Finite Group Databasebeta

← C3×#13310 #10631917

Subgroups — C3×#13310 #10631917

Order 768.

552 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#509C211yesmaximal, characteristic, complemented
2C3×#3067C211yesmaximal, characteristic, complemented
3C3×#3058C211yesmaximal, characteristic, complemented
4C2×C3×#361C211yesmaximal, characteristic, complemented
5C3×#2145C211yesmaximal, characteristic, complemented
6C3×#2117C211yesmaximal, characteristic, complemented
7C3×#1389C211yesmaximal, characteristic, complemented
8#13310C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#575—21no—
10C2×C3×#135C2211yescharacteristic
11C22×C3×D8C2211yescharacteristic
12C2×C3×#131C2211yescharacteristic
13C3×#535—21no—
14C3×#509—22no—
15C3×#509—22no—
16C2×C3×SD32—21no—
17C2×C3×D16—21no—
18C2×C3×M32C2211yescharacteristic
19C2×C3×M32C2211yescharacteristic
20C3×#473—21no—
21C3×#472—21no—
22C3×#471—22no—
23C3×#469—22no—
24C3×#466—22no—
25C2×C3×#108C2211yescharacteristic
26C3×#427C2211yescharacteristic
27C3×#418—21no—
28C3×#413—21no—
29C3×#361—22no—
30C3×#360—22no—
31C3×#350—22no—
32C2×#509C2×C311yescharacteristic, complemented
33#3067C2×C311yescharacteristic, complemented
34#3058C2×C311yescharacteristic, complemented
35C2×#361C2×C311yescharacteristic, complemented
36#2145C2×C311yescharacteristic, complemented
37#2117C2×C311yescharacteristic, complemented
38#1389C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C2×C3×#41D411yescharacteristic
41C22×C3×D4—21no—
42C3×#137—22no—
43C3×#137—41no—
44C3×AGL(1,ℤ/8)—22no—
45C3×AGL(1,ℤ/8)—41no—
46C3×#135—21no—
47C3×#135—21no—
48C3×#135—21no—
49C3×#135—21no—
50C3×#135—42no—
51C2×C3×Q16D411yescharacteristic, complemented
52C2×C3×SD16—21no—
53C2×C3×D8—21no—
54C2×C3×D8—21no—
55C2×C3×D8—21no—
56C2×C3×D8—21no—
57C2×C3×D8—21no—
58C2×C3×D8D412yes—
59C2×C3×D8D411yescharacteristic, complemented
60C3×#131—21no—
61C3×#131—21no—
62C3×#131—21no—
63C3×#131—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16D411yescharacteristic
67C22×C3×C8—21no—
68C22×C3×C8C2311yescharacteristic
69C3×#127—41no—
70C3×#124—41no—
71C3×#121—24no—
72C3×#120—42no—
73C3×#117—21no—
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×#108—21no—
83C3×#108D412yescomplemented
84C3×#107—21no—
85C3×#106D412yes—
86C3×#102—24no—
87C3×#101—22no—
88C3×#100—22no—
89#575—21no—
90C2×#135C22×C311yescharacteristic
91C22×D8C22×C311yescharacteristic
92C2×#131C22×C311yescharacteristic
93#535—21no—
94#509—22no—
95#509—22no—
96C2×SD32—21no—
97C2×D16—21no—
98C2×M32C22×C311yescharacteristic
99C2×M32C22×C311yescharacteristic
100#473—21no—
101#472—21no—
102#471—22no—
103#469—22no—
104#466—22no—
105C2×#108C22×C311yescharacteristic
106#427C22×C311yescharacteristic
107#418—21no—
108#413—21no—
109#361—22no—
110#360—22no—
111#350—22no—
112C24×C3—41no—
113C3×#41—21no—
114C3×#41—21no—
115C3×#41—21no—
116C3×#41—21no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C2×C3×Q8—21no—
122C2×C3×Q8D811yescharacteristic
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—22no—
126C2×C3×D4—41no—
127C2×C3×D4—44no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4—41no—
133C2×C3×D4D811yescharacteristic
134C22×C3×C4—21no—
135C22×C3×C4—41no—
136C22×C3×C4C2×D411yescharacteristic
137C3×Q16—21no—
138C3×Q16—21no—
139C3×Q16—42no—
140C3×SD16—42no—
141C3×SD16—42no—
142C3×SD16—41no—
143C3×SD16—41no—
144C3×D8—21no—
145C3×D8—22no—
146C3×D8—22no—
147C3×D8—21no—
148C3×D8—42no—
149C3×D8—41no—
150C3×D8—41no—
151C3×M16—21no—
152C3×M16—22no—
153C3×M16—21no—
154C3×M16—41no—
155C3×M16—41no—
156C3×M16—41no—
157C2×C3×C8—21no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C2×C3×C8—41no—
161C2×C3×C8C2×D411yescharacteristic
162C2×C3×C8C2×D411yescharacteristic
163C2×C3×C8C2×D411yescharacteristic
164C2×C3×C8C2×D411yescharacteristic
165C2×C3×C8C2×D411yescharacteristic
166C2×C3×C8#4111yescharacteristic
167C3×C4⋊C4—22no—
168C3×C22⋊C4—44no—
169C3×C22⋊C4—41no—
170C3×C42—41no—
171C3×C16—22no—
172C3×C16—41no—
173C2×#41—21no—
174C2×#41C3×D411yescharacteristic
175C22×D4—21no—
176#137—22no—
177#137—41no—
178AGL(1,ℤ/8)—22no—
179AGL(1,ℤ/8)—41no—
180#135—21no—
181#135—21no—
182#135—21no—
183#135—21no—
184#135—42no—
185C2×Q16C3×D411yescharacteristic, complemented
186C2×SD16—21no—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8—21no—
190C2×D8—21no—
191C2×D8—21no—
192C2×D8C3×D412yes—
193C2×D8C3×D411yescharacteristic, complemented
194#131—21no—
195#131—21no—
196#131—21no—
197#131—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16C3×D411yescharacteristic
201C22×C8—21no—
202C22×C8C23×C311yescharacteristic
203#127—41no—
204#124—41no—
205#121—24no—
206#120—42no—
207#117—21no—
208SD32—24no—
209D16—24no—
210M32—21no—
211M32—21no—
212M32—21no—
213M32C3×D412yescomplemented
214C2×C16—21no—
215C2×C16—21no—
216#108—21no—
217#108C3×D412yescomplemented
218#107—21no—
219#106C3×D412yes—
220#102—24no—
221#101—22no—
222#100—22no—
223C23×C3—21no—
224C23×C3—42no—
225C23×C3—41no—
226C23×C3—81no—
227C23×C3—81no—
228C23×C3—162no—
229C23×C3AGL(1,ℤ/8)11yescharacteristic
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—41no—
237C3×D4—41no—
238C3×D4—42no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—42no—
242C3×D4—81no—
243C3×D4—81no—
244C3×D4—81no—
245C3×D4—81no—
246C3×D4—164no—
247C2×C3×C4—21no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4C2×D811yescharacteristic
258C2×C3×C4#13511yescharacteristic
259C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
260C3×C8—41no—
261C3×C8—42no—
262C3×C8#12011yescharacteristic
263C3×C8#12111yescharacteristic
264C3×C8#12111yescharacteristic
265C3×C8#12711yescharacteristic
266C24—41no—
267#41—21no—
268#41—21no—
269#41—21no—
270#41—21no—
271#41—41no—
272#41—41no—
273#41—41no—
274#41—41no—
275C2×Q8—21no—
276C2×Q8C3×D811yescharacteristic
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—44no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4C3×D811yescharacteristic
288C22×C4—21no—
289C22×C4—41no—
290C22×C4C2×C3×D411yescharacteristic
291Q16—21no—
292Q16—21no—
293Q16—42no—
294SD16—42no—
295SD16—42no—
296SD16—41no—
297SD16—41no—
298D8—21no—
299D8—22no—
300D8—22no—
301D8—21no—
302D8—42no—
303D8—41no—
304D8—41no—
305M16—21no—
306M16—22no—
307M16—21no—
308M16—41no—
309M16—41no—
310M16—41no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8C2×C3×D411yescharacteristic
316C2×C8C2×C3×D411yescharacteristic
317C2×C8C2×C3×D411yescharacteristic
318C2×C8C2×C3×D411yescharacteristic
319C2×C8C2×C3×D411yescharacteristic
320C2×C8C3×#4111yescharacteristic
321C4⋊C4—22no—
322C22⋊C4—44no—
323C22⋊C4—41no—
324C42—41no—
325C16—22no—
326C16—41no—
327C22×C3—41no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—84no—
331C22×C3—81no—
332C22×C3—82no—
333C22×C3—161no—
334C22×C3—162no—
335C22×C3—162no—
336C22×C3#44711yescharacteristic
337C22×C3#44911yescharacteristic
338C22×C3#46911yescharacteristic
339C3×C4—41no—
340C3×C4—82no—
341C3×C4—81no—
342C3×C4#39211yescharacteristic
343C3×C4#44711yescharacteristic
344C3×C4#44911yescharacteristic
345C3×C4#46611yescharacteristic
346C23—21no—
347C23—42no—
348C23—41no—
349C23—81no—
350C23—81no—
351C23—162no—
352C23C3×AGL(1,ℤ/8)11yescharacteristic
353Q8—21no—
354Q8—21no—
355Q8—41no—
356Q8—41no—
357D4—21no—
358D4—21no—
359D4—41no—
360D4—41no—
361D4—42no—
362D4—41no—
363D4—41no—
364D4—42no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—164no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4C3×#12011yescharacteristic
378C2×C4C3×#12011yescharacteristic
379C2×C4C3×#12111yescharacteristic
380C2×C4C2×C3×D811yescharacteristic
381C2×C4C3×#13511yescharacteristic
382C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
383C8—41no—
384C8—42no—
385C8C3×#12011yescharacteristic
386C8C3×#12111yescharacteristic
387C8C3×#12111yescharacteristic
388C8C3×#12711yescharacteristic
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—81no—
393C2×C3—162no—
394C2×C3—161no—
395C2×C3#199911yescharacteristic
396C22—41no—
397C22—42no—
398C22—41no—
399C22—84no—
400C22—81no—
401C22—82no—
402C22—161no—
403C22—162no—
404C22—162no—
405C22C3×#44711yescharacteristic
406C22C3×#44911yescharacteristic
407C22C3×#46911yescharacteristic
408C4—41no—
409C4—82no—
410C4—81no—
411C4C3×#39211yescharacteristic
412C4C3×#44711yescharacteristic
413C4C3×#44911yescharacteristic
414C4C3×#46611yescharacteristic
415C3#1331011yescharacteristic, direct product factor, Sylow
416C2—21no—
417C2—21no—
418C2—21no—
419C2—81no—
420C2—162no—
421C2—161no—
422C2C3×#199911yescharacteristic
subgroupG/H
center, socleC2×C3#1999
commutator, Frattini, cosocleC22×C8C23×C3
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#509C2semidirect
2. C3×#3067C2semidirect
3. C3×#3058C2semidirect
4. C2×C3×#361C2semidirect
5. C3×#2145C2semidirect
6. C3×#2117C2semidirect
7. C3×#1389C2semidirect
8. #13310C3direct (N × H)
32. C2×#509C2×C3semidirect
33. #3067C2×C3semidirect
34. #3058C2×C3semidirect
35. C2×#361C2×C3semidirect
36. #2145C2×C3semidirect
37. #2117C2×C3semidirect
38. #1389C2×C3semidirect
51. C2×C3×Q16D4semidirect
59. C2×C3×D8D4semidirect
79. C3×M32D4semidirect
83. C3×#108D4semidirect
185. C2×Q16C3×D4semidirect
193. C2×D8C3×D4semidirect
213. M32C3×D4semidirect
217. #108C3×D4semidirect
415. C3#13310direct (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×#30672
C3×#30582
C2×C3×#3612
C3×#21452
C3×#21172
C3×#13892
#133103