Finite Group Databasebeta

← #391316

Subgroups — #391316

Order 512.

847 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23451C211yesmaximal, characteristic
2#21300C211yesmaximal, characteristic, complemented
3#21046C211yesmaximal, characteristic
4#11386C211yesmaximal, characteristic, complemented
5#11326C211yesmaximal, characteristic, complemented
6#11291C212yesmaximal, complemented
7#11258C212yesmaximal, complemented
8#11191C212yesmaximal
9#11125C212yesmaximal
10#11053C211yesmaximal, characteristic
11#10797C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C22×#108C2212yes—
14C2×#424C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C22×#104C2212yes—
17C2×#416C2212yes—
18C2×#408C2211yescharacteristic
19C2×#408C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2297C2211yescharacteristic
22#2026C412yescomplemented
23#2025C412yescomplemented
24#2024C411yescharacteristic, complemented
25#2024C411yescharacteristic, complemented
26#2023C411yescharacteristic, complemented
27#2023C411yescharacteristic, complemented
28#1984—22no—
29#1984—22no—
30#1923—22no—
31#1923—22no—
32C2×#341C2212yes—
33#1735C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36C2×#339C2212yes—
37#1729C2212yes—
38#1729C2212yes—
39#1729C2211yescharacteristic
40#1728C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C4×#95C2211yescharacteristic
45C22×#41C2311yescharacteristic
46C22×M16C2311yescharacteristic
47C22×M16C2311yescharacteristic
48C22×M16C2312yes—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52#533—42no—
53#533—42no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×#117C2312yes—
62C2×#117C2311yescharacteristic
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×#108—42no—
66C2×#108—42no—
67C2×#108—42no—
68#424—42no—
69#423—42no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×#104—42no—
74C2×#104—42no—
75C2×#104—42no—
76#416—42no—
77#416—42no—
78C2×#103C2×C412yes—
79C2×#103C2×C412yes—
80C2×#102C2×C412yes—
81C2×#102C2×C412yes—
82#408—42no—
83#408—42no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C4×C8—42no—
87C2×C4×C8D412yes—
88C2×C4×C8D412yes—
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#386—21no—
94#386—21no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97C2×#95—21no—
98C2×#95—21no—
99#341—42no—
100#340C2×C412yes—
101#340C2×C412yes—
102#340C2×C412yes—
103#340C2×C412yes—
104#339—42no—
105#339—42no—
106#339—42no—
107#336—22no—
108#336—24no—
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—22no—
120C2×#41—21no—
121C2×#41—21no—
122C2×#41—21no—
123C2×#41—22no—
124C2×#41—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8C22×C411yescharacteristic
128C22×D4—21no—
129C22×D4C22×C411yescharacteristic
130C23×C4—22no—
131C23×C4—21no—
132C23×C4#4111yescharacteristic
133C23×C4#4111yescharacteristic
134C23×C4C2411yescharacteristic
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—21no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8C22×C412yes—
167C22×C8C22×C412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172#125—42no—
173#125—42no—
174C22⋊Q8—44no—
175C22⋊Q8—44no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214#108—48no—
215C4⋊C8—42no—
216C4⋊C8—42no—
217C4⋊C8—44no—
218#104—48no—
219#103—42no—
220#103—42no—
221#102—42no—
222#102—42no—
223C22⋊C8—44no—
224C22⋊C8—44no—
225C4×C8—42no—
226C4×C8—42no—
227#95—24no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—42no—
233C24—21no—
234C24C2×#4111yescharacteristic
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—21no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4C4×D411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4H(5,𝔽2)12yes—
292C22×C4H(5,𝔽2)12yes—
293C22×C4H(5,𝔽2)12yes—
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C42−1+411yescharacteristic
296C22×C42−1+411yescharacteristic
297C22×C42−1+411yescharacteristic
298C22×C42−1+412yes—
299M16—42no—
300M16—42no—
301M16—42no—
302M16—42no—
303M16—42no—
304M16—42no—
305M16—42no—
306M16—42no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C42—22no—
354C42—22no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—42no—
360C42—41no—
361C42—42no—
362C42—41no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23C2×C4×D411yescharacteristic
379C23#53812yes—
380C23#54111yescharacteristic
381C23#54212yes—
382C23#54211yescharacteristic
383Q8—42no—
384Q8—42no—
385Q8—84no—
386Q8—84no—
387D4—42no—
388D4—42no—
389D4—82no—
390D4—82no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#44312yes—
430C2×C4#44412yes—
431C2×C4#51812yes—
432C2×C4#51911yescharacteristic
433C2×C4#52012yes—
434C2×C4#52011yescharacteristic
435C2×C4#53411yescharacteristic
436C2×C4#53511yescharacteristic
437C2×C4#53511yescharacteristic
438C2×C4#53611yescharacteristic
439C2×C4#53611yescharacteristic
440C2×C4#53711yescharacteristic
441C2×C4H(3,𝔽4)12yes—
442C2×C4#56212yes—
443C2×C4#56212yes—
444C2×C4#56212yes—
445C2×C4#56212yes—
446C2×C4#56312yes—
447C2×C4#56312yes—
448C2×C4#56312yes—
449C8—42no—
450C8—42no—
451C8—42no—
452C8—42no—
453C8—81no—
454C8—81no—
455C8—82no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—22no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22#232111yescharacteristic
472C22#232311yescharacteristic
473C22#232611yescharacteristic
474C22#283112yes—
475C22#283212yes—
476C22C2×#44311yescharacteristic
477C22C2×#44411yescharacteristic
478C22#295511yescharacteristic
479C22#295511yescharacteristic
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—84no—
485C4—81no—
486C4—82no—
487C4—82no—
488C4—81no—
489C4—81no—
490C4#237812yes—
491C4#237912yes—
492C4#281411yescharacteristic
493C4#281611yescharacteristic
494C4#281911yescharacteristic
495C4#282111yescharacteristic
496C2—22no—
497C2—22no—
498C2—161no—
499C2#1462511yescharacteristic
500C2#2206011yescharacteristic
501C2#2206111yescharacteristic
502C2#2213011yescharacteristic
503C2#2213111yescharacteristic
504C2#2223411yescharacteristic
505C2#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. #21300C2semidirect
4. #11386C2semidirect
5. #11326C2semidirect
6. #11291C2semidirect
7. #11258C2semidirect
11. #10797C2semidirect
22. #2026C4semidirect
23. #2025C4semidirect
24. #2024C4semidirect
25. #2024C4semidirect
26. #2023C4semidirect
27. #2023C4semidirect

Maximal subgroups

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

subgroupindex
#234512
#213002
#210462
#113862
#113262
#112912
#112582
#111912
#111252
#110532
#107972