Finite Group Databasebeta

← C2×C31×#1608 #21474862

Subgroups — C2×C31×#1608 #21474862

Order 7936.

1136 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#543C211yesmaximal, characteristic
2C2×C31×#483C212yesmaximal, complemented
3C2×C31×#397C211yesmaximal, characteristic
4C31×#1608C218yesmaximal, direct product factor
5C2×C31×#328C212yesmaximal, complemented
6C2×C31×#323C211yesmaximal, characteristic
7C31×#543—24no—
8C31×#543—22no—
9C31×#543C2214yes—
10C2×C31×C22⋊Q8—22no—
11C2×C31×C22⋊Q8—21no—
12C2×C31×C22⋊Q8—22no—
13C2×C31×C22⋊Q8C2212yes—
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C22⋊C4×C31—21no—
16C22×C22⋊C4×C31—21no—
17C22×C22⋊C4×C31C2211yescharacteristic
18C31×#483—24no—
19C31×#483C2218yescomplemented
20C2×C31×#106—22no—
21C2×C31×#103—22no—
22C2×C31×C22⋊C8C2212yes—
23C31×#397—22no—
24C31×#397C2214yes—
25C2×C31×#95—21no—
26C2×C31×#95C2211yescharacteristic
27C31×#328C2218yescomplemented
28C31×#323C2214yes—
29C25×C31D411yescharacteristic
30C22×Q8×C31—21no—
31C22×Q8×C31D412yes—
32C23×C4×C31—21no—
33C23×C4×C31—21no—
34C23×C4×C31—21no—
35C23×C4×C31—21no—
36C23×C4×C31D411yescharacteristic
37C23×C4×C31C2311yescharacteristic
38C22×C8×C31—22no—
39C31×C22⋊Q8—28no—
40C31×C22⋊Q8—28no—
41C31×C22⋊Q8—24no—
42C31×C22⋊Q8—28no—
43C31×C22⋊Q8—24no—
44C31×C22⋊Q8—22no—
45C31×C22⋊Q8—22no—
46C31×C22⋊Q8—28no—
47C31×C22⋊Q8D414yes—
48C31×C22⋊Q8C2314yes—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—22no—
51C2×C4⋊C4×C31—22no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31D412yes—
56C2×C4⋊C4×C31C2312yes—
57C2×C22⋊C4×C31—22no—
58C2×C22⋊C4×C31—24no—
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—22no—
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—44no—
65C2×C22⋊C4×C31D412yes—
66C2×C22⋊C4×C31C2312yes—
67C31×#106—28no—
68C31×#103—28no—
69C31×C22⋊C8—22no—
70C31×C22⋊C8C2314yes—
71C31×#95—21no—
72C31×#95—24no—
73C31×#95C2312yes—
74C24×C31—21no—
75C24×C31—21no—
76C24×C31—22no—
77C24×C31—42no—
78C24×C31—42no—
79C24×C31SD1614yes—
80C24×C31C2×D411yescharacteristic
81C24×C31C2×D412yes—
82C2×Q8×C31—24no—
83C2×Q8×C31—22no—
84C2×Q8×C31—44no—
85C2×Q8×C31—82no—
86C2×Q8×C31C2×D414yes—
87C22×C4×C31—22no—
88C22×C4×C31—21no—
89C22×C4×C31—22no—
90C22×C4×C31—24no—
91C22×C4×C31—24no—
92C22×C4×C31—21no—
93C22×C4×C31—22no—
94C22×C4×C31—22no—
95C22×C4×C31—42no—
96C22×C4×C31—42no—
97C22×C4×C31—42no—
98C22×C4×C31—42no—
99C22×C4×C31—41no—
100C22×C4×C31—41no—
101C22×C4×C31—42no—
102C22×C4×C31—42no—
103C22×C4×C31—42no—
104C22×C4×C31—82no—
105C22×C4×C31—82no—
106C22×C4×C31C2×D412yes—
107C22×C4×C31C2×D412yes—
108C22×C4×C31C2×D411yescharacteristic
109C22×C4×C31C2×D412yes—
110C22×C4×C31C2411yescharacteristic
111C2×C8×C31—24no—
112C2×C8×C31—44no—
113C4⋊C4×C31—24no—
114C4⋊C4×C31—28no—
115C4⋊C4×C31—24no—
116C4⋊C4×C31—22no—
117C4⋊C4×C31—24no—
118C4⋊C4×C31—24no—
119C4⋊C4×C31—42no—
120C4⋊C4×C31C2×D414yes—
121C22⋊C4×C31—28no—
122C22⋊C4×C31—24no—
123C22⋊C4×C31—24no—
124C22⋊C4×C31—24no—
125C22⋊C4×C31—24no—
126C22⋊C4×C31—48no—
127C2×#1608C3111yesmaximal, characteristic, direct product factor, Sylow
128C23×C31—22no—
129C23×C31—22no—
130C23×C31—22no—
131C23×C31—24no—
132C23×C31—22no—
133C23×C31—24no—
134C23×C31—48no—
135C23×C31—42no—
136C23×C31—42no—
137C23×C31—42no—
138C23×C31—42no—
139C23×C31—84no—
140C23×C31—81no—
141C23×C31—81no—
142C23×C31#12011yescharacteristic
143C23×C31#12011yescharacteristic
144C23×C31C2×SD1612yes—
145C23×C31C2×SD1614yes—
146C23×C31#13712yes—
147C23×C31C22×D411yescharacteristic
148Q8×C31—44no—
149Q8×C31—82no—
150Q8×C31—84no—
151C2×C4×C31—24no—
152C2×C4×C31—24no—
153C2×C4×C31—22no—
154C2×C4×C31—24no—
155C2×C4×C31—42no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—42no—
159C2×C4×C31—44no—
160C2×C4×C31—44no—
161C2×C4×C31—42no—
162C2×C4×C31—42no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—81no—
169C2×C4×C31—84no—
170C2×C4×C31—84no—
171C2×C4×C31—84no—
172C2×C4×C31—82no—
173C2×C4×C31—82no—
174C2×C4×C31#12012yes—
175C2×C4×C31C22×D412yes—
176C8×C31—44no—
177C2×#543C2×C3111yescharacteristic
178C2×#483C2×C3112yescomplemented
179C2×#397C2×C3111yescharacteristic
180#1608C2×C3118yesdirect product factor
181C2×#328C2×C3112yescomplemented
182C2×#323C2×C3111yescharacteristic
183C22×C31—24no—
184C22×C31—24no—
185C22×C31—21no—
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—24no—
190C22×C31—44no—
191C22×C31—48no—
192C22×C31—42no—
193C22×C31—42no—
194C22×C31—82no—
195C22×C31—84no—
196C22×C31#44812yes—
197C22×C31#44812yes—
198C22×C31#45512yes—
199C22×C31#45812yes—
200C22×C31C2×#12011yescharacteristic
201C22×C31C22×SD1611yescharacteristic
202C22×C31C2×#13711yescharacteristic
203C4×C31—22no—
204C4×C31—22no—
205C4×C31—42no—
206C4×C31—44no—
207C4×C31—44no—
208C4×C31—82no—
209C4×C31—82no—
210#543—24no—
211#543—22no—
212#543C22×C3114yes—
213C2×C22⋊Q8—22no—
214C2×C22⋊Q8—21no—
215C2×C22⋊Q8—22no—
216C2×C22⋊Q8C22×C3112yes—
217C22×C4⋊C4C22×C3111yescharacteristic
218C22×C22⋊C4—21no—
219C22×C22⋊C4—21no—
220C22×C22⋊C4C22×C3111yescharacteristic
221#483—24no—
222#483C22×C3118yescomplemented
223C2×#106—22no—
224C2×#103—22no—
225C2×C22⋊C8C22×C3112yes—
226#397—22no—
227#397C22×C3114yes—
228C2×#95—21no—
229C2×#95C22×C3111yescharacteristic
230#328C22×C3118yescomplemented
231#323C22×C3114yes—
232C2×C31—21no—
233C2×C31—21no—
234C2×C31—22no—
235C2×C31—44no—
236C2×C31#160814yesdirect product factor
237C2×C31C2×#44811yescharacteristic
238C2×C31C2×#45511yescharacteristic
239C2×C31C2×#45811yescharacteristic
240C25D4×C3111yescharacteristic
241C22×Q8—21no—
242C22×Q8D4×C3112yes—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4D4×C3111yescharacteristic
248C23×C4C23×C3111yescharacteristic
249C22×C8—22no—
250C22⋊Q8—28no—
251C22⋊Q8—28no—
252C22⋊Q8—24no—
253C22⋊Q8—28no—
254C22⋊Q8—24no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8—28no—
258C22⋊Q8D4×C3114yes—
259C22⋊Q8C23×C3114yes—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4D4×C3112yes—
267C2×C4⋊C4C23×C3112yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4D4×C3112yes—
277C2×C22⋊C4C23×C3112yes—
278#106—28no—
279#103—28no—
280C22⋊C8—22no—
281C22⋊C8C23×C3114yes—
282#95—24no—
283#95—21no—
284#95C23×C3112yes—
285C31C2×#160811yescharacteristic, direct product factor, Sylow
286C24—21no—
287C24—21no—
288C24—22no—
289C24—42no—
290C24—42no—
291C24SD16×C3114yes—
292C24C2×D4×C3111yescharacteristic
293C24C2×D4×C3112yes—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—44no—
297C2×Q8—82no—
298C2×Q8C2×D4×C3114yes—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4C2×D4×C3112yes—
319C22×C4C2×D4×C3112yes—
320C22×C4C2×D4×C3111yescharacteristic
321C22×C4C2×D4×C3112yes—
322C22×C4C24×C3111yescharacteristic
323C2×C8—24no—
324C2×C8—44no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4C2×D4×C3114yes—
333C22⋊C4—28no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—48no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—22no—
343C23—24no—
344C23—22no—
345C23—48no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—84no—
351C23—81no—
352C23—81no—
353C23C31×#12011yescharacteristic
354C23C31×#12011yescharacteristic
355C23C2×SD16×C3112yes—
356C23C2×SD16×C3114yes—
357C23C31×#13712yes—
358C23C22×D4×C3111yescharacteristic
359Q8—44no—
360Q8—82no—
361Q8—84no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—81no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4C31×#12012yes—
386C2×C4C22×D4×C3112yes—
387C8—44no—
388C22—24no—
389C22—24no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—24no—
395C22—44no—
396C22—48no—
397C22—42no—
398C22—42no—
399C22—84no—
400C22—82no—
401C22C31×#44812yes—
402C22C31×#44812yes—
403C22C31×#45512yes—
404C22C31×#45812yes—
405C22C2×C31×#12011yescharacteristic
406C22C22×SD16×C3111yescharacteristic
407C22C2×C31×#13711yescharacteristic
408C4—22no—
409C4—22no—
410C4—42no—
411C4—44no—
412C4—44no—
413C4—82no—
414C4—82no—
415C2—21no—
416C2—22no—
417C2—21no—
418C2—44no—
419C2C31×#160814yesdirect product factor
420C2C2×C31×#44811yescharacteristic
421C2C2×C31×#45511yescharacteristic
422C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC25×C31D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#483C2semidirect
4. C31×#1608C2direct (N × H)
5. C2×C31×#328C2semidirect
19. C31×#483C22semidirect
27. C31×#328C22semidirect
127. C2×#1608C31direct (N × H)
178. C2×#483C2×C31semidirect
180. #1608C2×C31direct (N × H)
181. C2×#328C2×C31semidirect
222. #483C22×C31semidirect
230. #328C22×C31semidirect
236. C2×C31#1608direct (N × H)
285. C31C2×#1608direct (N × H)
419. C2C31×#1608direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5432
C2×C31×#4832
C2×C31×#3972
C31×#16082
C2×C31×#3282
C2×C31×#3232
C2×#160831