Finite Group Databasebeta

← C31×#9174 #21465933

Subgroups — C31×#9174 #21465933

Order 7936.

516 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3038C211yesmaximal, characteristic
2C31×#2925C211yesmaximal, characteristic, complemented
3C31×#1545C211yesmaximal, characteristic, complemented
4C2×C31×#331C211yesmaximal, characteristic, complemented
5C31×#1477C211yesmaximal, characteristic, complemented
6C31×#1301C211yesmaximal, characteristic, complemented
7C31×#1292C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C2×C31×#127C2211yescharacteristic
10C4×#41×C31C2211yescharacteristic
11C31×#460—21no—
12C31×#460C412yes—
13C31×#459—21no—
14C31×#459C412yes—
15C31×#433—22no—
16C31×#432—21no—
17C31×#431—21no—
18C31×#431C2211yescharacteristic
19C31×#423C2211yescharacteristic
20C31×#423C2211yescharacteristic
21C2×C31×C4⋊C8C2211yescharacteristic
22C2×C31×C4⋊C8C2211yescharacteristic
23C31×#418—21no—
24C31×#408—21no—
25C31×#405—21no—
26C31×#331—22no—
27C31×#331—21no—
28C31×#331—21no—
29C31×Q8⋊C8—21no—
30C31×D4⋊C8—21no—
31C2×#41×C31D411yescharacteristic
32C22×D4×C31—21no—
33C22×D4×C31—21no—
34C22×D4×C31—21no—
35C31×AGL(1,ℤ/8)—42no—
36C31×AGL(1,ℤ/8)—42no—
37C2×SD16×C31—22no—
38C2×D8×C31—22no—
39C2×M16×C31—21no—
40C2×M16×C31—21no—
41C22×C8×C31—21no—
42C22×C8×C31—21no—
43C31×#127—21no—
44C31×#127—21no—
45C31×#127—21no—
46C31×#127C2×C412yes—
47C4×Q8×C31C2×C411yescharacteristic
48C4×Q8×C31D411yescharacteristic
49C4×D4×C31—21no—
50C4×D4×C31—21no—
51C4×D4×C31C2×C411yescharacteristic
52C4×D4×C31D411yescharacteristic
53C31×#117—21no—
54C31×#117D411yescharacteristic
55C2×C42×C31—21no—
56C2×C42×C31C2311yescharacteristic
57C31×C4⋊C8—21no—
58C31×C4⋊C8—21no—
59C31×C4⋊C8—21no—
60C31×C4⋊C8—21no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8C2×C412yes—
64C31×C4⋊C8D411yescharacteristic
65C31×C4⋊C8D411yescharacteristic
66C31×C4⋊C8D412yes—
67C31×#102—22no—
68C31×#102—22no—
69C31×C22⋊C8—22no—
70C31×C22⋊C8—22no—
71C31×C22⋊C8—21no—
72C31×#97—22no—
73C4×C8×C31—22no—
74C4×C8×C31—21no—
75C24×C31—41no—
76#41×C31—21no—
77#41×C31—21no—
78#41×C31—21no—
79#41×C31—21no—
80C2×Q8×C31C22⋊C411yescharacteristic
81C2×D4×C31—21no—
82C2×D4×C31—22no—
83C2×D4×C31—22no—
84C2×D4×C31—22no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C2×D4×C31—41no—
91C2×D4×C31—41no—
92C2×D4×C31—81no—
93C2×D4×C31—81no—
94C2×D4×C31—82no—
95C2×D4×C31C22⋊C411yescharacteristic
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31—21no—
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31#4111yescharacteristic
101C22×C4×C31#4111yescharacteristic
102SD16×C31—44no—
103D8×C31—44no—
104M16×C31—41no—
105M16×C31—41no—
106M16×C31—41no—
107M16×C31—41no—
108C2×C8×C31—22no—
109C2×C8×C31—21no—
110C2×C8×C31—22no—
111C2×C8×C31—21no—
112C2×C8×C31—41no—
113C2×C8×C31—41no—
114C2×C8×C31—41no—
115C2×C8×C31—41no—
116C2×C8×C31—81no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31C22⋊C411yescharacteristic
120C4⋊C4×C31C22⋊C411yescharacteristic
121C22⋊C4×C31—21no—
122C22⋊C4×C31—21no—
123C22⋊C4×C31—21no—
124C42×C31—21no—
125C42×C31—21no—
126C42×C31—21no—
127C42×C31C22×C411yescharacteristic
128C42×C31C2×D411yescharacteristic
129C42×C31C2×D411yescharacteristic
130C42×C31C2×D411yescharacteristic
131#9174C3111yesmaximal, characteristic, direct product factor, Sylow
132C23×C31—21no—
133C23×C31—42no—
134C23×C31—82no—
135C23×C31—82no—
136C23×C31—81no—
137C23×C31—81no—
138C23×C31#12311yescharacteristic
139Q8×C31—21no—
140Q8×C31—21no—
141D4×C31—21no—
142D4×C31—21no—
143D4×C31—41no—
144D4×C31—41no—
145D4×C31—42no—
146D4×C31—42no—
147D4×C31—82no—
148D4×C31—82no—
149D4×C31—82no—
150D4×C31—81no—
151D4×C31—81no—
152D4×C31—81no—
153D4×C31—81no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—81no—
165C2×C4×C31C2×C22⋊C411yescharacteristic
166C2×C4×C31C4×D411yescharacteristic
167C2×C4×C31C4×D411yescharacteristic
168C2×C4×C31#12011yescharacteristic
169C2×C4×C31#12111yescharacteristic
170C2×C4×C31#12111yescharacteristic
171C2×C4×C31#13511yescharacteristic
172C2×C4×C31#13511yescharacteristic
173C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
175C8×C31—42no—
176C8×C31—41no—
177C8×C31—41no—
178C8×C31—42no—
179C8×C31—81no—
180C8×C31—81no—
181#3038C2×C3111yescharacteristic
182#2925C2×C3111yescharacteristic, complemented
183#1545C2×C3111yescharacteristic, complemented
184C2×#331C2×C3111yescharacteristic, complemented
185#1477C2×C3111yescharacteristic, complemented
186#1301C2×C3111yescharacteristic, complemented
187#1292C2×C3111yescharacteristic, complemented
188C22×C31—21no—
189C22×C31—21no—
190C22×C31—41no—
191C22×C31—41no—
192C22×C31—41no—
193C22×C31—82no—
194C22×C31—82no—
195C22×C31—82no—
196C22×C31—162no—
197C22×C31—162no—
198C22×C31#38611yescharacteristic
199C22×C31#48211yescharacteristic
200C22×C31#48211yescharacteristic
201C4×C31—21no—
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—41no—
205C4×C31—41no—
206C4×C31—41no—
207C4×C31#41611yescharacteristic
208C4×C31#41811yescharacteristic
209C4×C31#44911yescharacteristic
210C4×C31#44911yescharacteristic
211C4×C31#47111yescharacteristic
212C4×C31#47211yescharacteristic
213C2×AGL(1,ℤ/8)—21no—
214C2×#127C22×C3111yescharacteristic
215C4×#41C22×C3111yescharacteristic
216#460—21no—
217#460C4×C3112yes—
218#459—21no—
219#459C4×C3112yes—
220#433—22no—
221#432—21no—
222#431—21no—
223#431C22×C3111yescharacteristic
224#423C22×C3111yescharacteristic
225#423C22×C3111yescharacteristic
226C2×C4⋊C8C22×C3111yescharacteristic
227C2×C4⋊C8C22×C3111yescharacteristic
228#418—21no—
229#408—21no—
230#405—21no—
231#331—21no—
232#331—21no—
233#331—22no—
234Q8⋊C8—21no—
235D4⋊C8—21no—
236C2×C31—21no—
237C2×C31—21no—
238C2×C31—81no—
239C2×C31—162no—
240C2×C31#187511yescharacteristic
241C2×C31#190311yescharacteristic
242C2×C31#190711yescharacteristic
243C2×#41D4×C3111yescharacteristic
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—21no—
247AGL(1,ℤ/8)—42no—
248AGL(1,ℤ/8)—42no—
249C2×SD16—22no—
250C2×D8—22no—
251C2×M16—21no—
252C2×M16—21no—
253C22×C8—21no—
254C22×C8—21no—
255#127—21no—
256#127—21no—
257#127—21no—
258#127C2×C4×C3112yes—
259C4×Q8C2×C4×C3111yescharacteristic
260C4×Q8D4×C3111yescharacteristic
261C4×D4—21no—
262C4×D4—21no—
263C4×D4C2×C4×C3111yescharacteristic
264C4×D4D4×C3111yescharacteristic
265#117—21no—
266#117D4×C3111yescharacteristic
267C2×C42—21no—
268C2×C42C23×C3111yescharacteristic
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8C2×C4×C3112yes—
276C4⋊C8D4×C3111yescharacteristic
277C4⋊C8D4×C3111yescharacteristic
278C4⋊C8D4×C3112yes—
279#102—22no—
280#102—22no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—21no—
284#97—22no—
285C4×C8—22no—
286C4×C8—21no—
287C31#917411yescharacteristic, direct product factor, Sylow
288C24—41no—
289#41—21no—
290#41—21no—
291#41—21no—
292#41—21no—
293C2×Q8C22⋊C4×C3111yescharacteristic
294C2×D4—21no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—81no—
306C2×D4—81no—
307C2×D4—82no—
308C2×D4C22⋊C4×C3111yescharacteristic
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4C2×D4×C3111yescharacteristic
313C22×C4#41×C3111yescharacteristic
314C22×C4#41×C3111yescharacteristic
315SD16—44no—
316D8—44no—
317M16—41no—
318M16—41no—
319M16—41no—
320M16—41no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—81no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4C22⋊C4×C3111yescharacteristic
333C4⋊C4C22⋊C4×C3111yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42C22×C4×C3111yescharacteristic
341C42C2×D4×C3111yescharacteristic
342C42C2×D4×C3111yescharacteristic
343C42C2×D4×C3111yescharacteristic
344C23—21no—
345C23—42no—
346C23—82no—
347C23—82no—
348C23—81no—
349C23—81no—
350C23C31×#12311yescharacteristic
351Q8—21no—
352Q8—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—41no—
357D4—42no—
358D4—42no—
359D4—82no—
360D4—82no—
361D4—82no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4C2×C22⋊C4×C3111yescharacteristic
378C2×C4C4×D4×C3111yescharacteristic
379C2×C4C4×D4×C3111yescharacteristic
380C2×C4C31×#12011yescharacteristic
381C2×C4C31×#12111yescharacteristic
382C2×C4C31×#12111yescharacteristic
383C2×C4C31×#13511yescharacteristic
384C2×C4C31×#13511yescharacteristic
385C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
386C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
387C8—42no—
388C8—41no—
389C8—41no—
390C8—42no—
391C8—81no—
392C8—81no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—162no—
402C22—162no—
403C22C31×#38611yescharacteristic
404C22C31×#48211yescharacteristic
405C22C31×#48211yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4C31×#41611yescharacteristic
413C4C31×#41811yescharacteristic
414C4C31×#44911yescharacteristic
415C4C31×#44911yescharacteristic
416C4C31×#47111yescharacteristic
417C4C31×#47211yescharacteristic
418C2—21no—
419C2—21no—
420C2—81no—
421C2—162no—
422C2C31×#187511yescharacteristic
423C2C31×#190311yescharacteristic
424C2C31×#190711yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3038C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2925C2semidirect
3. C31×#1545C2semidirect
4. C2×C31×#331C2semidirect
5. C31×#1477C2semidirect
6. C31×#1301C2semidirect
7. C31×#1292C2semidirect
131. #9174C31direct (N × H)
182. #2925C2×C31semidirect
183. #1545C2×C31semidirect
184. C2×#331C2×C31semidirect
185. #1477C2×C31semidirect
186. #1301C2×C31semidirect
187. #1292C2×C31semidirect
287. C31#9174direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30382
C31×#29252
C31×#15452
C2×C31×#3312
C31×#14772
C31×#13012
C31×#12922
#917431