Finite Group Databasebeta

← C3×#13322 #10631929

Subgroups — C3×#13322 #10631929

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Q32C211yesmaximal, characteristic, complemented
2C2×C3×#473C211yesmaximal, characteristic
3C3×#3059C211yesmaximal, characteristic
4C2×C3×#362C211yesmaximal, characteristic, complemented
5C3×#2141C211yesmaximal, characteristic, complemented
6C2×C3×#350C211yesmaximal, characteristic
7C3×#1382C211yesmaximal, characteristic, complemented
8#13322C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#137—21no—
10C22×C3×Q16C2211yescharacteristic
11C22×C3×Q16C2211yescharacteristic
12C2×C3×#131C2211yescharacteristic
13C3×#536—21no—
14C2×C3×Q32—22no—
15C2×C3×Q32—22no—
16C2×C3×Q32—22no—
17C2×C3×M32C2211yescharacteristic
18C22×C3×C16C2211yescharacteristic
19C3×#473—22no—
20C3×#473—22no—
21C3×#470—22no—
22C3×#467—22no—
23C2×C3×#108C2211yescharacteristic
24C3×#427C2211yescharacteristic
25C3×#419—21no—
26C2×C3×#101—21no—
27C3×#362—22no—
28C3×#358—22no—
29C3×#350—22no—
30C22×Q32C2×C311yescharacteristic, complemented
31C2×#473C2×C311yescharacteristic
32#3059C2×C311yescharacteristic
33C2×#362C2×C311yescharacteristic, complemented
34#2141C2×C311yescharacteristic, complemented
35C2×#350C2×C311yescharacteristic
36#1382C2×C311yescharacteristic, complemented
37C2×C3×#41D411yescharacteristic
38C22×C3×Q8—21no—
39C22×C3×Q8—21no—
40C3×#137—24no—
41C3×#137—42no—
42C2×C3×Q16—21no—
43C2×C3×Q16—21no—
44C2×C3×Q16—21no—
45C2×C3×Q16—21no—
46C2×C3×Q16—21no—
47C2×C3×Q16—21no—
48C2×C3×Q16—21no—
49C2×C3×Q16—21no—
50C2×C3×Q16—21no—
51C2×C3×Q16—21no—
52C2×C3×Q16—41no—
53C2×C3×Q16D412yes—
54C2×C3×Q16D412yes—
55C2×C3×SD16—41no—
56C3×#131—21no—
57C3×#131—21no—
58C3×#131—21no—
59C3×#131—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16D411yescharacteristic
63C22×C3×C8—21no—
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—41no—
66C3×#124—41no—
67C3×#123—42no—
68C3×C22⋊Q8—24no—
69C3×#117—21no—
70C3×Q32—28no—
71C3×M32—21no—
72C3×M32—21no—
73C2×C3×C16—21no—
74C2×C3×C16—21no—
75C2×C3×C16—21no—
76C2×C3×C16D412yes—
77C3×#108—21no—
78C3×#108D412yes—
79C3×#107D412yes—
80C3×#106—21no—
81C3×#103—24no—
82C3×#101—24no—
83C2×#137—21no—
84C22×Q16C22×C311yescharacteristic
85C22×Q16C22×C311yescharacteristic
86C2×#131C22×C311yescharacteristic
87#536—21no—
88C2×Q32—22no—
89C2×Q32—22no—
90C2×Q32—22no—
91C2×M32C22×C311yescharacteristic
92C22×C16C22×C311yescharacteristic
93#473—22no—
94#473—22no—
95#470—22no—
96#467—22no—
97C2×#108C22×C311yescharacteristic
98#427C22×C311yescharacteristic
99#419—21no—
100C2×#101—21no—
101#362—22no—
102#358—22no—
103#350—22no—
104C3×#41—21no—
105C3×#41—21no—
106C3×#41—21no—
107C3×#41—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8D811yescharacteristic
121C2×C3×D4—21no—
122C2×C3×D4D811yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C3×Q16—22no—
128C3×Q16—22no—
129C3×Q16—22no—
130C3×Q16—22no—
131C3×Q16—44no—
132C3×Q16—41no—
133C3×Q16—41no—
134C3×Q16—41no—
135C3×Q16—41no—
136C3×SD16—44no—
137C3×M16—21no—
138C3×M16—22no—
139C3×M16—21no—
140C3×M16—41no—
141C3×M16—41no—
142C3×M16—41no—
143C2×C3×C8—21no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8C2×D411yescharacteristic
148C2×C3×C8C2×D411yescharacteristic
149C2×C3×C8C2×D411yescharacteristic
150C2×C3×C8C2×D411yescharacteristic
151C2×C3×C8C2×D411yescharacteristic
152C2×C3×C8#4111yescharacteristic
153C3×C4⋊C4—22no—
154C3×C4⋊C4—44no—
155C3×C22⋊C4—44no—
156C3×C22⋊C4—41no—
157C3×C42—41no—
158C3×C16—22no—
159C3×C16—41no—
160C2×#41C3×D411yescharacteristic
161C22×Q8—21no—
162C22×Q8—21no—
163#137—24no—
164#137—42no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16—21no—
174C2×Q16—21no—
175C2×Q16—41no—
176C2×Q16C3×D412yes—
177C2×Q16C3×D412yes—
178C2×SD16—41no—
179#131—21no—
180#131—21no—
181#131—21no—
182#131—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16C3×D411yescharacteristic
186C22×C8—21no—
187C22×C8C23×C311yescharacteristic
188C4⋊Q8—41no—
189#124—41no—
190#123—42no—
191C22⋊Q8—24no—
192#117—21no—
193Q32—28no—
194M32—21no—
195M32—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16C3×D412yes—
200#108—21no—
201#108C3×D412yes—
202#107C3×D412yes—
203#106—21no—
204#103—24no—
205#101—24no—
206C23×C3—21no—
207C23×C3AGL(1,ℤ/8)11yescharacteristic
208C3×Q8—21no—
209C3×Q8—21no—
210C3×Q8—42no—
211C3×Q8—42no—
212C3×Q8—42no—
213C3×Q8—42no—
214C3×Q8—81no—
215C3×Q8—81no—
216C3×Q8—81no—
217C3×Q8—81no—
218C3×D4—21no—
219C3×D4—21no—
220C3×D4—41no—
221C3×D4—41no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—81no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4C2×D811yescharacteristic
236C2×C3×C4#13511yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238C3×C8—41no—
239C3×C8—42no—
240C3×C8#12011yescharacteristic
241C3×C8#12111yescharacteristic
242C3×C8#12111yescharacteristic
243C3×C8#12711yescharacteristic
244#41—21no—
245#41—21no—
246#41—21no—
247#41—21no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8C3×D811yescharacteristic
261C2×D4—21no—
262C2×D4C3×D811yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C2×C3×D411yescharacteristic
267Q16—22no—
268Q16—22no—
269Q16—22no—
270Q16—22no—
271Q16—44no—
272Q16—41no—
273Q16—41no—
274Q16—41no—
275Q16—41no—
276SD16—44no—
277M16—21no—
278M16—22no—
279M16—21no—
280M16—41no—
281M16—41no—
282M16—41no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8C2×C3×D411yescharacteristic
288C2×C8C2×C3×D411yescharacteristic
289C2×C8C2×C3×D411yescharacteristic
290C2×C8C2×C3×D411yescharacteristic
291C2×C8C2×C3×D411yescharacteristic
292C2×C8C3×#4111yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—41no—
297C42—41no—
298C16—22no—
299C16—41no—
300C22×C3—22no—
301C22×C3—42no—
302C22×C3—41no—
303C22×C3#44711yescharacteristic
304C22×C3#44911yescharacteristic
305C22×C3#46911yescharacteristic
306C3×C4—41no—
307C3×C4—82no—
308C3×C4—82no—
309C3×C4—82no—
310C3×C4#39211yescharacteristic
311C3×C4#44711yescharacteristic
312C3×C4#44911yescharacteristic
313C3×C4#46611yescharacteristic
314C23—21no—
315C23C3×AGL(1,ℤ/8)11yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—42no—
319Q8—42no—
320Q8—42no—
321Q8—42no—
322Q8—81no—
323Q8—81no—
324Q8—81no—
325Q8—81no—
326D4—21no—
327D4—21no—
328D4—41no—
329D4—41no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4C3×#12011yescharacteristic
341C2×C4C3×#12011yescharacteristic
342C2×C4C3×#12111yescharacteristic
343C2×C4C2×C3×D811yescharacteristic
344C2×C4C3×#13511yescharacteristic
345C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
346C8—41no—
347C8—42no—
348C8C3×#12011yescharacteristic
349C8C3×#12111yescharacteristic
350C8C3×#12111yescharacteristic
351C8C3×#12711yescharacteristic
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—81no—
355C2×C3#199911yescharacteristic
356C2×C3#219512yes—
357C22—22no—
358C22—41no—
359C22—42no—
360C22C3×#44711yescharacteristic
361C22C3×#44911yescharacteristic
362C22C3×#46911yescharacteristic
363C4—41no—
364C4—82no—
365C4—82no—
366C4—82no—
367C4C3×#39211yescharacteristic
368C4C3×#44711yescharacteristic
369C4C3×#44911yescharacteristic
370C4C3×#46611yescharacteristic
371C3#1332211yescharacteristic, direct product factor, Sylow
372C2—21no—
373C2—21no—
374C2—81no—
375C2C3×#199911yescharacteristic
376C2C3×#219512yes—
subgroupG/H
center, socleC22×C3#447
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC2×C3×#41D4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q32C2semidirect
4. C2×C3×#362C2semidirect
5. C3×#2141C2semidirect
7. C3×#1382C2semidirect
8. #13322C3direct (N × H)
30. C22×Q32C2×C3semidirect
33. C2×#362C2×C3semidirect
34. #2141C2×C3semidirect
36. #1382C2×C3semidirect
371. C3#13322direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q322
C2×C3×#4732
C3×#30592
C2×C3×#3622
C3×#21412
C2×C3×#3502
C3×#13822
#133223