Finite Group Databasebeta

← #393150

Subgroups — #393150

Order 512.

866 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21181C211yesmaximal, characteristic
2C2×#1945C211yesmaximal, characteristic
3#21041C212yesmaximal
4C4×#405C211yesmaximal, characteristic
5#10914C212yesmaximal
6#10895C212yesmaximal
7#10873C211yesmaximal, characteristic
8#10872C211yesmaximal, characteristic
9#10871C211yesmaximal, characteristic
10#10859C211yesmaximal, characteristic
11#10826C212yesmaximal
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14#2911C2212yes—
15C2×#427C2211yescharacteristic
16C2×#427C2211yescharacteristic
17C2×#424C2212yes—
18C2×#405C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1952C414yescomplemented
26#1951C414yescomplemented
27#1945—21no—
28#1945—21no—
29#1945—21no—
30#1945—21no—
31#1922—28no—
32C2×#344C2212yes—
33C2×#344C2212yes—
34#1742C2212yes—
35#1740C2212yes—
36#1738C2212yes—
37#1733C2212yes—
38#1733C2212yes—
39C2×#339C2212yes—
40C2×#339C2212yes—
41#1729C2212yes—
42#1726—21no—
43#1726—21no—
44C4×#97—21no—
45C42×C8—21no—
46C22×M16C2312yes—
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#117C2312yes—
53C2×#117C2312yes—
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×#108—22no—
58C2×#108—22no—
59C2×#108—21no—
60C2×#108—21no—
61C2×#108—22no—
62C2×#108—22no—
63C2×#108C2×C412yes—
64C2×#108C2×C412yes—
65#428—24no—
66#428C2×C418yes—
67#427—21no—
68#427—21no—
69#427—21no—
70#427—21no—
71#427—24no—
72#427—22no—
73#427—22no—
74C2×#107—21no—
75C2×#107C2×C412yes—
76C2×#106—21no—
77C2×#106C2×C412yes—
78#424—24no—
79#424—22no—
80#424—22no—
81C2×C4⋊C8—24no—
82#405—22no—
83#405—21no—
84#405—21no—
85#405—22no—
86#405—22no—
87#405—22no—
88#405C2×C412yes—
89#405C2×C412yes—
90#405D414yes—
91#405Q814yes—
92C4×M16—22no—
93C4×M16Q814yes—
94C2×#97—21no—
95C2×#97—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—21no—
106C2×C4×C8D411yescharacteristic
107C2×C4×C8D411yescharacteristic
108C2×C4×C8D411yescharacteristic
109C2×C4×C8D411yescharacteristic
110C4×C4⋊C4—22no—
111C4×C22⋊C4—22no—
112#376—21no—
113C43—21no—
114#344—24no—
115#344—24no—
116#343C2×C418yes—
117#342—24no—
118#341—24no—
119#339—24no—
120#339—24no—
121#339—22no—
122#339—22no—
123#339—24no—
124#336—24no—
125C23×C4—21no—
126C23×C4C2411yescharacteristic
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—21no—
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16C22×C414yes—
145C2×M16C22×C412yes—
146C2×M16C2×D412yes—
147C2×M16C2×Q812yes—
148C2×M16C2×Q812yes—
149C2×M16C2×Q812yes—
150C2×M16#4112yes—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8C22×C411yescharacteristic
160C22×C8C22×C411yescharacteristic
161C22×C8C2×D412yes—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×Q812yes—
167C22×C8#4112yes—
168C22×C8#4112yes—
169C22×C8#4112yes—
170C22×C8#4111yescharacteristic
171C22×C8#4111yescharacteristic
172C22×C8#4112yes—
173#117—21no—
174#117—21no—
175#117—21no—
176#117—21no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117C22×C412yes—
186#117C2×D412yes—
187#117C2×Q812yes—
188#117#4112yes—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4C22×C414yes—
192C2×C4⋊C4#4112yes—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4#4112yes—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×Q812yes—
208C2×C42#4112yes—
209#108—24no—
210#108—24no—
211#108—42no—
212#108—42no—
213#107—22no—
214#107—22no—
215#107—28no—
216#107—42no—
217#106—22no—
218#106—22no—
219#106—28no—
220#106—42no—
221C4⋊C8—44no—
222C4⋊C8—44no—
223#97—24no—
224#97—22no—
225#97—22no—
226#97—24no—
227#97—24no—
228#97—24no—
229C4×C8—24no—
230C4×C8—22no—
231C4×C8—22no—
232C4×C8—24no—
233C4×C8—24no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—24no—
239#95—24no—
240C24C2×#4111yescharacteristic
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4C23×C411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×Q811yescharacteristic
254C22×C4C22×Q811yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4111yescharacteristic
264M16—22no—
265M16—22no—
266M16—22no—
267M16—22no—
268M16—22no—
269M16—22no—
270M16—24no—
271M16—24no—
272M16—42no—
273M16—42no—
274M16—42no—
275M16—42no—
276M16—42no—
277M16—42no—
278M16C4×Q814yes—
279M16C4⋊Q814yes—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8#11712yes—
301C2×C8#11712yes—
302C2×C8C4×D411yescharacteristic
303C2×C8C4×D411yescharacteristic
304C2×C8C4×D411yescharacteristic
305C2×C8C4×D411yescharacteristic
306C2×C8C4×Q812yes—
307C2×C8C4×Q812yes—
308C2×C8#12112yes—
309C2×C8#12112yes—
310C2×C8C22⋊Q812yes—
311C2×C8C22⋊Q812yes—
312C2×C8#12312yes—
313C2×C8#12312yes—
314C2×C8#12512yes—
315C2×C8#12512yes—
316C2×C8#12612yes—
317C2×C8#12612yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4C4×D414yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4#12514yes—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—48no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42C22⋊Q812yes—
348C42C22⋊Q812yes—
349C23—42no—
350C23C4×#4111yescharacteristic
351C23#52511yescharacteristic
352C23C2×#12511yescharacteristic
353C23#52911yescharacteristic
354C23#52911yescharacteristic
355C23#53211yescharacteristic
356C23#53311yescharacteristic
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4C2×#11711yescharacteristic
372C2×C4C2×C4×D411yescharacteristic
373C2×C4C2×C4×D411yescharacteristic
374C2×C4C2×C4×Q811yescharacteristic
375C2×C4C2×C4×Q811yescharacteristic
376C2×C4C4×#4111yescharacteristic
377C2×C4C2×#12111yescharacteristic
378C2×C4C2×C22⋊Q811yescharacteristic
379C2×C4C2×C22⋊Q811yescharacteristic
380C2×C4C2×#12311yescharacteristic
381C2×C4#52511yescharacteristic
382C2×C4C2×#12511yescharacteristic
383C2×C4C2×#12611yescharacteristic
384C2×C4#52911yescharacteristic
385C2×C4#52911yescharacteristic
386C2×C4#52911yescharacteristic
387C2×C4#52911yescharacteristic
388C2×C4#52911yescharacteristic
389C2×C4#52911yescharacteristic
390C2×C4C2×C4⋊Q811yescharacteristic
391C2×C4#53211yescharacteristic
392C2×C4#53211yescharacteristic
393C2×C4#53211yescharacteristic
394C2×C4#53311yescharacteristic
395C2×C4#53311yescharacteristic
396C2×C4#53311yescharacteristic
397C2×C4#53311yescharacteristic
398C2×C4#53311yescharacteristic
399C8—22no—
400C8—22no—
401C8—24no—
402C8—24no—
403C8—44no—
404C8—82no—
405C8#38712yes—
406C8#38712yes—
407C8#38712yes—
408C8#38712yes—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22#229511yescharacteristic
417C22C4×#12111yescharacteristic
418C22C4×#12511yescharacteristic
419C22#285111yescharacteristic
420C22#285511yescharacteristic
421C22#285911yescharacteristic
422C22#286011yescharacteristic
423C4—24no—
424C4—44no—
425C4—42no—
426C4—44no—
427C4—84no—
428C4—162no—
429C4C2×#38711yescharacteristic
430C4#229311yescharacteristic
431C4C4×C22⋊Q811yescharacteristic
432C4C4×C4⋊Q811yescharacteristic
433C4#285411yescharacteristic
434C4#285711yescharacteristic
435C4#286311yescharacteristic
436C4#286411yescharacteristic
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2#1469911yescharacteristic
444C2#2155812yes—
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C4×#121
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1952C4semidirect
26. #1951C4semidirect

Maximal subgroups

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

subgroupindex
#211812
C2×#19452
#210412
C4×#4052
#109142
#108952
#108732
#108722
#108712
#108592
#108262