Finite Group Databasebeta

← C2×#11722 #481495

Subgroups — C2×#11722 #481495

Order 512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C22×#343C211yesmaximal, characteristic
3C22×#324C211yesmaximal, characteristic
4#11722C218yesmaximal, direct product factor
5C2×#1354C212yesmaximal, complemented
6C2×#1353C212yesmaximal, complemented
7C23×M16—21no—
8C22×C22⋊Q8C411yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C22×C22⋊C8—21no—
11#2627—22no—
12#2627—22no—
13#2627C414yes—
14#2627C2214yes—
15C2×#400C411yescharacteristic
16C2×#400C2211yescharacteristic
17C2×#398—21no—
18C2×#398—21no—
19C2×#395C411yescharacteristic
20C2×#395C2211yescharacteristic
21C2×#385—21no—
22C2×#381—21no—
23C2×#380C411yescharacteristic
24C2×#343—22no—
25C2×#343—21no—
26C2×#343C2212yes—
27C2×#343C2214yes—
28C2×#324—22no—
29C2×#324—21no—
30C2×#324C2212yes—
31C2×#324C2214yes—
32#1354C2218yescomplemented
33#1353C2218yescomplemented
34C23×Q8C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C22×M16—28no—
39C22×M16—22no—
40C22×M16—24no—
41C22×M16—42no—
42C22×M16—41no—
43C22×M16—44no—
44C23×C8—41no—
45C23×C8—41no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—42no—
48C2×C22⋊Q8—41no—
49C2×C22⋊Q8—41no—
50C2×C22⋊Q8C2×C414yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4—42no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—41no—
73#400—22no—
74#400—21no—
75#400—21no—
76#400C2×C412yes—
77#400C2×C414yes—
78#400C2312yes—
79#398—24no—
80#398—24no—
81#398—24no—
82#398—24no—
83#395—22no—
84#395—21no—
85#395—21no—
86#395C2×C412yes—
87#395C2×C414yes—
88#395C2312yes—
89#385—24no—
90#385—24no—
91#381—24no—
92#381—24no—
93#380—22no—
94#380C2×C414yes—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103#343—22no—
104#343—24no—
105#343C2314yes—
106#324—22no—
107#324—24no—
108#324C2314yes—
109C25C22⋊C411yescharacteristic
110C22×Q8—21no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—82no—
114C22×Q8C22×C412yes—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—24no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—82no—
130C23×C4—82no—
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4SD1614yes—
138C23×C4Q1614yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C2×M16—28no—
146C2×M16—216no—
147C2×M16—24no—
148C2×M16—48no—
149C2×M16—416no—
150C2×M16—44no—
151C2×M16—42no—
152C2×M16—44no—
153C2×M16—48no—
154C22×C8—48no—
155C22×C8—44no—
156C22×C8—42no—
157C22×C8—44no—
158C22×C8—42no—
159C22×C8—84no—
160C22⋊Q8—44no—
161C22⋊Q8—44no—
162C22⋊Q8—44no—
163C22⋊Q8—44no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2411yescharacteristic
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—44no—
210C22⋊C8—24no—
211C22⋊C8—42no—
212C22⋊C8—44no—
213#95—24no—
214#95—22no—
215#95—24no—
216#95—22no—
217#95—24no—
218#95—22no—
219#95—21no—
220#95—21no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95C22×C412yes—
225#95C22×C412yes—
226C24—42no—
227C24—42no—
228C24#9911yescharacteristic
229C24#9911yescharacteristic
230C24#9912yes—
231C24#10314yes—
232C24#10414yes—
233C24C2×C22⋊C411yescharacteristic
234C24C2×C22⋊C412yes—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—84no—
240C2×Q8—84no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—28no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4#10314yes—
307C22×C4#10318yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×SD1612yes—
315C22×C4C2×SD1614yes—
316C22×C4C2×Q1612yes—
317C22×C4C2×Q1614yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321M16—28no—
322M16—416no—
323M16—48no—
324M16—44no—
325C2×C8—44no—
326C2×C8—416no—
327C2×C8—48no—
328C2×C8—44no—
329C2×C8—88no—
330C2×C8—84no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—48no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—48no—
361C23—48no—
362C23—42no—
363C23#32814yes—
364C23#32814yes—
365C23#32818yes—
366C23C2×#9912yes—
367C23C2×#9912yes—
368C23C2×#9911yescharacteristic
369C23C2×#9911yescharacteristic
370C23C2×#10312yes—
371C23C2×#10314yes—
372C23C2×#10412yes—
373C23C2×#10414yes—
374C23C22×C22⋊C411yescharacteristic
375Q8—84no—
376Q8—84no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4C2×#10314yes—
441C2×C4C2×#10312yes—
442C2×C4C22×SD1611yescharacteristic
443C2×C4C22×Q1611yescharacteristic
444C8—48no—
445C8—84no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—22no—
450C22—21no—
451C22—21no—
452C22—22no—
453C22—22no—
454C22—24no—
455C22—28no—
456C22—28no—
457C22—24no—
458C22—48no—
459C22—44no—
460C22—44no—
461C22C2×#32814yes—
462C22C2×#32814yes—
463C22C2×#32812yes—
464C22C2×#32812yes—
465C22#212112yes—
466C22#213312yes—
467C22C22×#9911yescharacteristic
468C22C22×#10311yescharacteristic
469C22C22×#10411yescharacteristic
470C4—21no—
471C4—21no—
472C4—24no—
473C4—22no—
474C4—41no—
475C4—42no—
476C4—41no—
477C4—42no—
478C4—41no—
479C4—41no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—82no—
484C2—21no—
485C2—21no—
486C2—22no—
487C2—24no—
488C2—24no—
489C2#1172214yesdirect product factor
490C2C22×#32811yescharacteristic
491C2C2×#212111yescharacteristic
492C2C2×#213311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11722C2direct (N × H)
5. C2×#1354C2semidirect
6. C2×#1353C2semidirect
32. #1354C22semidirect
33. #1353C22semidirect
489. C2#11722direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26272
C22×#3432
C22×#3242
#117222
C2×#13542
C2×#13532