Finite Group Databasebeta

← #239214

Subgroups — #239214

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4#21311C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6C22×#333C211yesmaximal, characteristic
7#10543C214yesmaximal, complemented
8#10541C214yesmaximal, complemented
9#7877C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—24no—
13#3230—24no—
14C2×#500C2212yes—
15C2×#487C2212yes—
16C2×#484—22no—
17C2×#477—21no—
18C2×#477—21no—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1989—22no—
25#1988—22no—
26#1988C2212yes—
27#1987C2214yes—
28#1987C2212yes—
29#1939C2214yes—
30#1850—22no—
31#1842—21no—
32#1831—21no—
33#1736—21no—
34#1736—21no—
35#1735—21no—
36#1735—21no—
37#1550—24no—
38C2×#333—24no—
39C2×#333C2214yes—
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1277C2214yes—
43C23×Q8—21no—
44C22×M16—21no—
45C23×C8—21no—
46C23×C8—21no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533D414yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×C4⋊Q8C2312yes—
60C2×C4⋊Q8C2311yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117—21no—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C42C2311yescharacteristic
72#500—24no—
73#500—24no—
74#487—24no—
75#487—24no—
76#484—216no—
77#477—28no—
78#477—28no—
79#431—22no—
80#431—22no—
81C2×#107—22no—
82C2×#107—22no—
83#423—22no—
84#423—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×#103—22no—
99C2×#103—22no—
100C2×C22⋊C8D412yes—
101C2×C4×C8C2312yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4C2312yes—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106#376—21no—
107C43—21no—
108#340—24no—
109#340—24no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#336—24no—
115#333—28no—
116#333—24no—
117#332—24no—
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—41no—
123C22×Q8—42no—
124C22×Q8—84no—
125C23×C4—41no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4111yescharacteristic
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—82no—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—22no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—22no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—22no—
153C4⋊Q8—21no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8C2×D414yes—
158C4⋊Q8C2×D412yes—
159C4⋊Q8C2×D412yes—
160#125—24no—
161#125—42no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—42no—
169C4×Q8—42no—
170#117—22no—
171#117—24no—
172#117—22no—
173#117—22no—
174#117—42no—
175#117Q1614yes—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—41no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208#107—28no—
209#107—28no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—42no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221C4⋊C8—84no—
222#103—28no—
223#103—28no—
224C22⋊C8—24no—
225C4×C8—22no—
226C4×C8—22no—
227#95—22no—
228#95—22no—
229C24#12111yescharacteristic
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—41no—
234C2×Q8—42no—
235C2×Q8—41no—
236C2×Q8—44no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—84no—
240C2×Q8—82no—
241C2×Q8—84no—
242C2×Q8—88no—
243C2×Q8—88no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—84no—
254C22×C4#12012yes—
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12112yes—
258C22×C4#12112yes—
259C22×C4#12111yescharacteristic
260C22×C4#12312yes—
261C22×C4#12312yes—
262C22×C4#12412yes—
263C22×C4C2×Q1612yes—
264C22×C4#13512yes—
265C22×C4#13712yes—
266C22×C4#13712yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270M16—84no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—44no—
283C2×C8—84no—
284C2×C8—84no—
285C4⋊C4—22no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4C2×Q1614yes—
305C22⋊C4—24no—
306C22⋊C4—44no—
307C22⋊C4—42no—
308C42—24no—
309C42—42no—
310C42—42no—
311C42—44no—
312C42—42no—
313C42—82no—
314C42#12012yes—
315C42#12112yes—
316C42#12112yes—
317C42#12312yes—
318C42#12312yes—
319C42#12412yes—
320C42C22×D411yescharacteristic
321C42C22×D411yescharacteristic
322C42C2×#4111yescharacteristic
323C42C2×#4111yescharacteristic
324C23—22no—
325C23—22no—
326C23#39412yes—
327C23#46712yes—
328C23#47012yes—
329C23C2×#12111yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—82no—
333Q8—84no—
334Q8—82no—
335Q8—84no—
336Q8—88no—
337Q8—82no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—88no—
356C2×C4—82no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4#39412yes—
360C2×C4#39412yes—
361C2×C4#39412yes—
362C2×C4#46214yes—
363C2×C4#46214yes—
364C2×C4#46414yes—
365C2×C4#46414yes—
366C2×C4#46712yes—
367C2×C4#47012yes—
368C2×C4C2×#12011yescharacteristic
369C2×C4C2×#12111yescharacteristic
370C2×C4C2×#12111yescharacteristic
371C2×C4C2×#12311yescharacteristic
372C2×C4C2×#12311yescharacteristic
373C2×C4C2×#12411yescharacteristic
374C2×C4C22×Q1611yescharacteristic
375C2×C4C2×#13511yescharacteristic
376C2×C4C2×#13711yescharacteristic
377C2×C4C2×#13711yescharacteristic
378C2×C4#57712yes—
379C2×C4#57812yes—
380C8—42no—
381C8—42no—
382C8—82no—
383C8—82no—
384C8—82no—
385C22—22no—
386C22—24no—
387C22—22no—
388C22—24no—
389C22#205614yes—
390C22C2×#39411yescharacteristic
391C22#304612yes—
392C22C2×#46711yescharacteristic
393C22C2×#47011yescharacteristic
394C22#306812yes—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—44no—
400C4—82no—
401C4—84no—
402C4—82no—
403C4—82no—
404C4#205614yes—
405C4#301011yescharacteristic
406C4#301111yescharacteristic
407C4C2×#46211yescharacteristic
408C4C2×#46211yescharacteristic
409C4C2×#46411yescharacteristic
410C4C2×#46411yescharacteristic
411C4#313712yes—
412C2—24no—
413C2#1941712yes—
414C2#1945312yes—
415C2#2351011yescharacteristic
416C2#2352111yescharacteristic
417C2C2×#205611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10543C2semidirect
8. #10541C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#219722
#213132
#213112
C2×#15502
C22×#3332
#105432
#105412
#78772