Finite Group Databasebeta

← #127404

Subgroups — #127404

Order 512.

620 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21303C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3C2×#1801C211yesmaximal, characteristic
4#7238C214yesmaximal, complemented
5C22×C4×Q8C2211yescharacteristic
6#2920—21no—
7C2×#423—21no—
8C2×#423C411yescharacteristic
9C22×#101—21no—
10C2×C4×M16C411yescharacteristic
11#1985—21no—
12#1985—21no—
13#1984C412yes—
14#1984C2212yes—
15#1821C412yes—
16#1801—21no—
17#1801—21no—
18#1801—21no—
19#1801C412yes—
20#1801C2212yes—
21#1801C2212yes—
22#1784—21no—
23C4×#101—21no—
24C4×#101C412yes—
25C2×#343C412yes—
26#1728—21no—
27#1728—21no—
28C23×Q8D411yescharacteristic
29C22×M16—21no—
30C22×M16—21no—
31C22×M16—22no—
32C2×C4×Q8—41no—
33C2×C4×Q8—41no—
34C2×C4×Q8—41no—
35C2×C4×Q8—41no—
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8C2×C411yescharacteristic
38C2×C4×Q8C2×C411yescharacteristic
39C2×C4×Q8D411yescharacteristic
40C2×C4×Q8D411yescharacteristic
41C2×C4×Q8D411yescharacteristic
42C2×C4×Q8Q811yescharacteristic
43C2×C4×Q8C2311yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4Q811yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50#432—41no—
51#432—41no—
52#431—41no—
53#431—41no—
54#423—22no—
55#423—22no—
56#423—22no—
57#423—41no—
58#423—41no—
59#423—41no—
60#423—41no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—41no—
63C2×C4⋊C8—41no—
64C2×C4⋊C8C2×C412yes—
65C2×#101—22no—
66C2×#101—22no—
67C2×#101—22no—
68C2×#101—22no—
69C2×#101—44no—
70#407—22no—
71C4×M16—22no—
72C4×M16—41no—
73C4×M16—41no—
74C2×#97—21no—
75C2×C4×C8C2×C412yes—
76#343—22no—
77#343—22no—
78#343C2×C414yes—
79#343C2×C414yes—
80#336—22no—
81#336—22no—
82#336—22no—
83#336—22no—
84C22×Q8—22no—
85C22×Q8—22no—
86C22×Q8—41no—
87C22×Q8—41no—
88C22×Q8—42no—
89C22×Q8C22⋊C411yescharacteristic
90C22×Q8C22⋊C411yescharacteristic
91C22×Q8C4⋊C411yescharacteristic
92C22×Q8C2×D411yescharacteristic
93C23×C4C4211yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—24no—
103C2×M16—42no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—44no—
107C2×M16—42no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—44no—
111C2×M16—44no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—42no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8—41no—
118C4×Q8—41no—
119C4×Q8—41no—
120C4×Q8—41no—
121C4×Q8—41no—
122C4×Q8—41no—
123C4×Q8—41no—
124C4×Q8—41no—
125C4×Q8—41no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4C4211yescharacteristic
138C2×C4⋊C4C4211yescharacteristic
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C22⋊C411yescharacteristic
141C2×C4⋊C4C4⋊C411yescharacteristic
142C2×C4⋊C4C4⋊C411yescharacteristic
143C2×C4⋊C4C4⋊C411yescharacteristic
144C2×C4⋊C4C4⋊C411yescharacteristic
145C2×C4⋊C4C4⋊C411yescharacteristic
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C4⋊C4C22×C411yescharacteristic
148C2×C4⋊C4C2×Q811yescharacteristic
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×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42C4211yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C4⋊C411yescharacteristic
168C2×C42C4⋊C411yescharacteristic
169C2×C42C4⋊C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C4⋊C8—42no—
174C4⋊C8—42no—
175C4⋊C8—42no—
176C4⋊C8—42no—
177C4⋊C8—42no—
178C4⋊C8—42no—
179C4⋊C8—42no—
180C4⋊C8—42no—
181#101—44no—
182#101—44no—
183#101—42no—
184#101—44no—
185#101—42no—
186C22⋊C8—42no—
187C22⋊C8—42no—
188#97—42no—
189#97—42no—
190C4×C8—42no—
191C4×C8—42no—
192C24#9511yescharacteristic
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8AGL(1,ℤ/8)12yes—
210C2×Q8#13712yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
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—41no—
231C22×C4—41no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4#9511yescharacteristic
236C22×C4#9511yescharacteristic
237C22×C4#9511yescharacteristic
238C22×C4#9511yescharacteristic
239C22×C4#9511yescharacteristic
240C22×C4#9511yescharacteristic
241C22×C4#9511yescharacteristic
242C22×C4#9911yescharacteristic
243C22×C4#9911yescharacteristic
244C22×C4#9911yescharacteristic
245C22×C4#9911yescharacteristic
246C22×C4C2×C4211yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C4⋊C411yescharacteristic
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C2×C4⋊C411yescharacteristic
253M16—44no—
254M16—44no—
255M16—44no—
256M16—44no—
257M16—44no—
258M16—44no—
259M16—84no—
260M16—82no—
261M16—82no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—84no—
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—
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—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C23—22no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23#34211yescharacteristic
311C23#34211yescharacteristic
312C23C2×#9511yescharacteristic
313Q8—42no—
314Q8—42no—
315Q8—44no—
316Q8—44no—
317Q8—42no—
318Q8—42no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
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—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4H(3,ℤ/4)12yes—
367C2×C4#34211yescharacteristic
368C2×C4#34211yescharacteristic
369C2×C4#35512yes—
370C2×C4#35612yes—
371C2×C4C2×#9911yescharacteristic
372C2×C4C2×#9911yescharacteristic
373C2×C4#41712yes—
374C2×C4#41811yescharacteristic
375C2×C4#41911yescharacteristic
376C2×C4#42812yes—
377C8—44no—
378C8—84no—
379C8—84no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22#139412yes—
391C22C2×H(3,ℤ/4)11yescharacteristic
392C22#173611yescharacteristic
393C22C2×#34211yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—22no—
398C4—22no—
399C4—21no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4#139412yes—
410C4C2×#35511yescharacteristic
411C4C2×#35611yescharacteristic
412C2—22no—
413C2—21no—
414C2—21no—
415C2—41no—
416C2#798311yescharacteristic
417C2#1257911yescharacteristic
418C2C2×#139411yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
prime radicalC24#95
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ 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
#213032
#207272
C2×#18012
#72382