Finite Group Databasebeta

← C3×#61828 #10680435

Subgroups — C3×#61828 #10680435

Order 768.

1914 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
1C3×#3497C211yesmaximal, characteristic
2C2×C3×#557C211yesmaximal, characteristic, complemented
3C3×#3386C218yesmaximal, complemented
4C3×#3368C218yesmaximal, complemented
5C3×#3234C214yesmaximal, complemented
6C3×#3226C212yesmaximal, complemented
7C3×#3112C214yesmaximal, complemented
8C3×#3103C212yesmaximal, complemented
9C3×#2983C211yesmaximal, characteristic, complemented
10#61828C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×2−1+4—21no—
12C3×#557—24no—
13C3×#557C2218yes—
14C3×#550C2214yes—
15C3×D4×Q8C2214yes—
16C3×#548—24no—
17C3×#537—21no—
18C3×#537—22no—
19C3×#532C2212yes—
20C3×#532C2214yes—
21C2×C3×C4⋊Q8—21no—
22C2×C3×C4⋊Q8C2211yescharacteristic
23C2×C3×#125—22no—
24C2×C3×#125C2212yes—
25C2×C3×C4×Q8—21no—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×Q8C2212yes—
28C2×C3×C4×Q8C2211yescharacteristic
29C3×#501C2218yes—
30C3×#500C2214yes—
31C3×#490C2218yes—
32C3×#486C2214yes—
33C3×#481C2214yes—
34C3×#480C2218yes—
35C3×#478C22116yes—
36C3×#477C2218yes—
37C3×#475C2214yes—
38C3×#474C2218yes—
39C3×#460C2214yes—
40C3×#460C2218yes—
41C3×#448C2214yes—
42C3×#446C2218yes—
43C3×#440C22116yes—
44C3×C4×SD16C2218yes—
45C3×#432C2214yes—
46C3×#431C2212yes—
47#3497C2×C311yescharacteristic
48C2×#557C2×C311yescharacteristic, complemented
49#3386C2×C318yescomplemented
50#3368C2×C318yescomplemented
51#3234C2×C314yescomplemented
52#3226C2×C312yescomplemented
53#3112C2×C314yescomplemented
54#3103C2×C312yescomplemented
55#2983C2×C311yescharacteristic, complemented
56C3×2−1+4—48no—
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—22no—
60C22×C3×Q8—24no—
61C22×C3×Q8—21no—
62C22×C3×Q8D411yescharacteristic
63C22×C3×Q8C2311yescharacteristic
64C2×C3×SD16C2318yes—
65C3×C4⋊Q8—28no—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8C2314yes—
68C3×C4⋊Q8C2312yes—
69C3×C4⋊Q8C2314yes—
70C3×#127C2312yes—
71C3×#127C2314yes—
72C3×#125—216no—
73C3×#125—24no—
74C3×#125C2318yes—
75C3×#124—24no—
76C3×#124—22no—
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8—28no—
79C3×#121—24no—
80C3×#121—22no—
81C3×#121—24no—
82C3×#121—28no—
83C3×#121C2314yes—
84C3×C4×Q8—28no—
85C3×C4×Q8—24no—
86C3×C4×Q8—22no—
87C3×C4×Q8D414yes—
88C3×C4×Q8C2314yes—
89C3×C4×Q8C2318yes—
90C3×C4×Q8C2314yes—
91C3×C4×D4—24no—
92C3×C4×D4—22no—
93C3×C4×D4C2314yes—
94C3×C4×D4C2318yes—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4D412yes—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C42—22no—
105C2×C3×C42—21no—
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2312yes—
108C3×#107C23116yes—
109C3×#106C2318yes—
110C3×C4⋊C8C2314yes—
111C3×C4⋊C8C2318yes—
112C3×#103C2318yes—
113C3×#102C2318yes—
114C3×#102C23116yes—
115C3×C22⋊C8C2314yes—
116C3×#97C2318yes—
117C3×C4×C8C2314yes—
118C2×2−1+4—21no—
119#557—24no—
120#557C22×C318yes—
121#550C22×C314yes—
122D4×Q8C22×C314yes—
123#548—24no—
124#537—21no—
125#537—22no—
126#532C22×C312yes—
127#532C22×C314yes—
128C2×C4⋊Q8—21no—
129C2×C4⋊Q8C22×C311yescharacteristic
130C2×#125—22no—
131C2×#125C22×C312yes—
132C2×C4×Q8—21no—
133C2×C4×Q8C22×C311yescharacteristic
134C2×C4×Q8C22×C312yes—
135C2×C4×Q8C22×C311yescharacteristic
136#501C22×C318yes—
137#500C22×C314yes—
138#490C22×C318yes—
139#486C22×C314yes—
140#481C22×C314yes—
141#480C22×C318yes—
142#478C22×C3116yes—
143#477C22×C318yes—
144#475C22×C314yes—
145#474C22×C318yes—
146#460C22×C314yes—
147#460C22×C318yes—
148#448C22×C314yes—
149#446C22×C318yes—
150#440C22×C3116yes—
151C4×SD16C22×C318yes—
152#432C22×C314yes—
153#431C22×C312yes—
154C3×#41—416no—
155C3×#41—416no—
156C3×#41—48no—
157C2×C3×Q8—24no—
158C2×C3×Q8—24no—
159C2×C3×Q8—21no—
160C2×C3×Q8—24no—
161C2×C3×Q8—24no—
162C2×C3×Q8—22no—
163C2×C3×Q8—416no—
164C2×C3×Q8—44no—
165C2×C3×Q8C2×D414yes—
166C2×C3×Q8C2412yes—
167C2×C3×D4—24no—
168C2×C3×D4—24no—
169C2×C3×D4—22no—
170C2×C3×D4—24no—
171C2×C3×D4—22no—
172C2×C3×D4C2414yes—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—28no—
177C22×C3×C4—24no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C3×SD16—216no—
184C2×C3×C8C2418yes—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4C2×D414yes—
195C3×C4⋊C4C2×D418yes—
196C3×C4⋊C4C2412yes—
197C3×C4⋊C4C2414yes—
198C3×C4⋊C4C2414yes—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—24no—
202C3×C42—28no—
203C3×C42—24no—
204C3×C42C2×D414yes—
205C3×C42C2×D412yes—
206C3×C42C2412yes—
207C3×C42C2414yes—
2082−1+4—48no—
209C2×#41—24no—
210C2×#41—24no—
211C2×#41—22no—
212C22×Q8—24no—
213C22×Q8—21no—
214C22×Q8C3×D411yescharacteristic
215C22×Q8C23×C311yescharacteristic
216C2×SD16C23×C318yes—
217C4⋊Q8—28no—
218C4⋊Q8—22no—
219C4⋊Q8C23×C314yes—
220C4⋊Q8C23×C312yes—
221C4⋊Q8C23×C314yes—
222#127C23×C312yes—
223#127C23×C314yes—
224#125—216no—
225#125—24no—
226#125C23×C318yes—
227#124—24no—
228#124—22no—
229C22⋊Q8—24no—
230C22⋊Q8—28no—
231#121—24no—
232#121—22no—
233#121—24no—
234#121—28no—
235#121C23×C314yes—
236C4×Q8—28no—
237C4×Q8—24no—
238C4×Q8—22no—
239C4×Q8C3×D414yes—
240C4×Q8C23×C314yes—
241C4×Q8C23×C318yes—
242C4×Q8C23×C314yes—
243C4×D4—24no—
244C4×D4—22no—
245C4×D4C23×C314yes—
246C4×D4C23×C318yes—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4C3×D411yescharacteristic
253C2×C4⋊C4C3×D412yes—
254C2×C4⋊C4C23×C311yescharacteristic
255C2×C4⋊C4C23×C312yes—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42C23×C312yes—
259C2×C42C23×C311yescharacteristic
260#107C23×C3116yes—
261#106C23×C318yes—
262C4⋊C8C23×C314yes—
263C4⋊C8C23×C318yes—
264#103C23×C318yes—
265#102C23×C318yes—
266#102C23×C3116yes—
267C22⋊C8C23×C314yes—
268#97C23×C318yes—
269C4×C8C23×C314yes—
270C23×C3—24no—
271C23×C3C22×D411yescharacteristic
272C3×Q8—24no—
273C3×Q8—28no—
274C3×Q8—22no—
275C3×Q8—48no—
276C3×Q8—48no—
277C3×Q8—44no—
278C3×Q82−1+414yescomplemented
279C3×D4—28no—
280C3×D4—48no—
281C3×D4—48no—
282C3×D4—44no—
283C3×D4—84no—
284C3×D4—82no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—416no—
295C2×C3×C4—42no—
296C2×C3×C4—44no—
297C2×C3×C4—48no—
298C2×C3×C4AGL(1,ℤ/8)14yes—
299C2×C3×C4C22×D414yes—
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×D412yes—
302C2×C3×C4C22×D414yes—
303C2×C3×C4C22×D412yes—
304C2×C3×C4C2511yescharacteristic
305C3×C8—28no—
306#41—416no—
307#41—416no—
308#41—48no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—21no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—416no—
316C2×Q8—44no—
317C2×Q8C2×C3×D414yes—
318C2×Q8C24×C312yes—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4C24×C314yes—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—28no—
329C22×C4—24no—
330C22×C4C2×C3×D412yes—
331C22×C4C2×C3×D412yes—
332C22×C4C2×C3×D411yescharacteristic
333C22×C4C2×C3×D411yescharacteristic
334C22×C4C24×C311yescharacteristic
335SD16—216no—
336C2×C8C24×C318yes—
337C4⋊C4—28no—
338C4⋊C4—24no—
339C4⋊C4—28no—
340C4⋊C4—24no—
341C4⋊C4—28no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4C2×C3×D414yes—
347C4⋊C4C2×C3×D418yes—
348C4⋊C4C24×C312yes—
349C4⋊C4C24×C314yes—
350C4⋊C4C24×C314yes—
351C22⋊C4—24no—
352C22⋊C4—28no—
353C22⋊C4—24no—
354C42—28no—
355C42—24no—
356C42C2×C3×D414yes—
357C42C2×C3×D412yes—
358C42C24×C312yes—
359C42C24×C314yes—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—44no—
363C22×C3—44no—
364C22×C3—44no—
365C22×C3C2×AGL(1,ℤ/8)12yes—
366C22×C3C23×D411yescharacteristic
367C3×C4—24no—
368C3×C4—22no—
369C3×C4—24no—
370C3×C4—21no—
371C3×C4—24no—
372C3×C4—42no—
373C3×C4—44no—
374C3×C4—44no—
375C3×C4C2×AGL(1,ℤ/8)14yes—
376C3×C4C2×2−1+412yes—
377C23—24no—
378C23C22×C3×D411yescharacteristic
379Q8—24no—
380Q8—28no—
381Q8—22no—
382Q8—48no—
383Q8—44no—
384Q8—48no—
385Q8C3×2−1+414yescomplemented
386D4—28no—
387D4—48no—
388D4—48no—
389D4—44no—
390D4—84no—
391D4—82no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—416no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4C3×AGL(1,ℤ/8)14yes—
406C2×C4C22×C3×D414yes—
407C2×C4C22×C3×D411yescharacteristic
408C2×C4C22×C3×D412yes—
409C2×C4C22×C3×D414yes—
410C2×C4C22×C3×D412yes—
411C2×C4C25×C311yescharacteristic
412C8—28no—
413C2×C3—22no—
414C2×C3—84no—
415C2×C3#348611yescharacteristic
416C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
417C2×C3#358611yescharacteristic
418C22—21no—
419C22—21no—
420C22—44no—
421C22—44no—
422C22—44no—
423C22C2×C3×AGL(1,ℤ/8)12yes—
424C22C23×C3×D411yescharacteristic
425C4—24no—
426C4—21no—
427C4—24no—
428C4—24no—
429C4—22no—
430C4—44no—
431C4—44no—
432C4—42no—
433C4C2×C3×AGL(1,ℤ/8)14yes—
434C4C2×C3×2−1+412yes—
435C3#6182811yescharacteristic, direct product factor, Sylow
436C2—22no—
437C2—84no—
438C2C3×#348611yescharacteristic
439C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
440C2C3×#358611yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#3497C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#557C2semidirect
3. C3×#3386C2semidirect
4. C3×#3368C2semidirect
5. C3×#3234C2semidirect
6. C3×#3226C2semidirect
7. C3×#3112C2semidirect
8. C3×#3103C2semidirect
9. C3×#2983C2semidirect
10. #61828C3direct (N × H)
48. C2×#557C2×C3semidirect
49. #3386C2×C3semidirect
50. #3368C2×C3semidirect
51. #3234C2×C3semidirect
52. #3226C2×C3semidirect
53. #3112C2×C3semidirect
54. #3103C2×C3semidirect
55. #2983C2×C3semidirect
278. C3×Q82−1+4semidirect
385. Q8C3×2−1+4semidirect
435. C3#61828direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34972
C2×C3×#5572
C3×#33862
C3×#33682
C3×#32342
C3×#32262
C3×#31122
C3×#31032
C3×#29832
#618283