Finite Group Databasebeta

← C3×#13386 #10631993

Subgroups — C3×#13386 #10631993

Order 768.

432 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3054C211yesmaximal, characteristic
2C3×#2159C211yesmaximal, characteristic, complemented
3C2×C3×#360C211yesmaximal, characteristic, complemented
4C3×#2139C211yesmaximal, characteristic, complemented
5C2×C3×#349C211yesmaximal, characteristic, complemented
6C3×#1809C211yesmaximal, characteristic, complemented
7C3×#1386C211yesmaximal, characteristic, complemented
8#13386C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#135C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C3×#525—21no—
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C3×#470—21no—
15C3×#469—21no—
16C3×#468—21no—
17C3×#468—21no—
18C3×#427C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C3×#416—21no—
22C3×#380—21no—
23C3×#367—21no—
24C3×#366—21no—
25C3×#360—22no—
26C3×#358—21no—
27C3×#357—21no—
28C3×#349—22no—
29C3×#340—21no—
30#3054C2×C311yescharacteristic
31#2159C2×C311yescharacteristic, complemented
32C2×#360C2×C311yescharacteristic, complemented
33#2139C2×C311yescharacteristic, complemented
34C2×#349C2×C311yescharacteristic, complemented
35#1809C2×C311yescharacteristic, complemented
36#1386C2×C311yescharacteristic, complemented
37C2×C3×#41—21no—
38C23×C3×C4D411yescharacteristic
39C3×#135—21no—
40C3×#135—21no—
41C3×#135—21no—
42C3×#135—21no—
43C2×C3×Q16D411yescharacteristic
44C2×C3×SD16—21no—
45C2×C3×D8D411yescharacteristic
46C2×C3×M16—21no—
47C2×C3×M16—21no—
48C2×C3×M16—21no—
49C2×C3×M16—21no—
50C2×C3×M16—21no—
51C2×C3×M16—21no—
52C22×C3×C8D411yescharacteristic
53C22×C3×C8C2311yescharacteristic
54C3×#123—41no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#120—41no—
60C3×C4×D4—41no—
61C3×C4×D4—41no—
62C3×#117—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C22⋊C4—41no—
66C3×M32—21no—
67C3×M32—21no—
68C3×M32—21no—
69C3×M32D412yescomplemented
70C2×C3×C16—21no—
71C2×C3×C16—21no—
72C3×#107—21no—
73C3×#107—21no—
74C3×#107D411yescharacteristic
75C3×#107D411yescharacteristic
76C3×#106—21no—
77C3×#106—21no—
78C3×#106—21no—
79C3×#103—21no—
80C3×#103—21no—
81C3×#102—21no—
82C3×#102—21no—
83C3×#95—41no—
84C2×#135C22×C311yescharacteristic
85C22×M16C22×C311yescharacteristic
86#525—21no—
87C2×M32C22×C311yescharacteristic
88C2×M32C22×C311yescharacteristic
89#470—21no—
90#469—21no—
91#468—21no—
92#468—21no—
93#427C22×C311yescharacteristic
94C2×#107C22×C311yescharacteristic
95C2×#106C22×C311yescharacteristic
96#416—21no—
97#380—21no—
98#367—21no—
99#366—21no—
100#360—22no—
101#358—21no—
102#357—21no—
103#349—22no—
104#340—21no—
105C24×C3D811yescharacteristic
106C3×#41—41no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—41no—
110C2×C3×Q8—21no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4D811yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C3×Q16—21no—
127C3×Q16—21no—
128C3×SD16—41no—
129C3×SD16—41no—
130C3×D8—21no—
131C3×D8—21no—
132C3×M16—41no—
133C3×M16—41no—
134C3×M16—41no—
135C3×M16—41no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8C2×D411yescharacteristic
140C2×C3×C8C2×D411yescharacteristic
141C2×C3×C8C2×D411yescharacteristic
142C2×C3×C8#4111yescharacteristic
143C2×C3×C8#4111yescharacteristic
144C2×C3×C8#4111yescharacteristic
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—82no—
159C3×C42—41no—
160C3×C16—22no—
161C3×C16—41no—
162C2×#41—21no—
163C23×C4C3×D411yescharacteristic
164#135—21no—
165#135—21no—
166#135—21no—
167#135—21no—
168C2×Q16C3×D411yescharacteristic
169C2×SD16—21no—
170C2×D8C3×D411yescharacteristic
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C22×C8C3×D411yescharacteristic
178C22×C8C23×C311yescharacteristic
179#123—41no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182#121—21no—
183#121—21no—
184#120—41no—
185C4×D4—41no—
186C4×D4—41no—
187#117—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C22⋊C4—41no—
191M32—21no—
192M32—21no—
193M32—21no—
194M32C3×D412yescomplemented
195C2×C16—21no—
196C2×C16—21no—
197#107—21no—
198#107—21no—
199#107C3×D411yescharacteristic
200#107C3×D411yescharacteristic
201#106—21no—
202#106—21no—
203#106—21no—
204#103—21no—
205#103—21no—
206#102—21no—
207#102—21no—
208#95—41no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3C2×D811yescharacteristic
216C3×Q8—41no—
217C3×Q8—41no—
218C3×D4—41no—
219C3×D4—41no—
220C3×D4—82no—
221C3×D4—82no—
222C3×D4—81no—
223C3×D4—81no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—81no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12311yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4#13711yescharacteristic
248C3×C8—41no—
249C3×C8#12111yescharacteristic
250C3×C8#12111yescharacteristic
251C3×C8#12311yescharacteristic
252C3×C8#12411yescharacteristic
253C24C3×D811yescharacteristic
254#41—41no—
255#41—41no—
256#41—41no—
257#41—41no—
258C2×Q8—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4C3×D811yescharacteristic
273C22×C4C2×C3×D411yescharacteristic
274Q16—21no—
275Q16—21no—
276SD16—41no—
277SD16—41no—
278D8—21no—
279D8—21no—
280M16—41no—
281M16—41no—
282M16—41no—
283M16—41no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8C2×C3×D411yescharacteristic
288C2×C8C2×C3×D411yescharacteristic
289C2×C8C2×C3×D411yescharacteristic
290C2×C8C3×#4111yescharacteristic
291C2×C8C3×#4111yescharacteristic
292C2×C8C3×#4111yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—82no—
307C42—41no—
308C16—22no—
309C16—41no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—42no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3#44711yescharacteristic
321C22×C3#46611yescharacteristic
322C22×C3#48011yescharacteristic
323C3×C4—41no—
324C3×C4—81no—
325C3×C4—81no—
326C3×C4—81no—
327C3×C4—161no—
328C3×C4—161no—
329C3×C4#39411yescharacteristic
330C3×C4#45211yescharacteristic
331C3×C4#46811yescharacteristic
332C3×C4#48211yescharacteristic
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23C2×C3×D811yescharacteristic
340Q8—41no—
341Q8—41no—
342D4—41no—
343D4—41no—
344D4—82no—
345D4—82no—
346D4—81no—
347D4—81no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4C3×#12011yescharacteristic
367C2×C4C3×#12111yescharacteristic
368C2×C4C3×#12311yescharacteristic
369C2×C4C3×#13511yescharacteristic
370C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
371C2×C4C3×#13711yescharacteristic
372C8—41no—
373C8C3×#12111yescharacteristic
374C8C3×#12111yescharacteristic
375C8C3×#12311yescharacteristic
376C8C3×#12411yescharacteristic
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—81no—
380C2×C3—161no—
381C2×C3#203311yescharacteristic
382C2×C3#219011yescharacteristic
383C2×C3#219111yescharacteristic
384C22—21no—
385C22—21no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22C3×#44711yescharacteristic
395C22C3×#46611yescharacteristic
396C22C3×#48011yescharacteristic
397C4—41no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—161no—
402C4—161no—
403C4C3×#39411yescharacteristic
404C4C3×#45211yescharacteristic
405C4C3×#46811yescharacteristic
406C4C3×#48211yescharacteristic
407C3#1338611yescharacteristic, direct product factor, Sylow
408C2—21no—
409C2—21no—
410C2—81no—
411C2—161no—
412C2C3×#203311yescharacteristic
413C2C3×#219011yescharacteristic
414C2C3×#219111yescharacteristic
subgroupG/H
center, socleC22×C3#447
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC3×#3054C2
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
2. C3×#2159C2semidirect
3. C2×C3×#360C2semidirect
4. C3×#2139C2semidirect
5. C2×C3×#349C2semidirect
6. C3×#1809C2semidirect
7. C3×#1386C2semidirect
8. #13386C3direct (N × H)
31. #2159C2×C3semidirect
32. C2×#360C2×C3semidirect
33. #2139C2×C3semidirect
34. C2×#349C2×C3semidirect
35. #1809C2×C3semidirect
36. #1386C2×C3semidirect
69. C3×M32D4semidirect
194. M32C3×D4semidirect
407. C3#13386direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30542
C3×#21592
C2×C3×#3602
C3×#21392
C2×C3×#3492
C3×#18092
C3×#13862
#133863