Finite Group Databasebeta

← #508410

Subgroups — #508410

Order 512.

603 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic, complemented
2C2×#2210C211yesmaximal, characteristic
3#33240C211yesmaximal, characteristic, complemented
4#21275C211yesmaximal, characteristic, complemented
5#13477C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic, complemented
7#13475C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13449C211yesmaximal, characteristic, complemented
10#13449C211yesmaximal, characteristic, complemented
11#13448C211yesmaximal, characteristic
12#13448C211yesmaximal, characteristic
13#13277C211yesmaximal, characteristic, complemented
14#13254C211yesmaximal, characteristic, complemented
15#12898C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#495C2211yescharacteristic
20C2×#488—21no—
21C2×C4×D8C2211yescharacteristic
22#2861—21no—
23#2293—21no—
24#2228—24no—
25#2210—24no—
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic
46#1969C2211yescharacteristic
47#1969C2211yescharacteristic
48#1968—21no—
49#1968—21no—
50#1942C2211yescharacteristic
51#1942C2211yescharacteristic
52#1934—21no—
53#1934—21no—
54C4×#107C2211yescharacteristic
55C4×#102—21no—
56C42×C8C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C22×D8C2311yescharacteristic
65C2×#125—21no—
66C2×#124—21no—
67C2×C4×D4—21no—
68C2×#117—41no—
69C2×SD32—22no—
70C2×SD32—41no—
71C2×SD32—41no—
72C2×D16—22no—
73C2×D16—41no—
74C2×D16—41no—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#499—22no—
80#499—22no—
81#495—22no—
82#495—22no—
83#488—44no—
84C4×D8—22no—
85C4×D8D414yes—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#106—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×C4×C8D411yescharacteristic
100C2×C4×C8D411yescharacteristic
101C2×C4×C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#391—21no—
106#391—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#388—41no—
114#388—41no—
115#388—41no—
116#388—41no—
117#388—41no—
118#388—41no—
119C4×C4⋊C4—21no—
120C4×C22⋊C4—41no—
121C43D411yescharacteristic
122#367—22no—
123#366—22no—
124C4⋊C16—22no—
125C4⋊C16—22no—
126C4⋊C16—41no—
127C4⋊C16—41no—
128C4⋊C16—41no—
129C4⋊C16—41no—
130#358—22no—
131#358—41no—
132#358—41no—
133#357—22no—
134#357—22no—
135#357—22no—
136#357—41no—
137#357—41no—
138#340—21no—
139#340—21no—
140#340—21no—
141C22×Q8—21no—
142C22×D4—21no—
143C22×D4—21no—
144C23×C4—41no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×SD16—42no—
152C2×SD16—42no—
153C2×SD16—42no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8C2×D412yes—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170#125—42no—
171#125—42no—
172#124—42no—
173#124—42no—
174C4×D4—24no—
175C4×D4—42no—
176#117—84no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—41no—
198C2×C42D811yescharacteristic
199C2×C42D811yescharacteristic
200C2×C42D811yescharacteristic
201C2×C42D811yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205SD32—44no—
206D16—44no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107C2×D412yes—
229#106—42no—
230#103—42no—
231#103—42no—
232#102—24no—
233#102—42no—
234#102—42no—
235#102—42no—
236#102—42no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—22no—
242C4×C8—41no—
243C4×C8—41no—
244C4×C8C2×D412yes—
245C4×C8#4112yes—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24—41no—
254C24—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—22no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C22×C4—41no—
274C22×C4—44no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C22×D411yescharacteristic
289Q16—41no—
290Q16—41no—
291Q16—41no—
292Q16—41no—
293SD16—84no—
294D8—22no—
295D8—22no—
296D8—41no—
297D8—41no—
298D8—41no—
299D8—41no—
300D8—41no—
301D8—41no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8#12112yes—
311C2×C8C22×D411yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C2×C8H(5,𝔽2)11yescharacteristic
315C2×C82−1+411yescharacteristic
316C2×C82−1+411yescharacteristic
317C4⋊C4—22no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C22⋊C4—44no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—22no—
347C42—22no—
348C42—41no—
349C42—41no—
350C42—82no—
351C42#12112yes—
352C42AGL(1,ℤ/8)12yes—
353C16—41no—
354C16—41no—
355C16—41no—
356C16—41no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23C22×D811yescharacteristic
372Q8—81no—
373Q8—81no—
374Q8—81no—
375Q8—81no—
376D4—42no—
377D4—42no—
378D4—81no—
379D4—81no—
380D4—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—42no—
399C2×C4—88no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4#45912yes—
418C2×C4#45912yes—
419C2×C4#50812yes—
420C2×C4#50912yes—
421C2×C4C2×#12111yescharacteristic
422C2×C4#53711yescharacteristic
423C2×C4#53711yescharacteristic
424C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
425C2×C4#57511yescharacteristic
426C2×C4#57511yescharacteristic
427C8—22no—
428C8—41no—
429C8—41no—
430C8#53511yescharacteristic
431C8#53611yescharacteristic
432C8#54211yescharacteristic
433C8#54211yescharacteristic
434C22—82no—
435C22—82no—
436C22—82no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22C2×#45911yescharacteristic
449C22#324511yescharacteristic
450C22#324511yescharacteristic
451C22C2×#50811yescharacteristic
452C22C2×#50911yescharacteristic
453C22#341411yescharacteristic
454C22#341411yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—41no—
460C4—41no—
461C4—82no—
462C4—161no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4#262011yescharacteristic
468C4#304011yescharacteristic
469C4#325111yescharacteristic
470C4#325111yescharacteristic
471C2—162no—
472C2—321no—
473C2#2982411yescharacteristic
474C2#3357911yescharacteristic
475C2#3358011yescharacteristic
476C2#3381311yescharacteristic
477C2#3381611yescharacteristic
478C2#3381711yescharacteristic
479C2#3382211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2228C2semidirect
3. #33240C2semidirect
4. #21275C2semidirect
5. #13477C2semidirect
6. #13476C2semidirect
7. #13475C2semidirect
8. #13474C2semidirect
9. #13449C2semidirect
10. #13449C2semidirect
13. #13277C2semidirect
14. #13254C2semidirect
15. #12898C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×#22102
#332402
#212752
#134772
#134762
#134752
#134742
#134492
#134482
#132772
#132542
#128982