Finite Group Databasebeta

← #355821

Subgroups — #355821

Order 512.

1228 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21224C211yesmaximal, characteristic
2C2×#1876C212yesmaximal
3#20384C212yesmaximal, complemented
4#10934C212yesmaximal, complemented
5#10924C212yesmaximal, complemented
6#10909C212yesmaximal, complemented
7#10822C211yesmaximal, characteristic, complemented
8#10791C212yesmaximal, complemented
9#10694C211yesmaximal, characteristic, complemented
10C22×#135C2211yescharacteristic
11C22×#131C2212yes—
12C2×#427C2211yescharacteristic
13C22×#104C2212yes—
14C22×#104C2212yes—
15C2×#416C2212yes—
16C2×#408C2212yes—
17C2×#405C2212yes—
18C22×#97C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2297C2212yes—
21C2×#376C2211yescharacteristic
22#1970C412yescomplemented
23#1969C412yescomplemented
24#1968C414yescomplemented
25#1876C4116yescomplemented
26#1753—22no—
27#1753—22no—
28#1752—22no—
29#1752—22no—
30C2×#341C2211yescharacteristic
31C2×#341C2211yescharacteristic
32C2×#341C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35#1729C2212yes—
36#1728C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C22×#41C2312yes—
40C2×#135—41no—
41C2×#135—41no—
42C2×#135—42no—
43C2×#135C2×C418yes—
44C22×Q16C2×C411yescharacteristic
45C22×SD16C2×C412yes—
46C22×D8C2×C411yescharacteristic
47C2×#131—24no—
48C2×#131—42no—
49C2×#131—42no—
50C2×#131—42no—
51C2×#131—42no—
52C2×#131C2×C418yes—
53C22×M16—22no—
54C22×M16C2312yes—
55C22×M16C2312yes—
56C23×C8—22no—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2312yes—
60C2×#117C2312yes—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63#427—42no—
64C2×#107C2×C412yes—
65C2×#106C2×C412yes—
66C2×#104—42no—
67C2×#104—42no—
68C2×#104—42no—
69C2×#104—42no—
70C2×#104C2×C418yes—
71C2×#104C2×C418yes—
72#416—42no—
73#416—42no—
74C2×#103C2×C414yes—
75C2×#102C2×C414yes—
76#408—24no—
77#408C2×C418yes—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80#405—42no—
81#405—42no—
82C2×#97—22no—
83C2×#97—41no—
84C2×#97—41no—
85C2×#97D411yescharacteristic
86C2×#97D412yes—
87C2×#97D411yescharacteristic
88C2×C4×C8—22no—
89C2×C4×C8—41no—
90C2×C4×C8—41no—
91C2×C4×C8C2×C412yes—
92C2×C4×C8C2×C412yes—
93#391C2×C414yes—
94#390C2×C414yes—
95#386—24no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98#376—41no—
99#376—41no—
100C2×#95—22no—
101#341—41no—
102#341—41no—
103#341C2×C418yes—
104#339—42no—
105#339—42no—
106#336—22no—
107#336—22no—
108#336—24no—
109#336—22no—
110#336—22no—
111#336C2×C418yes—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—22no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41C22×C414yes—
123C2×#41C22×C414yes—
124C22×Q8C22×C412yes—
125C22×D4—22no—
126C22×D4C22×C412yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4#4112yes—
130C23×C4C2411yescharacteristic
131#135—48no—
132#135—48no—
133C2×Q16—41no—
134C2×Q16—41no—
135C2×Q16C4214yes—
136C2×SD16—42no—
137C2×SD16—42no—
138C2×SD16C4218yes—
139C2×D8—41no—
140C2×D8—41no—
141C2×D8C4214yes—
142#131—48no—
143#131—48no—
144#131—48no—
145#131—48no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—28no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—44no—
158C2×M16—44no—
159C2×M16C22×C414yes—
160C2×M16C22×C414yes—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—28no—
175C22×C8—44no—
176C22×C8—44no—
177C22×C8C22×C412yes—
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C414yes—
181C22×C8C22×C414yes—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C22×C8#4114yes—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—22no—
189C2×C4⋊C4C22×C414yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C22×C414yes—
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207#107—42no—
208#106—42no—
209#104—48no—
210#104—48no—
211#103—44no—
212#102—44no—
213C22⋊C8—28no—
214C22⋊C8—28no—
215#97—41no—
216#97—42no—
217#97—41no—
218#97—44no—
219C4×C8—42no—
220C4×C8—42no—
221C4×C8—44no—
222#95—24no—
223#95—24no—
224#95—44no—
225C24—22no—
226C24C2×#4111yescharacteristic
227#41—24no—
228#41—24no—
229#41—24no—
230#41—24no—
231#41—24no—
232#41—24no—
233#41—24no—
234#41—24no—
235#41—42no—
236#41—42no—
237#41—42no—
238#41—42no—
239#41—42no—
240#41—42no—
241#41—42no—
242#41—42no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8C2×C4214yes—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4C2×C4214yes—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C4×D412yes—
279C22×C4C4×D412yes—
280C22×C4C23×C412yes—
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C2×#4112yes—
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)12yes—
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C4H(5,𝔽2)12yes—
289C22×C42−1+411yescharacteristic
290Q16—44no—
291SD16—48no—
292D8—44no—
293M16—24no—
294M16—24no—
295M16—24no—
296M16—24no—
297M16—416no—
298M16—42no—
299M16—42no—
300M16—42no—
301M16—42no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—416no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8C2×C4212yes—
329C2×C8C2×C4212yes—
330C2×C8C4×D414yes—
331C2×C8C4×D414yes—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C42—22no—
339C42—22no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—24no—
345C42—24no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C23—22no—
351C23—22no—
352C23—24no—
353C23—42no—
354C23—42no—
355C23—44no—
356C23—44no—
357C23C2×C4×D411yescharacteristic
358C23C4×#4112yes—
359C23#53811yescharacteristic
360C23#53911yescharacteristic
361C23#54012yes—
362Q8—24no—
363Q8—24no—
364Q8—42no—
365Q8—42no—
366D4—24no—
367D4—24no—
368D4—44no—
369D4—44no—
370D4—44no—
371D4—44no—
372D4—42no—
373D4—42no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4C22×C4211yescharacteristic
398C2×C4C2×C4×D412yes—
399C2×C4C4×#4111yescharacteristic
400C2×C4#51812yes—
401C2×C4#51812yes—
402C2×C4#51812yes—
403C2×C4#51811yescharacteristic
404C2×C4#51811yescharacteristic
405C2×C4#51811yescharacteristic
406C2×C4#52012yes—
407C2×C4#52011yescharacteristic
408C2×C4#53411yescharacteristic
409C2×C4#53512yes—
410C2×C4#53711yescharacteristic
411C2×C4#53812yes—
412C2×C4#53812yes—
413C2×C4#54012yes—
414C2×C4#54212yes—
415C8—24no—
416C8—24no—
417C8—22no—
418C8—22no—
419C8—44no—
420C8—44no—
421C8—42no—
422C22—24no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—44no—
428C22—44no—
429C22—44no—
430C22—84no—
431C22—84no—
432C22C42×D411yescharacteristic
433C22#232511yescharacteristic
434C22#232811yescharacteristic
435C22#233312yes—
436C22#233512yes—
437C22#295712yes—
438C22#295712yes—
439C4—42no—
440C4—42no—
441C4—44no—
442C4—84no—
443C4—84no—
444C4#227311yescharacteristic
445C4#227511yescharacteristic
446C4#232112yes—
447C4#232412yes—
448C4#233311yescharacteristic
449C4#233711yescharacteristic
450C2—22no—
451C2—22no—
452C2—84no—
453C2#1379111yescharacteristic
454C2#2051311yescharacteristic
455C2#2051411yescharacteristic
456C2#2220612yes—
457C2#2228212yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#135C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20384C2semidirect
4. #10934C2semidirect
5. #10924C2semidirect
6. #10909C2semidirect
7. #10822C2semidirect
8. #10791C2semidirect
9. #10694C2semidirect
22. #1970C4semidirect
23. #1969C4semidirect
24. #1968C4semidirect
25. #1876C4semidirect

Maximal subgroups

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

subgroupindex
#212242
C2×#18762
#203842
#109342
#109242
#109092
#108222
#107912
#106942