Finite Group Databasebeta

← C3×#13318 #10631925

Subgroups — C3×#13318 #10631925

Order 768.

532 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3414C211yesmaximal, characteristic, complemented
2C3×#3067C211yesmaximal, characteristic, complemented
3C3×#3060C211yesmaximal, characteristic, complemented
4C2×C3×#360C211yesmaximal, characteristic, complemented
5C3×#2143C211yesmaximal, characteristic, complemented
6C3×#2117C211yesmaximal, characteristic, complemented
7C3×#1383C211yesmaximal, characteristic, complemented
8#13318C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#575—21no—
10C2×C3×#135C2211yescharacteristic
11C2×C3×#135C2211yescharacteristic
12C2×C3×#131C2211yescharacteristic
13C3×#537—21no—
14C3×#510—21no—
15C3×#510—21no—
16C3×#509—21no—
17C3×#509—21no—
18C3×#508—22no—
19C2×C3×M32C2211yescharacteristic
20C2×C3×M32C2211yescharacteristic
21C3×#473—21no—
22C3×#472—21no—
23C3×#471—22no—
24C3×#470—21no—
25C3×#469—21no—
26C3×#465—22no—
27C2×C3×#108C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C3×#417—21no—
30C3×#413—21no—
31C3×#360—22no—
32C3×#359—22no—
33C3×#350—22no—
34#3414C2×C311yescharacteristic, complemented
35#3067C2×C311yescharacteristic, complemented
36#3060C2×C311yescharacteristic, complemented
37C2×#360C2×C311yescharacteristic, complemented
38#2143C2×C311yescharacteristic, complemented
39#2117C2×C311yescharacteristic, complemented
40#1383C2×C311yescharacteristic, complemented
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41D411yescharacteristic
44C3×#137—22no—
45C3×#137—41no—
46C3×AGL(1,ℤ/8)—22no—
47C3×AGL(1,ℤ/8)—41no—
48C3×#135—21no—
49C3×#135—21no—
50C3×#135—21no—
51C3×#135—21no—
52C3×#135—21no—
53C3×#135—21no—
54C3×#135—21no—
55C3×#135—21no—
56C3×#135—42no—
57C2×C3×Q16D411yescharacteristic
58C2×C3×Q16D411yescharacteristic, complemented
59C2×C3×SD16—21no—
60C2×C3×SD16—21no—
61C2×C3×D8D411yescharacteristic
62C2×C3×D8D411yescharacteristic, complemented
63C3×#131—21no—
64C3×#131—21no—
65C3×#131—21no—
66C3×#131—21no—
67C2×C3×M16—21no—
68C2×C3×M16—21no—
69C2×C3×M16D411yescharacteristic
70C22×C3×C8—21no—
71C22×C3×C8C2311yescharacteristic
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—41no—
74C3×#121—22no—
75C3×#121—42no—
76C3×#121—41no—
77C2×C3×C4⋊C4—21no—
78C3×Q32—22no—
79C3×SD32—22no—
80C3×SD32—22no—
81C3×D16—22no—
82C3×M32—21no—
83C3×M32—21no—
84C3×M32—21no—
85C3×M32—21no—
86C2×C3×C16—21no—
87C2×C3×C16D412yescomplemented
88C3×#108—21no—
89C3×#108D412yescomplemented
90C3×#106—21no—
91C3×#106D412yes—
92C3×#103—22no—
93C3×#102—22no—
94C3×#101—22no—
95C3×#100—22no—
96#575—21no—
97C2×#135C22×C311yescharacteristic
98C2×#135C22×C311yescharacteristic
99C2×#131C22×C311yescharacteristic
100#537—21no—
101#510—21no—
102#510—21no—
103#509—21no—
104#509—21no—
105#508—22no—
106C2×M32C22×C311yescharacteristic
107C2×M32C22×C311yescharacteristic
108#473—21no—
109#472—21no—
110#471—22no—
111#470—21no—
112#469—21no—
113#465—22no—
114C2×#108C22×C311yescharacteristic
115C2×#106C22×C311yescharacteristic
116#417—21no—
117#413—21no—
118#360—22no—
119#359—22no—
120#350—22no—
121C3×#41—21no—
122C3×#41—21no—
123C3×#41—21no—
124C3×#41—21no—
125C3×#41—41no—
126C3×#41—41no—
127C3×#41—41no—
128C3×#41—41no—
129C3×#41—41no—
130C3×#41—41no—
131C3×#41—41no—
132C3×#41—41no—
133C2×C3×Q8—21no—
134C2×C3×Q8—21no—
135C2×C3×Q8D811yescharacteristic
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—21no—
139C2×C3×D4—41no—
140C2×C3×D4—41no—
141C2×C3×D4—42no—
142C2×C3×D4D811yescharacteristic
143C22×C3×C4—21no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4C2×D411yescharacteristic
148C3×Q16—21no—
149C3×Q16—21no—
150C3×Q16—21no—
151C3×Q16—21no—
152C3×Q16—42no—
153C3×SD16—42no—
154C3×SD16—42no—
155C3×SD16—41no—
156C3×SD16—41no—
157C3×SD16—41no—
158C3×SD16—41no—
159C3×D8—21no—
160C3×D8—21no—
161C3×D8—21no—
162C3×D8—21no—
163C3×D8—42no—
164C3×M16—21no—
165C3×M16—22no—
166C3×M16—21no—
167C3×M16—41no—
168C3×M16—41no—
169C3×M16—41no—
170C2×C3×C8—21no—
171C2×C3×C8—41no—
172C2×C3×C8—41no—
173C2×C3×C8—41no—
174C2×C3×C8C2×D411yescharacteristic
175C2×C3×C8C2×D411yescharacteristic
176C2×C3×C8C2×D411yescharacteristic
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8C2×D411yescharacteristic
179C2×C3×C8#4111yescharacteristic
180C3×C4⋊C4—22no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—41no—
183C3×C22⋊C4—42no—
184C3×C22⋊C4—42no—
185C3×C16—22no—
186C3×C16—41no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41C3×D411yescharacteristic
190#137—22no—
191#137—41no—
192AGL(1,ℤ/8)—22no—
193AGL(1,ℤ/8)—41no—
194#135—21no—
195#135—21no—
196#135—21no—
197#135—21no—
198#135—21no—
199#135—21no—
200#135—21no—
201#135—21no—
202#135—42no—
203C2×Q16C3×D411yescharacteristic
204C2×Q16C3×D411yescharacteristic, complemented
205C2×SD16—21no—
206C2×SD16—21no—
207C2×D8C3×D411yescharacteristic
208C2×D8C3×D411yescharacteristic, complemented
209#131—21no—
210#131—21no—
211#131—21no—
212#131—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16C3×D411yescharacteristic
216C22×C8—21no—
217C22×C8C23×C311yescharacteristic
218C22⋊Q8—22no—
219C22⋊Q8—41no—
220#121—22no—
221#121—42no—
222#121—41no—
223C2×C4⋊C4—21no—
224Q32—22no—
225SD32—22no—
226SD32—22no—
227D16—22no—
228M32—21no—
229M32—21no—
230M32—21no—
231M32—21no—
232C2×C16—21no—
233C2×C16C3×D412yescomplemented
234#108—21no—
235#108C3×D412yescomplemented
236#106—21no—
237#106C3×D412yes—
238#103—22no—
239#102—22no—
240#101—22no—
241#100—22no—
242C23×C3—21no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3AGL(1,ℤ/8)11yescharacteristic
246C3×Q8—21no—
247C3×Q8—21no—
248C3×Q8—41no—
249C3×Q8—41no—
250C3×Q8—41no—
251C3×Q8—41no—
252C3×D4—21no—
253C3×D4—21no—
254C3×D4—41no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—81no—
261C3×D4—81no—
262C3×D4—81no—
263C3×D4—81no—
264C3×D4—162no—
265C2×C3×C4—21no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—81no—
272C2×C3×C4—81no—
273C2×C3×C4—81no—
274C2×C3×C4—81no—
275C2×C3×C4—161no—
276C2×C3×C4#12011yescharacteristic
277C2×C3×C4#12011yescharacteristic
278C2×C3×C4#12111yescharacteristic
279C2×C3×C4C2×D811yescharacteristic
280C2×C3×C4#13511yescharacteristic
281C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
282C3×C8—41no—
283C3×C8—42no—
284C3×C8#12011yescharacteristic
285C3×C8#12111yescharacteristic
286C3×C8#12111yescharacteristic
287C3×C8#12711yescharacteristic
288#41—21no—
289#41—21no—
290#41—21no—
291#41—21no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8C3×D811yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—42no—
309C2×D4C3×D811yescharacteristic
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4C2×C3×D411yescharacteristic
315Q16—21no—
316Q16—21no—
317Q16—21no—
318Q16—21no—
319Q16—42no—
320SD16—42no—
321SD16—42no—
322SD16—41no—
323SD16—41no—
324SD16—41no—
325SD16—41no—
326D8—21no—
327D8—21no—
328D8—21no—
329D8—21no—
330D8—42no—
331M16—21no—
332M16—22no—
333M16—21no—
334M16—41no—
335M16—41no—
336M16—41no—
337C2×C8—21no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8C2×C3×D411yescharacteristic
342C2×C8C2×C3×D411yescharacteristic
343C2×C8C2×C3×D411yescharacteristic
344C2×C8C2×C3×D411yescharacteristic
345C2×C8C2×C3×D411yescharacteristic
346C2×C8C3×#4111yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C16—22no—
353C16—41no—
354C22×C3—41no—
355C22×C3—42no—
356C22×C3—41no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—161no—
362C22×C3#44711yescharacteristic
363C22×C3#44911yescharacteristic
364C22×C3#46911yescharacteristic
365C3×C4—41no—
366C3×C4—82no—
367C3×C4—81no—
368C3×C4—81no—
369C3×C4#39211yescharacteristic
370C3×C4#44711yescharacteristic
371C3×C4#44911yescharacteristic
372C3×C4#46611yescharacteristic
373C23—21no—
374C23—41no—
375C23—41no—
376C23C3×AGL(1,ℤ/8)11yescharacteristic
377Q8—21no—
378Q8—21no—
379Q8—41no—
380Q8—41no—
381Q8—41no—
382Q8—41no—
383D4—21no—
384D4—21no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—81no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—162no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—161no—
407C2×C4C3×#12011yescharacteristic
408C2×C4C3×#12011yescharacteristic
409C2×C4C3×#12111yescharacteristic
410C2×C4C2×C3×D811yescharacteristic
411C2×C4C3×#13511yescharacteristic
412C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
413C8—41no—
414C8—42no—
415C8C3×#12011yescharacteristic
416C8C3×#12111yescharacteristic
417C8C3×#12111yescharacteristic
418C8C3×#12711yescharacteristic
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—81no—
423C2×C3—161no—
424C2×C3—161no—
425C2×C3#199911yescharacteristic
426C22—41no—
427C22—42no—
428C22—41no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—161no—
434C22C3×#44711yescharacteristic
435C22C3×#44911yescharacteristic
436C22C3×#46911yescharacteristic
437C4—41no—
438C4—82no—
439C4—81no—
440C4—81no—
441C4C3×#39211yescharacteristic
442C4C3×#44711yescharacteristic
443C4C3×#44911yescharacteristic
444C4C3×#46611yescharacteristic
445C3#1331811yescharacteristic, direct product factor, Sylow
446C2—21no—
447C2—21no—
448C2—21no—
449C2—81no—
450C2—161no—
451C2—161no—
452C2C3×#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. C3×#3414C2semidirect
2. C3×#3067C2semidirect
3. C3×#3060C2semidirect
4. C2×C3×#360C2semidirect
5. C3×#2143C2semidirect
6. C3×#2117C2semidirect
7. C3×#1383C2semidirect
8. #13318C3direct (N × H)
34. #3414C2×C3semidirect
35. #3067C2×C3semidirect
36. #3060C2×C3semidirect
37. C2×#360C2×C3semidirect
38. #2143C2×C3semidirect
39. #2117C2×C3semidirect
40. #1383C2×C3semidirect
58. C2×C3×Q16D4semidirect
62. C2×C3×D8D4semidirect
87. C2×C3×C16D4semidirect
89. C3×#108D4semidirect
204. C2×Q16C3×D4semidirect
208. C2×D8C3×D4semidirect
233. C2×C16C3×D4semidirect
235. #108C3×D4semidirect
445. C3#13318direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34142
C3×#30672
C3×#30602
C2×C3×#3602
C3×#21432
C3×#21172
C3×#13832
#133183