×C4⋊C4C4211yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C4⋊C411yescharacteristic
133C2×C4⋊C4C4⋊C411yescharacteristic
134C2×C4⋊C4C4⋊C411yescharacteristic
135C2×C4⋊C4C4⋊C411yescharacteristic
136C2×C4⋊C4C22×C411yescharacteristic
137C2×C4⋊C4C22×C411yescharacteristic
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C4⋊C4C2×Q811yescharacteristic
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—24no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42—42no—
156C2×C42—82no—
157C2×C42C4211yescharacteristic
158C2×C42C4211yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C4⋊C411yescharacteristic
164C2×C42C4⋊C411yescharacteristic
165C2×C42C4⋊C411yescharacteristic
166C2×C42C4⋊C411yescharacteristic
167C2×C42C4⋊C411yescharacteristic
168C2×C42C4⋊C411yescharacteristic
169C4⋊C8—42no—
170C4⋊C8—42no—
171C4⋊C8—42no—
172C4⋊C8—42no—
173#101—44no—
174#101—44no—
175C22⋊C8—24no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—42no—
179#97—42no—
180#97—42no—
181C4×C8—42no—
182C4×C8—42no—
183#95—28no—
184#95—28no—
185#95—42no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—82no—
191#95—82no—
192C24C2×C22⋊C411yescharacteristic
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—48no—
205C2×Q8—48no—
206C2×Q8#10414yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4#9511yescharacteristic
247C22×C4#9511yescharacteristic
248C22×C4#9511yescharacteristic
249C22×C4#9511yescharacteristic
250C22×C4#9511yescharacteristic
251C22×C4#9511yescharacteristic
252C22×C4#9511yescharacteristic
253C22×C4#9511yescharacteristic
254C22×C4#9911yescharacteristic
255C22×C4#9911yescharacteristic
256C22×C4#9911yescharacteristic
257C22×C4#9911yescharacteristic
258C22×C4C2×C4211yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C4⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4C2×C4⋊C411yescharacteristic
264M16—44no—
265M16—44no—
266M16—82no—
267M16—82no—
268C2×C8—44no—
269C2×C8—44no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—48no—
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—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—48no—
308C42—84no—
309C23—21no—
310C23—21no—
311C23—22no—
312C23—21no—
313C23—21no—
314C23C2×#9511yescharacteristic
315C23C2×#9911yescharacteristic
316C23C2×#9911yescharacteristic
317Q8—44no—
318Q8—44no—
319Q8—44no—
320Q8—44no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
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—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4H(3,ℤ/4)12yes—
383C2×C4#33912yes—
384C2×C4#33912yes—
385C2×C4#34211yescharacteristic
386C2×C4#34211yescharacteristic
387C2×C4#34211yescharacteristic
388C2×C4#34211yescharacteristic
389C2×C4C2×#10412yes—
390C8—84no—
391C8—84no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22C2×H(3,ℤ/4)11yescharacteristic
403C22C2×#33911yescharacteristic
404C22C2×#34211yescharacteristic
405C22#212211yescharacteristic
406C22#212311yescharacteristic
407C22#213312yes—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—84no—
425C4—84no—
426C2—21no—
427C2—22no—
428C2—21no—
429C2—21no—
430C2—21no—
431C2#794211yescharacteristic
432C2#1257511yescharacteristic
433C2#1258611yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1801C4semidirect
16. C4×#101C4semidirect
18. #1467C4semidirect
22. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#207272
#173652
#137862
#72152