Finite Group Databasebeta

← C31×#18293 #21475052

Subgroups — C31×#18293 #21475052

Order 7936.

1220 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#537C211yesmaximal, characteristic
2C31×#3187C212yesmaximal, complemented
3C31×#2627C211yesmaximal, characteristic, complemented
4C31×#1616C214yesmaximal, complemented
5C31×#1604C214yesmaximal, complemented
6C31×#1519C212yesmaximal, complemented
7C2×C31×#324C211yesmaximal, characteristic, complemented
8C22×#41×C31—21no—
9C31×#537—24no—
10C31×#537—22no—
11C31×#537C2214yes—
12C2×C31×C22⋊Q8—22no—
13C2×C31×C22⋊Q8C2211yescharacteristic
14C2×C31×#121—22no—
15C2×C31×#121—21no—
16C2×C31×#121C2212yes—
17C22×C4⋊C4×C31C2211yescharacteristic
18C31×#483—22no—
19C31×#481C2214yescomplemented
20C31×#465—24no—
21C31×#450—22no—
22C31×#448C2214yescomplemented
23C2×C31×C22⋊C8C2212yes—
24C31×#400C2212yescomplemented
25C31×#398—22no—
26C31×#395C2212yescomplemented
27C31×#385—21no—
28C31×#381—21no—
29C31×#380C2211yescharacteristic
30C31×#327C2218yescomplemented
31C31×#324C2214yescomplemented
32C2×#41×C31—24no—
33C2×#41×C31—24no—
34C2×#41×C31—44no—
35C2×#41×C31—44no—
36C22×Q8×C31—21no—
37C22×Q8×C31C2311yescharacteristic
38C22×D4×C31—21no—
39C22×D4×C31—22no—
40C22×D4×C31—21no—
41C22×D4×C31D412yes—
42C23×C4×C31—22no—
43C23×C4×C31D411yescharacteristic
44C23×C4×C31D411yescharacteristic
45C23×C4×C31C2311yescharacteristic
46C2×SD16×C31—24no—
47C22×C8×C31—22no—
48C31×C22⋊Q8—28no—
49C31×C22⋊Q8—28no—
50C31×C22⋊Q8—22no—
51C31×C22⋊Q8—22no—
52C31×#121—24no—
53C31×#121—28no—
54C31×#121—28no—
55C31×#121—24no—
56C31×#121—22no—
57C31×#121—22no—
58C31×#121D414yes—
59C31×#121C2314yes—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—21no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—21no—
66C2×C4⋊C4×C31D412yes—
67C2×C4⋊C4×C31D412yes—
68C2×C4⋊C4×C31C2311yescharacteristic
69C2×C4⋊C4×C31C2312yes—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—21no—
73C2×C22⋊C4×C31—21no—
74C31×#106—24no—
75C31×#103—24no—
76C31×#102—24no—
77C31×C22⋊C8—22no—
78C31×C22⋊C8C2314yes—
79C31×#95—22no—
80C31×#95—22no—
81C31×#95—22no—
82C31×#95C2312yes—
83C24×C31—22no—
84C24×C31C2×D411yescharacteristic
85#41×C31—48no—
86#41×C31—416no—
87#41×C31—48no—
88C2×Q8×C31—24no—
89C2×Q8×C31—22no—
90C2×Q8×C31—44no—
91C2×Q8×C31—81no—
92C2×D4×C31—24no—
93C2×D4×C31—24no—
94C2×D4×C31—24no—
95C2×D4×C31—24no—
96C2×D4×C31—22no—
97C2×D4×C31—44no—
98C2×D4×C31—44no—
99C2×D4×C31—44no—
100C2×D4×C31—82no—
101C2×D4×C31—82no—
102C2×D4×C31C2×D414yes—
103C22×C4×C31—21no—
104C22×C4×C31—21no—
105C22×C4×C31—22no—
106C22×C4×C31—28no—
107C22×C4×C31—21no—
108C22×C4×C31—22no—
109C22×C4×C31—21no—
110C22×C4×C31—21no—
111C22×C4×C31—22no—
112C22×C4×C31—21no—
113C22×C4×C31—48no—
114C22×C4×C31—81no—
115C22×C4×C31—81no—
116C22×C4×C31SD1614yescomplemented
117C22×C4×C31C2×D412yes—
118C22×C4×C31C2×D412yes—
119C22×C4×C31C2×D411yescharacteristic
120C22×C4×C31C2×D412yes—
121C22×C4×C31C2×D412yes—
122C22×C4×C31C2411yescharacteristic
123SD16×C31—84no—
124C2×C8×C31—24no—
125C2×C8×C31—82no—
126C4⋊C4×C31—24no—
127C4⋊C4×C31—24no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—24no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—42no—
132C4⋊C4×C31—81no—
133C4⋊C4×C31—82no—
134C4⋊C4×C31C2×D414yes—
135C22⋊C4×C31—28no—
136C22⋊C4×C31—24no—
137C22⋊C4×C31—24no—
138C22⋊C4×C31—41no—
139C22⋊C4×C31—41no—
140C22⋊C4×C31—42no—
141#18293C3111yesmaximal, characteristic, direct product factor, Sylow
142C23×C31—24no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31—44no—
146C23×C31—44no—
147C23×C31—44no—
148C23×C31#12011yescharacteristic
149C23×C31#12011yescharacteristic
150C23×C31C2×SD1612yes—
151C23×C31AGL(1,ℤ/8)12yes—
152C23×C31C22×D411yescharacteristic
153Q8×C31—48no—
154Q8×C31—44no—
155D4×C31—44no—
156D4×C31—48no—
157D4×C31—44no—
158D4×C31—44no—
159D4×C31—48no—
160D4×C31—82no—
161D4×C31—84no—
162C2×C4×C31—22no—
163C2×C4×C31—22no—
164C2×C4×C31—24no—
165C2×C4×C31—24no—
166C2×C4×C31—22no—
167C2×C4×C31—24no—
168C2×C4×C31—41no—
169C2×C4×C31—42no—
170C2×C4×C31—42no—
171C2×C4×C31—41no—
172C2×C4×C31—416no—
173C2×C4×C31—42no—
174C2×C4×C31—44no—
175C2×C4×C31—42no—
176C2×C4×C31—42no—
177C2×C4×C31—42no—
178C2×C4×C31—42no—
179C2×C4×C31—42no—
180C2×C4×C31—41no—
181C2×C4×C31—41no—
182C2×C4×C31—42no—
183C2×C4×C31—81no—
184C2×C4×C31#12012yes—
185C2×C4×C31C2×SD1614yes—
186C2×C4×C31C22×D412yes—
187C8×C31—44no—
188C2×#537C2×C3111yescharacteristic
189#3187C2×C3112yescomplemented
190#2627C2×C3111yescharacteristic, complemented
191#1616C2×C3114yescomplemented
192#1604C2×C3114yescomplemented
193#1519C2×C3112yescomplemented
194C2×#324C2×C3111yescharacteristic, complemented
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—21no—
199C22×C31—21no—
200C22×C31—21no—
201C22×C31—42no—
202C22×C31—42no—
203C22×C31—44no—
204C22×C31—44no—
205C22×C31—44no—
206C22×C31—88no—
207C22×C31#45012yes—
208C22×C31#45012yes—
209C22×C31C2×#12011yescharacteristic
210C22×C31C22×SD1611yescharacteristic
211C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
212C4×C31—24no—
213C4×C31—22no—
214C4×C31—22no—
215C4×C31—44no—
216C4×C31—42no—
217C4×C31—82no—
218C4×C31—81no—
219C4×C31—81no—
220C22×#41—21no—
221#537—24no—
222#537—22no—
223#537C22×C3114yes—
224C2×C22⋊Q8—22no—
225C2×C22⋊Q8C22×C3111yescharacteristic
226C2×#121—22no—
227C2×#121—21no—
228C2×#121C22×C3112yes—
229C22×C4⋊C4C22×C3111yescharacteristic
230#483—22no—
231#481C22×C3114yescomplemented
232#465—24no—
233#450—22no—
234#448C22×C3114yescomplemented
235C2×C22⋊C8C22×C3112yes—
236#400C22×C3112yescomplemented
237#398—22no—
238#395C22×C3112yescomplemented
239#385—21no—
240#381—21no—
241#380C22×C3111yescharacteristic
242#327C22×C3118yescomplemented
243#324C22×C3114yescomplemented
244C2×C31—21no—
245C2×C31—22no—
246C2×C31—21no—
247C2×C31—22no—
248C2×C31—84no—
249C2×C31C2×#45011yescharacteristic
250C2×C31#302011yescharacteristic
251C2×C31#302611yescharacteristic
252C2×#41—24no—
253C2×#41—24no—
254C2×#41—44no—
255C2×#41—44no—
256C22×Q8—21no—
257C22×Q8C23×C3111yescharacteristic
258C22×D4—21no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4D4×C3112yes—
262C23×C4—22no—
263C23×C4D4×C3111yescharacteristic
264C23×C4D4×C3111yescharacteristic
265C23×C4C23×C3111yescharacteristic
266C2×SD16—24no—
267C22×C8—22no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272#121—24no—
273#121—28no—
274#121—28no—
275#121—24no—
276#121—22no—
277#121—22no—
278#121D4×C3114yes—
279#121C23×C3114yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4D4×C3112yes—
287C2×C4⋊C4D4×C3112yes—
288C2×C4⋊C4C23×C3111yescharacteristic
289C2×C4⋊C4C23×C3112yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294#106—24no—
295#103—24no—
296#102—24no—
297C22⋊C8—22no—
298C22⋊C8C23×C3114yes—
299#95—22no—
300#95—22no—
301#95—22no—
302#95C23×C3112yes—
303C31#1829311yescharacteristic, direct product factor, Sylow
304C24—22no—
305C24C2×D4×C3111yescharacteristic
306#41—416no—
307#41—48no—
308#41—48no—
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—44no—
312C2×Q8—81no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4C2×D4×C3114yes—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—48no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4SD16×C3114yescomplemented
338C22×C4C2×D4×C3112yes—
339C22×C4C2×D4×C3112yes—
340C22×C4C2×D4×C3111yescharacteristic
341C22×C4C2×D4×C3112yes—
342C22×C4C2×D4×C3112yes—
343C22×C4C24×C3111yescharacteristic
344SD16—84no—
345C2×C8—24no—
346C2×C8—82no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—81no—
354C4⋊C4—82no—
355C4⋊C4C2×D4×C3114yes—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—42no—
362C23—24no—
363C23—41no—
364C23—41no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23C31×#12011yescharacteristic
369C23C31×#12011yescharacteristic
370C23C2×SD16×C3112yes—
371C23C31×AGL(1,ℤ/8)12yes—
372C23C22×D4×C3111yescharacteristic
373Q8—48no—
374Q8—44no—
375D4—44no—
376D4—48no—
377D4—48no—
378D4—44no—
379D4—44no—
380D4—82no—
381D4—84no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—416no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—81no—
404C2×C4C31×#12012yes—
405C2×C4C2×SD16×C3114yes—
406C2×C4C22×D4×C3112yes—
407C8—44no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22C31×#45012yes—
421C22C31×#45012yes—
422C22C2×C31×#12011yescharacteristic
423C22C22×SD16×C3111yescharacteristic
424C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
425C4—24no—
426C4—22no—
427C4—22no—
428C4—44no—
429C4—42no—
430C4—82no—
431C4—81no—
432C4—81no—
433C2—21no—
434C2—22no—
435C2—22no—
436C2—21no—
437C2—84no—
438C2C2×C31×#45011yescharacteristic
439C2C31×#302011yescharacteristic
440C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3187C2semidirect
3. C31×#2627C2semidirect
4. C31×#1616C2semidirect
5. C31×#1604C2semidirect
6. C31×#1519C2semidirect
7. C2×C31×#324C2semidirect
19. C31×#481C22semidirect
22. C31×#448C22semidirect
24. C31×#400C22semidirect
26. C31×#395C22semidirect
30. C31×#327C22semidirect
31. C31×#324C22semidirect
116. C22×C4×C31SD16semidirect
141. #18293C31direct (N × H)
189. #3187C2×C31semidirect
190. #2627C2×C31semidirect
191. #1616C2×C31semidirect
192. #1604C2×C31semidirect
193. #1519C2×C31semidirect
194. C2×#324C2×C31semidirect
231. #481C22×C31semidirect
234. #448C22×C31semidirect
236. #400C22×C31semidirect
238. #395C22×C31semidirect
242. #327C22×C31semidirect
243. #324C22×C31semidirect
303. C31#18293direct (N × H)
337. C22×C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5372
C31×#31872
C31×#26272
C31×#16162
C31×#16042
C31×#15192
C2×C31×#3242
#1829331