Finite Group Databasebeta

← #415289

Subgroups — #415289

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1997C211yesmaximal, characteristic
2#21324C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4C2×#1950C211yesmaximal, characteristic
5#21170C211yesmaximal, characteristic
6#21145C211yesmaximal, characteristic
7#11303C212yesmaximal
8#11249C212yesmaximal
9#11242C212yesmaximal
10#11216C212yesmaximal
11C4×#343C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#424C2212yes—
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1997—24no—
23#1993—22no—
24#1990—21no—
25#1990—21no—
26#1987—22no—
27#1987—22no—
28#1950—24no—
29#1949—22no—
30#1948—21no—
31#1947—21no—
32#1940—22no—
33#1939—22no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2211yescharacteristic
37C2×#343C2211yescharacteristic
38C2×#343C2211yescharacteristic
39C2×#343C2211yescharacteristic
40C2×#341C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46#1729C2212yes—
47C23×Q8—21no—
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8Q814yes—
62C2×#117C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C2×#108—24no—
70C2×#108—24no—
71C2×#108—42no—
72#428—24no—
73#428—42no—
74C2×#107—22no—
75C2×#106—22no—
76#424—24no—
77#424—42no—
78#423—22no—
79#423D414yes—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8D412yes—
86C2×C4⋊C8D412yes—
87C4×M16—22no—
88C4×M16—22no—
89C4×M16—22no—
90C4×M16—22no—
91C2×#97—21no—
92C2×#97—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95#384D411yescharacteristic
96#384D411yescharacteristic
97#382D412yes—
98#380—21no—
99#380—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C2×#95—21no—
107#343—24no—
108#343—24no—
109#343—22no—
110#343—22no—
111#343—22no—
112#343Q814yescomplemented
113#341—24no—
114#340—24no—
115#340—24no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#336—24no—
123C22×Q8—24no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—42no—
127C22×Q8—82no—
128C22×Q8—82no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4#4111yescharacteristic
133C23×C4#4111yescharacteristic
134C23×C4C2411yescharacteristic
135C2×M16—22no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—22no—
159C2×M16—21no—
160C2×M16C2×D414yes—
161C2×M16C2×Q812yes—
162C2×M16C2×Q812yes—
163C2×M16#4112yes—
164C2×M16#4112yes—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8C2×D414yes—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8—42no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C4⋊C4C2×Q812yes—
216C2×C4⋊C4#4112yes—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—82no—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230#108—48no—
231#107—44no—
232#106—44no—
233C4⋊C8—24no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C4⋊C8—48no—
238#97—42no—
239#97—42no—
240C4×C8—42no—
241C4×C8—42no—
242#95—22no—
243#95—22no—
244#95—41no—
245#95—41no—
246C24C2×#4111yescharacteristic
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×Q8—88no—
254C2×Q8—88no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4C22⋊Q812yes—
281C22×C4C22⋊Q812yes—
282C22×C4#12612yes—
283C22×C4#12612yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×Q811yescharacteristic
288C22×C4C22×Q811yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C42−1+411yescharacteristic
296C22×C42−1+411yescharacteristic
297C22×C42−1+411yescharacteristic
298M16—22no—
299M16—22no—
300M16—22no—
301M16—22no—
302M16—22no—
303M16—24no—
304M16—24no—
305M16—22no—
306M16—22no—
307M16—22no—
308M16—42no—
309M16—41no—
310M16—41no—
311M16—41no—
312M16—42no—
313M16—41no—
314M16—41no—
315M16—41no—
316M16—41no—
317M16—41no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—41no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C42—42no—
365C42—42no—
366C42—82no—
367C42—82no—
368C42#12112yes—
369C42#12112yes—
370C42#12112yes—
371C42#12112yes—
372C42C22⋊Q812yes—
373C42C22⋊Q812yes—
374C42C22⋊Q812yes—
375C42C22⋊Q812yes—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23C2×#12111yescharacteristic
381C23C2×C22⋊Q811yescharacteristic
382C23C2×#12611yescharacteristic
383C23#52911yescharacteristic
384C23#53311yescharacteristic
385C23#54111yescharacteristic
386C23#54211yescharacteristic
387Q8—88no—
388Q8—88no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—84no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—84no—
420C2×C4#39612yes—
421C2×C4#39612yes—
422C2×C4#39612yes—
423C2×C4#39612yes—
424C2×C4#47112yes—
425C2×C4#47312yes—
426C2×C4C2×#12111yescharacteristic
427C2×C4C2×#12111yescharacteristic
428C2×C4C2×C22⋊Q811yescharacteristic
429C2×C4C2×C22⋊Q811yescharacteristic
430C2×C4#52911yescharacteristic
431C2×C4#52911yescharacteristic
432C2×C4#53311yescharacteristic
433C2×C4#53311yescharacteristic
434C2×C4#54111yescharacteristic
435C2×C4#54111yescharacteristic
436C2×C4#54211yescharacteristic
437C2×C4#54211yescharacteristic
438C2×C4#54611yescharacteristic
439C2×C4#54611yescharacteristic
440C2×C4#54611yescharacteristic
441C2×C4#54711yescharacteristic
442C2×C4#54711yescharacteristic
443C2×C4#54711yescharacteristic
444C2×C4#54811yescharacteristic
445C2×C4#54811yescharacteristic
446C2×C4#54811yescharacteristic
447C2×C4D4×Q811yescharacteristic
448C2×C4D4×Q811yescharacteristic
449C2×C4D4×Q811yescharacteristic
450C2×C4#55711yescharacteristic
451C2×C4#55711yescharacteristic
452C2×C4#55711yescharacteristic
453C2×C4Q8211yescharacteristic
454C8—44no—
455C8—44no—
456C8—42no—
457C8—42no—
458C8—42no—
459C8—42no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22C2×#39611yescharacteristic
473C22#242711yescharacteristic
474C22#242911yescharacteristic
475C22#245111yescharacteristic
476C22#245311yescharacteristic
477C22#264511yescharacteristic
478C22#286311yescharacteristic
479C22C2×#47111yescharacteristic
480C22C2×#47311yescharacteristic
481C22#306811yescharacteristic
482C22#306811yescharacteristic
483C4—22no—
484C4—22no—
485C4—22no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4—84no—
490C4—82no—
491C4#206414yes—
492C4#242711yescharacteristic
493C4#242711yescharacteristic
494C4#242911yescharacteristic
495C4#242911yescharacteristic
496C4#245111yescharacteristic
497C4#245111yescharacteristic
498C4#245311yescharacteristic
499C4#245311yescharacteristic
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2#1518911yescharacteristic
505C2C2×#206411yescharacteristic
506C2#2360211yescharacteristic
507C2#2535911yescharacteristic
508C2#2536011yescharacteristic
509C2#2549011yescharacteristic
510C2#2549111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×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
112. #343Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19972
#213242
C2×#19872
C2×#19502
#211702
#211452
#113032
#112492
#112422
#112162
C4×#3432