Finite Group Databasebeta

← C3×#13324 #10631931

Subgroups — C3×#13324 #10631931

Order 768.

514 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#510C211yesmaximal, characteristic, complemented
2C3×#3067C211yesmaximal, characteristic
3C2×C3×#473C211yesmaximal, characteristic, complemented
4C2×C3×#362C211yesmaximal, characteristic, complemented
5C3×#2143C211yesmaximal, characteristic, complemented
6C3×#2115C211yesmaximal, characteristic, complemented
7C3×#1388C211yesmaximal, characteristic, complemented
8#13324C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#575—21no—
10C2×C3×#137—21no—
11C2×C3×#135C2211yescharacteristic
12C22×C3×Q16C2211yescharacteristic
13C2×C3×#131C2211yescharacteristic
14C3×#510—22no—
15C3×#510—22no—
16C2×C3×Q32—21no—
17C2×C3×SD32—21no—
18C2×C3×M32C2211yescharacteristic
19C2×C3×M32C2211yescharacteristic
20C3×#473—22no—
21C3×#473—21no—
22C3×#473—22no—
23C3×#472—21no—
24C3×#471—22no—
25C2×C3×#108C2211yescharacteristic
26C2×C3×#108C2211yescharacteristic
27C3×#413—21no—
28C2×C3×#101—21no—
29C3×#362—22no—
30C3×#359—22no—
31C3×#349—22no—
32C2×#510C2×C311yescharacteristic, complemented
33#3067C2×C311yescharacteristic
34C2×#473C2×C311yescharacteristic, complemented
35C2×#362C2×C311yescharacteristic, complemented
36#2143C2×C311yescharacteristic, complemented
37#2115C2×C311yescharacteristic, complemented
38#1388C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C2×C3×#41D411yescharacteristic
41C22×C3×Q8—21no—
42C3×#137—24no—
43C3×#137—22no—
44C3×#137—42no—
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—42no—
53C2×C3×Q16—21no—
54C2×C3×Q16—21no—
55C2×C3×Q16—21no—
56C2×C3×Q16—21no—
57C2×C3×Q16—21no—
58C2×C3×Q16—41no—
59C2×C3×Q16D411yescharacteristic
60C2×C3×Q16D412yes—
61C2×C3×SD16—21no—
62C2×C3×SD16—41no—
63C2×C3×D8D411yescharacteristic
64C3×#131—21no—
65C3×#131—21no—
66C3×#131—21no—
67C3×#131—21no—
68C2×C3×M16—21no—
69C2×C3×M16—21no—
70C2×C3×M16—21no—
71C2×C3×M16D411yescharacteristic
72C22×C3×C8—21no—
73C22×C3×C8C2311yescharacteristic
74C3×Q32—24no—
75C3×SD32—24no—
76C3×M32—21no—
77C3×M32—21no—
78C3×M32—21no—
79C3×M32D412yes—
80C2×C3×C16—21no—
81C2×C3×C16—21no—
82C3×#108—21no—
83C3×#108—21no—
84C3×#108D412yes—
85C3×#108D412yes—
86C3×#101—24no—
87C3×#101—22no—
88C3×#100—22no—
89#575—21no—
90C2×#137—21no—
91C2×#135C22×C311yescharacteristic
92C22×Q16C22×C311yescharacteristic
93C2×#131C22×C311yescharacteristic
94#510—22no—
95#510—22no—
96C2×Q32—21no—
97C2×SD32—21no—
98C2×M32C22×C311yescharacteristic
99C2×M32C22×C311yescharacteristic
100#473—22no—
101#473—21no—
102#473—22no—
103#472—21no—
104#471—22no—
105C2×#108C22×C311yescharacteristic
106C2×#108C22×C311yescharacteristic
107#413—21no—
108C2×#101—21no—
109#362—22no—
110#359—22no—
111#349—22no—
112C3×#41—21no—
113C3×#41—21no—
114C3×#41—21no—
115C3×#41—21no—
116C3×#41—41no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C2×C3×Q8—21no—
121C2×C3×Q8—22no—
122C2×C3×Q8—41no—
123C2×C3×Q8—41no—
124C2×C3×Q8—41no—
125C2×C3×Q8—41no—
126C2×C3×Q8—41no—
127C2×C3×Q8D811yescharacteristic
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—41no—
131C2×C3×D4D811yescharacteristic
132C22×C3×C4—21no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4C2×D411yescharacteristic
136C3×Q16—22no—
137C3×Q16—21no—
138C3×Q16—22no—
139C3×Q16—21no—
140C3×Q16—44no—
141C3×Q16—42no—
142C3×Q16—41no—
143C3×Q16—41no—
144C3×SD16—44no—
145C3×SD16—42no—
146C3×SD16—42no—
147C3×SD16—41no—
148C3×SD16—41no—
149C3×D8—21no—
150C3×D8—21no—
151C3×D8—42no—
152C3×M16—21no—
153C3×M16—22no—
154C3×M16—21no—
155C3×M16—22no—
156C3×M16—41no—
157C3×M16—41no—
158C3×M16—41no—
159C3×M16—41no—
160C2×C3×C8—21no—
161C2×C3×C8—41no—
162C2×C3×C8—41no—
163C2×C3×C8—41no—
164C2×C3×C8—41no—
165C2×C3×C8C2×D411yescharacteristic
166C2×C3×C8C2×D411yescharacteristic
167C2×C3×C8C2×D411yescharacteristic
168C2×C3×C8C2×D411yescharacteristic
169C2×C3×C8C2×D411yescharacteristic
170C2×C3×C8#4111yescharacteristic
171C3×C16—22no—
172C3×C16—41no—
173C2×#41—21no—
174C2×#41C3×D411yescharacteristic
175C22×Q8—21no—
176#137—24no—
177#137—22no—
178#137—42no—
179#137—41no—
180AGL(1,ℤ/8)—22no—
181AGL(1,ℤ/8)—41no—
182#135—21no—
183#135—21no—
184#135—21no—
185#135—21no—
186#135—42no—
187C2×Q16—21no—
188C2×Q16—21no—
189C2×Q16—21no—
190C2×Q16—21no—
191C2×Q16—21no—
192C2×Q16—41no—
193C2×Q16C3×D412yes—
194C2×Q16C3×D411yescharacteristic
195C2×SD16—21no—
196C2×SD16—41no—
197C2×D8C3×D411yescharacteristic
198#131—21no—
199#131—21no—
200#131—21no—
201#131—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16C3×D411yescharacteristic
206C22×C8—21no—
207C22×C8C23×C311yescharacteristic
208Q32—24no—
209SD32—24no—
210M32—21no—
211M32—21no—
212M32—21no—
213M32C3×D412yes—
214C2×C16—21no—
215C2×C16—21no—
216#108—21no—
217#108—21no—
218#108C3×D412yes—
219#108C3×D412yes—
220#101—24no—
221#101—22no—
222#100—22no—
223C23×C3—21no—
224C23×C3—41no—
225C23×C3#13511yescharacteristic
226C3×Q8—21no—
227C3×Q8—21no—
228C3×Q8—41no—
229C3×Q8—42no—
230C3×Q8—42no—
231C3×Q8—41no—
232C3×Q8—81no—
233C3×Q8—81no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—41no—
237C3×D4—41no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—81no—
241C3×D4—81no—
242C2×C3×C4—21no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4C2×D811yescharacteristic
255C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
256C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
257C3×C8—41no—
258C3×C8—42no—
259C3×C8—42no—
260C3×C8#12011yescharacteristic
261C3×C8#12111yescharacteristic
262C3×C8#12111yescharacteristic
263C3×C8#12711yescharacteristic
264#41—21no—
265#41—21no—
266#41—21no—
267#41—21no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8C3×D811yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4C3×D811yescharacteristic
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4C2×C3×D411yescharacteristic
288Q16—22no—
289Q16—21no—
290Q16—22no—
291Q16—21no—
292Q16—44no—
293Q16—42no—
294Q16—41no—
295Q16—41no—
296SD16—44no—
297SD16—42no—
298SD16—42no—
299SD16—41no—
300SD16—41no—
301D8—21no—
302D8—21no—
303D8—42no—
304M16—21no—
305M16—22no—
306M16—21no—
307M16—22no—
308M16—41no—
309M16—41no—
310M16—41no—
311M16—41no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8C2×C3×D411yescharacteristic
318C2×C8C2×C3×D411yescharacteristic
319C2×C8C2×C3×D411yescharacteristic
320C2×C8C2×C3×D411yescharacteristic
321C2×C8C2×C3×D411yescharacteristic
322C2×C8C3×#4111yescharacteristic
323C16—22no—
324C16—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—161no—
330C22×C3#44911yescharacteristic
331C22×C3#44911yescharacteristic
332C22×C3#46611yescharacteristic
333C3×C4—41no—
334C3×C4—81no—
335C3×C4—82no—
336C3×C4#39211yescharacteristic
337C3×C4#44711yescharacteristic
338C3×C4#44711yescharacteristic
339C3×C4#46911yescharacteristic
340C23—21no—
341C23—41no—
342C23C3×#13511yescharacteristic
343Q8—21no—
344Q8—21no—
345Q8—41no—
346Q8—42no—
347Q8—42no—
348Q8—41no—
349Q8—81no—
350Q8—81no—
351D4—21no—
352D4—21no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—81no—
358D4—81no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4C3×#12011yescharacteristic
369C2×C4C3×#12011yescharacteristic
370C2×C4C3×#12111yescharacteristic
371C2×C4C2×C3×D811yescharacteristic
372C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
373C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
374C8—41no—
375C8—42no—
376C8—42no—
377C8C3×#12011yescharacteristic
378C8C3×#12111yescharacteristic
379C8C3×#12111yescharacteristic
380C8C3×#12711yescharacteristic
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—81no—
385C2×C3—161no—
386C2×C3#199911yescharacteristic
387C22—41no—
388C22—41no—
389C22—81no—
390C22—81no—
391C22—161no—
392C22C3×#44911yescharacteristic
393C22C3×#44911yescharacteristic
394C22C3×#46611yescharacteristic
395C4—41no—
396C4—81no—
397C4—82no—
398C4C3×#39211yescharacteristic
399C4C3×#44711yescharacteristic
400C4C3×#44711yescharacteristic
401C4C3×#46911yescharacteristic
402C3#1332411yescharacteristic, direct product factor, Sylow
403C2—21no—
404C2—21no—
405C2—21no—
406C2—81no—
407C2—161no—
408C2C3×#199911yescharacteristic
subgroupG/H
center, socleC2×C3#1999
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC3×#3067C2
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×#510C2semidirect
3. C2×C3×#473C2semidirect
4. C2×C3×#362C2semidirect
5. C3×#2143C2semidirect
6. C3×#2115C2semidirect
7. C3×#1388C2semidirect
8. #13324C3direct (N × H)
32. C2×#510C2×C3semidirect
34. C2×#473C2×C3semidirect
35. C2×#362C2×C3semidirect
36. #2143C2×C3semidirect
37. #2115C2×C3semidirect
38. #1388C2×C3semidirect
402. C3#13324direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5102
C3×#30672
C2×C3×#4732
C2×C3×#3622
C3×#21432
C3×#21152
C3×#13882
#133243