Finite Group Databasebeta

← C2×#17627 #6343470

Subgroups — C2×#17627 #6343470

Order 512.

3148 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#2975C211yesmaximal, characteristic
3C2×#2357C211yesmaximal, characteristic
4#17627C2116yesmaximal, direct product factor
5C2×#1518C212yesmaximal, complemented
6C2×#1511C214yesmaximal, complemented
7C2×#1508C214yesmaximal, complemented
8C2×#1503C212yesmaximal, complemented
9#3501—24no—
10#3501—24no—
11#3501C418yescomplemented
12#3501C2218yes—
13C2×#554—24no—
14C2×#554C414yescomplemented
15C2×D4×Q8C414yescomplemented
16C2×#544—21no—
17C2×#543—22no—
18C2×#533C2212yes—
19C22×C22⋊Q8C2214yes—
20C2×#518—21no—
21C22×C4×D4C2211yescharacteristic
22#2975—24no—
23#2975C2218yes—
24C2×#436—24no—
25C2×#431C2212yes—
26C2×#407C2214yes—
27#2357—24no—
28#2357C2218yes—
29C2×#390—21no—
30C2×#388—22no—
31C2×#382C2211yescharacteristic
32C2×#381C2212yes—
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2212yes—
36#1518C22116yescomplemented
37#1511C22132yescomplemented
38#1508C22132yescomplemented
39#1503C22116yescomplemented
40C2×#328C2214yes—
41C2×#328C2214yes—
42C2×#328C2218yes—
43C23×Q8C2×C414yes—
44C23×D4D411yescharacteristic
45C24×C4C2312yes—
46C22×M16—24no—
47#554—232no—
48#554—216no—
49#554—232no—
50#554C2×C4132yescomplemented
51D4×Q8—216no—
52D4×Q8C2×C4132yescomplemented
53#544—28no—
54#544—28no—
55#543—216no—
56#543—216no—
57#533—24no—
58#533—24no—
59#533C2×C418yes—
60#533C2318yes—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C412yes—
64C2×#125—24no—
65C2×#125—22no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8C2×C418yes—
74C2×C22⋊Q8C2×C4116yes—
75C2×C22⋊Q8C2314yes—
76C2×C22⋊Q8C23116yes—
77#518—28no—
78#518—28no—
79C2×C4×Q8—22no—
80C2×C4×Q8C2×C414yes—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4C2×C414yes—
84C2×C4×D4C2×C414yes—
85C2×C4×D4D414yes—
86C2×C4×D4D414yes—
87C2×C4×D4D414yes—
88C2×C4×D4C2314yes—
89C2×#117—22no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4C2×C414yes—
94C22×C4⋊C4D411yescharacteristic
95C22×C22⋊C4—24no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4D412yes—
99C22×C42—21no—
100C22×C42—22no—
101C22×C42C2311yescharacteristic
102#436—232no—
103#431—24no—
104#431C2318yes—
105C2×C4⋊C8—22no—
106#407—28no—
107#407C23116yes—
108C2×C22⋊C8—24no—
109C2×C22⋊C8C2314yes—
110C2×C4×C8—22no—
111#390—28no—
112#388—216no—
113#382—22no—
114#382C2314yes—
115#381—24no—
116#381C2318yes—
117C4×C4⋊C4—28no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2314yes—
120C4×C22⋊C4—24no—
121C4×C22⋊C4C2318yes—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95C2311yescharacteristic
125C2×#95C2312yes—
126C2×#95C2311yescharacteristic
127#328C23116yes—
128#328C23116yes—
129#328C23132yes—
130C25C2×D412yes—
131C22×Q8—22no—
132C22×Q8—24no—
133C22×Q8—28no—
134C22×Q8—88no—
135C22×Q8C22×C414yes—
136C22×Q8C22×C4116yes—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4C22⋊C414yes—
140C22×D4C22⋊C414yes—
141C22×D4C2×D414yes—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—84no—
148C23×C4C22×C414yes—
149C23×C4C22×C414yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D414yes—
153C23×C4C2×D414yes—
154C23×C4C2×D414yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C23×C4C2414yes—
158C2×M16—216no—
159C2×M16—88no—
160C22×C8—24no—
161C4⋊Q8—232no—
162C4⋊Q8—24no—
163C4⋊Q8—24no—
164C4⋊Q8C22×C418yes—
165C4⋊Q8C22×C418yes—
166#125—232no—
167#125—28no—
168#125—28no—
169C22⋊Q8—216no—
170C22⋊Q8—216no—
171C22⋊Q8—216no—
172C22⋊Q8—28no—
173C22⋊Q8—24no—
174C22⋊Q8—24no—
175C22⋊Q8—216no—
176C22⋊Q8—216no—
177C22⋊Q8—28no—
178C22⋊Q8—264no—
179C22⋊Q8—264no—
180C22⋊Q8C22×C418yes—
181C22⋊Q8C22×C4132yes—
182C22⋊Q8C2418yes—
183C4×Q8—28no—
184C4×Q8—28no—
185C4×Q8—28no—
186C4×Q8C22×C4116yes—
187C4×D4—232no—
188C4×D4—232no—
189C4×D4C22⋊C4116yes—
190C4×D4C22⋊C4116yes—
191C4×D4C22×C4116yes—
192C4×D4C2×D4116yes—
193#117—28no—
194#117—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—216no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—216no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C22⋊C414yes—
210C2×C4⋊C4C22⋊C414yes—
211C2×C4⋊C4C22×C414yes—
212C2×C4⋊C4C22×C4116yes—
213C2×C4⋊C4C2×D414yes—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—216no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—216no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—84no—
222C2×C22⋊C4C22⋊C418yes—
223C2×C22⋊C4C22⋊C418yes—
224C2×C22⋊C4C2×D418yes—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—28no—
228C2×C42—44no—
229C2×C42—84no—
230C2×C42C22×C412yes—
231C2×C42C22×C412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2412yes—
236C4⋊C8—28no—
237C22⋊C8—216no—
238C22⋊C8—24no—
239C22⋊C8C2418yes—
240C4×C8—28no—
241#95—28no—
242#95—24no—
243#95—21no—
244#95—22no—
245#95—21no—
246#95C2412yes—
247#95C2414yes—
248#95C2412yes—
249C24—24no—
250C24—24no—
251C24—44no—
252C24—44no—
253C24C2×C22⋊C414yes—
254C24C2×C22⋊C414yes—
255C24C22×D411yescharacteristic
256C24C22×D414yes—
257C2×Q8—24no—
258C2×Q8—28no—
259C2×Q8—216no—
260C2×Q8—48no—
261C2×Q8—84no—
262C2×Q8—88no—
263C2×Q8—816no—
264C2×Q8—816no—
265C2×Q8C23×C418yes—
266C2×D4—28no—
267C2×D4—28no—
268C2×D4—216no—
269C2×D4—216no—
270C2×D4C2×C22⋊C4116yes—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—216no—
276C22×C4—216no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—48no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—88no—
297C22×C4—88no—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C414yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C414yes—
302C22×C4C2×C22⋊C418yes—
303C22×C4C2×C22⋊C418yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4#13714yes—
307C22×C4#13714yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C23×C411yescharacteristic
310C22×C4C23×C418yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D414yes—
314C22×C4C22×D412yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D418yes—
317C22×C4C22×D412yes—
318C22×C4C2511yescharacteristic
319M16—816no—
320C2×C8—28no—
321C2×C8—48no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—232no—
325C4⋊C4—216no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—28no—
329C4⋊C4—216no—
330C4⋊C4—28no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C4⋊C4C2×C22⋊C4116yes—
334C4⋊C4C23×C418yes—
335C22⋊C4—28no—
336C22⋊C4—232no—
337C22⋊C4—28no—
338C22⋊C4—416no—
339C22⋊C4—88no—
340C22⋊C4C2×C22⋊C4132yes—
341C42—216no—
342C42—48no—
343C42—88no—
344C42C2×C22⋊C414yes—
345C42C2×C22⋊C414yes—
346C42C23×C414yes—
347C42C22×D414yes—
348C23—22no—
349C23—22no—
350C23—28no—
351C23—28no—
352C23—28no—
353C23—28no—
354C23—24no—
355C23—24no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—88no—
361C23—88no—
362C23C22×C22⋊C418yes—
363C23C22×C22⋊C411yescharacteristic
364C23C22×C22⋊C411yescharacteristic
365C23C2×#13712yes—
366C23C2×#13712yes—
367C23C23×D411yescharacteristic
368Q8—48no—
369Q8—48no—
370Q8—88no—
371Q8—816no—
372D4—232no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—28no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—28no—
379C2×C4—28no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—48no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—816no—
404C2×C4—816no—
405C2×C4—84no—
406C2×C4—88no—
407C2×C4#41918yes—
408C2×C4#41918yes—
409C2×C4C22×C22⋊C411yescharacteristic
410C2×C4C22×C22⋊C411yescharacteristic
411C2×C4C22×C22⋊C414yes—
412C2×C4C22×C22⋊C418yes—
413C2×C4C22×C22⋊C414yes—
414C2×C4C2×#13714yes—
415C2×C4C2×#13714yes—
416C2×C4C24×C411yescharacteristic
417C2×C4C23×D411yescharacteristic
418C8—48no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—24no—
424C22—24no—
425C22—28no—
426C22—28no—
427C22—28no—
428C22—416no—
429C22—84no—
430C22—84no—
431C22—88no—
432C22#288312yes—
433C22C2×#41912yes—
434C22C2×#41912yes—
435C22#290112yes—
436C22C23×C22⋊C411yescharacteristic
437C22C22×#13711yescharacteristic
438C22C22×#13711yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—28no—
444C4—24no—
445C4—44no—
446C4—48no—
447C4—48no—
448C4—82no—
449C4—84no—
450C4C2×#41918yes—
451C2—21no—
452C2—21no—
453C2—22no—
454C2—48no—
455C2#1762714yesdirect product factor
456C2C2×#288311yescharacteristic
457C2C22×#41911yescharacteristic
458C2C2×#290111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17627C2direct (N × H)
5. C2×#1518C2semidirect
6. C2×#1511C2semidirect
7. C2×#1508C2semidirect
8. C2×#1503C2semidirect
11. #3501C4semidirect
14. C2×#554C4semidirect
15. C2×D4×Q8C4semidirect
36. #1518C22semidirect
37. #1511C22semidirect
38. #1508C22semidirect
39. #1503C22semidirect
50. #554C2×C4semidirect
52. D4×Q8C2×C4semidirect
455. C2#17627direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#29752
C2×#23572
#176272
C2×#15182
C2×#15112
C2×#15082
C2×#15032