Finite Group Databasebeta

← #201610

Subgroups — #201610

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60123C211yesmaximal, characteristic
2#34971C211yesmaximal, characteristic
3C2×#1491C212yesmaximal, complemented
4#17363C211yesmaximal, characteristic
5#9493C218yesmaximal, complemented
6#7972C212yesmaximal, complemented
7C2×C4×#41C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2923—24no—
10#2922—24no—
11#2921C2214yes—
12#2920—22no—
13C2×#432C2212yes—
14#2905—21no—
15#2879—21no—
16C2×C4×M16C2212yes—
17#2599C2214yes—
18#2581—21no—
19#2536—21no—
20#2486C412yes—
21#2484—21no—
22#2409C412yes—
23#2406—21no—
24#2399C412yes—
25#2398C412yes—
26C4×C22⋊Q8—22no—
27C4×#123—22no—
28#2283C2211yescharacteristic
29#1491—24no—
30#1491C2218yes—
31C2×Q8⋊C8—24no—
32#1467—21no—
33#1466—21no—
34#1464C2214yes—
35#1459—22no—
36#1288C2218yes—
37C22×#41D411yescharacteristic
38C23×Q8C2×C411yescharacteristic
39C22×M16—22no—
40C2×#123—22no—
41C2×C22⋊Q8—22no—
42C4×#41—22no—
43C4×#41—22no—
44C4×#41—22no—
45C4×#41—22no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2312yes—
56C2×C4×D4C2×C412yes—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D412yes—
59C2×C4×D4D411yescharacteristic
60C2×#117D411yescharacteristic
61C2×#117C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C42—21no—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67#433—216no—
68#432—28no—
69#432—22no—
70#432—22no—
71#432—22no—
72#432C2314yes—
73#431—28no—
74#424—28no—
75#423—24no—
76C2×C4⋊C8—22no—
77#408—28no—
78#407—24no—
79C2×C22⋊C8—22no—
80C4×M16—22no—
81C4×M16—22no—
82C4×M16D414yes—
83C4×M16C2314yes—
84C2×#97—22no—
85C2×#97—22no—
86C2×C4×C8D414yes—
87#400—22no—
88#398—22no—
89#398C2×C414yes—
90#397C2×C414yes—
91#395—22no—
92#395C2×C414yes—
93#394—22no—
94#393C2×C414yes—
95#391—22no—
96#389—22no—
97#388—24no—
98#387—24no—
99#387—24no—
100#386—22no—
101#385—24no—
102#385—22no—
103#383C2312yes—
104#382—21no—
105#381—21no—
106#381—22no—
107#380—22no—
108#380C2×C412yes—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—21no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4—21no—
113#376C2×C412yes—
114Q8⋊C8—216no—
115#324—24no—
116#323—24no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—44no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C412yes—
126C22×D4C22⋊C412yes—
127C22×D4C22⋊C411yescharacteristic
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—48no—
139C22×C8—24no—
140C22×C8—42no—
141#123—48no—
142C22⋊Q8—48no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—42no—
154C4×Q8—42no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165#117—22no—
166#117—22no—
167#117—24no—
168#117—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22×C412yes—
189C2×C22⋊C4C2×D412yes—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42C22⋊C412yes—
200C2×C42C22⋊C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42C2411yescharacteristic
210C4⋊C8—216no—
211C4⋊C8—24no—
212C4⋊C8—42no—
213C22⋊C8—216no—
214C22⋊C8—24no—
215C22⋊C8—42no—
216#97—24no—
217#97—24no—
218#97—24no—
219#97—42no—
220C4×C8—24no—
221C4×C8C2×D418yes—
222#95—22no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—22no—
228#95—22no—
229#95C22×C414yes—
230C24C2×C22⋊C412yes—
231C24C2×C22⋊C411yescharacteristic
232#41—48no—
233#41—48no—
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—28no—
237C2×Q8—24no—
238C2×Q8—42no—
239C2×Q8—44no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—48no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—42no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C4×D412yes—
277C22×C4C4×D412yes—
278C22×C4#12012yes—
279C22×C4#12112yes—
280C22×C4#12112yes—
281C22×C4#12312yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—48no—
286M16—48no—
287C2×C8—28no—
288C2×C8—48no—
289C2×C8—44no—
290C2×C8—84no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—44no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—44no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—48no—
308C22⋊C4—48no—
309C22⋊C4—44no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—44no—
317C42—44no—
318C42—42no—
319C42—42no—
320C42C4×D412yes—
321C42C4×D412yes—
322C42#12011yescharacteristic
323C42#12011yescharacteristic
324C42#12112yes—
325C42#12112yes—
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42C22×D412yes—
329C42C2×#4112yes—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23#38612yes—
338C23#42114yes—
339C23C22×C22⋊C411yescharacteristic
340Q8—28no—
341Q8—44no—
342Q8—44no—
343Q8—44no—
344Q8—44no—
345D4—44no—
346D4—48no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—48no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—84no—
382C2×C4#38611yescharacteristic
383C2×C4#38611yescharacteristic
384C2×C4#38612yes—
385C2×C4#38611yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4C2×C4×D412yes—
388C2×C4C2×#12011yescharacteristic
389C2×C4C2×#12112yes—
390C2×C4C2×#12311yescharacteristic
391C8—48no—
392C8—84no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—44no—
398C22—44no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—82no—
403C22#189714yes—
404C22C2×#38611yescharacteristic
405C22C2×#42112yes—
406C4—22no—
407C4—21no—
408C4—21no—
409C4—22no—
410C4—21no—
411C4—21no—
412C4—22no—
413C4—42no—
414C4—44no—
415C4—42no—
416C4—42no—
417C4—84no—
418C4—82no—
419C4#189112yes—
420C4#190912yes—
421C2—22no—
422C2—24no—
423C2—82no—
424C2C2×#189111yescharacteristic
425C2C2×#189711yescharacteristic
426C2C2×#190911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1491C2semidirect
5. #9493C2semidirect
6. #7972C2semidirect

Maximal subgroups

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

subgroupindex
#601232
#349712
C2×#14912
#173632
#94932
#79722