Finite Group Databasebeta

← C2×#13201 #495111

Subgroups — C2×#13201 #495111

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2930C211yesmaximal, characteristic
2C2×#2153C211yesmaximal, characteristic, complemented
3C2×#2117C212yesmaximal, complemented
4C22×#349C212yesmaximal, complemented
5C2×#2110C211yesmaximal, characteristic, complemented
6#13201C218yesmaximal, direct product factor
7C22×#131C2211yescharacteristic
8C23×M16C2211yescharacteristic
9C2×#525C411yescharacteristic
10C22×M32C2212yes—
11C22×M32C2212yes—
12#2930—22no—
13#2930—22no—
14#2930C414yes—
15#2930C2214yes—
16C2×#436—22no—
17C2×#435—22no—
18C2×#424C411yescharacteristic
19C2×#408C411yescharacteristic
20C2×#407C411yescharacteristic
21C2×#405C2211yescharacteristic
22#2153—22no—
23#2153C2214yescomplemented
24C2×#364—22no—
25#2117—24no—
26#2117C2218yescomplemented
27C2×#350—24no—
28C2×#349—24no—
29C2×#349—24no—
30C2×#349C2218yescomplemented
31#2110—22no—
32#2110C2214yescomplemented
33C2×#347—22no—
34C22×#41C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C2×#131—21no—
37C2×#131—21no—
38C2×#131—22no—
39C2×#131—22no—
40C2×#131—22no—
41C2×#131—22no—
42C2×#131C2×C412yes—
43C2×#131C2312yes—
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—21no—
47C22×M16—21no—
48C22×M16—21no—
49C22×M16—21no—
50C22×M16—21no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16C2×C411yescharacteristic
54C22×M16C2×C411yescharacteristic
55C22×M16C2×C412yes—
56C22×M16C2312yes—
57C23×C8—21no—
58C23×C8C2×C411yescharacteristic
59C23×C8C2311yescharacteristic
60#525—22no—
61#525—22no—
62#525—22no—
63#525C2×C414yes—
64C2×#123—21no—
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×#120—21no—
68C2×C4×D4—22no—
69C2×#117C2×C411yescharacteristic
70C2×M32—22no—
71C2×M32—22no—
72C2×M32—22no—
73C2×M32—22no—
74C2×M32—22no—
75C2×M32—22no—
76C2×M32—22no—
77C2×M32—22no—
78C2×M32D414yescomplemented
79C2×M32D414yescomplemented
80C2×M32C2314yes—
81C2×M32C2314yes—
82C22×C16—22no—
83C22×C16—22no—
84#436—28no—
85#436—28no—
86#435—28no—
87#435—28no—
88#424—22no—
89#424C2×C414yes—
90C2×C4⋊C8—22no—
91#408—22no—
92#408C2×C414yes—
93#407—22no—
94#407C2×C414yes—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97#405C2×C412yes—
98#405D412yes—
99#405D412yes—
100#405C2312yes—
101C2×#97D412yes—
102C2×C4×C8D412yes—
103#364—28no—
104#350—216no—
105#349—216no—
106#347—28no—
107C25M1611yescharacteristic
108C2×#41—21no—
109C2×#41—21no—
110C2×#41—22no—
111C2×#41—22no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41M1612yes—
115C2×#41C22×C412yes—
116C22×Q8M1611yescharacteristic
117C22×D4—21no—
118C22×D4—22no—
119C22×D4M1611yescharacteristic
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4M1611yescharacteristic
130C23×C4M1612yes—
131C23×C4C22×C412yes—
132C23×C4C22×C411yescharacteristic
133#131—24no—
134#131—24no—
135#131—24no—
136#131—24no—
137#131—42no—
138#131—42no—
139#131—42no—
140#131—42no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—24no—
148C2×M16—24no—
149C2×M16—24no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—24no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—42no—
157C2×M16—42no—
158C2×M16—42no—
159C2×M16—42no—
160C2×M16—41no—
161C2×M16—42no—
162C2×M16—41no—
163C2×M16—41no—
164C2×M16—41no—
165C2×M16—42no—
166C2×M16—41no—
167C2×M16—41no—
168C2×M16—41no—
169C2×M16—41no—
170C2×M16C22⋊C412yes—
171C2×M16C22×C412yes—
172C2×M16C22×C412yes—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—42no—
181C22×C8—42no—
182C22×C8C22⋊C412yes—
183C22×C8C22×C412yes—
184C22×C8C22×C411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C22×C8#4112yes—
192C22×C8#4112yes—
193C22×C8C2411yescharacteristic
194#123—24no—
195#123—24no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198#121—24no—
199#121—24no—
200#120—24no—
201#120—24no—
202C4×D4—28no—
203C4×D4—28no—
204#117—21no—
205#117—21no—
206#117C22⋊C412yes—
207#117C22×C412yes—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C42C22⋊C412yes—
214M32—24no—
215M32—24no—
216M32—24no—
217M32—24no—
218M32—24no—
219M32—42no—
220M32—42no—
221M32C2×D418yescomplemented
222C2×C16—24no—
223C2×C16—24no—
224C2×C16—24no—
225C2×C16—24no—
226C2×C16—42no—
227C2×C16—42no—
228C4⋊C8—28no—
229C22⋊C8—28no—
230C22⋊C8—28no—
231#97C22⋊C414yes—
232#97C2×D414yes—
233C4×C8C22⋊C414yes—
234C4×C8C2×D414yes—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—22no—
239C24—22no—
240C24—82no—
241C24C2×M1612yes—
242C24C2×M1611yescharacteristic
243#41—24no—
244#41—24no—
245#41—24no—
246#41—24no—
247#41—42no—
248#41—42no—
249#41—42no—
250#41—42no—
251C2×Q8—21no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8C2×M1612yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—28no—
258C2×D4—21no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—84no—
264C2×D4C2×M1612yes—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C4×D412yes—
298C22×C4C4×D412yes—
299C22×C4C2×M1612yes—
300C22×C4C2×M1611yescharacteristic
301C22×C4C2×M1611yescharacteristic
302C22×C4C2×M1611yescharacteristic
303C22×C4#13112yes—
304C22×C4#13112yes—
305C22×C4C23×C411yescharacteristic
306M16—24no—
307M16—24no—
308M16—24no—
309M16—42no—
310M16—44no—
311M16—42no—
312M16—42no—
313M16—42no—
314M16—42no—
315M16—44no—
316M16—42no—
317M16—42no—
318M16—42no—
319M16—42no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8C2×C22⋊C414yes—
333C2×C8C2×C22⋊C411yescharacteristic
334C2×C8C2×C22⋊C411yescharacteristic
335C2×C8C4×D412yes—
336C2×C8C4×D412yes—
337C2×C8#12112yes—
338C2×C8#12112yes—
339C2×C8#12112yes—
340C2×C8#12112yes—
341C2×C8#12411yescharacteristic
342C2×C8#12412yes—
343C2×C8#12411yescharacteristic
344C2×C8#12712yes—
345C2×C8#12711yescharacteristic
346C2×C8#12711yescharacteristic
347C2×C8C22×D411yescharacteristic
348C2×C8C22×D412yes—
349C2×C8C22×D411yescharacteristic
350C2×C8C2×#4112yes—
351C4⋊C4—28no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C22⋊C4—28no—
355C22⋊C4—28no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C42—22no—
359C42C2×C22⋊C414yes—
360C16—28no—
361C16—44no—
362C23—22no—
363C23—22no—
364C23—24no—
365C23—42no—
366C23—42no—
367C23—44no—
368C23—42no—
369C23—41no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—84no—
375C23—82no—
376C23—82no—
377C23—84no—
378C23#40711yescharacteristic
379C23#40711yescharacteristic
380C23#43512yes—
381C23#43512yes—
382C23C22×M1611yescharacteristic
383Q8—24no—
384Q8—42no—
385D4—24no—
386D4—44no—
387D4—44no—
388D4—42no—
389D4—88no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—44no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4#39011yescharacteristic
411C2×C4#39012yes—
412C2×C4#39011yescharacteristic
413C2×C4#40711yescharacteristic
414C2×C4#40711yescharacteristic
415C2×C4#40811yescharacteristic
416C2×C4#40811yescharacteristic
417C2×C4#40812yes—
418C2×C4#43512yes—
419C2×C4#43512yes—
420C2×C4#43512yes—
421C2×C4#43512yes—
422C2×C4#43512yes—
423C2×C4#43512yes—
424C2×C4C22×C22⋊C411yescharacteristic
425C2×C4C2×C4×D412yes—
426C2×C4C22×M1611yescharacteristic
427C2×C4C2×#13112yes—
428C8—24no—
429C8—42no—
430C8—42no—
431C8#39012yes—
432C8#39011yescharacteristic
433C8#39011yescharacteristic
434C8C2×#12112yes—
435C8C2×#12411yescharacteristic
436C8C2×#12711yescharacteristic
437C22—22no—
438C22—24no—
439C22—41no—
440C22—41no—
441C22—42no—
442C22—42no—
443C22—44no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—84no—
448C22—82no—
449C22—82no—
450C22—84no—
451C22—82no—
452C22—82no—
453C22#195511yescharacteristic
454C22#195512yes—
455C22#195511yescharacteristic
456C22C2×#40711yescharacteristic
457C22C2×#43512yes—
458C4—44no—
459C4—42no—
460C4—42no—
461C4#195511yescharacteristic
462C4#195511yescharacteristic
463C4#195512yes—
464C4C2×#39011yescharacteristic
465C4C2×#40811yescharacteristic
466C4C2×#43512yes—
467C2—21no—
468C2—21no—
469C2—22no—
470C2—22no—
471C2—82no—
472C2—82no—
473C2#1320112yesdirect product factor
474C2C2×#195511yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC22×C4C23×C4
Frattini, cosocleC22×C8C24
socleC22#1955
prime radicalC2×#525C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2153C2semidirect
3. C2×#2117C2semidirect
4. C22×#349C2semidirect
5. C2×#2110C2semidirect
6. #13201C2direct (N × H)
23. #2153C22semidirect
26. #2117C22semidirect
30. C2×#349C22semidirect
32. #2110C22semidirect
78. C2×M32D4semidirect
79. C2×M32D4semidirect
221. M32C2×D4semidirect
473. C2#13201direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29302
C2×#21532
C2×#21172
C22×#3492
C2×#21102
#132012