Finite Group Databasebeta

← #120119

Subgroups — #120119

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21930C211yesmaximal, characteristic
2#13908C211yesmaximal, characteristic
3#10846C211yesmaximal, characteristic
4#8023C212yesmaximal
5C2×#1294C212yesmaximal
6C22×C22⋊C8C2212yes—
7C2×#380—21no—
8C2×#380C2212yes—
9C2×#380C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11C22×#95C2211yescharacteristic
12#1850—24no—
13#1797—22no—
14C2×#342—21no—
15C2×#342—21no—
16#1444C412yescomplemented
17#1442C412yescomplemented
18#1436—22no—
19C22⋊#95—21no—
20#1294—24no—
21#1294—24no—
22C24×C4—21no—
23C24×C4C2311yescharacteristic
24C23×C8—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4C2×C412yes—
28C22×C22⋊C4—22no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42D412yes—
33C2×C4⋊C8—24no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8D414yes—
40#380—22no—
41#380—22no—
42#380—22no—
43#380—22no—
44#380—22no—
45#380—44no—
46#380Q814yes—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—21no—
56C2×#95—22no—
57C2×#95C2×C412yes—
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95D412yes—
61C2×#95Q812yes—
62#342—42no—
63#342—42no—
64#336—24no—
65C25C2×D411yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—44no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—82no—
76C23×C4—82no—
77C23×C4C22⋊C412yes—
78C23×C4C22⋊C412yes—
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D412yes—
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×Q811yescharacteristic
84C23×C4C2×Q812yes—
85C23×C4#4112yes—
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4112yes—
89C22×C8—24no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8—84no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—82no—
99C2×C4⋊C4—84no—
100C2×C4⋊C4—82no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4SD1612yes—
118C2×C22⋊C4Q1612yes—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—82no—
122C4⋊C8—84no—
123C22⋊C8—24no—
124C22⋊C8—24no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127#95—21no—
128#95—21no—
129#95—24no—
130#95—22no—
131#95—22no—
132#95—24no—
133#95—44no—
134#95—44no—
135#95—41no—
136#95—42no—
137#95—41no—
138#95—82no—
139#95C4⋊C412yes—
140#95C4⋊C412yes—
141C24—42no—
142C24—42no—
143C24C2×C22⋊C411yescharacteristic
144C24#12112yes—
145C24#12111yescharacteristic
146C24C22⋊Q811yescharacteristic
147C24C22⋊Q812yes—
148C24C2×SD1611yescharacteristic
149C24C2×Q1611yescharacteristic
150C24#13512yes—
151C24AGL(1,ℤ/8)11yescharacteristic
152C24AGL(1,ℤ/8)12yes—
153C24#13711yescharacteristic
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—81no—
190C22×C4—82no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—82no—
194C22×C4—81no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—88no—
198C22×C4—88no—
199C22×C4—84no—
200C22×C4—84no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4#10314yes—
204C22×C4C2×C4⋊C411yescharacteristic
205C22×C4#11711yescharacteristic
206C22×C4C4×D412yes—
207C22×C4C4×Q812yes—
208C22×C4#12011yescharacteristic
209C22×C4#12112yes—
210C22×C4#12111yescharacteristic
211C22×C4C22⋊Q812yes—
212C22×C4C22⋊Q812yes—
213C22×C4C22⋊Q811yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4#12312yes—
216C22×C4#12411yescharacteristic
217C22×C4#12512yes—
218C22×C4#12512yes—
219C22×C4#12612yes—
220C22×C4C4⋊Q812yes—
221C22×C4C4⋊Q811yescharacteristic
222C2×C8—48no—
223C2×C8—44no—
224C2×C8—84no—
225C2×C8—84no—
226C4⋊C4—84no—
227C4⋊C4—88no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C42—44no—
241C42—82no—
242C23—22no—
243C23—24no—
244C23—22no—
245C23—24no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—44no—
252C23—44no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23#38011yescharacteristic
259C23#38611yescharacteristic
260C23#39111yescharacteristic
261C23#39612yes—
262C23#39812yes—
263C23C2×#10311yescharacteristic
264C23#41611yescharacteristic
265C23#41711yescharacteristic
266C23#41811yescharacteristic
267C23#46011yescharacteristic
268C23#46211yescharacteristic
269C23#46312yes—
270C23#46512yes—
271C23#46712yes—
272C23#46812yes—
273C23#46912yes—
274C23#47511yescharacteristic
275C23#47612yes—
276C23#47711yescharacteristic
277C23#47712yes—
278C23#47812yes—
279C23#47812yes—
280C23#47912yes—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—82no—
316C2×C4—88no—
317C2×C4—88no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—164no—
332C2×C4—164no—
333C2×C4#38712yes—
334C2×C4#38912yes—
335C2×C4#39411yescharacteristic
336C2×C4#39511yescharacteristic
337C2×C4#39811yescharacteristic
338C2×C4#40011yescharacteristic
339C8—84no—
340C22—22no—
341C22—22no—
342C22—24no—
343C22—24no—
344C22—22no—
345C22—22no—
346C22—24no—
347C22—24no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—22no—
354C22—21no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—41no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22#144111yescharacteristic
367C22#180311yescharacteristic
368C22#180411yescharacteristic
369C22#188412yes—
370C22#189211yescharacteristic
371C22#189511yescharacteristic
372C22#191312yes—
373C22#198711yescharacteristic
374C22#198911yescharacteristic
375C22#206512yes—
376C22#206812yes—
377C22#208712yes—
378C22#208812yes—
379C4—21no—
380C4—22no—
381C4—22no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—44no—
386C4—44no—
387C4—82no—
388C4—82no—
389C4—164no—
390C4—162no—
391C2—21no—
392C2—22no—
393C2—21no—
394C2—22no—
395C2—22no—
396C2—22no—
397C2—21no—
398C2—21no—
399C2#938312yes—
400C2#938612yes—
401C2#1120511yescharacteristic
402C2#1121211yescharacteristic
403C2#1122811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1444C4semidirect
17. #1442C4semidirect

Maximal subgroups

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

subgroupindex
#219302
#139082
#108462
#80232
C2×#12942