Finite Group Databasebeta

← C2×#11700 #481457

Subgroups — C2×#11700 #481457

Order 512.

1252 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic
2C2×#1742C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#11700C218yesmaximal, direct product factor
5C2×#1346C214yesmaximal, complemented
6C23×M16—21no—
7C22×C4⋊Q8C411yescharacteristic
8C22×C22⋊Q8—21no—
9C22×#117C2211yescharacteristic
10C2×#431—21no—
11#2621—22no—
12#2621—22no—
13#2621C414yes—
14#2621C2214yes—
15C2×#398—21no—
16C2×#398C2212yes—
17C2×#397—21no—
18C2×#397C412yes—
19C2×#391—21no—
20C2×#382C411yescharacteristic
21#1742—22no—
22#1742C2214yes—
23C2×#343—21no—
24C2×#343C2212yes—
25#1458—22no—
26#1458C2214yes—
27C2×#324C2212yes—
28C2×#323—21no—
29#1346C22116yescomplemented
30C23×Q8—21no—
31C24×C4D411yescharacteristic
32C22×M16—28no—
33C22×M16—22no—
34C22×M16—24no—
35C22×M16—41no—
36C22×M16—42no—
37C22×M16—44no—
38C23×C8—41no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—42no—
41C2×C4⋊Q8—41no—
42C2×C4⋊Q8—41no—
43C2×C4⋊Q8C2×C414yes—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—44no—
47C2×C22⋊Q8—44no—
48C2×#117—41no—
49C2×#117—41no—
50C2×#117—42no—
51C2×#117C2×C412yes—
52C2×#117D412yes—
53C2×#117D412yes—
54C2×#117C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—22no—
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42D411yescharacteristic
66#431—24no—
67#431—42no—
68C2×C4⋊C8—41no—
69C2×C22⋊C8—22no—
70C2×C4×C8—41no—
71#398—24no—
72#398—22no—
73#398—22no—
74#398—24no—
75#398C2×C414yes—
76#398C2314yes—
77#397—24no—
78#397—24no—
79#397—24no—
80#397C2×C418yes—
81#391—24no—
82#391—24no—
83#382—22no—
84#382C2×C414yes—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95C2×C412yes—
89#343—24no—
90#343—22no—
91#343C2314yes—
92#324—22no—
93#324C2314yes—
94#323—24no—
95C25C22⋊C411yescharacteristic
96C22×Q8—24no—
97C22×Q8—42no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—44no—
101C22×Q8—44no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—24no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4D814yes—
113C23×C4SD1614yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C2×M16—28no—
119C2×M16—216no—
120C2×M16—24no—
121C2×M16—42no—
122C2×M16—48no—
123C2×M16—416no—
124C2×M16—44no—
125C2×M16—44no—
126C2×M16—48no—
127C22×C8—42no—
128C22×C8—48no—
129C22×C8—44no—
130C22×C8—42no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C22⋊Q8—416no—
136C22⋊Q8—48no—
137C22⋊Q8—48no—
138#117—42no—
139#117—42no—
140#117—42no—
141#117—42no—
142#117—42no—
143#117—42no—
144#117—42no—
145#117—42no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2411yescharacteristic
169C2×C22⋊C4—28no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—48no—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22×C412yes—
178C2×C22⋊C4C2×D412yes—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22×C412yes—
188C2×C42C2×D412yes—
189C4⋊C8—44no—
190C22⋊C8—24no—
191C22⋊C8—42no—
192C4×C8—44no—
193#95—24no—
194#95—24no—
195#95—28no—
196#95—22no—
197#95—24no—
198#95C22×C414yes—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—42no—
203C24—42no—
204C24#10214yes—
205C24C2×C22⋊C411yescharacteristic
206C24C2×C22⋊C412yes—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—48no—
210C2×Q8—48no—
211C2×Q8—88no—
212C2×Q8—84no—
213C2×Q8—84no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—28no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—28no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4#9912yes—
251C22×C4#9911yescharacteristic
252C22×C4#9911yescharacteristic
253C22×C4#10218yes—
254C22×C4#10214yes—
255C22×C4#10414yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×D812yes—
263C22×C4C2×D814yes—
264C22×C4C2×SD1612yes—
265C22×C4C2×SD1614yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269M16—28no—
270M16—48no—
271M16—416no—
272M16—44no—
273C2×C8—44no—
274C2×C8—48no—
275C2×C8—416no—
276C2×C8—44no—
277C2×C8—84no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—416no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—24no—
308C23—22no—
309C23—24no—
310C23—48no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—84no—
318C23C2×#9911yescharacteristic
319C23C2×#9911yescharacteristic
320C23C2×#10212yes—
321C23C2×#10214yes—
322C23C2×#10412yes—
323C23C22×C22⋊C411yescharacteristic
324Q8—88no—
325Q8—88no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—88no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4#32718yes—
370C2×C4#32714yes—
371C2×C4#32714yes—
372C2×C4C2×#9912yes—
373C2×C4C2×#9912yes—
374C2×C4C2×#10212yes—
375C2×C4C2×#10214yes—
376C2×C4C2×#10414yes—
377C2×C4C22×D811yescharacteristic
378C2×C4C22×SD1611yescharacteristic
379C8—48no—
380C8—84no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—24no—
386C22—24no—
387C22—24no—
388C22—44no—
389C22—42no—
390C22—42no—
391C22—48no—
392C22—82no—
393C22—84no—
394C22C2×#32712yes—
395C22C2×#32712yes—
396C22#212312yes—
397C22#213512yes—
398C22C22×#9911yescharacteristic
399C22C22×#10211yescharacteristic
400C22C22×#10411yescharacteristic
401C4—21no—
402C4—21no—
403C4—22no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—42no—
409C4—42no—
410C4—84no—
411C4—84no—
412C4C2×#32714yes—
413C4C2×#32714yes—
414C2—21no—
415C2—21no—
416C2—22no—
417C2—44no—
418C2#1170014yesdirect product factor
419C2C22×#32711yescharacteristic
420C2C2×#212311yescharacteristic
421C2C2×#213511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11700C2direct (N × H)
5. C2×#1346C2semidirect
29. #1346C22semidirect
418. C2#11700direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26212
C2×#17422
C2×#14582
#117002
C2×#13462