Finite Group Databasebeta

← #391374

Subgroups — #391374

Order 512.

807 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23730C211yesmaximal, characteristic
2#21143C211yesmaximal, characteristic, complemented
3#21046C211yesmaximal, characteristic
4#11370C211yesmaximal, characteristic, complemented
5#11269C211yesmaximal, characteristic, complemented
6#11221C212yesmaximal, complemented
7#11221C212yesmaximal, complemented
8#11191C212yesmaximal
9#11068C212yesmaximal
10#11053C211yesmaximal, characteristic
11#10797C211yesmaximal, characteristic, complemented
12C2×#427C2212yes—
13C2×#424C2211yescharacteristic
14C2×#424C2212yes—
15C2×#423C2211yescharacteristic
16C22×#104C2212yes—
17C2×#416C2212yes—
18C2×#408C2211yescharacteristic
19C2×#408C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2297C2211yescharacteristic
22#2285C2211yescharacteristic
23#2097C412yescomplemented
24#2096C411yescharacteristic, complemented
25#2095C411yescharacteristic, complemented
26#2094C411yescharacteristic, complemented
27#2093C411yescharacteristic, complemented
28#2092C412yescomplemented
29#1939—22no—
30#1939—22no—
31#1923—22no—
32#1923—22no—
33C2×#341C2212yes—
34C2×#341C2212yes—
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
56#427—42no—
57#427—42no—
58C2×#107C2×C412yes—
59C2×#107C2×C412yes—
60C2×#106C2×C412yes—
61C2×#106C2×C412yes—
62#424—42no—
63#424—42no—
64#424—42no—
65#423—42no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×#104—42no—
71C2×#104—42no—
72C2×#104—42no—
73#416—42no—
74#416—42no—
75C2×#103C2×C412yes—
76C2×#103C2×C412yes—
77C2×#102C2×C412yes—
78C2×#102C2×C412yes—
79#408—42no—
80#408—42no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C4×C8—42no—
84C2×C4×C8D412yes—
85C2×C4×C8D412yes—
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#390C2×C411yescharacteristic
89#390C2×C411yescharacteristic
90#386—21no—
91#386—21no—
92#384C2×C411yescharacteristic
93#384C2×C411yescharacteristic
94#382C2×C412yes—
95#380—21no—
96#380—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101C2×#95—21no—
102C2×#95—21no—
103#341—42no—
104#341—42no—
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×C412yes—
115#336C2×C411yescharacteristic
116#336C2×C411yescharacteristic
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×Q8C22×C411yescharacteristic
126C22×D4—21no—
127C22×D4C22×C411yescharacteristic
128C23×C4—22no—
129C23×C4—21no—
130C23×C4#4111yescharacteristic
131C23×C4#4111yescharacteristic
132C23×C4C2411yescharacteristic
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—21no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—22no—
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—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
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×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
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#107—42no—
203#107—42no—
204#106—42no—
205#106—42no—
206C4⋊C8—42no—
207C4⋊C8—42no—
208C4⋊C8—44no—
209C4⋊C8—48no—
210#104—48no—
211#103—42no—
212#103—42no—
213#102—42no—
214#102—42no—
215C22⋊C8—44no—
216C22⋊C8—44no—
217C4×C8—42no—
218C4×C8—42no—
219#95—24no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—42no—
227C24—21no—
228C24C2×#4111yescharacteristic
229#41—42no—
230#41—42no—
231#41—42no—
232#41—42no—
233#41—42no—
234#41—42no—
235#41—42no—
236#41—42no—
237C2×Q8—22no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—21no—
241C2×D4—22no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—22no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C4×D411yescharacteristic
274C22×C4C4×D411yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4C4×D411yescharacteristic
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4H(5,𝔽2)12yes—
280C22×C4H(5,𝔽2)12yes—
281C22×C4H(5,𝔽2)12yes—
282C22×C4H(5,𝔽2)11yescharacteristic
283C22×C42−1+411yescharacteristic
284C22×C42−1+411yescharacteristic
285C22×C42−1+411yescharacteristic
286C22×C42−1+412yes—
287M16—42no—
288M16—42no—
289M16—42no—
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294M16—42no—
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—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—42no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C42—22no—
342C42—22no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—41no—
348C42—42no—
349C42—42no—
350C42—41no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23C2×C4×D411yescharacteristic
367C23#53912yes—
368C23#54012yes—
369C23#54111yescharacteristic
370C23#54211yescharacteristic
371Q8—42no—
372Q8—42no—
373D4—42no—
374D4—42no—
375D4—82no—
376D4—82no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4#44312yes—
416C2×C4#44412yes—
417C2×C4#51812yes—
418C2×C4#51911yescharacteristic
419C2×C4#52011yescharacteristic
420C2×C4#52012yes—
421C2×C4#53411yescharacteristic
422C2×C4#53511yescharacteristic
423C2×C4#53511yescharacteristic
424C2×C4#53611yescharacteristic
425C2×C4#53611yescharacteristic
426C2×C4#53711yescharacteristic
427C2×C4#56012yes—
428C2×C4#56212yes—
429C2×C4#56212yes—
430C2×C4#56212yes—
431C2×C4#56212yes—
432C2×C4#56312yes—
433C2×C4#56312yes—
434C2×C4#56312yes—
435C8—42no—
436C8—42no—
437C8—42no—
438C8—42no—
439C8—81no—
440C8—81no—
441C8—82no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22#232211yescharacteristic
458C22#232311yescharacteristic
459C22#232811yescharacteristic
460C22#283612yes—
461C22#283712yes—
462C22C2×#44311yescharacteristic
463C22C2×#44411yescharacteristic
464C22#295511yescharacteristic
465C22#295511yescharacteristic
466C4—22no—
467C4—22no—
468C4—41no—
469C4—41no—
470C4—84no—
471C4—81no—
472C4—82no—
473C4—81no—
474C4—82no—
475C4—81no—
476C4#237612yes—
477C4#237812yes—
478C4#281311yescharacteristic
479C4#281611yescharacteristic
480C4#282011yescharacteristic
481C4#282111yescharacteristic
482C2—22no—
483C2—22no—
484C2—161no—
485C2#1462811yescharacteristic
486C2#2209511yescharacteristic
487C2#2209611yescharacteristic
488C2#2213011yescharacteristic
489C2#2213111yescharacteristic
490C2#2228011yescharacteristic
491C2#2228111yescharacteristic
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. #11370C2semidirect
5. #11269C2semidirect
6. #11221C2semidirect
7. #11221C2semidirect
11. #10797C2semidirect
23. #2097C4semidirect
24. #2096C4semidirect
25. #2095C4semidirect
26. #2094C4semidirect
27. #2093C4semidirect
28. #2092C4semidirect

Maximal subgroups

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

subgroupindex
#237302
#211432
#210462
#113702
#112692
#112212
#111912
#110682
#110532
#107972