Finite Group Databasebeta

← C3×#13419 #10632026

Subgroups — C3×#13419 #10632026

Order 768.

776 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2679C211yesmaximal, characteristic
2C2×C3×AΣL(1,𝔽16)C212yesmaximal, complemented
3C3×#1889C212yesmaximal, complemented
4C3×#1856C211yesmaximal, characteristic, complemented
5C3×#1393C211yesmaximal, characteristic, complemented
6#13419C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#120—22no—
8C2×C3×#120C2212yes—
9C3×#518C2211yescharacteristic
10C3×#428—21no—
11C2×C3×#100C2211yescharacteristic
12C2×C3×#99C2212yes—
13C3×#394—21no—
14C3×#394—22no—
15C3×#390—21no—
16C3×#390C2211yescharacteristic
17C3×#379—22no—
18C3×AΣL(1,𝔽16)—24no—
19C3×#342—22no—
20C3×#339—21no—
21#2679C2×C311yescharacteristic
22C2×AΣL(1,𝔽16)C2×C312yescomplemented
23#1889C2×C312yescomplemented
24#1856C2×C311yescharacteristic, complemented
25#1393C2×C311yescharacteristic, complemented
26C25×C3D412yes—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4C2311yescharacteristic
31C2×C3×M16—21no—
32C3×#120—22no—
33C3×#120—22no—
34C3×#120—22no—
35C3×#120—22no—
36C3×#120—48no—
37C3×C4×D4—41no—
38C3×C4×D4—41no—
39C3×#117—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—42no—
49C2×C3×C22⋊C4—42no—
50C2×C3×C22⋊C4D412yes—
51C2×C3×C42D411yescharacteristic
52C3×#107—41no—
53C3×#106—41no—
54C3×#100—21no—
55C3×#100D412yescomplemented
56C3×#99—22no—
57C3×#99—22no—
58C3×#95—21no—
59C2×#120—22no—
60C2×#120C22×C312yes—
61#518C22×C311yescharacteristic
62#428—21no—
63C2×#100C22×C311yescharacteristic
64C2×#99C22×C312yes—
65#394—21no—
66#394—22no—
67#390—21no—
68#390C22×C311yescharacteristic
69#379—22no—
70AΣL(1,𝔽16)—24no—
71#342—22no—
72#339—21no—
73C24×C3—22no—
74C24×C3—22no—
75C24×C3—22no—
76C24×C3—42no—
77C24×C3—42no—
78C24×C3—44no—
79C24×C3—82no—
80C24×C3C2×D412yes—
81C2×C3×D4—24no—
82C2×C3×D4—41no—
83C2×C3×D4—41no—
84C2×C3×D4—44no—
85C2×C3×D4—44no—
86C2×C3×D4—42no—
87C2×C3×D4—42no—
88C2×C3×D4—81no—
89C2×C3×D4—81no—
90C2×C3×D4—82no—
91C2×C3×D4—81no—
92C2×C3×D4C2×D411yescharacteristic
93C2×C3×D4#4111yescharacteristic
94C2×C3×D4#4112yes—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—42no—
101C22×C3×C4C2×D411yescharacteristic
102C3×M16—22no—
103C3×M16—41no—
104C2×C3×C8—41no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—81no—
109C3×C22⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—44no—
112C3×C22⋊C4—44no—
113C3×C22⋊C4—42no—
114C3×C22⋊C4—44no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—42no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—84no—
121C3×C22⋊C4—84no—
122C3×C42—41no—
123C3×C42—41no—
124C25C3×D412yes—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4C23×C311yescharacteristic
129C2×M16—21no—
130#120—22no—
131#120—22no—
132#120—22no—
133#120—22no—
134#120—48no—
135C4×D4—41no—
136C4×D4—41no—
137#117—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4C3×D411yescharacteristic
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4C3×D412yes—
149C2×C42C3×D411yescharacteristic
150#107—41no—
151#106—41no—
152#100—21no—
153#100C3×D412yescomplemented
154#99—22no—
155#99—22no—
156#95—21no—
157C23×C3—22no—
158C23×C3—22no—
159C23×C3—24no—
160C23×C3—42no—
161C23×C3—42no—
162C23×C3—42no—
163C23×C3—42no—
164C23×C3—42no—
165C23×C3—44no—
166C23×C3—44no—
167C23×C3—44no—
168C23×C3—42no—
169C23×C3—42no—
170C23×C3—42no—
171C23×C3—82no—
172C23×C3—84no—
173C23×C3—82no—
174C23×C3—84no—
175C23×C3—88no—
176C23×C3—82no—
177C23×C3#12011yescharacteristic
178C23×C3#12112yes—
179C23×C3#12312yes—
180C3×D4—41no—
181C3×D4—41no—
182C3×D4—42no—
183C3×D4—84no—
184C3×D4—84no—
185C3×D4—84no—
186C3×D4—84no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—82no—
200C2×C3×C4—82no—
201C2×C3×C4—82no—
202C2×C3×C4—82no—
203C2×C3×C4—82no—
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12411yescharacteristic
206C3×C8—42no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—42no—
211C24—44no—
212C24—42no—
213C24—82no—
214C24C2×C3×D412yes—
215C2×D4—24no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—81no—
223C2×D4—81no—
224C2×D4—82no—
225C2×D4—81no—
226C2×D4C2×C3×D411yescharacteristic
227C2×D4C3×#4111yescharacteristic
228C2×D4C3×#4112yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4C2×C3×D411yescharacteristic
236M16—22no—
237M16—41no—
238C2×C8—41no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—81no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—84no—
255C22⋊C4—84no—
256C42—41no—
257C42—41no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—44no—
265C22×C3—44no—
266C22×C3—42no—
267C22×C3—42no—
268C22×C3—48no—
269C22×C3—44no—
270C22×C3—82no—
271C22×C3—82no—
272C22×C3—82no—
273C22×C3—84no—
274C22×C3—84no—
275C22×C3—84no—
276C22×C3—82no—
277C22×C3—82no—
278C22×C3#39411yescharacteristic
279C22×C3B(4,𝔽2)11yescharacteristic
280C22×C3#45811yescharacteristic
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—42no—
284C3×C4—82no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4—81no—
288C3×C4—162no—
289C23—22no—
290C23—22no—
291C23—24no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—44no—
298C23—44no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—82no—
304C23—84no—
305C23—82no—
306C23—84no—
307C23—88no—
308C23—82no—
309C23C3×#12011yescharacteristic
310C23C3×#12112yes—
311C23C3×#12312yes—
312D4—41no—
313D4—41no—
314D4—42no—
315D4—84no—
316D4—84no—
317D4—84no—
318D4—84no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4C3×#12111yescharacteristic
337C2×C4C3×#12411yescharacteristic
338C8—42no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—42no—
342C2×C3—42no—
343C2×C3—84no—
344C2×C3#204411yescharacteristic
345C2×C3#219611yescharacteristic
346C2×C3#219911yescharacteristic
347C22—21no—
348C22—21no—
349C22—22no—
350C22—22no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—48no—
358C22—44no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—84no—
363C22—84no—
364C22—84no—
365C22—82no—
366C22—82no—
367C22C3×#39411yescharacteristic
368C22C3×B(4,𝔽2)11yescharacteristic
369C22C3×#45811yescharacteristic
370C4—21no—
371C4—21no—
372C4—42no—
373C4—82no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4—162no—
378C3#1341911yescharacteristic, direct product factor, Sylow
379C2—21no—
380C2—21no—
381C2—42no—
382C2—42no—
383C2—84no—
384C2C3×#204411yescharacteristic
385C2C3×#219611yescharacteristic
386C2C3×#219911yescharacteristic
subgroupG/H
center, socleC22×C3B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C3
prime radicalC3×#2679C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×AΣL(1,𝔽16)C2semidirect
3. C3×#1889C2semidirect
4. C3×#1856C2semidirect
5. C3×#1393C2semidirect
6. #13419C3direct (N × H)
22. C2×AΣL(1,𝔽16)C2×C3semidirect
23. #1889C2×C3semidirect
24. #1856C2×C3semidirect
25. #1393C2×C3semidirect
55. C3×#100D4semidirect
153. #100C3×D4semidirect
378. C3#13419direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26792
C2×C3×AΣL(1,𝔽16)2
C3×#18892
C3×#18562
C3×#13932
#134193