Finite Group Databasebeta

← C2×#8554 #188074

Subgroups — C2×#8554 #188074

Order 512.

2534 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
1C23×#125C211yesmaximal, characteristic
2C2×#1821C212yesmaximal
3C2×#1463C212yesmaximal
4C2×#1463C212yesmaximal
5#8554C218yesmaximal, direct product factor
6C22×#125—24no—
7C22×#125—24no—
8C22×#125—24no—
9C22×#125—24no—
10C22×#125—24no—
11C22×#125—24no—
12C22×#125C414yes—
13C22×#125C2214yes—
14C23×C4⋊C4C411yescharacteristic
15C23×C4⋊C4C411yescharacteristic
16C23×C4⋊C4C411yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C411yescharacteristic
21C2×#407C2214yes—
22#1821—24no—
23#1821C2218yes—
24C2×#343—24no—
25#1463—24no—
26#1463—24no—
27#1463C2218yes—
28#1463C2218yes—
29C2×#324—24no—
30C2×#324—24no—
31C24×C4C2×C411yescharacteristic
32C24×C4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2311yescharacteristic
38C22×M16—24no—
39C2×#125—232no—
40C2×#125—232no—
41C2×#125—232no—
42C2×#125—232no—
43C2×#125—216no—
44C2×#125—216no—
45C2×#125—216no—
46C2×#125—216no—
47C2×#125—216no—
48C2×#125—216no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4C2×C414yes—
80C22×C4⋊C4C2×C414yes—
81C22×C4⋊C4C2×C414yes—
82C22×C4⋊C4C2×C412yes—
83C22×C4⋊C4C2×C412yes—
84C22×C4⋊C4C2×C412yes—
85C22×C4⋊C4D414yes—
86C22×C4⋊C4D414yes—
87C22×C4⋊C4D414yes—
88C22×C4⋊C4D414yes—
89C22×C4⋊C4D414yes—
90C22×C4⋊C4D414yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C42—22no—
95C22×C42—22no—
96C22×C42—22no—
97C22×C42—22no—
98C22×C42—22no—
99C22×C42—22no—
100C22×C42C2×C414yes—
101#407—24no—
102#407—24no—
103#407—24no—
104#407C2318yes—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107#343—216no—
108#324—216no—
109#324—216no—
110C25C22×C411yescharacteristic
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—28no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C412yes—
151C23×C4C22⋊C412yes—
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2411yescharacteristic
168C2×M16—28no—
169C2×M16—44no—
170C2×M16—416no—
171C22×C8—44no—
172#125—2128no—
173#125—2128no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4C22⋊C418yes—
220C2×C4⋊C4C22⋊C418yes—
221C2×C4⋊C4C22⋊C418yes—
222C2×C4⋊C4C2×D418yes—
223C2×C4⋊C4C2×D418yes—
224C2×C4⋊C4C2×D418yes—
225C2×C42—28no—
226C2×C42—28no—
227C2×C42—28no—
228C2×C42—28no—
229C2×C42—28no—
230C2×C42—28no—
231C2×C42—28no—
232C22⋊C8—28no—
233C22⋊C8—28no—
234C22⋊C8—44no—
235C22⋊C8—44no—
236C24—24no—
237C24—24no—
238C24—24no—
239C24C2×C22⋊C411yescharacteristic
240C24C2×C22⋊C411yescharacteristic
241C24C2×C22⋊C411yescharacteristic
242C24C2×C22⋊C411yescharacteristic
243C24C2×C22⋊C411yescharacteristic
244C24C2×C22⋊C411yescharacteristic
245C24C23×C411yescharacteristic
246C22×C4—28no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—28no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4#10118yes—
333C22×C4C2×C22⋊C414yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C414yes—
336C22×C4C2×C22⋊C414yes—
337C22×C4C2×C22⋊C414yes—
338C22×C4C2×C22⋊C414yes—
339C22×C4C2×C22⋊C414yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4#12014yes—
343C22×C4#12014yes—
344C22×C4#12014yes—
345C22×C4#12014yes—
346C22×C4C22×D412yes—
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349M16—416no—
350M16—88no—
351C2×C8—48no—
352C2×C8—88no—
353C4⋊C4—232no—
354C4⋊C4—232no—
355C4⋊C4—232no—
356C4⋊C4—216no—
357C4⋊C4—216no—
358C4⋊C4—216no—
359C4⋊C4—216no—
360C4⋊C4—216no—
361C4⋊C4—216no—
362C42—232no—
363C23—232no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23#37911yescharacteristic
383C23#37911yescharacteristic
384C23#37911yescharacteristic
385C23#37911yescharacteristic
386C23#41014yes—
387C23C2×#10114yes—
388C23#41314yes—
389C23C22×C22⋊C411yescharacteristic
390C23C22×C22⋊C411yescharacteristic
391C23C22×C22⋊C411yescharacteristic
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4#37914yes—
451C2×C4C2×#10118yes—
452C2×C4C2×#12014yes—
453C8—88no—
454C22—28no—
455C22—28no—
456C22—28no—
457C22—28no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—28no—
463C22—28no—
464C22—28no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22#178212yes—
474C22#178212yes—
475C22#178412yes—
476C22#178412yes—
477C22#178512yes—
478C22#178512yes—
479C22C2×#37911yescharacteristic
480C22C2×#41012yes—
481C22C22×#10112yes—
482C22C2×#41312yes—
483C4—24no—
484C4—24no—
485C4—28no—
486C4—44no—
487C4—44no—
488C4—44no—
489C4—44no—
490C4—44no—
491C2—22no—
492C2—22no—
493C2—22no—
494C2—22no—
495C2—22no—
496C2—22no—
497C2#855414yesdirect product factor
498C2C2×#178211yescharacteristic
499C2C2×#178411yescharacteristic
500C2C2×#178511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8554C2direct (N × H)
497. C2#8554direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1252
C2×#18212
C2×#14632
#85542