Finite Group Databasebeta

← C3×#13326 #10631933

Subgroups — C3×#13326 #10631933

Order 768.

614 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×SD32C211yesmaximal, characteristic, complemented
2C22×C3×D16C211yesmaximal, characteristic
3C2×C3×#495C211yesmaximal, characteristic, complemented
4C2×C3×#358C211yesmaximal, characteristic, complemented
5C2×C3×#357C211yesmaximal, characteristic, complemented
6C3×#1970C211yesmaximal, characteristic, complemented
7C3×#1374C211yesmaximal, characteristic, complemented
8#13326C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×Q16C2211yescharacteristic
10C22×C3×SD16—21no—
11C22×C3×D8C2211yescharacteristic
12C22×C3×D8C2211yescharacteristic
13C2×C3×#124—21no—
14C2×C3×SD32—22no—
15C2×C3×SD32—22no—
16C2×C3×SD32—22no—
17C2×C3×D16—22no—
18C2×C3×D16—22no—
19C2×C3×D16—22no—
20C22×C3×C16C2211yescharacteristic
21C22×C3×C16C2211yescharacteristic
22C3×#495—22no—
23C3×#495—22no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#102—21no—
26C2×C3×C4×C8C2211yescharacteristic
27C3×#390—21no—
28C3×#358—22no—
29C3×#357—22no—
30C22×SD32C2×C311yescharacteristic, complemented
31C22×D16C2×C311yescharacteristic
32C2×#495C2×C311yescharacteristic, complemented
33C2×#358C2×C311yescharacteristic, complemented
34C2×#357C2×C311yescharacteristic, complemented
35#1970C2×C311yescharacteristic, complemented
36#1374C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C2×C3×Q16—21no—
41C2×C3×Q16—21no—
42C2×C3×Q16—21no—
43C2×C3×Q16—21no—
44C2×C3×Q16—21no—
45C2×C3×Q16D412yes—
46C2×C3×SD16—24no—
47C2×C3×SD16—42no—
48C2×C3×SD16—42no—
49C2×C3×D8—21no—
50C2×C3×D8—21no—
51C2×C3×D8—21no—
52C2×C3×D8—21no—
53C2×C3×D8—21no—
54C2×C3×D8—21no—
55C2×C3×D8—21no—
56C2×C3×D8—21no—
57C2×C3×D8—21no—
58C2×C3×D8—21no—
59C2×C3×D8D412yes—
60C2×C3×D8D412yes—
61C22×C3×C8D411yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×#124—24no—
64C3×#124—42no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C22⋊C4—42no—
67C2×C3×C22⋊C4—42no—
68C2×C3×C42D411yescharacteristic
69C3×SD32—28no—
70C3×D16—28no—
71C2×C3×C16—21no—
72C2×C3×C16—21no—
73C2×C3×C16—21no—
74C2×C3×C16—21no—
75C2×C3×C16D412yes—
76C2×C3×C16D412yes—
77C3×#107—21no—
78C3×#107—21no—
79C3×#102—42no—
80C3×C4×C8—21no—
81C3×C4×C8—21no—
82C22×Q16C22×C311yescharacteristic
83C22×SD16—21no—
84C22×D8C22×C311yescharacteristic
85C22×D8C22×C311yescharacteristic
86C2×#124—21no—
87C2×SD32—22no—
88C2×SD32—22no—
89C2×SD32—22no—
90C2×D16—22no—
91C2×D16—22no—
92C2×D16—22no—
93C22×C16C22×C311yescharacteristic
94C22×C16C22×C311yescharacteristic
95#495—22no—
96#495—22no—
97C2×#107C22×C311yescharacteristic
98C2×#102—21no—
99C2×C4×C8C22×C311yescharacteristic
100#390—21no—
101#358—22no—
102#357—22no—
103C24×C3—41no—
104C24×C3—41no—
105C2×C3×Q8—22no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4D812yes—
126C22×C3×C4C2×D411yescharacteristic
127C3×Q16—22no—
128C3×Q16—22no—
129C3×Q16—41no—
130C3×Q16—41no—
131C3×SD16—48no—
132C3×D8—22no—
133C3×D8—22no—
134C3×D8—22no—
135C3×D8—22no—
136C3×D8—41no—
137C3×D8—41no—
138C3×D8—41no—
139C3×D8—41no—
140C2×C3×C8—21no—
141C2×C3×C8—21no—
142C2×C3×C8D812yes—
143C2×C3×C8C2×D411yescharacteristic
144C2×C3×C8C2×D411yescharacteristic
145C2×C3×C8C2×D411yescharacteristic
146C2×C3×C8C2×D411yescharacteristic
147C2×C3×C8C2×D411yescharacteristic
148C2×C3×C8#4111yescharacteristic
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C22⋊C4—48no—
152C3×C22⋊C4—84no—
153C3×C42—21no—
154C3×C42—21no—
155C3×C16—22no—
156C3×C16—22no—
157C22×Q8—21no—
158C22×D4—21no—
159C22×D4—21no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16C3×D412yes—
166C2×SD16—24no—
167C2×SD16—42no—
168C2×SD16—42no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8C3×D412yes—
180C2×D8C3×D412yes—
181C22×C8C3×D411yescharacteristic
182C22×C8C23×C311yescharacteristic
183#124—24no—
184#124—42no—
185C2×C4⋊C4—21no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C42C3×D411yescharacteristic
189SD32—28no—
190D16—28no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16C3×D412yes—
196C2×C16C3×D412yes—
197#107—21no—
198#107—21no—
199#102—42no—
200C4×C8—21no—
201C4×C8—21no—
202C23×C3—42no—
203C23×C3—42no—
204C23×C3—81no—
205C23×C3—81no—
206C23×C3—81no—
207C23×C3—81no—
208C23×C3—82no—
209C23×C3—82no—
210C23×C3—82no—
211C23×C3—82no—
212C23×C3C2×D811yescharacteristic
213C3×Q8—42no—
214C3×Q8—42no—
215C3×Q8—81no—
216C3×Q8—81no—
217C3×D4—42no—
218C3×D4—42no—
219C3×D4—42no—
220C3×D4—42no—
221C3×D4—81no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—81no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—42no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4—82no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12711yescharacteristic
236C2×C3×C4C2×D811yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
239C3×C8—22no—
240C3×C8#12011yescharacteristic
241C3×C8#12011yescharacteristic
242C3×C8#12111yescharacteristic
243C3×C8#12111yescharacteristic
244C24—41no—
245C24—41no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C3×D812yes—
267C22×C4C2×C3×D411yescharacteristic
268Q16—22no—
269Q16—22no—
270Q16—41no—
271Q16—41no—
272SD16—48no—
273D8—22no—
274D8—22no—
275D8—22no—
276D8—22no—
277D8—41no—
278D8—41no—
279D8—41no—
280D8—41no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8C3×D812yes—
284C2×C8C2×C3×D411yescharacteristic
285C2×C8C2×C3×D411yescharacteristic
286C2×C8C2×C3×D411yescharacteristic
287C2×C8C2×C3×D411yescharacteristic
288C2×C8C2×C3×D411yescharacteristic
289C2×C8C3×#4111yescharacteristic
290C4⋊C4—41no—
291C4⋊C4—41no—
292C22⋊C4—48no—
293C22⋊C4—84no—
294C42—21no—
295C42—21no—
296C16—22no—
297C16—22no—
298C22×C3—84no—
299C22×C3—84no—
300C22×C3—82no—
301C22×C3—82no—
302C22×C3—162no—
303C22×C3—162no—
304C22×C3—161no—
305C22×C3—161no—
306C22×C3—161no—
307C22×C3—161no—
308C22×C3#44711yescharacteristic
309C22×C3#45911yescharacteristic
310C22×C3#49311yescharacteristic
311C22×C3#50812yes—
312C22×C3#50912yes—
313C3×C4—42no—
314C3×C4—82no—
315C3×C4—161no—
316C3×C4#39211yescharacteristic
317C3×C4#44711yescharacteristic
318C3×C4#45911yescharacteristic
319C3×C4#49611yescharacteristic
320C23—42no—
321C23—42no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23—82no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23C2×C3×D811yescharacteristic
331Q8—42no—
332Q8—42no—
333Q8—81no—
334Q8—81no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—42no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4C3×#12011yescharacteristic
352C2×C4C3×#12111yescharacteristic
353C2×C4C3×#12711yescharacteristic
354C2×C4C2×C3×D811yescharacteristic
355C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
356C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
357C8—22no—
358C8C3×#12011yescharacteristic
359C8C3×#12011yescharacteristic
360C8C3×#12111yescharacteristic
361C8C3×#12111yescharacteristic
362C2×C3—162no—
363C2×C3—162no—
364C2×C3#201111yescharacteristic
365C2×C3#218612yes—
366C2×C3#221012yes—
367C2×C3#224911yescharacteristic
368C2×C3#225011yescharacteristic
369C22—82no—
370C22—82no—
371C22—84no—
372C22—84no—
373C22—161no—
374C22—161no—
375C22—161no—
376C22—161no—
377C22—162no—
378C22—162no—
379C22C3×#44711yescharacteristic
380C22C3×#45911yescharacteristic
381C22C3×#49311yescharacteristic
382C22C3×#50812yes—
383C22C3×#50912yes—
384C4—42no—
385C4—82no—
386C4—161no—
387C4C3×#39211yescharacteristic
388C4C3×#44711yescharacteristic
389C4C3×#45911yescharacteristic
390C4C3×#49611yescharacteristic
391C3#1332611yescharacteristic, direct product factor, Sylow
392C2—162no—
393C2—162no—
394C2C3×#201111yescharacteristic
395C2C3×#218612yes—
396C2C3×#221012yes—
397C2C3×#224911yescharacteristic
398C2C3×#225011yescharacteristic
subgroupG/H
center, socleC23×C3C2×D8
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC22×C3×D16C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×SD32C2semidirect
3. C2×C3×#495C2semidirect
4. C2×C3×#358C2semidirect
5. C2×C3×#357C2semidirect
6. C3×#1970C2semidirect
7. C3×#1374C2semidirect
8. #13326C3direct (N × H)
30. C22×SD32C2×C3semidirect
32. C2×#495C2×C3semidirect
33. C2×#358C2×C3semidirect
34. C2×#357C2×C3semidirect
35. #1970C2×C3semidirect
36. #1374C2×C3semidirect
391. C3#13326direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×SD322
C22×C3×D162
C2×C3×#4952
C2×C3×#3582
C2×C3×#3572
C3×#19702
C3×#13742
#133263