Finite Group Databasebeta

← #391416

Subgroups — #391416

Order 512.

807 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23627C211yesmaximal, characteristic
2#21143C211yesmaximal, characteristic, complemented
3#21046C211yesmaximal, characteristic
4#11359C211yesmaximal, characteristic, complemented
5#11329C212yesmaximal, complemented
6#11326C211yesmaximal, characteristic, complemented
7#11317C212yesmaximal, complemented
8#11125C212yesmaximal
9#11068C212yesmaximal
10#11053C211yesmaximal, characteristic
11#10797C211yesmaximal, characteristic, complemented
12C22×#108C2212yes—
13C2×#424C2211yescharacteristic
14C2×#424C2212yes—
15C2×#423C2211yescharacteristic
16C22×#104C2212yes—
17C2×#416C2212yes—
18C2×#408C2211yescharacteristic
19C2×#408C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2297C2211yescharacteristic
22#2285C2211yescharacteristic
23#2074C412yescomplemented
24#2073C412yescomplemented
25#2072C411yescharacteristic, complemented
26#2072C411yescharacteristic, complemented
27#2071C411yescharacteristic, complemented
28#2071C411yescharacteristic, complemented
29#1939—22no—
30#1939—22no—
31#1923—22no—
32#1923—22no—
33C2×#341C2212yes—
34#1735C2212yes—
35C2×#339C2212yes—
36C2×#339C2212yes—
37C2×#339C2212yes—
38#1729C2211yescharacteristic
39#1729C2212yes—
40#1728C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C4×#95C2211yescharacteristic
45C22×#41C2311yescharacteristic
46C22×M16C2311yescharacteristic
47C22×M16C2311yescharacteristic
48C22×M16C2312yes—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#117C2312yes—
53C2×#117C2311yescharacteristic
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56C2×#108—42no—
57C2×#108—42no—
58C2×#108—42no—
59#424—42no—
60#424—42no—
61#424—42no—
62#423—42no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—24no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×#104—42no—
68C2×#104—42no—
69C2×#104—42no—
70#416—42no—
71#416—42no—
72C2×#103C2×C412yes—
73C2×#103C2×C412yes—
74C2×#102C2×C412yes—
75C2×#102C2×C412yes—
76#408—42no—
77#408—42no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C4×C8—42no—
81C2×C4×C8D412yes—
82C2×C4×C8D412yes—
83#391C2×C411yescharacteristic
84#391C2×C411yescharacteristic
85#390C2×C411yescharacteristic
86#390C2×C411yescharacteristic
87#386—21no—
88#386—21no—
89#384C2×C411yescharacteristic
90#384C2×C411yescharacteristic
91#382C2×C412yes—
92#380—21no—
93#380—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C2×#95—21no—
99C2×#95—21no—
100#341—42no—
101#340C2×C412yes—
102#340C2×C412yes—
103#340C2×C412yes—
104#340C2×C412yes—
105#339—42no—
106#339—42no—
107#339—42no—
108#336—22no—
109#336—24no—
110#336—21no—
111#336—21no—
112#336—42no—
113#336—42no—
114#336C2×C411yescharacteristic
115#336C2×C411yescharacteristic
116#336C2×C412yes—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—21no—
120C2×#41—21no—
121C2×#41—21no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—21no—
125C22×Q8C22×C411yescharacteristic
126C22×D4—21no—
127C22×D4C22×C411yescharacteristic
128C23×C4—22no—
129C23×C4—21no—
130C23×C4#4111yescharacteristic
131C23×C4#4111yescharacteristic
132C23×C4C2411yescharacteristic
133C2×M16—21no—
134C2×M16—21no—
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×M16—21no—
145C2×M16—22no—
146C2×M16—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8C22×C412yes—
165C22×C8C22×C412yes—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202#108—48no—
203C4⋊C8—42no—
204C4⋊C8—42no—
205C4⋊C8—44no—
206C4⋊C8—48no—
207#104—48no—
208#103—42no—
209#103—42no—
210#102—42no—
211#102—42no—
212C22⋊C8—44no—
213C22⋊C8—44no—
214C4×C8—42no—
215C4×C8—42no—
216#95—24no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—22no—
222#95—22no—
223#95—42no—
224C24—21no—
225C24C2×#4111yescharacteristic
226#41—42no—
227#41—42no—
228#41—42no—
229#41—42no—
230#41—42no—
231#41—42no—
232#41—42no—
233#41—42no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—42no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4C4×D411yescharacteristic
271C22×C4C4×D411yescharacteristic
272C22×C4C4×D411yescharacteristic
273C22×C4C4×D411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C4H(5,𝔽2)11yescharacteristic
278C22×C4H(5,𝔽2)12yes—
279C22×C4H(5,𝔽2)12yes—
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C42−1+412yes—
282C22×C42−1+412yes—
283C22×C42−1+411yescharacteristic
284M16—42no—
285M16—42no—
286M16—42no—
287M16—42no—
288M16—42no—
289M16—42no—
290M16—42no—
291M16—42no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C42—22no—
339C42—22no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—42no—
345C42—41no—
346C42—41no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C2×C4×D411yescharacteristic
364C23#53912yes—
365C23#54011yescharacteristic
366C23#54211yescharacteristic
367C23#54212yes—
368Q8—42no—
369Q8—42no—
370D4—42no—
371D4—42no—
372D4—82no—
373D4—82no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4#44312yes—
413C2×C4#44412yes—
414C2×C4#51811yescharacteristic
415C2×C4#52012yes—
416C2×C4#52012yes—
417C2×C4#52011yescharacteristic
418C2×C4#53411yescharacteristic
419C2×C4#53511yescharacteristic
420C2×C4#53511yescharacteristic
421C2×C4#53611yescharacteristic
422C2×C4#53611yescharacteristic
423C2×C4#53711yescharacteristic
424C2×C4#56012yes—
425C2×C4#56012yes—
426C2×C4#56212yes—
427C2×C4#56212yes—
428C2×C4#56312yes—
429C2×C4#56312yes—
430C2×C4#56312yes—
431C2×C4#56312yes—
432C8—42no—
433C8—42no—
434C8—42no—
435C8—42no—
436C8—81no—
437C8—81no—
438C8—82no—
439C22—22no—
440C22—22no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—22no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22#232211yescharacteristic
455C22#232411yescharacteristic
456C22#232611yescharacteristic
457C22#283712yes—
458C22#283812yes—
459C22C2×#44311yescharacteristic
460C22C2×#44411yescharacteristic
461C22#295511yescharacteristic
462C22#295511yescharacteristic
463C4—22no—
464C4—22no—
465C4—41no—
466C4—41no—
467C4—84no—
468C4—81no—
469C4—82no—
470C4—82no—
471C4—81no—
472C4—81no—
473C4#237612yes—
474C4#238312yes—
475C4#281411yescharacteristic
476C4#281511yescharacteristic
477C4#282011yescharacteristic
478C4#282111yescharacteristic
479C2—22no—
480C2—22no—
481C2—161no—
482C2#1463011yescharacteristic
483C2#2209511yescharacteristic
484C2#2209611yescharacteristic
485C2#2216511yescharacteristic
486C2#2216611yescharacteristic
487C2#2223411yescharacteristic
488C2#2223511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21143C2semidirect
4. #11359C2semidirect
5. #11329C2semidirect
6. #11326C2semidirect
7. #11317C2semidirect
11. #10797C2semidirect
23. #2074C4semidirect
24. #2073C4semidirect
25. #2072C4semidirect
26. #2072C4semidirect
27. #2071C4semidirect
28. #2071C4semidirect

Maximal subgroups

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

subgroupindex
#236272
#211432
#210462
#113592
#113292
#113262
#113172
#111252
#110682
#110532
#107972