Finite Group Databasebeta

← #117598

Subgroups — #117598

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21940C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4#13924C211yesmaximal, characteristic
5#10854C211yesmaximal, characteristic
6#8023C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C2×#380C2211yescharacteristic
12C2×#380C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1852—22no—
16#1852—22no—
17#1824—22no—
18#1824—22no—
19#1824—22no—
20#1824C414yescomplemented
21#1808—22no—
22C2×#342—22no—
23C2×#340—22no—
24C2×#340—22no—
25#1445—22no—
26C4⋊#95—22no—
27C22⋊#95—22no—
28#1294—22no—
29#1282—22no—
30C24×C4—21no—
31C24×C4C2311yescharacteristic
32C23×C8—21no—
33C23×C8D412yes—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4D412yes—
50C22×C4⋊C4Q812yes—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4D412yes—
55C22×C42—21no—
56C2×#107—24no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8C2×C412yes—
67C2×C22⋊C8D412yes—
68C2×C22⋊C8D412yes—
69#380—21no—
70#380—21no—
71#380—21no—
72#380—21no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380—21no—
79#380—21no—
80#380C2×C412yes—
81C4×C22⋊C4—21no—
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—28no—
90#342—44no—
91#340—24no—
92#340—28no—
93#340—28no—
94C25C2×D411yescharacteristic
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—44no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—84no—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×Q811yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—44no—
126C22×C8—44no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8C22⋊C414yes—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4C4⋊C414yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4Q1614yes—
185C2×C42—44no—
186C2×C42—41no—
187C2×C42—41no—
188#107—84no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#95—22no—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—88no—
206C24—42no—
207C24—42no—
208C24C4×D411yescharacteristic
209C24#12011yescharacteristic
210C24#12111yescharacteristic
211C24#12111yescharacteristic
212C24C22⋊Q811yescharacteristic
213C24#12311yescharacteristic
214C24#12711yescharacteristic
215C24C2×Q1612yes—
216C24#13512yes—
217C24AGL(1,ℤ/8)12yes—
218C24#13712yes—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—84no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4—84no—
258C22×C4—84no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C4⋊C411yescharacteristic
267C22×C4#11711yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4C4×D411yescharacteristic
270C22×C4C4×D411yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4C22⋊Q811yescharacteristic
280C22×C4C22⋊Q811yescharacteristic
281C22×C4C22⋊Q811yescharacteristic
282C22×C4C22⋊Q811yescharacteristic
283C22×C4C22⋊Q811yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4#12311yescharacteristic
286C22×C4#12311yescharacteristic
287C22×C4#12311yescharacteristic
288C22×C4#12411yescharacteristic
289C22×C4#12411yescharacteristic
290C22×C4#12411yescharacteristic
291C22×C4#12611yescharacteristic
292C22×C4#12611yescharacteristic
293C2×C8—24no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—44no—
297C2×C8—42no—
298C2×C8—82no—
299C2×C8—82no—
300C2×C8—82no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—81no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C22⋊C4—81no—
327C42—82no—
328C42—82no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23#38511yescharacteristic
348C23#38811yescharacteristic
349C23#39011yescharacteristic
350C23#39211yescharacteristic
351C23#39411yescharacteristic
352C23#39611yescharacteristic
353C23#39911yescharacteristic
354C23C4×Q1612yes—
355C23#44012yes—
356C23#44912yes—
357C23#45112yes—
358C23#46212yes—
359C23#46312yes—
360C23#46712yes—
361C23#46812yes—
362C23#47712yes—
363C23#47912yes—
364C23#48412yes—
365C23#48512yes—
366C23#49411yescharacteristic
367C23#49511yescharacteristic
368C23#49611yescharacteristic
369C23#49711yescharacteristic
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—82no—
411C2×C4—88no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—162no—
421C2×C4—162no—
422C2×C4—164no—
423C2×C4#38011yescharacteristic
424C2×C4#38511yescharacteristic
425C2×C4#38611yescharacteristic
426C2×C4#38811yescharacteristic
427C2×C4#39311yescharacteristic
428C2×C4#39411yescharacteristic
429C2×C4#39611yescharacteristic
430C2×C4#39711yescharacteristic
431C2×C4#45814yes—
432C8—44no—
433C8—82no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—24no—
447C22—24no—
448C22—22no—
449C22—22no—
450C22—24no—
451C22—22no—
452C22—24no—
453C22—22no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22#143911yescharacteristic
463C22#186312yes—
464C22#192812yes—
465C22#196911yescharacteristic
466C22#197311yescharacteristic
467C22#200812yes—
468C22#201211yescharacteristic
469C22#201611yescharacteristic
470C22#202212yes—
471C22#203612yes—
472C22#206812yes—
473C22#209511yescharacteristic
474C22#209711yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—84no—
482C4—84no—
483C4—162no—
484C4—162no—
485C4—162no—
486C4#189012yes—
487C4#203012yes—
488C2—22no—
489C2—21no—
490C2—22no—
491C2—21no—
492C2—21no—
493C2—22no—
494C2—22no—
495C2—21no—
496C2#995012yes—
497C2#995912yes—
498C2#1107011yescharacteristic
499C2#1112411yescharacteristic
500C2#1112911yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1824C4semidirect

Maximal subgroups

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

subgroupindex
#219402
C2×#18242
#139242
#108542
#80232
C2×#12822