Finite Group Databasebeta

← C3×#13323 #10631930

Subgroups — C3×#13323 #10631930

Order 768.

534 conjugacy classes of subgroups.

The subgroup lattice has 420 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
3C2×C3×#472C211yesmaximal, characteristic, complemented
4C2×C3×#361C211yesmaximal, characteristic, complemented
5C3×#2143C211yesmaximal, characteristic, complemented
6C3×#2115C211yesmaximal, characteristic, complemented
7C3×#1388C211yesmaximal, characteristic, complemented
8#13323C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#575—21no—
10C2×C3×AGL(1,ℤ/8)—21no—
11C2×C3×#135C2211yescharacteristic
12C22×C3×D8C2211yescharacteristic
13C2×C3×#131C2211yescharacteristic
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—22no—
22C3×#472—21no—
23C3×#472—22no—
24C3×#471—22no—
25C2×C3×#108C2211yescharacteristic
26C2×C3×#108C2211yescharacteristic
27C3×#413—21no—
28C2×C3×#100—21no—
29C3×#361—22no—
30C3×#359—22no—
31C3×#349—22no—
32C2×#509C2×C311yescharacteristic, complemented
33#3067C2×C311yescharacteristic, complemented
34C2×#472C2×C311yescharacteristic, complemented
35C2×#361C2×C311yescharacteristic, complemented
36#2143C2×C311yescharacteristic, complemented
37#2115C2×C311yescharacteristic, complemented
38#1388C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C2×C3×#41D411yescharacteristic
41C22×C3×D4—21no—
42C3×#137—22no—
43C3×#137—41no—
44C3×AGL(1,ℤ/8)—24no—
45C3×AGL(1,ℤ/8)—22no—
46C3×AGL(1,ℤ/8)—42no—
47C3×AGL(1,ℤ/8)—41no—
48C3×#135—21no—
49C3×#135—21no—
50C3×#135—21no—
51C3×#135—21no—
52C3×#135—42no—
53C2×C3×Q16D411yescharacteristic
54C2×C3×SD16—21no—
55C2×C3×SD16—41no—
56C2×C3×D8—21no—
57C2×C3×D8—21no—
58C2×C3×D8—21no—
59C2×C3×D8—21no—
60C2×C3×D8—21no—
61C2×C3×D8—41no—
62C2×C3×D8D411yescharacteristic
63C2×C3×D8D412yes—
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×SD32—24no—
75C3×D16—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—22no—
87C3×#100—24no—
88C3×#100—22no—
89#575—21no—
90C2×AGL(1,ℤ/8)—21no—
91C2×#135C22×C311yescharacteristic
92C22×D8C22×C311yescharacteristic
93C2×#131C22×C311yescharacteristic
94#509—22no—
95#509—22no—
96C2×SD32—21no—
97C2×D16—21no—
98C2×M32C22×C311yescharacteristic
99C2×M32C22×C311yescharacteristic
100#473—21no—
101#472—22no—
102#472—21no—
103#472—22no—
104#471—22no—
105C2×#108C22×C311yescharacteristic
106C2×#108C22×C311yescharacteristic
107#413—21no—
108C2×#100—21no—
109#361—22no—
110#359—22no—
111#349—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—22no—
125C2×C3×D4—21no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4D811yescharacteristic
133C22×C3×C4—21no—
134C22×C3×C4—41no—
135C22×C3×C4C2×D411yescharacteristic
136C3×Q16—21no—
137C3×Q16—21no—
138C3×Q16—42no—
139C3×SD16—42no—
140C3×SD16—44no—
141C3×SD16—42no—
142C3×SD16—41no—
143C3×SD16—41no—
144C3×D8—21no—
145C3×D8—22no—
146C3×D8—22no—
147C3×D8—21no—
148C3×D8—44no—
149C3×D8—42no—
150C3×D8—41no—
151C3×D8—41no—
152C3×M16—21no—
153C3×M16—21no—
154C3×M16—22no—
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×D4—21no—
176#137—22no—
177#137—41no—
178AGL(1,ℤ/8)—24no—
179AGL(1,ℤ/8)—22no—
180AGL(1,ℤ/8)—42no—
181AGL(1,ℤ/8)—41no—
182#135—21no—
183#135—21no—
184#135—21no—
185#135—21no—
186#135—42no—
187C2×Q16C3×D411yescharacteristic
188C2×SD16—21no—
189C2×SD16—41no—
190C2×D8—21no—
191C2×D8—21no—
192C2×D8—21no—
193C2×D8—21no—
194C2×D8—21no—
195C2×D8—41no—
196C2×D8C3×D412yes—
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
208SD32—24no—
209D16—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—22no—
221#100—24no—
222#100—22no—
223C23×C3—21no—
224C23×C3—41no—
225C23×C3—42no—
226C23×C3—81no—
227C23×C3—81no—
228C23×C3—162no—
229C23×C3#13511yescharacteristic
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—42no—
241C3×D4—41no—
242C3×D4—81no—
243C3×D4—81no—
244C3×D4—81no—
245C3×D4—81no—
246C2×C3×C4—21no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4C2×D811yescharacteristic
256C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
257C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
258C3×C8—41no—
259C3×C8—42no—
260C3×C8—42no—
261C3×C8#12011yescharacteristic
262C3×C8#12111yescharacteristic
263C3×C8#12111yescharacteristic
264C3×C8#12711yescharacteristic
265C24—41no—
266#41—21no—
267#41—21no—
268#41—21no—
269#41—21no—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274C2×Q8—21no—
275C2×Q8C3×D811yescharacteristic
276C2×D4—21no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4C3×D811yescharacteristic
286C22×C4—21no—
287C22×C4—41no—
288C22×C4C2×C3×D411yescharacteristic
289Q16—21no—
290Q16—21no—
291Q16—42no—
292SD16—42no—
293SD16—44no—
294SD16—42no—
295SD16—41no—
296SD16—41no—
297D8—21no—
298D8—22no—
299D8—22no—
300D8—21no—
301D8—44no—
302D8—42no—
303D8—41no—
304D8—41no—
305M16—21no—
306M16—21no—
307M16—22no—
308M16—22no—
309M16—41no—
310M16—41no—
311M16—41no—
312M16—41no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8C2×C3×D411yescharacteristic
319C2×C8C2×C3×D411yescharacteristic
320C2×C8C2×C3×D411yescharacteristic
321C2×C8C2×C3×D411yescharacteristic
322C2×C8C2×C3×D411yescharacteristic
323C2×C8C3×#4111yescharacteristic
324C16—22no—
325C16—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—82no—
331C22×C3—161no—
332C22×C3—162no—
333C22×C3—162no—
334C22×C3—162no—
335C22×C3#44911yescharacteristic
336C22×C3#44911yescharacteristic
337C22×C3#46611yescharacteristic
338C3×C4—41no—
339C3×C4—81no—
340C3×C4#39211yescharacteristic
341C3×C4#44711yescharacteristic
342C3×C4#44711yescharacteristic
343C3×C4#46911yescharacteristic
344C23—21no—
345C23—41no—
346C23—42no—
347C23—81no—
348C23—81no—
349C23—162no—
350C23C3×#13511yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—41no—
354Q8—41no—
355D4—21no—
356D4—21no—
357D4—41no—
358D4—41no—
359D4—42no—
360D4—41no—
361D4—42no—
362D4—41no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4C3×#12011yescharacteristic
374C2×C4C3×#12011yescharacteristic
375C2×C4C3×#12111yescharacteristic
376C2×C4C2×C3×D811yescharacteristic
377C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
378C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
379C8—41no—
380C8—42no—
381C8—42no—
382C8C3×#12011yescharacteristic
383C8C3×#12111yescharacteristic
384C8C3×#12111yescharacteristic
385C8C3×#12711yescharacteristic
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—81no—
390C2×C3—161no—
391C2×C3—162no—
392C2×C3#199911yescharacteristic
393C22—41no—
394C22—41no—
395C22—81no—
396C22—81no—
397C22—82no—
398C22—162no—
399C22—161no—
400C22—162no—
401C22—162no—
402C22C3×#44911yescharacteristic
403C22C3×#44911yescharacteristic
404C22C3×#46611yescharacteristic
405C4—41no—
406C4—81no—
407C4C3×#39211yescharacteristic
408C4C3×#44711yescharacteristic
409C4C3×#44711yescharacteristic
410C4C3×#46911yescharacteristic
411C3#1332311yescharacteristic, direct product factor, Sylow
412C2—21no—
413C2—21no—
414C2—21no—
415C2—81no—
416C2—162no—
417C2—161no—
418C2C3×#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. C2×C3×#472C2semidirect
4. C2×C3×#361C2semidirect
5. C3×#2143C2semidirect
6. C3×#2115C2semidirect
7. C3×#1388C2semidirect
8. #13323C3direct (N × H)
32. C2×#509C2×C3semidirect
33. #3067C2×C3semidirect
34. C2×#472C2×C3semidirect
35. C2×#361C2×C3semidirect
36. #2143C2×C3semidirect
37. #2115C2×C3semidirect
38. #1388C2×C3semidirect
411. C3#13323direct (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
C2×C3×#4722
C2×C3×#3612
C3×#21432
C3×#21152
C3×#13882
#133233