Finite Group Databasebeta

← C3×#71916 #11780185

Subgroups — C3×#71916 #11780185

Order 1152.

608 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D4×Q8C211yesmaximal, characteristic
2C3×#5078C211yesmaximal, characteristic, complemented
3C3×#5031C211yesmaximal, characteristic, complemented
4C3×#5007C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic, complemented
6C3×#4535C211yesmaximal, characteristic, complemented
7C3×D4⋊(C3⋊C8)C211yesmaximal, characteristic
8C3×#1649—31nomaximal
9#71916C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8—21no—
11C32×C4⋊Q8—21no—
12C32×C4⋊Q8C2211yescharacteristic
13C32×C22⋊Q8—21no—
14C32×C22⋊Q8—22no—
15C32×C4×Q8C2211yescharacteristic
16C32×C4×D4—21no—
17C32×C4×D4C2211yescharacteristic
18C2×C3×#216—21no—
19C3×#869C2211yescharacteristic
20C3×#832—21no—
21C3×#810—21no—
22C3×#807—21no—
23C3×#807—22no—
24C3×C4⋊(C3⋊C8)C2211yescharacteristic
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27C3×D4×Q8—21no—
28C3×D4×Q8D311yescharacteristic
29C3×D4×Q8C2×C311yescharacteristic
30C3×#500—31no—
31C3×#474—31no—
32C3×#462—31no—
33#5078C2×C311yescharacteristic, complemented
34#5031C2×C311yescharacteristic, complemented
35#5007C2×C311yescharacteristic
36C3×#332—31no—
37C3×Q8⋊C8—31no—
38C3×D4⋊C8—31no—
39#4547C2×C311yescharacteristic, complemented
40#4535C2×C311yescharacteristic, complemented
41D4⋊(C3⋊C8)C2×C311yescharacteristic
42C2×C32×Q8—22no—
43C2×C32×Q8—21no—
44C2×C32×Q8—42no—
45C2×C32×Q8—42no—
46C2×C32×Q8D411yescharacteristic
47C2×C32×D4D411yescharacteristic
48C22×C32×C4—22no—
49C22×C32×C4—21no—
50C2×C3×Dic6—21no—
51C32×C4⋊C4—22no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4D411yescharacteristic
55C32×C4⋊C4D412yes—
56C32×C4⋊C4D411yescharacteristic
57C32×C22⋊C4—22no—
58C32×C22⋊C4—21no—
59C32×C42—21no—
60C32×C42C2311yescharacteristic
61C3×#216—42no—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—21no—
67#1649—31noSylow
68C22×C3×Q8—22no—
69C22×C3×Q8—21no—
70C22×C3×Q8—21no—
71C2×C3×Q16—61no—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8—21no—
76C3×C4⋊Q8—31no—
77C3×C4⋊Q8C2×D311yescharacteristic
78C3×C4⋊Q8C22×C311yescharacteristic
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—21no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8—22no—
85C3×C4×Q8—21no—
86C3×C4×Q8C2×D311yescharacteristic
87C3×C4×Q8C22×C311yescharacteristic
88C3×C4×D4—22no—
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C3×C4×D4C2×D311yescharacteristic
93C3×C4×D4C22×C311yescharacteristic
94C2×#216—21no—
95#869C22×C311yescharacteristic
96C3×#107—61no—
97C3×#107—62no—
98C3×C4⋊C8—31no—
99C3×C4⋊C8—31no—
100C3×#103—61no—
101C3×#102—61no—
102C3×C4×C8—31no—
103#832—21no—
104#810—21no—
105#807—21no—
106#807—22no—
107C4⋊(C3⋊C8)C22×C311yescharacteristic
108C4⋊(C3⋊C8)C22×C311yescharacteristic
109C4×C3⋊C8C22×C311yescharacteristic
110C23×C32—21no—
111C32×Q8—21no—
112C32×Q8—44no—
113C32×Q8—41no—
114C32×Q8—41no—
115C32×Q8D812yes—
116C32×D4—21no—
117C32×D4Q1612yescomplemented
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—44no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C2×C3×Dic3—41no—
129C3×Dic6—41no—
130C3×Dic6—41no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8—41no—
133C3×C3⋊C8—41no—
134D4×Q8C3×D311yescharacteristic
135#500—31no—
136#474—31no—
137#462—31no—
138#332—31no—
139Q8⋊C8—31no—
140D4⋊C8—31no—
141C2×C3×Q8—24no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—44no—
149C2×C3×Q8—42no—
150C2×C3×Q8—42no—
151C2×C3×Q8—44no—
152C2×C3×Q8—42no—
153C2×C3×Q8—42no—
154C2×C3×Q8—61no—
155C2×C3×Q8#6711yescharacteristic
156C2×C3×Q8C3×D411yescharacteristic
157C2×C3×D4—21no—
158C2×C3×D4#6711yescharacteristic
159C2×C3×D4C3×D411yescharacteristic
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C2×Dic6—21no—
167C3×Q16—122no—
168C2×C3×C8—61no—
169C2×C3×C8—61no—
170C2×C3×C8—61no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—61no—
184C3×C4⋊C4—61no—
185C3×C4⋊C4#6711yescharacteristic
186C3×C4⋊C4#6711yescharacteristic
187C3×C4⋊C4#6712yes—
188C3×C4⋊C4C3×D411yescharacteristic
189C3×C4⋊C4C3×D412yes—
190C3×C4⋊C4C3×D411yescharacteristic
191C3×C22⋊C4—24no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—21no—
196C3×C22⋊C4—21no—
197C3×C42—22no—
198C3×C42—21no—
199C3×C42—21no—
200C3×C42—21no—
201C3×C42C22×D311yescharacteristic
202C3×C42C23×C311yescharacteristic
203#216—42no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C22×C32—22no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32#12011yescharacteristic
213C32×C4—22no—
214C32×C4—21no—
215C32×C4—42no—
216C32×C4—41no—
217C32×C4—41no—
218C32×C4C2×D811yescharacteristic
219C32×C4C2×Q1611yescharacteristic
220C32×C4AGL(1,ℤ/8)11yescharacteristic
221C32×C4#13711yescharacteristic
222C3×Dic3—81no—
223C22×Q8—21no—
224C2×Q16—61no—
225C4⋊Q8—21no—
226C4⋊Q8—31no—
227C4⋊Q8C2×C3×D311yescharacteristic
228C22⋊Q8—21no—
229C22⋊Q8—22no—
230C4×Q8C2×C3×D311yescharacteristic
231C4×D4—21no—
232C4×D4C2×C3×D311yescharacteristic
233#107—61no—
234#107—62no—
235C4⋊C8—31no—
236C4⋊C8—31no—
237#103—61no—
238#102—61no—
239C4×C8—31no—
240C23×C3—22no—
241C23×C3—21no—
242C23×C3—21no—
243C3×Q8—22no—
244C3×Q8—21no—
245C3×Q8—21no—
246C3×Q8—22no—
247C3×Q8—48no—
248C3×Q8—44no—
249C3×Q8—44no—
250C3×Q8—42no—
251C3×Q8—42no—
252C3×Q8—41no—
253C3×Q8—41no—
254C3×Q8—41no—
255C3×Q8—41no—
256C3×Q8—121no—
257C3×Q8—121no—
258C3×Q8#21312yes—
259C3×Q8C3×D812yes—
260C3×D4—22no—
261C3×D4—21no—
262C3×D4—21no—
263C3×D4—22no—
264C3×D4#21612yescomplemented
265C3×D4C3×Q1612yescomplemented
266C2×C3×C4—22no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—22no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—21no—
273C2×C3×C4—22no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—48no—
282C2×C3×C4—44no—
283C2×C3×C4—44no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—121no—
291C2×C3×C4C2×#6711yescharacteristic
292C2×C3×C4C2×#6711yescharacteristic
293C2×C3×C4C2×#6711yescharacteristic
294C2×C3×C4C2×C3×D411yescharacteristic
295C2×C3×C4C2×C3×D411yescharacteristic
296C2×C3×C4C2×C3×D411yescharacteristic
297C2×Dic3—41no—
298Dic6—41no—
299Dic6—41no—
300C3×C8—62no—
301C3×C8—121no—
302C3×C8—121no—
303C3⋊C8—22no—
304C3⋊C8—41no—
305C3⋊C8—41no—
306C2×C32—42no—
307C2×C32#44711yescharacteristic
308C2×C32#45111yescharacteristic
309C2×C32#45611yescharacteristic
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—61no—
315C2×Q8C3×#6711yescharacteristic
316C2×D4C3×#6711yescharacteristic
317C22×C4—22no—
318C22×C4—21no—
319Q16—122no—
320C2×C8—61no—
321C2×C8—61no—
322C2×C8—61no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4C3×#6711yescharacteristic
329C4⋊C4C3×#6711yescharacteristic
330C4⋊C4C3×#6712yes—
331C22⋊C4—22no—
332C22⋊C4—21no—
333C42—21no—
334C42C22×C3×D311yescharacteristic
335C22×C3—24no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3#95311yescharacteristic
346C22×C3C3×#12011yescharacteristic
347C3×C4—24no—
348C3×C4—22no—
349C3×C4—21no—
350C3×C4—22no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—41no—
357C3×C4—44no—
358C3×C4—42no—
359C3×C4—42no—
360C3×C4—42no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4—42no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—241no—
367C3×C4C2×#21311yescharacteristic
368C3×C4#93211yescharacteristic
369C3×C4#94211yescharacteristic
370C3×C4C2×#21611yescharacteristic
371C3×C4C2×C3×D811yescharacteristic
372C3×C4C2×C3×Q1611yescharacteristic
373C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
374C3×C4C3×#13711yescharacteristic
375Dic3—81no—
376C32#164911yescharacteristic, complemented, Sylow
377C23—21no—
378Q8—21no—
379Q8—44no—
380Q8—41no—
381Q8—41no—
382Q8—121no—
383Q8—121no—
384Q8C3×#21312yes—
385D4—21no—
386D4C3×#21612yescomplemented
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—121no—
395C2×C4C2×C3×#6711yescharacteristic
396C2×C4C2×C3×#6711yescharacteristic
397C2×C4C2×C3×#6711yescharacteristic
398C8—62no—
399C8—121no—
400C8—121no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—44no—
405C2×C3—42no—
406C2×C3—42no—
407C2×C3#521711yescharacteristic
408C2×C3#522811yescharacteristic
409C2×C3#524611yescharacteristic
410C2×C3C3×#44711yescharacteristic
411C2×C3C3×#45111yescharacteristic
412C2×C3C3×#45611yescharacteristic
413C22—22no—
414C22—41no—
415C22—41no—
416C22C3×#95311yescharacteristic
417C4—22no—
418C4—21no—
419C4—42no—
420C4—41no—
421C4—41no—
422C4—241no—
423C4C2×C3×#21311yescharacteristic
424C4C3×#93211yescharacteristic
425C4C3×#94211yescharacteristic
426C4C2×C3×#21611yescharacteristic
427C3—21no—
428C3#7191611yescharacteristic, direct product factor
429C3C3×#164911yescharacteristic, complemented
430C2—42no—
431C2C3×#521711yescharacteristic
432C2C3×#522811yescharacteristic
433C2C3×#524611yescharacteristic
subgroupG/H
centerC22×C3#953
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×D4×Q8C2
socleC22×C32#120
prime radicalC2×C32×D4D4
hypercenterC3×D4×Q8D3
hypocenterC3C3×#1649
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5078C2semidirect
3. C3×#5031C2semidirect
5. C3×#4547C2semidirect
6. C3×#4535C2semidirect
9. #71916C3direct (N × H)
33. #5078C2×C3semidirect
34. #5031C2×C3semidirect
39. #4547C2×C3semidirect
40. #4535C2×C3semidirect
117. C32×D4Q16semidirect
264. C3×D4#216semidirect
265. C3×D4C3×Q16semidirect
376. C32#1649semidirect
386. D4C3×#216semidirect
428. C3#71916direct (N × H)
429. C3C3×#1649semidirect

Maximal subgroups

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

subgroupindex
C32×D4×Q82
C3×#50782
C3×#50312
C3×#50072
C3×#45472
C3×#45352
C3×D4⋊(C3⋊C8)2
C3×#16493
#719163