Finite Group Databasebeta

← #120215

Subgroups — #120215

Order 512.

739 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21940C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic
3#10854C212yesmaximal
4C2×#1289C211yesmaximal, characteristic
5C2×#1289C212yesmaximal
6C22×C22⋊C8C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8C2×#380C2212yes—
9C2×#380C2212yes—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1852—24no—
12#1850—22no—
13#1850C414yescomplemented
14#1808—22no—
15#1797—22no—
16C2×#342—22no—
17C2×#342—22no—
18#1289—22no—
19#1289—24no—
20C24×C4C2311yescharacteristic
21C23×C8C2×C412yes—
22C23×C8D412yes—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—22no—
25C22×C22⋊C4—22no—
26C22×C22⋊C4—22no—
27C22×C22⋊C4D411yescharacteristic
28C22×C22⋊C4D411yescharacteristic
29C22×C42—21no—
30C2×#107—24no—
31C2×C4⋊C8—24no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8C2×C412yes—
39C2×C22⋊C8D412yes—
40#380—22no—
41#380—22no—
42#380—22no—
43#380—22no—
44#380—22no—
45#380—22no—
46#380—22no—
47#380D414yes—
48C4×C22⋊C4—21no—
49C4×C22⋊C4—21no—
50C4×C22⋊C4C2×C412yes—
51C4×C22⋊C4Q812yes—
52C2×#95—21no—
53C2×#95—22no—
54C2×#95Q814yes—
55#342—44no—
56#342—44no—
57#340—24no—
58#336—24no—
59C25C2×D411yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—24no—
63C23×C4—42no—
64C23×C4—82no—
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D412yes—
69C23×C4C2×Q812yes—
70C23×C4C2×Q811yescharacteristic
71C23×C4#4112yes—
72C23×C4#4111yescharacteristic
73C23×C4#4111yescharacteristic
74C23×C4#4112yes—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8—82no—
80C22×C8—82no—
81C22×C8C22⋊C414yes—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—84no—
87C2×C4⋊C4—84no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4C4⋊C412yes—
107C2×C22⋊C4C4⋊C412yes—
108C2×C22⋊C4D812yes—
109C2×C22⋊C4Q1612yes—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—82no—
113#107—84no—
114C4⋊C8—84no—
115C22⋊C8—24no—
116C22⋊C8—24no—
117C22⋊C8—41no—
118C22⋊C8—42no—
119C22⋊C8—41no—
120#95—22no—
121#95—22no—
122#95—24no—
123#95—24no—
124#95—24no—
125#95—42no—
126C24—81no—
127C24—81no—
128C24C2×C4⋊C411yescharacteristic
129C24#12111yescharacteristic
130C24#12111yescharacteristic
131C24#12112yes—
132C24C22⋊Q812yes—
133C24C2×D811yescharacteristic
134C24C2×Q1611yescharacteristic
135C24#13511yescharacteristic
136C24#13511yescharacteristic
137C24AGL(1,ℤ/8)12yes—
138C24#13712yes—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—81no—
154C22×C4—82no—
155C22×C4—81no—
156C22×C4—82no—
157C22×C4—81no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4—84no—
163C22×C4—84no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—84no—
167C22×C4—84no—
168C22×C4C2×C22⋊C411yescharacteristic
169C22×C4#11711yescharacteristic
170C22×C4C4×D412yes—
171C22×C4C4×Q812yes—
172C22×C4#12011yescharacteristic
173C22×C4#12112yes—
174C22×C4C22⋊Q812yes—
175C22×C4C22⋊Q812yes—
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4#12311yescharacteristic
179C22×C4#12312yes—
180C22×C4#12411yescharacteristic
181C22×C4#12512yes—
182C22×C4#12512yes—
183C22×C4#12612yes—
184C22×C4C4⋊Q811yescharacteristic
185C22×C4C4⋊Q812yes—
186C2×C8—24no—
187C2×C8—44no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—42no—
191C2×C8—82no—
192C2×C8—84no—
193C4⋊C4—48no—
194C4⋊C4—84no—
195C4⋊C4—84no—
196C4⋊C4—84no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—44no—
204C22⋊C4—44no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C22⋊C4—82no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4#10714yescomplemented
212C42—44no—
213C42—82no—
214C23—41no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23—41no—
223C23—41no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23—82no—
229C23—82no—
230C23#38011yescharacteristic
231C23#38912yes—
232C23#39411yescharacteristic
233C23#39612yes—
234C23#39811yescharacteristic
235C23C2×#10711yescharacteristic
236C23#42711yescharacteristic
237C23#42812yes—
238C23#45912yes—
239C23#46212yes—
240C23#46312yes—
241C23#46412yes—
242C23#46611yescharacteristic
243C23#46611yescharacteristic
244C23#46711yescharacteristic
245C23#46711yescharacteristic
246C23#46812yes—
247C23#46911yescharacteristic
248C23#47011yescharacteristic
249C23#47412yes—
250C23#47712yes—
251C23#47812yes—
252C23#47912yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4—164no—
287C2×C4—164no—
288C2×C4—164no—
289C2×C4—164no—
290C2×C4#38611yescharacteristic
291C2×C4#38712yes—
292C2×C4#39111yescharacteristic
293C2×C4#39511yescharacteristic
294C2×C4#39812yes—
295C2×C4#40011yescharacteristic
296C8—44no—
297C8—82no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22—22no—
307C22—21no—
308C22—41no—
309C22—41no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—42no—
325C22—42no—
326C22#144111yescharacteristic
327C22#180811yescharacteristic
328C22#180911yescharacteristic
329C22#194712yes—
330C22#194912yes—
331C22#205411yescharacteristic
332C22#205611yescharacteristic
333C22#206712yes—
334C22#206812yes—
335C22#208611yescharacteristic
336C22#208711yescharacteristic
337C4—22no—
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—44no—
343C4—82no—
344C4—164no—
345C4—164no—
346C4#191312yes—
347C2—21no—
348C2—21no—
349C2—21no—
350C2—22no—
351C2—21no—
352C2—22no—
353C2—21no—
354C2—21no—
355C2—22no—
356C2—22no—
357C2#1120511yescharacteristic
358C2#1123211yescharacteristic
359C2#1123411yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1850C4semidirect
211. C22⋊C4#107semidirect

Maximal subgroups

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

subgroupindex
#219402
#219302
#108542
C2×#12892