Finite Group Databasebeta

← #197520

Subgroups — #197520

Order 512.

1537 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35165C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#17574C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic, complemented
5#9205C218yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C2×#535—21no—
8C2×#532C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#431C2212yes—
11#2847—22no—
12#2617C2214yes—
13#2547C412yescomplemented
14#2544—22no—
15#2541C412yescomplemented
16#2537—22no—
17#2535C414yescomplemented
18#2357—24no—
19#2354—24no—
20#2312C2214yes—
21#2311—21no—
22#2309—21no—
23#2285C2211yescharacteristic
24#2285C2212yes—
25#2280—22no—
26#2274—21no—
27#1519C2218yes—
28#1517—24no—
29#1502—24no—
30#1460—22no—
31#1460—22no—
32C2×#323C2214yes—
33#1289C2218yes—
34C22×#41—21no—
35C23×D4D411yescharacteristic
36C24×C4D412yes—
37#535—48no—
38#532—22no—
39#532—22no—
40#532—22no—
41#532—22no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#127—22no—
44C2×#127C2×C411yescharacteristic
45C2×#124—22no—
46C2×#124—21no—
47C2×#121—24no—
48C2×#121—21no—
49C2×#121—24no—
50C2×#121C2312yes—
51C2×#120—24no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—21no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4C2×C412yes—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C412yes—
60C2×C4×D4D412yes—
61C2×C4×D4D412yes—
62C2×#117—22no—
63C2×#117—21no—
64C2×#117—24no—
65C22×C4⋊C4D411yescharacteristic
66C22×C22⋊C4C2312yes—
67C22×C42C2311yescharacteristic
68#431—22no—
69#431—22no—
70#431—22no—
71#431—22no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8C2314yes—
74C2×C4×C8—22no—
75#397C2×C418yes—
76#394—28no—
77#392C2×C418yes—
78#391—22no—
79#390—28no—
80#390—22no—
81#388—216no—
82#386—28no—
83#386—24no—
84#384—22no—
85#384—21no—
86#382C2×C412yes—
87#382D414yescomplemented
88#381—24no—
89#380—24no—
90#380—24no—
91#380—21no—
92#380—22no—
93#380—24no—
94#380C2312yes—
95#380C2314yes—
96#379—24no—
97C4×C4⋊C4—28no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D414yescomplemented
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104#327—216no—
105#323—28no—
106C25C2×D412yes—
107C2×#41—42no—
108C2×#41—42no—
109C2×#41—44no—
110C2×#41—44no—
111C22×Q8—21no—
112C22×D4—22no—
113C22×D4—28no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—24no—
119C22×D4—24no—
120C22×D4C22⋊C412yes—
121C22×D4C22⋊C412yes—
122C22×D4C22×C412yes—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—84no—
129C23×C4C22⋊C414yes—
130C23×C4C22⋊C414yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—24no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138#127—22no—
139#127—22no—
140#127—48no—
141#124—48no—
142#124—42no—
143#124—42no—
144#121—24no—
145#121—24no—
146#121—416no—
147#121—41no—
148#121—41no—
149#121—41no—
150#121—41no—
151#121—88no—
152#120—416no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—42no—
164C4×D4—42no—
165#117—44no—
166#117—44no—
167#117—44no—
168#117—88no—
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—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4C22×C414yes—
192C2×C22⋊C4C2×D414yes—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—28no—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42#4112yes—
203C2×C42#4112yes—
204C4⋊C8—42no—
205C4⋊C8—42no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C4×C8—42no—
209C4×C8—42no—
210#95—28no—
211#95—22no—
212#95—28no—
213#95—24no—
214#95C22×C414yes—
215#95C2×D418yes—
216C24—22no—
217C24—22no—
218C24—24no—
219C24—24no—
220C24—22no—
221C24—82no—
222C24—82no—
223C24C2×C22⋊C414yes—
224C24C22×D411yescharacteristic
225#41—84no—
226#41—84no—
227#41—88no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—44no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—416no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—48no—
252C2×D4—82no—
253C2×D4—88no—
254C2×D4—88no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—42no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—84no—
271C22×C4—81no—
272C22×C4—84no—
273C22×C4—82no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C414yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12012yes—
284C22×C4#12311yescharacteristic
285C22×C4#12311yescharacteristic
286C22×C4#12312yes—
287C22×C4#13514yes—
288C22×C4AGL(1,ℤ/8)14yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D412yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4112yes—
293C2×C8—44no—
294C2×C8—84no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—48no—
299C4⋊C4—44no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—84no—
305C4⋊C4—82no—
306C22⋊C4—28no—
307C22⋊C4—28no—
308C22⋊C4—416no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—88no—
316C22⋊C4—88no—
317C22⋊C4—88no—
318C42—44no—
319C42—48no—
320C42—88no—
321C42C4×D412yes—
322C42C4×D412yes—
323C42C4×D412yes—
324C42C4×D412yes—
325C42#12112yes—
326C42#12112yes—
327C42#12112yes—
328C42#12112yes—
329C23—24no—
330C23—24no—
331C23—24no—
332C23—41no—
333C23—41no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—44no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—82no—
346C23—82no—
347C23—88no—
348C23—88no—
349C23C22×C22⋊C411yescharacteristic
350C23C2×#12011yescharacteristic
351C23C2×#12311yescharacteristic
352C23C2×#13512yes—
353C23C2×AGL(1,ℤ/8)12yes—
354Q8—84no—
355D4—48no—
356D4—48no—
357D4—84no—
358D4—82no—
359D4—84no—
360D4—84no—
361D4—84no—
362D4—82no—
363D4—1616no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—48no—
387C2×C4—42no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—82no—
391C2×C4—88no—
392C2×C4#38611yescharacteristic
393C2×C4#38611yescharacteristic
394C2×C4#38611yescharacteristic
395C2×C4#38612yes—
396C2×C4#38612yes—
397C2×C4#38611yescharacteristic
398C2×C4#41612yes—
399C2×C4#41612yes—
400C2×C4#41812yes—
401C2×C4#41812yes—
402C2×C4#44914yes—
403C2×C4#44914yes—
404C2×C4#48218yes—
405C2×C4C2×C4×D412yes—
406C2×C4C2×#12112yes—
407C8—84no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—48no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22—84no—
421C22—88no—
422C22—88no—
423C22—84no—
424C22—84no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22C2×#38611yescharacteristic
429C22C2×#41611yescharacteristic
430C22C2×#41811yescharacteristic
431C22C2×#44912yes—
432C22C2×#48212yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—22no—
438C4—44no—
439C4—88no—
440C4—84no—
441C4—82no—
442C4#187514yes—
443C4#187514yes—
444C4#191112yes—
445C2—21no—
446C2—22no—
447C2—21no—
448C2—22no—
449C2—84no—
450C2—162no—
451C2C2×#187511yescharacteristic
452C2C2×#191111yescharacteristic
453C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35165C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34186C2semidirect
3. #17574C2semidirect
4. C2×#1460C2semidirect
5. #9205C2semidirect
6. #7981C2semidirect
13. #2547C4semidirect
15. #2541C4semidirect
17. #2535C4semidirect
87. #382D4semidirect
101. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#351652
#341862
#175742
C2×#14602
#92052
#79812