Finite Group Databasebeta

← C3×#13401 #10632008

Subgroups — C3×#13401 #10632008

Order 768.

390 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3067C211yesmaximal, characteristic
2C3×#2155C211yesmaximal, characteristic, complemented
3C3×#2145C211yesmaximal, characteristic, complemented
4C3×#2138C211yesmaximal, characteristic, complemented
5C2×C3×#350C211yesmaximal, characteristic, complemented
6C3×#1812C211yesmaximal, characteristic, complemented
7C3×#1382C211yesmaximal, characteristic, complemented
8#13401C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#575—21no—
10C2×C3×#135C2211yescharacteristic
11C2×C3×#131C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14C3×#473—21no—
15C3×#472—21no—
16C3×#471—22no—
17C2×C3×#108C2211yescharacteristic
18C3×#427C2211yescharacteristic
19C3×#427C2211yescharacteristic
20C3×#413—21no—
21C3×#410—21no—
22C3×#365—22no—
23C3×#360—22no—
24C3×#358—21no—
25C3×#357—21no—
26C3×#350—22no—
27C3×H(3,ℤ/4)—21no—
28#3067C2×C311yescharacteristic
29#2155C2×C311yescharacteristic, complemented
30#2145C2×C311yescharacteristic, complemented
31#2138C2×C311yescharacteristic, complemented
32C2×#350C2×C311yescharacteristic, complemented
33#1812C2×C311yescharacteristic, complemented
34#1382C2×C311yescharacteristic, complemented
35C2×C3×#41—21no—
36C2×C3×#41D411yescharacteristic
37C3×#137—22no—
38C3×#137—41no—
39C3×AGL(1,ℤ/8)—22no—
40C3×AGL(1,ℤ/8)—41no—
41C3×#135—21no—
42C3×#135—21no—
43C3×#135—21no—
44C3×#135—21no—
45C3×#135—42no—
46C2×C3×Q16D411yescharacteristic
47C2×C3×SD16—21no—
48C2×C3×D8D411yescharacteristic
49C3×#131—21no—
50C3×#131—21no—
51C3×#131—21no—
52C3×#131—21no—
53C2×C3×M16—21no—
54C2×C3×M16—21no—
55C2×C3×M16D411yescharacteristic
56C22×C3×C8—21no—
57C22×C3×C8C2311yescharacteristic
58C3×#117—21no—
59C3×#117—21no—
60C3×M32—21no—
61C3×M32D412yes—
62C2×C3×C16—21no—
63C2×C3×C16—21no—
64C2×C3×C16—21no—
65C2×C3×C16—21no—
66C3×#108—21no—
67C3×#108D412yes—
68C3×#107—21no—
69C3×#107—21no—
70C3×#106—21no—
71C3×#106—21no—
72C3×#101—22no—
73C3×#100—22no—
74C3×#99—42no—
75#575—21no—
76C2×#135C22×C311yescharacteristic
77C2×#131C22×C311yescharacteristic
78C2×M32C22×C311yescharacteristic
79C22×C16C22×C311yescharacteristic
80#473—21no—
81#472—21no—
82#471—22no—
83C2×#108C22×C311yescharacteristic
84#427C22×C311yescharacteristic
85#427C22×C311yescharacteristic
86#413—21no—
87#410—21no—
88#365—22no—
89#360—22no—
90#358—21no—
91#357—21no—
92#350—22no—
93H(3,ℤ/4)—21no—
94C3×#41—21no—
95C3×#41—21no—
96C3×#41—21no—
97C3×#41—21no—
98C3×#41—41no—
99C3×#41—41no—
100C3×#41—41no—
101C3×#41—41no—
102C2×C3×Q8—21no—
103C2×C3×Q8SD1611yescharacteristic
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—41no—
107C2×C3×D4SD1611yescharacteristic
108C22×C3×C4—21no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C3×Q16—21no—
112C3×Q16—21no—
113C3×Q16—42no—
114C3×SD16—42no—
115C3×SD16—42no—
116C3×SD16—41no—
117C3×SD16—41no—
118C3×D8—21no—
119C3×D8—21no—
120C3×D8—42no—
121C3×M16—21no—
122C3×M16—22no—
123C3×M16—21no—
124C3×M16—41no—
125C3×M16—41no—
126C3×M16—41no—
127C2×C3×C8—21no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8C2×D411yescharacteristic
132C2×C3×C8C2×D411yescharacteristic
133C2×C3×C8C2×D411yescharacteristic
134C2×C3×C8#4111yescharacteristic
135C2×C3×C8#4111yescharacteristic
136C2×C3×C8#4111yescharacteristic
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—41no—
142C3×C42—41no—
143C3×C16—22no—
144C3×C16—41no—
145C2×#41—21no—
146C2×#41C3×D411yescharacteristic
147#137—22no—
148#137—41no—
149AGL(1,ℤ/8)—22no—
150AGL(1,ℤ/8)—41no—
151#135—21no—
152#135—21no—
153#135—21no—
154#135—21no—
155#135—42no—
156C2×Q16C3×D411yescharacteristic
157C2×SD16—21no—
158C2×D8C3×D411yescharacteristic
159#131—21no—
160#131—21no—
161#131—21no—
162#131—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16C3×D411yescharacteristic
166C22×C8—21no—
167C22×C8C23×C311yescharacteristic
168#117—21no—
169#117—21no—
170M32—21no—
171M32C3×D412yes—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176#108—21no—
177#108C3×D412yes—
178#107—21no—
179#107—21no—
180#106—21no—
181#106—21no—
182#101—22no—
183#100—22no—
184#99—42no—
185C23×C3—21no—
186C23×C3—41no—
187C23×C3AGL(1,ℤ/8)11yescharacteristic
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—41no—
191C3×Q8—41no—
192C3×D4—21no—
193C3×D4—21no—
194C3×D4—41no—
195C3×D4—41no—
196C3×D4—41no—
197C3×D4—41no—
198C3×D4—81no—
199C3×D4—81no—
200C2×C3×C4—21no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12311yescharacteristic
211C2×C3×C4C2×SD1611yescharacteristic
212C2×C3×C4#13511yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C3×C8—41no—
215C3×C8—42no—
216C3×C8#12111yescharacteristic
217C3×C8#12111yescharacteristic
218C3×C8#12311yescharacteristic
219C3×C8#12411yescharacteristic
220#41—21no—
221#41—21no—
222#41—21no—
223#41—21no—
224#41—41no—
225#41—41no—
226#41—41no—
227#41—41no—
228C2×Q8—21no—
229C2×Q8C3×SD1611yescharacteristic
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—41no—
233C2×D4C3×SD1611yescharacteristic
234C22×C4—21no—
235C22×C4—41no—
236C22×C4C2×C3×D411yescharacteristic
237Q16—21no—
238Q16—21no—
239Q16—42no—
240SD16—42no—
241SD16—42no—
242SD16—41no—
243SD16—41no—
244D8—21no—
245D8—21no—
246D8—42no—
247M16—21no—
248M16—22no—
249M16—21no—
250M16—41no—
251M16—41no—
252M16—41no—
253C2×C8—21no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8C2×C3×D411yescharacteristic
258C2×C8C2×C3×D411yescharacteristic
259C2×C8C2×C3×D411yescharacteristic
260C2×C8C3×#4111yescharacteristic
261C2×C8C3×#4111yescharacteristic
262C2×C8C3×#4111yescharacteristic
263C4⋊C4—41no—
264C4⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C42—41no—
268C42—41no—
269C16—22no—
270C16—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—81no—
274C22×C3—81no—
275C22×C3—161no—
276C22×C3#44911yescharacteristic
277C22×C3#46911yescharacteristic
278C22×C3#48111yescharacteristic
279C3×C4—41no—
280C3×C4—81no—
281C3×C4—161no—
282C3×C4—161no—
283C3×C4#39411yescharacteristic
284C3×C4#45011yescharacteristic
285C3×C4#46511yescharacteristic
286C3×C4#48211yescharacteristic
287C23—21no—
288C23—41no—
289C23C3×AGL(1,ℤ/8)11yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—41no—
293Q8—41no—
294D4—21no—
295D4—21no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—81no—
301D4—81no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4C3×#12011yescharacteristic
311C2×C4C3×#12111yescharacteristic
312C2×C4C3×#12311yescharacteristic
313C2×C4C2×C3×SD1611yescharacteristic
314C2×C4C3×#13511yescharacteristic
315C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
316C8—41no—
317C8—42no—
318C8C3×#12111yescharacteristic
319C8C3×#12111yescharacteristic
320C8C3×#12311yescharacteristic
321C8C3×#12411yescharacteristic
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—81no—
326C2×C3—161no—
327C2×C3#203411yescharacteristic
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22—161no—
333C22C3×#44911yescharacteristic
334C22C3×#46911yescharacteristic
335C22C3×#48111yescharacteristic
336C4—41no—
337C4—81no—
338C4—161no—
339C4—161no—
340C4C3×#39411yescharacteristic
341C4C3×#45011yescharacteristic
342C4C3×#46511yescharacteristic
343C4C3×#48211yescharacteristic
344C3#1340111yescharacteristic, direct product factor, Sylow
345C2—21no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—161no—
350C2C3×#203411yescharacteristic
subgroupG/H
centerC3×C4#482
commutator, Frattini, cosocleC22×C8C23×C3
socleC2×C3#2034
prime radicalC3×#3067C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2155C2semidirect
3. C3×#2145C2semidirect
4. C3×#2138C2semidirect
5. C2×C3×#350C2semidirect
6. C3×#1812C2semidirect
7. C3×#1382C2semidirect
8. #13401C3direct (N × H)
29. #2155C2×C3semidirect
30. #2145C2×C3semidirect
31. #2138C2×C3semidirect
32. C2×#350C2×C3semidirect
33. #1812C2×C3semidirect
34. #1382C2×C3semidirect
344. C3#13401direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30672
C3×#21552
C3×#21452
C3×#21382
C2×C3×#3502
C3×#18122
C3×#13822
#134013