Finite Group Databasebeta

← #370281

Subgroups — #370281

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1952C211yesmaximal, characteristic
2C2×#1952C211yesmaximal, characteristic
3#21159C212yesmaximal
4C2×#1942C211yesmaximal, characteristic
5C2×C4×#108C211yesmaximal, characteristic
6#20435C211yesmaximal, characteristic
7#11290C212yesmaximal
8#11216C212yesmaximal
9#10913C212yesmaximal
10#10870C211yesmaximal, characteristic
11#10870C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#428C2212yes—
15C2×#428C2212yes—
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C2×C4×M16C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2296C2211yescharacteristic
24C2×#389C2212yes—
25#1952—24no—
26#1952C418yescomplemented
27#1944—22no—
28#1943—22no—
29#1942—21no—
30#1942—21no—
31#1942—21no—
32#1942—21no—
33#1942—22no—
34#1941—22no—
35C4×#108—24no—
36#1771—21no—
37#1771—21no—
38C8×C4⋊C4—21no—
39C8×C4⋊C4—21no—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#341C2212yes—
45#1736C2212yes—
46#1736C2212yes—
47C2×#340C2212yes—
48#1729C2211yescharacteristic
49#1728C2211yescharacteristic
50C22×M16C2311yescharacteristic
51C22×M16C2312yes—
52C22×M16C2311yescharacteristic
53C22×M16C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C42C2311yescharacteristic
62C2×#108—22no—
63C2×#108—22no—
64C2×#108C2×C414yes—
65C2×#108D414yes—
66C2×#108Q814yes—
67C2×#108Q814yes—
68#428—24no—
69#428—24no—
70#428—42no—
71#428C2×C418yes—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107C2×C414yes—
82C2×#106—22no—
83C2×#106—22no—
84#424—22no—
85#424—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88#405—22no—
89#405—22no—
90#405—22no—
91#405C2×C414yes—
92C4×M16—24no—
93C4×M16—42no—
94C2×#97—21no—
95C2×#97—21no—
96C2×#97—22no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8—22no—
100C2×C4×C8—21no—
101C2×C4×C8—22no—
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104C2×C4×C8D411yescharacteristic
105C2×C4×C8D411yescharacteristic
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389D411yescharacteristic
111#389D411yescharacteristic
112#389D411yescharacteristic
113#389D411yescharacteristic
114#385—21no—
115#385—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C2×#95—22no—
119#343—24no—
120#343—24no—
121#343—42no—
122#343C2×C418yes—
123#341—24no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#336—21no—
131#336—21no—
132#336—21no—
133#336—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4#4111yescharacteristic
137C23×C4#4111yescharacteristic
138C23×C4C2411yescharacteristic
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—21no—
147C2×M16—22no—
148C2×M16—21no—
149C2×M16—22no—
150C2×M16—21no—
151C2×M16—22no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16C22×C412yes—
158C2×M16C22×C414yes—
159C2×M16C2×D414yes—
160C2×M16C2×Q814yes—
161C2×M16C2×Q814yes—
162C2×M16#4112yes—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8C22×C412yes—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D412yes—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×Q812yes—
180C22×C8C2×Q812yes—
181C22×C8#4112yes—
182C22×C8#4112yes—
183#117—21no—
184#117—21no—
185#117—21no—
186#117—21no—
187#117—21no—
188#117—21no—
189#117C22×C412yes—
190#117#4112yes—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—84no—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226#108—28no—
227#108C4⋊C4116yescomplemented
228#107—24no—
229#107—24no—
230#107—28no—
231#107—44no—
232#107—42no—
233#107—42no—
234#106—28no—
235#106—44no—
236C4⋊C8—24no—
237C4⋊C8—24no—
238#97—24no—
239#97—44no—
240#97—42no—
241C4×C8—24no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8—22no—
246C4×C8—44no—
247C4×C8—42no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—42no—
253#95—42no—
254C24C2×#4111yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×Q811yescharacteristic
286C22×C4C22×Q811yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4H(5,𝔽2)11yescharacteristic
292C22×C42−1+411yescharacteristic
293C22×C42−1+411yescharacteristic
294C22×C42−1+411yescharacteristic
295M16—24no—
296M16—24no—
297M16—24no—
298M16—22no—
299M16—22no—
300M16—24no—
301M16—22no—
302M16—22no—
303M16—24no—
304M16—41no—
305M16—41no—
306M16—42no—
307M16—41no—
308M16—42no—
309M16—41no—
310M16C2×C4⋊C418yes—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—24no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8C2×C4⋊C414yes—
334C2×C8C4×D412yes—
335C2×C8C4×D412yes—
336C2×C8C4×Q814yes—
337C2×C8#12112yes—
338C2×C8#12112yes—
339C2×C8C22⋊Q814yes—
340C2×C8#12514yes—
341C2×C8C4⋊Q812yes—
342C2×C8C4⋊Q812yes—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4C4×D412yes—
364C4⋊C4C4×D412yes—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C42—22no—
370C42—21no—
371C42—21no—
372C42—22no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—41no—
380C42—41no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23C4×#4111yescharacteristic
386C23C2×#12111yescharacteristic
387C23#52911yescharacteristic
388C23#54811yescharacteristic
389C23#55311yescharacteristic
390C23#55711yescharacteristic
391C23#55711yescharacteristic
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—22no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
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—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—84no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—162no—
427C2×C4—162no—
428C2×C4#47212yes—
429C2×C4#47312yes—
430C2×C4C22×C4⋊C411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×C4×Q811yescharacteristic
434C2×C4#51911yescharacteristic
435C2×C4#52011yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4#52511yescharacteristic
439C2×C4C2×#12511yescharacteristic
440C2×C4C2×C4⋊Q811yescharacteristic
441C2×C4#53211yescharacteristic
442C2×C4#53611yescharacteristic
443C2×C4#53711yescharacteristic
444C2×C4#54111yescharacteristic
445C2×C4#54211yescharacteristic
446C2×C4D4211yescharacteristic
447C2×C4#54711yescharacteristic
448C2×C4#54711yescharacteristic
449C2×C4#54811yescharacteristic
450C2×C4D4×Q811yescharacteristic
451C2×C4D4×Q811yescharacteristic
452C2×C4#55211yescharacteristic
453C2×C4#55411yescharacteristic
454C2×C4#55411yescharacteristic
455C2×C4#55611yescharacteristic
456C2×C4#55711yescharacteristic
457C2×C4#55711yescharacteristic
458C8—28no—
459C8—42no—
460C8—41no—
461C8—41no—
462C8—42no—
463C8—44no—
464C8#38914yes—
465C8#38914yes—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22C4×#12111yescharacteristic
479C22#235111yescharacteristic
480C22#237011yescharacteristic
481C22#241611yescharacteristic
482C22#245311yescharacteristic
483C22#250911yescharacteristic
484C22#254611yescharacteristic
485C22C2×#47211yescharacteristic
486C22C2×#47311yescharacteristic
487C22#306711yescharacteristic
488C22#306711yescharacteristic
489C4—22no—
490C4—22no—
491C4—42no—
492C4—42no—
493C4—41no—
494C4—41no—
495C4—84no—
496C4—81no—
497C4—81no—
498C4—162no—
499C4C2×#38911yescharacteristic
500C4#232311yescharacteristic
501C4D4×C4⋊C411yescharacteristic
502C4#236511yescharacteristic
503C4#241511yescharacteristic
504C4#245211yescharacteristic
505C4#250611yescharacteristic
506C4#258411yescharacteristic
507C2—21no—
508C2—21no—
509C2—21no—
510C2—21no—
511C2#1404611yescharacteristic
512C2C4×#47211yescharacteristic
513C2C4×#47311yescharacteristic
514C2#2480311yescharacteristic
515C2#2480411yescharacteristic
516C2#2869911yescharacteristic
517C2#2870211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1952C4semidirect
227. #108C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19522
#211592
C2×#19422
C2×C4×#1082
#204352
#112902
#112162
#109132
#108702