Finite Group Databasebeta

← #502349

Subgroups — #502349

Order 512.

590 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2207C211yesmaximal, characteristic
2C2×#2183C211yesmaximal, characteristic, complemented
3#28736C211yesmaximal, characteristic
4#28268C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic, complemented
7#13449C211yesmaximal, characteristic, complemented
8#13449C211yesmaximal, characteristic, complemented
9#13238C211yesmaximal, characteristic, complemented
10#13237C211yesmaximal, characteristic, complemented
11#13206C211yesmaximal, characteristic
12#13178C211yesmaximal, characteristic, complemented
13#13149C211yesmaximal, characteristic, complemented
14#13147C211yesmaximal, characteristic, complemented
15#12638C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21#2581—21no—
22#2548—21no—
23#2207—24no—
24#2183—24no—
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#346C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41#2066—21no—
42#2066—21no—
43#2054—21no—
44#2054—21no—
45#1982—21no—
46#1970C2211yescharacteristic
47#1970C2211yescharacteristic
48#1970C2211yescharacteristic
49#1970C2211yescharacteristic
50#1969C2211yescharacteristic
51#1969C2211yescharacteristic
52#1942C2211yescharacteristic
53#1942C2211yescharacteristic
54#1846C2211yescharacteristic
55#1805—21no—
56C4×#107C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8—21no—
63C22×D8C2311yescharacteristic
64C22×D8C2311yescharacteristic
65C2×#127—21no—
66C2×#124—41no—
67C2×C4×Q8—21no—
68C2×C4×Q8D411yescharacteristic
69C2×SD32—22no—
70C2×SD32—22no—
71C2×SD32—22no—
72C2×SD32—22no—
73C2×SD32—41no—
74C2×SD32—41no—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#493—22no—
80#493—22no—
81C8×Q8—22no—
82C8×Q8—22no—
83C4×Q16—22no—
84C4×Q16—22no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8D411yescharacteristic
93C2×C4⋊C8D411yescharacteristic
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#400—41no—
105#395—41no—
106#394—41no—
107#394—41no—
108#391—21no—
109#391—21no—
110#390—21no—
111#390—21no—
112#390—21no—
113#390—21no—
114#390—41no—
115#389—21no—
116#389—21no—
117#387—41no—
118#387—41no—
119#382—21no—
120C4×C4⋊C4—21no—
121#367—41no—
122#366—41no—
123C4⋊C16—22no—
124C4⋊C16—22no—
125C4⋊C16D414yes—
126#358—22no—
127#358—22no—
128#357—22no—
129#357—22no—
130#357—22no—
131#357—22no—
132#357—41no—
133#357—41no—
134#346—22no—
135#346—22no—
136C4×C16—22no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×Q8—21no—
141C22×Q8D811yescharacteristic
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—41no—
145C22×D4—41no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16#4112yes—
152C2×D8—21no—
153C2×D8—21no—
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—42no—
165C2×D8—81no—
166C2×D8—81no—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2411yescharacteristic
171#127—42no—
172#127—42no—
173#124—84no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—42no—
177C4×Q8—42no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4D811yescharacteristic
187C2×C4⋊C4D811yescharacteristic
188C2×C4⋊C4D811yescharacteristic
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×C42—41no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202SD32—44no—
203SD32—44no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16C2×D412yes—
214C2×C16C2×D412yes—
215C2×C16#4112yes—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#106—81no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230#103—42no—
231#103—81no—
232#103—81no—
233#102—81no—
234#102—81no—
235#102—81no—
236#102—81no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8C2×D412yes—
242C4×C8#4112yes—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24—41no—
253C24—41no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—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—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—161no—
280C2×D4—161no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C22×D411yescharacteristic
294Q16—22no—
295Q16—22no—
296Q16—41no—
297Q16—41no—
298D8—41no—
299D8—41no—
300D8—41no—
301D8—41no—
302D8—41no—
303D8—41no—
304D8—41no—
305D8—41no—
306D8—84no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8#12112yes—
316C2×C8C22×D411yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8C2×#4111yescharacteristic
320C2×C8H(5,𝔽2)11yescharacteristic
321C2×C82−1+411yescharacteristic
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—161no—
343C22⋊C4—161no—
344C22⋊C4—161no—
345C22⋊C4—161no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—82no—
351C42—82no—
352C42#12112yes—
353C42#13512yes—
354C16—22no—
355C16—22no—
356C16—22no—
357C16—41no—
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—
371C23—81no—
372C23C22×D811yescharacteristic
373Q8—44no—
374Q8—42no—
375Q8—42no—
376Q8—81no—
377Q8—81no—
378D4—81no—
379D4—81no—
380D4—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—164no—
387D4—164no—
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—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—42no—
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—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4#46612yes—
422C2×C4#46612yes—
423C2×C4#50912yes—
424C2×C4C2×#12111yescharacteristic
425C2×C4#54811yescharacteristic
426C2×C4#55011yescharacteristic
427C2×C4C2×#13511yescharacteristic
428C2×C4#57611yescharacteristic
429C2×C4#57711yescharacteristic
430C8—22no—
431C8—41no—
432C8—41no—
433C8#52911yescharacteristic
434C8#54211yescharacteristic
435C8#54611yescharacteristic
436C8#54811yescharacteristic
437C22—161no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22C2×#46611yescharacteristic
452C22#335811yescharacteristic
453C22#339111yescharacteristic
454C22C2×#50911yescharacteristic
455C22#341411yescharacteristic
456C22#341611yescharacteristic
457C22#341611yescharacteristic
458C4—22no—
459C4—22no—
460C4—22no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—82no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4#242611yescharacteristic
471C4#306211yescharacteristic
472C4#336111yescharacteristic
473C4#339411yescharacteristic
474C2—321no—
475C2—321no—
476C2#2529211yescharacteristic
477C2#3360511yescharacteristic
478C2#3360911yescharacteristic
479C2#3393511yescharacteristic
480C2#3393811yescharacteristic
481C2#3396411yescharacteristic
482C2#3396811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#493C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2183C2semidirect
4. #28268C2semidirect
5. #13476C2semidirect
6. #13474C2semidirect
7. #13449C2semidirect
8. #13449C2semidirect
9. #13238C2semidirect
10. #13237C2semidirect
12. #13178C2semidirect
13. #13149C2semidirect
14. #13147C2semidirect
15. #12638C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22072
C2×#21832
#287362
#282682
#134762
#134742
#134492
#132382
#132372
#132062
#131782
#131492
#131472
#126382