Finite Group Databasebeta

← C3×#13316 #10631923

Subgroups — C3×#13316 #10631923

Order 768.

604 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×C3×#509C211yesmaximal, characteristic, complemented
2C3×#3054C211yesmaximal, characteristic, complemented
3C2×C3×#469C211yesmaximal, characteristic, complemented
4C2×C3×#360C211yesmaximal, characteristic, complemented
5C3×#2140C211yesmaximal, characteristic, complemented
6C2×C3×#349C211yesmaximal, characteristic, complemented
7C3×#1386C211yesmaximal, characteristic, complemented
8#13316C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#135C2211yescharacteristic
10C22×C3×D8C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C3×#525—21no—
13C2×C3×#121—21no—
14C3×#509—22no—
15C3×#509—22no—
16C2×C3×SD32—21no—
17C2×C3×D16—21no—
18C2×C3×M32C2211yescharacteristic
19C2×C3×M32C2211yescharacteristic
20C3×#470—21no—
21C3×#469—22no—
22C3×#469—21no—
23C3×#469—22no—
24C3×#468—22no—
25C3×#427C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C3×#416—21no—
28C2×C3×#102—21no—
29C3×#360—22no—
30C3×#357—22no—
31C3×#349—22no—
32C2×#509C2×C311yescharacteristic, complemented
33#3054C2×C311yescharacteristic, complemented
34C2×#469C2×C311yescharacteristic, complemented
35C2×#360C2×C311yescharacteristic, complemented
36#2140C2×C311yescharacteristic, complemented
37C2×#349C2×C311yescharacteristic, complemented
38#1386C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C22×C3×D4—21no—
41C22×C3×D4—41no—
42C23×C3×C4D411yescharacteristic
43C3×#135—21no—
44C3×#135—21no—
45C3×#135—21no—
46C3×#135—21no—
47C2×C3×Q16D411yescharacteristic, complemented
48C2×C3×SD16—21no—
49C2×C3×D8—21no—
50C2×C3×D8—21no—
51C2×C3×D8—21no—
52C2×C3×D8—21no—
53C2×C3×D8—21no—
54C2×C3×D8D411yescharacteristic, complemented
55C2×C3×D8D412yescomplemented
56C2×C3×M16—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C22×C3×C8D411yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×#123—41no—
65C3×C22⋊Q8—22no—
66C3×#121—24no—
67C3×#121—22no—
68C3×#121—42no—
69C3×#120—41no—
70C3×C4×D4—42no—
71C3×#117—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C22⋊C4—41no—
74C3×SD32—24no—
75C3×D16—24no—
76C3×M32—21no—
77C3×M32—21no—
78C3×M32—21no—
79C3×M32D412yescomplemented
80C2×C3×C16—21no—
81C2×C3×C16—21no—
82C3×#107D412yescomplemented
83C3×#106—21no—
84C3×#106—21no—
85C3×#106D412yescomplemented
86C3×#103—22no—
87C3×#102—24no—
88C3×#102—22no—
89C2×#135C22×C311yescharacteristic
90C22×D8C22×C311yescharacteristic
91C22×M16C22×C311yescharacteristic
92#525—21no—
93C2×#121—21no—
94#509—22no—
95#509—22no—
96C2×SD32—21no—
97C2×D16—21no—
98C2×M32C22×C311yescharacteristic
99C2×M32C22×C311yescharacteristic
100#470—21no—
101#469—22no—
102#469—21no—
103#469—22no—
104#468—22no—
105#427C22×C311yescharacteristic
106C2×#106C22×C311yescharacteristic
107#416—21no—
108C2×#102—21no—
109#360—22no—
110#357—22no—
111#349—22no—
112C24×C3—41no—
113C24×C3D811yescharacteristic
114C3×#41—41no—
115C3×#41—41no—
116C3×#41—41no—
117C3×#41—41no—
118C2×C3×Q8—21no—
119C2×C3×D4—22no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C2×C3×D4—44no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—84no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4D811yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C3×Q16—21no—
141C3×Q16—21no—
142C3×SD16—41no—
143C3×SD16—41no—
144C3×D8—21no—
145C3×D8—22no—
146C3×D8—22no—
147C3×D8—21no—
148C3×D8—41no—
149C3×D8—41no—
150C3×M16—41no—
151C3×M16—41no—
152C3×M16—41no—
153C3×M16—41no—
154C2×C3×C8—21no—
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8C2×D411yescharacteristic
158C2×C3×C8C2×D411yescharacteristic
159C2×C3×C8C2×D411yescharacteristic
160C2×C3×C8C2×D411yescharacteristic
161C2×C3×C8C2×D411yescharacteristic
162C2×C3×C8#4111yescharacteristic
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—41no—
167C3×C22⋊C4—44no—
168C3×C22⋊C4—42no—
169C3×C22⋊C4—42no—
170C3×C22⋊C4—41no—
171C3×C42—41no—
172C3×C16—22no—
173C3×C16—41no—
174C2×#41—21no—
175C22×D4—21no—
176C22×D4—41no—
177C23×C4C3×D411yescharacteristic
178#135—21no—
179#135—21no—
180#135—21no—
181#135—21no—
182C2×Q16C3×D411yescharacteristic, complemented
183C2×SD16—21no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8—21no—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8C3×D412yescomplemented
190C2×D8C3×D411yescharacteristic, complemented
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C22×C8C3×D411yescharacteristic
198C22×C8C23×C311yescharacteristic
199#123—41no—
200C22⋊Q8—22no—
201#121—24no—
202#121—22no—
203#121—42no—
204#120—41no—
205C4×D4—42no—
206#117—21no—
207C2×C4⋊C4—21no—
208C2×C22⋊C4—41no—
209SD32—24no—
210D16—24no—
211M32—21no—
212M32—21no—
213M32—21no—
214M32C3×D412yescomplemented
215C2×C16—21no—
216C2×C16—21no—
217#107C3×D412yescomplemented
218#106—21no—
219#106—21no—
220#106C3×D412yescomplemented
221#103—22no—
222#102—24no—
223#102—22no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—41no—
230C23×C3—82no—
231C23×C3—82no—
232C23×C3—81no—
233C23×C3—81no—
234C23×C3C2×D811yescharacteristic
235C3×Q8—41no—
236C3×Q8—41no—
237C3×D4—42no—
238C3×D4—41no—
239C3×D4—42no—
240C3×D4—41no—
241C3×D4—84no—
242C3×D4—88no—
243C3×D4—81no—
244C3×D4—81no—
245C3×D4—81no—
246C3×D4—81no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—82no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#13511yescharacteristic
264C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
265C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
266C3×C8—41no—
267C3×C8#12011yescharacteristic
268C3×C8#12111yescharacteristic
269C3×C8#12111yescharacteristic
270C3×C8#12711yescharacteristic
271C24—41no—
272C24C3×D811yescharacteristic
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×D4—22no—
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—44no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—84no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4C3×D811yescharacteristic
298C22×C4C2×C3×D411yescharacteristic
299Q16—21no—
300Q16—21no—
301SD16—41no—
302SD16—41no—
303D8—21no—
304D8—22no—
305D8—22no—
306D8—21no—
307D8—41no—
308D8—41no—
309M16—41no—
310M16—41no—
311M16—41no—
312M16—41no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8C2×C3×D411yescharacteristic
317C2×C8C2×C3×D411yescharacteristic
318C2×C8C2×C3×D411yescharacteristic
319C2×C8C2×C3×D411yescharacteristic
320C2×C8C2×C3×D411yescharacteristic
321C2×C8C3×#4111yescharacteristic
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—41no—
330C42—41no—
331C16—22no—
332C16—41no—
333C22×C3—22no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3—41no—
337C22×C3—81no—
338C22×C3—82no—
339C22×C3—82no—
340C22×C3—82no—
341C22×C3—81no—
342C22×C3—82no—
343C22×C3—162no—
344C22×C3—162no—
345C22×C3#44711yescharacteristic
346C22×C3#44711yescharacteristic
347C22×C3#46611yescharacteristic
348C3×C4—41no—
349C3×C4—82no—
350C3×C4—82no—
351C3×C4—81no—
352C3×C4#39211yescharacteristic
353C3×C4#44911yescharacteristic
354C3×C4#44911yescharacteristic
355C3×C4#46911yescharacteristic
356C23—21no—
357C23—21no—
358C23—21no—
359C23—42no—
360C23—41no—
361C23—42no—
362C23—82no—
363C23—82no—
364C23—81no—
365C23—81no—
366C23C2×C3×D811yescharacteristic
367Q8—41no—
368Q8—41no—
369D4—42no—
370D4—41no—
371D4—42no—
372D4—41no—
373D4—88no—
374D4—84no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4C3×#12011yescharacteristic
393C2×C4C3×#12011yescharacteristic
394C2×C4C3×#12111yescharacteristic
395C2×C4C3×#13511yescharacteristic
396C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
397C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
398C8—41no—
399C8C3×#12011yescharacteristic
400C8C3×#12111yescharacteristic
401C8C3×#12111yescharacteristic
402C8C3×#12711yescharacteristic
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—81no—
406C2×C3—161no—
407C2×C3—162no—
408C2×C3#199911yescharacteristic
409C2×C3#219012yes—
410C22—22no—
411C22—42no—
412C22—41no—
413C22—42no—
414C22—81no—
415C22—82no—
416C22—82no—
417C22—81no—
418C22—82no—
419C22—82no—
420C22—162no—
421C22—162no—
422C22C3×#44711yescharacteristic
423C22C3×#44711yescharacteristic
424C22C3×#46611yescharacteristic
425C4—41no—
426C4—82no—
427C4—81no—
428C4—82no—
429C4C3×#39211yescharacteristic
430C4C3×#44911yescharacteristic
431C4C3×#44911yescharacteristic
432C4C3×#46911yescharacteristic
433C3#1331611yescharacteristic, direct product factor, Sylow
434C2—21no—
435C2—21no—
436C2—81no—
437C2—162no—
438C2—161no—
439C2C3×#199911yescharacteristic
440C2C3×#219012yes—
subgroupG/H
center, socleC22×C3#447
commutator, Frattini, cosocleC22×C8C23×C3
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×M16 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#509C2semidirect
2. C3×#3054C2semidirect
3. C2×C3×#469C2semidirect
4. C2×C3×#360C2semidirect
5. C3×#2140C2semidirect
6. C2×C3×#349C2semidirect
7. C3×#1386C2semidirect
8. #13316C3direct (N × H)
32. C2×#509C2×C3semidirect
33. #3054C2×C3semidirect
34. C2×#469C2×C3semidirect
35. C2×#360C2×C3semidirect
36. #2140C2×C3semidirect
37. C2×#349C2×C3semidirect
38. #1386C2×C3semidirect
47. C2×C3×Q16D4semidirect
54. C2×C3×D8D4semidirect
55. C2×C3×D8D4semidirect
79. C3×M32D4semidirect
82. C3×#107D4semidirect
85. C3×#106D4semidirect
182. C2×Q16C3×D4semidirect
189. C2×D8C3×D4semidirect
190. C2×D8C3×D4semidirect
214. M32C3×D4semidirect
217. #107C3×D4semidirect
220. #106C3×D4semidirect
433. C3#13316direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5092
C3×#30542
C2×C3×#4692
C2×C3×#3602
C3×#21402
C2×C3×#3492
C3×#13862
#133163