Finite Group Databasebeta

← C2×#13302 #495424

Subgroups — C2×#13302 #495424

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#1801C211yesmaximal, characteristic
3C2×C4×#101C211yesmaximal, characteristic
4#13302C218yesmaximal, direct product factor
5C2×#1394C214yesmaximal
6C22×C4⋊Q8C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C2×#423—21no—
9C22×#101C2212yes—
10C22×#101C2212yes—
11C2×C4×M16—21no—
12#2567—22no—
13#2567—22no—
14#2567C414yes—
15#2567C2214yes—
16C2×#395—21no—
17C2×#395—21no—
18C2×#391C2212yes—
19C2×#389—21no—
20C2×#389—21no—
21C2×#384C412yes—
22C2×C4×C4⋊C4C411yescharacteristic
23#1801—22no—
24#1801C2214yes—
25C4×#101—22no—
26C4×#101C2214yes—
27C2×#343—21no—
28C2×#343—21no—
29#1394C22116yes—
30C23×Q8—21no—
31C23×Q8C2311yescharacteristic
32C22×M16—22no—
33C22×M16—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—42no—
36C2×C4⋊Q8—41no—
37C2×C4⋊Q8—41no—
38C2×C4⋊Q8C2×C414yes—
39C2×C4×Q8—41no—
40C2×C4×Q8—41no—
41C2×C4×Q8—42no—
42C2×C4×Q8C2×C412yes—
43C2×C4×Q8D412yes—
44C2×C4×Q8D412yes—
45C2×C4×Q8C2312yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C412yes—
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58#423—24no—
59#423—42no—
60C2×C4⋊C8—41no—
61C2×C4⋊C8—41no—
62C2×#101—22no—
63C2×#101—24no—
64C2×#101—28no—
65C2×#101—22no—
66C2×#101Q818yescomplemented
67C2×#101C2314yes—
68C2×#101C2314yes—
69C4×M16—24no—
70C4×M16—42no—
71C2×#97—41no—
72C2×C4×C8—41no—
73#395—24no—
74#395—24no—
75#395—24no—
76#395—24no—
77#391—22no—
78#391—22no—
79#391C2×C414yes—
80#391C2314yes—
81#389—24no—
82#389—24no—
83#389—24no—
84#389—24no—
85#384—24no—
86#384C2×C418yes—
87C4×C4⋊C4—22no—
88C4×C4⋊C4C2×C414yes—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92#343—24no—
93#343—24no—
94C22×Q8—24no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—42no—
100C22×Q8—84no—
101C22×Q8C22×C411yescharacteristic
102C22×Q8C2×D411yescharacteristic
103C22×Q8C2×D411yescharacteristic
104C22×Q8C2×Q814yes—
105C22×Q8#4114yes—
106C22×Q8C2411yescharacteristic
107C23×C4—22no—
108C23×C4—22no—
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×M16—28no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—44no—
120C2×M16—42no—
121C2×M16—42no—
122C2×M16—44no—
123C2×M16—44no—
124C22×C8—42no—
125C22×C8—42no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4×Q8—42no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—84no—
156C2×C4⋊C4—84no—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22×C414yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—44no—
175C2×C42C22⋊C412yes—
176C2×C42C22⋊C412yes—
177C2×C42C22⋊C412yes—
178C2×C42C22×C412yes—
179C2×C42C22×C414yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C4⋊C8—44no—
183C4⋊C8—44no—
184#101—28no—
185#101—24no—
186#101—28no—
187#101—44no—
188#101C2×Q818yescomplemented
189#97—44no—
190C4×C8—44no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196C24C2×C22⋊C411yescharacteristic
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—44no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—44no—
208C2×Q8—41no—
209C2×Q8—48no—
210C2×Q8—88no—
211C2×Q8—88no—
212C2×Q8—88no—
213C2×Q8C4×Q814yes—
214C2×Q8C22⋊Q814yes—
215C2×Q8#12412yes—
216C2×Q8C4⋊Q812yes—
217C2×Q8C22×Q812yes—
218C2×Q8C2×#4112yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4#9912yes—
248C22×C4#9911yescharacteristic
249C22×C4#9911yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C4×Q814yes—
258C22×C4C22⋊Q814yes—
259C22×C4#12412yes—
260C22×C4C4⋊Q812yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264M16—28no—
265M16—44no—
266M16—48no—
267M16—44no—
268M16—44no—
269M16—84no—
270C2×C8—44no—
271C2×C8—48no—
272C2×C8—44no—
273C2×C8—84no—
274C4⋊C4—216no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—88no—
286C4⋊C4—88no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—44no—
300C42—44no—
301C23—22no—
302C23—22no—
303C23#39112yes—
304C23C2×#9911yescharacteristic
305C23C2×#9911yescharacteristic
306C23#41012yes—
307C23C22×C22⋊C411yescharacteristic
308Q8—44no—
309Q8—44no—
310Q8—42no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314Q8—816no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—28no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—44no—
343C2×C4—416no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4#35614yes—
360C2×C4#39112yes—
361C2×C4#39112yes—
362C2×C4#39112yes—
363C2×C4C2×#9912yes—
364C2×C4C2×#9912yes—
365C2×C4C2×C4×Q812yes—
366C2×C4C2×C22⋊Q812yes—
367C2×C4C2×#12411yescharacteristic
368C2×C4C2×C4⋊Q811yescharacteristic
369C8—48no—
370C8—84no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—24no—
376C22—24no—
377C22#199612yes—
378C22#199612yes—
379C22C2×#35612yes—
380C22#213412yes—
381C22C2×#39111yescharacteristic
382C22C22×#9911yescharacteristic
383C22C2×#41011yescharacteristic
384C4—22no—
385C4—21no—
386C4—22no—
387C4—21no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—48no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—84no—
397C4C2×#35614yes—
398C2—21no—
399C2—22no—
400C2—21no—
401C2#1330214yesdirect product factor
402C2C2×#199611yescharacteristic
403C2C22×#35611yescharacteristic
404C2C2×#213411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13302C2direct (N × H)
66. C2×#101Q8semidirect
188. #101C2×Q8semidirect
401. C2#13302direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#18012
C2×C4×#1012
#133022
C2×#13942