Finite Group Databasebeta

← #120118

Subgroups — #120118

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 414 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—42no—
46#380—42no—
47#380Q814yes—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95C2×C412yes—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95D412yes—
62C2×#95Q812yes—
63#342—42no—
64#342—42no—
65#336—24no—
66C25C2×D411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—44no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4—42no—
78C23×C4—82no—
79C23×C4—82no—
80C23×C4C22⋊C412yes—
81C23×C4C22⋊C412yes—
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D412yes—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×Q811yescharacteristic
87C23×C4C2×Q812yes—
88C23×C4#4112yes—
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4112yes—
92C22×C8—24no—
93C22×C8—42no—
94C22×C8—42no—
95C22×C8—84no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—82no—
102C2×C4⋊C4—84no—
103C2×C4⋊C4—82no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4D812yes—
121C2×C22⋊C4SD1612yes—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—82no—
125C4⋊C8—84no—
126C22⋊C8—24no—
127C22⋊C8—24no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130#95—21no—
131#95—21no—
132#95—24no—
133#95—22no—
134#95—22no—
135#95—24no—
136#95—44no—
137#95—44no—
138#95—41no—
139#95—42no—
140#95—41no—
141#95—82no—
142#95C4⋊C412yes—
143#95C4⋊C412yes—
144C24—42no—
145C24—42no—
146C24C2×C22⋊C411yescharacteristic
147C24#12112yes—
148C24#12111yescharacteristic
149C24C22⋊Q811yescharacteristic
150C24C22⋊Q812yes—
151C24C2×D811yescharacteristic
152C24C2×SD1611yescharacteristic
153C24#13512yes—
154C24AGL(1,ℤ/8)11yescharacteristic
155C24#13711yescharacteristic
156C24#13712yes—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—81no—
195C22×C4—82no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—82no—
199C22×C4—81no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4#10214yes—
211C22×C4C2×C4⋊C411yescharacteristic
212C22×C4#11711yescharacteristic
213C22×C4C4×D412yes—
214C22×C4C4×Q812yes—
215C22×C4#12011yescharacteristic
216C22×C4#12112yes—
217C22×C4#12111yescharacteristic
218C22×C4C22⋊Q812yes—
219C22×C4C22⋊Q812yes—
220C22×C4C22⋊Q811yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12312yes—
223C22×C4#12411yescharacteristic
224C22×C4#12512yes—
225C22×C4#12512yes—
226C22×C4#12612yes—
227C22×C4C4⋊Q812yes—
228C22×C4C4⋊Q811yescharacteristic
229C2×C8—48no—
230C2×C8—44no—
231C2×C8—84no—
232C2×C8—84no—
233C4⋊C4—84no—
234C4⋊C4—88no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C42—44no—
248C42—82no—
249C23—22no—
250C23—24no—
251C23—22no—
252C23—24no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—44no—
264C23—44no—
265C23#38011yescharacteristic
266C23#38611yescharacteristic
267C23#39111yescharacteristic
268C23#39612yes—
269C23#39812yes—
270C23C2×#10211yescharacteristic
271C23#41611yescharacteristic
272C23#41711yescharacteristic
273C23#41911yescharacteristic
274C23#45911yescharacteristic
275C23#46111yescharacteristic
276C23#46412yes—
277C23#46512yes—
278C23#46612yes—
279C23#46812yes—
280C23#47012yes—
281C23#47411yescharacteristic
282C23#47412yes—
283C23#47512yes—
284C23#47611yescharacteristic
285C23#47812yes—
286C23#47912yes—
287C23#47912yes—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—84no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—164no—
341C2×C4—164no—
342C2×C4#38712yes—
343C2×C4#38912yes—
344C2×C4#39411yescharacteristic
345C2×C4#39511yescharacteristic
346C2×C4#39811yescharacteristic
347C2×C4#40011yescharacteristic
348C8—84no—
349C22—22no—
350C22—22no—
351C22—24no—
352C22—24no—
353C22—22no—
354C22—22no—
355C22—24no—
356C22—24no—
357C22—21no—
358C22—22no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#144111yescharacteristic
376C22#180211yescharacteristic
377C22#180511yescharacteristic
378C22#188412yes—
379C22#189311yescharacteristic
380C22#189411yescharacteristic
381C22#191312yes—
382C22#198811yescharacteristic
383C22#198911yescharacteristic
384C22#206612yes—
385C22#206712yes—
386C22#208612yes—
387C22#208812yes—
388C4—21no—
389C4—22no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—44no—
395C4—44no—
396C4—82no—
397C4—82no—
398C4—164no—
399C4—162no—
400C2—21no—
401C2—22no—
402C2—21no—
403C2—22no—
404C2—22no—
405C2—22no—
406C2—21no—
407C2—21no—
408C2#938412yes—
409C2#938512yes—
410C2#1120511yescharacteristic
411C2#1121211yescharacteristic
412C2#1122711yescharacteristic
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