Finite Group Databasebeta

← C3×#12701 #10631308

Subgroups — C3×#12701 #10631308

Order 768.

1070 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#356C212yesmaximal, complemented
3C2×C3×#355C212yesmaximal, complemented
4C3×#1785C212yesmaximal, complemented
5#12701C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×#41C411yescharacteristic
7C22×C3×#41C2211yescharacteristic
8C3×#532—24no—
9C3×#532—24no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#127C411yescharacteristic
12C2×C3×#124C411yescharacteristic
13C2×C3×#124C2211yescharacteristic
14C2×C3×#121—22no—
15C2×C3×C4×D4—22no—
16C22×C3×C42C411yescharacteristic
17C2×C3×#101C2214yes—
18C2×C3×#100—22no—
19C3×#407—22no—
20C3×#356—24no—
21C3×#355—24no—
22C2×#532C2×C311yescharacteristic
23C2×#356C2×C312yescomplemented
24C2×#355C2×C312yescomplemented
25#1785C2×C312yescomplemented
26C2×C3×#41—22no—
27C2×C3×#41—22no—
28C2×C3×#41—22no—
29C2×C3×#41—22no—
30C2×C3×#41—42no—
31C2×C3×#41—42no—
32C2×C3×#41—42no—
33C2×C3×#41—42no—
34C22×C3×Q8C2×C411yescharacteristic
35C22×C3×Q8C2311yescharacteristic
36C22×C3×D4—21no—
37C22×C3×D4—22no—
38C22×C3×D4C2×C411yescharacteristic
39C22×C3×D4C2×C411yescharacteristic
40C22×C3×D4D412yes—
41C23×C3×C4—21no—
42C23×C3×C4—21no—
43C23×C3×C4C2×C411yescharacteristic
44C23×C3×C4D412yes—
45C2×C3×M16—24no—
46C3×C4⋊Q8—22no—
47C3×C4⋊Q8D414yes—
48C3×#127—22no—
49C3×#127—22no—
50C3×#124—22no—
51C3×#124—22no—
52C3×#124—22no—
53C3×#124D414yes—
54C3×#121—216no—
55C3×C4×D4—216no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C42—21no—
60C2×C3×C42—22no—
61C2×C3×C42—22no—
62C2×C3×C42C2×C411yescharacteristic
63C2×C3×C42C2×C411yescharacteristic
64C3×#101—24no—
65C3×#101—24no—
66C3×#100—44no—
67C3×C22⋊C8—44no—
68C22×#41C3×C411yescharacteristic
69C22×#41C22×C311yescharacteristic
70#532—24no—
71#532—24no—
72C2×C4⋊Q8C22×C311yescharacteristic
73C2×#127C3×C411yescharacteristic
74C2×#124C3×C411yescharacteristic
75C2×#124C22×C311yescharacteristic
76C2×#121—22no—
77C2×C4×D4—22no—
78C22×C42C3×C411yescharacteristic
79C2×#101C22×C314yes—
80C2×#100—22no—
81#407—22no—
82#356—24no—
83#355—24no—
84C24×C3—21no—
85C24×C3C22⋊C412yes—
86C24×C3C22⋊C411yescharacteristic
87C3×#41—48no—
88C3×#41—48no—
89C3×#41—48no—
90C3×#41—48no—
91C2×C3×Q8—21no—
92C2×C3×Q8—41no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8C22⋊C412yes—
97C2×C3×Q8C2×D412yes—
98C2×C3×Q8C2×D412yes—
99C2×C3×D4—28no—
100C2×C3×D4—24no—
101C2×C3×D4—24no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—24no—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4—42no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—48no—
112C2×C3×D4—42no—
113C2×C3×D4—42no—
114C2×C3×D4C22⋊C412yes—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—24no—
121C22×C3×C4—24no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—24no—
125C22×C3×C4—24no—
126C22×C3×C4—42no—
127C22×C3×C4—44no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4C22⋊C411yescharacteristic
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C2×D412yes—
133C3×M16—44no—
134C3×M16—44no—
135C2×C3×C8—44no—
136C3×C4⋊C4—28no—
137C3×C22⋊C4—28no—
138C3×C22⋊C4—28no—
139C3×C42—24no—
140C3×C42—21no—
141C3×C42—21no—
142C3×C42C22⋊C412yes—
143C3×C42C22⋊C412yes—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—42no—
151C2×#41—42no—
152C22×Q8C2×C3×C411yescharacteristic
153C22×Q8C23×C311yescharacteristic
154C22×D4—22no—
155C22×D4—21no—
156C22×D4C2×C3×C411yescharacteristic
157C22×D4C2×C3×C411yescharacteristic
158C22×D4C3×D412yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2×C3×C411yescharacteristic
162C23×C4C3×D412yes—
163C2×M16—24no—
164C4⋊Q8—22no—
165C4⋊Q8C3×D414yes—
166#127—22no—
167#127—22no—
168#124—22no—
169#124—22no—
170#124—22no—
171#124C3×D414yes—
172#121—216no—
173C4×D4—216no—
174C2×C4⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C2×C3×C411yescharacteristic
181C2×C42C2×C3×C411yescharacteristic
182#101—24no—
183#101—24no—
184#100—44no—
185C22⋊C8—44no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—22no—
189C23×C3—22no—
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—42no—
193C23×C3—44no—
194C23×C3—44no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3#9912yes—
198C23×C3C2×C22⋊C411yescharacteristic
199C3×Q8—42no—
200C3×Q8—42no—
201C3×Q8—42no—
202C3×Q8—42no—
203C3×D4—48no—
204C3×D4—42no—
205C3×D4—416no—
206C3×D4—42no—
207C3×D4—42no—
208C3×D4—42no—
209C3×D4—48no—
210C2×C3×C4—24no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—21no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—48no—
223C2×C3×C4—48no—
224C2×C3×C4—48no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4#9912yes—
229C2×C3×C4C2×C22⋊C411yescharacteristic
230C2×C3×C4C2×C22⋊C411yescharacteristic
231C2×C3×C4#12014yes—
232C3×C8—84no—
233C24—21no—
234C24C3×C22⋊C412yes—
235C24C3×C22⋊C411yescharacteristic
236#41—48no—
237#41—48no—
238#41—48no—
239#41—48no—
240C2×Q8—21no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8C3×C22⋊C412yes—
246C2×Q8C2×C3×D412yes—
247C2×Q8C2×C3×D412yes—
248C2×D4—28no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—24no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—48no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4C3×C22⋊C412yes—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4C3×C22⋊C411yescharacteristic
280C22×C4C22×C3×C411yescharacteristic
281C22×C4C2×C3×D412yes—
282M16—44no—
283M16—44no—
284C2×C8—44no—
285C4⋊C4—28no—
286C22⋊C4—28no—
287C22⋊C4—28no—
288C42—24no—
289C42—21no—
290C42—21no—
291C42C3×C22⋊C412yes—
292C42C3×C22⋊C412yes—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3—44no—
298C22×C3—44no—
299C22×C3—42no—
300C22×C3—44no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—82no—
304C22×C3—82no—
305C22×C3—82no—
306C22×C3#37911yescharacteristic
307C22×C3C2×#9911yescharacteristic
308C22×C3C2×#9911yescharacteristic
309C3×C4—24no—
310C3×C4—24no—
311C3×C4—22no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—42no—
315C3×C4—42no—
316C23—21no—
317C23—21no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—42no—
323C23—44no—
324C23—44no—
325C23—41no—
326C23—41no—
327C23C3×#9912yes—
328C23C2×C3×C22⋊C411yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—42no—
332Q8—42no—
333D4—48no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—416no—
339D4—48no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—48no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4C3×#9912yes—
359C2×C4C2×C3×C22⋊C411yescharacteristic
360C2×C4C2×C3×C22⋊C411yescharacteristic
361C2×C4C3×#12014yes—
362C8—84no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—42no—
366C2×C3—82no—
367C2×C3—82no—
368C2×C3#178111yescharacteristic
369C2×C3#213212yes—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—42no—
374C22—44no—
375C22—44no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—44no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22C3×#37911yescharacteristic
384C22C2×C3×#9911yescharacteristic
385C22C2×C3×#9911yescharacteristic
386C4—24no—
387C4—24no—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—42no—
392C4—42no—
393C3#1270111yescharacteristic, direct product factor, Sylow
394C2—21no—
395C2—21no—
396C2—42no—
397C2—82no—
398C2—82no—
399C2C3×#178111yescharacteristic
400C2C3×#213212yes—
subgroupG/H
center, socleC22×C3C2×#99
commutatorC22×C4C22×C3×C4
Frattini, cosocleC22×Q8C23×C3
prime radicalC2×C3×#532C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#356C2semidirect
3. C2×C3×#355C2semidirect
4. C3×#1785C2semidirect
5. #12701C3direct (N × H)
23. C2×#356C2×C3semidirect
24. C2×#355C2×C3semidirect
25. #1785C2×C3semidirect
393. C3#12701direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#3562
C2×C3×#3552
C3×#17852
#127013