Finite Group Databasebeta

← #498310

Subgroups — #498310

Order 512.

702 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2210C211yesmaximal, characteristic
2C2×#2177C211yesmaximal, characteristic
3C2×#2175C211yesmaximal, characteristic, complemented
4#23186C211yesmaximal, characteristic, complemented
5#22682C211yesmaximal, characteristic, complemented
6#13204C211yesmaximal, characteristic, complemented
7#13178C211yesmaximal, characteristic, complemented
8#13150C211yesmaximal, characteristic, complemented
9#13149C211yesmaximal, characteristic, complemented
10#13014C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic
12#12938C211yesmaximal, characteristic, complemented
13#12935C211yesmaximal, characteristic, complemented
14#12662C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24#2367—21no—
25#2349—21no—
26#2210C418yescomplemented
27#2177—24no—
28#2175—24no—
29C2×#367C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#346C2211yescharacteristic
41C2×#346C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1983—21no—
44#1970C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1880—21no—
49#1877—21no—
50#1864—21no—
51#1864—21no—
52#1846C2211yescharacteristic
53#1804—21no—
54C4×#107C2211yescharacteristic
55C8×C4⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×SD16—21no—
64C22×D8C2311yescharacteristic
65C22×D8C2311yescharacteristic
66C2×#124—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×SD32—22no—
73C2×SD32—22no—
74C2×SD32C2×C414yescomplemented
75C2×D16—22no—
76C2×D16—22no—
77C2×D16C2×C414yes—
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#495—22no—
83#495C2×C414yescomplemented
84C4×Q16—22no—
85C4×Q16—41no—
86C4×D8—22no—
87C4×D8—22no—
88C4×D8—22no—
89C4×D8—22no—
90C4×D8—41no—
91C4×D8—41no—
92C4×D8—41no—
93C4×D8C2×C414yescomplemented
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#106—21no—
100C2×C4⋊C8D411yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#388—41no—
116#388—41no—
117#385—41no—
118#385—41no—
119#385—41no—
120#382—21no—
121#380—41no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—41no—
125C2×#95—41no—
126#367—22no—
127#366—22no—
128C4⋊C16C2×C414yescomplemented
129#358—22no—
130#358—41no—
131#358C2×C414yescomplemented
132#357—22no—
133#357—22no—
134#357—41no—
135#357C2×C414yescomplemented
136#357D414yes—
137#346—22no—
138#346—22no—
139C4×C16—22no—
140#340—21no—
141#340—21no—
142#340—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145C22×Q8—21no—
146C22×D4—21no—
147C22×D4—21no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4—41no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16C22×C412yes—
158C2×SD16—24no—
159C2×SD16—81no—
160C2×SD16—81no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8C22×C412yes—
170C2×D8C22×C412yes—
171C2×D8C2×D412yes—
172C2×D8#4112yes—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2411yescharacteristic
177#124—24no—
178#124—42no—
179C4×Q8—42no—
180C4×Q8—81no—
181C4×D4—24no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—81no—
187C4×D4—81no—
188C4×D4—81no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4D811yescharacteristic
198C2×C4⋊C4D811yescharacteristic
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212SD32—28no—
213D16—28no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16C22×C412yes—
224C2×C16C22×C412yes—
225C2×C16C2×D412yes—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107C22×C412yes—
235#107C2×D412yes—
236#106—81no—
237C4⋊C8—22no—
238#103—42no—
239#103—81no—
240#102—24no—
241#102—42no—
242#102—42no—
243#102—42no—
244#102—81no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8C22×C412yes—
250C4×C8#4112yes—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95D811yescharacteristic
259#95D811yescharacteristic
260C24—41no—
261C24—41no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—44no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—161no—
294C22×C4—161no—
295C22×C4—161no—
296C22×C4—161no—
297C22×C4—161no—
298C22×C4C2×D811yescharacteristic
299C22×C4C2×D811yescharacteristic
300C22×C4C2×D811yescharacteristic
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D811yescharacteristic
304C22×C4C22×D411yescharacteristic
305Q16—22no—
306Q16—22no—
307Q16—41no—
308Q16—41no—
309SD16—84no—
310D8—22no—
311D8—22no—
312D8—22no—
313D8—22no—
314D8—22no—
315D8—41no—
316D8—41no—
317D8C4×D414yes—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8C4×D412yes—
327C2×C8C23×C411yescharacteristic
328C2×C8C22×D411yescharacteristic
329C2×C8C2×#4111yescharacteristic
330C2×C8C2×#4111yescharacteristic
331C2×C8H(5,𝔽2)11yescharacteristic
332C2×C82−1+411yescharacteristic
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—82no—
346C4⋊C4—161no—
347C4⋊C4—161no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—161no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—82no—
361C42—161no—
362C42C4×D412yes—
363C42#13512yes—
364C16—22no—
365C16—22no—
366C16—22no—
367C16—41no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23C22×D811yescharacteristic
383Q8—42no—
384Q8—42no—
385Q8—81no—
386Q8—81no—
387D4—24no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—81no—
394D4—81no—
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—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—164no—
430C2×C4—164no—
431C2×C4—164no—
432C2×C4—164no—
433C2×C4C4×D812yes—
434C2×C4C4×D812yes—
435C2×C4C2×C4×D411yescharacteristic
436C2×C4D4211yescharacteristic
437C2×C4#54811yescharacteristic
438C2×C4C2×#13511yescharacteristic
439C2×C4#57611yescharacteristic
440C2×C4#57711yescharacteristic
441C8—22no—
442C8—41no—
443C8—41no—
444C8C2×C4×D411yescharacteristic
445C8#52011yescharacteristic
446C8#54711yescharacteristic
447C8#55311yescharacteristic
448C22—44no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22C2×C4×D811yescharacteristic
463C22D4×D811yescharacteristic
464C22#335811yescharacteristic
465C22#341411yescharacteristic
466C22#341411yescharacteristic
467C22#341511yescharacteristic
468C22#341611yescharacteristic
469C4—22no—
470C4—22no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4#236111yescharacteristic
482C4#294611yescharacteristic
483C4#328411yescharacteristic
484C4#336111yescharacteristic
485C2—84no—
486C2—162no—
487C2#2299511yescharacteristic
488C2#3345111yescharacteristic
489C2#3345411yescharacteristic
490C2#3383911yescharacteristic
491C2#3384111yescharacteristic
492C2#3393711yescharacteristic
493C2#3394311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2175C2semidirect
4. #23186C2semidirect
5. #22682C2semidirect
6. #13204C2semidirect
7. #13178C2semidirect
8. #13150C2semidirect
9. #13149C2semidirect
10. #13014C2semidirect
12. #12938C2semidirect
13. #12935C2semidirect
14. #12662C2semidirect
15. C4×#357C2semidirect
26. #2210C4semidirect
74. C2×SD32C2×C4semidirect
83. #495C2×C4semidirect
93. C4×D8C2×C4semidirect
128. C4⋊C16C2×C4semidirect
131. #358C2×C4semidirect
135. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22102
C2×#21772
C2×#21752
#231862
#226822
#132042
#131782
#131502
#131492
#130142
#130122
#129382
#129352
#126622
C4×#3572