Finite Group Databasebeta

← C3×#29715 #10648322

Subgroups — C3×#29715 #10648322

Order 768.

1012 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#451C212yesmaximal
2C3×#2857C211yesmaximal, characteristic
3C2×C3×#391C211yesmaximal, characteristic
4C3×#2072C212yesmaximal, complemented
5C3×#2056C214yesmaximal, complemented
6C3×#2016C212yesmaximal, complemented
7C3×#1872C212yesmaximal
8C3×#1797C211yesmaximal, characteristic
9#29715C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×C3×Q16C2214yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C22⋊Q8C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C3×#451—28no—
17C2×C3×C4⋊C8C2212yes—
18C2×C3×#103C2214yes—
19C2×C3×#103C2214yes—
20C2×C3×C22⋊C8C2212yes—
21C3×#391—21no—
22C3×#391—21no—
23C3×#391C2212yes—
24C3×#391C2212yes—
25C3×#391C2212yes—
26C3×#389—21no—
27C3×#389—21no—
28C3×#384C2211yescharacteristic
29C3×#380—21no—
30C3×#380—21no—
31C2×C3×#95—21no—
32C2×C3×#95—21no—
33C3×#340C2214yes—
34C3×#336C2212yes—
35C2×#451C2×C312yes—
36#2857C2×C311yescharacteristic
37C2×#391C2×C311yescharacteristic
38#2072C2×C312yescomplemented
39#2056C2×C314yescomplemented
40#2016C2×C312yescomplemented
41#1872C2×C312yes—
42#1797C2×C311yescharacteristic
43C22×C3×Q8—21no—
44C22×C3×Q8—22no—
45C22×C3×Q8—42no—
46C22×C3×Q8—42no—
47C22×C3×Q8D414yes—
48C22×C3×Q8C2312yes—
49C22×C3×Q8C2312yes—
50C23×C3×C4—21no—
51C23×C3×C4—21no—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C2×C3×Q16—28no—
56C2×C3×Q16—44no—
57C2×C3×Q16—44no—
58C22×C3×C8C2314yes—
59C3×C4⋊Q8—42no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—24no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—22no—
72C2×C3×C42—42no—
73C2×C3×C42D411yescharacteristic
74C2×C3×C42D411yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C3×C4⋊C8—42no—
78C3×#103—28no—
79C3×#103—44no—
80C3×C22⋊C8—24no—
81C3×#95—22no—
82C3×#95—22no—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—22no—
86C3×#95—22no—
87C23×Q8C22×C311yescharacteristic
88C22×Q16C22×C314yes—
89C2×C4⋊Q8C22×C311yescharacteristic
90C2×C22⋊Q8C22×C312yes—
91C22×C4⋊C4C22×C311yescharacteristic
92C22×C42C22×C311yescharacteristic
93#451—28no—
94C2×C4⋊C8C22×C312yes—
95C2×#103C22×C314yes—
96C2×#103C22×C314yes—
97C2×C22⋊C8C22×C312yes—
98#391—21no—
99#391—21no—
100#391C22×C312yes—
101#391C22×C312yes—
102#391C22×C312yes—
103#389—21no—
104#389—21no—
105#384C22×C311yescharacteristic
106#380—21no—
107#380—21no—
108C2×#95—21no—
109C2×#95—21no—
110#340C22×C314yes—
111#336C22×C312yes—
112C24×C3C2×D411yescharacteristic
113C2×C3×Q8—24no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C2×C3×Q8—24no—
118C2×C3×Q8—24no—
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C2×C3×Q8—48no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×Q8—48no—
125C2×C3×Q8C2×D414yes—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—24no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4Q1614yescomplemented
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4#4112yes—
151C22×C3×C4C2411yescharacteristic
152C3×Q16—416no—
153C2×C3×C8—24no—
154C2×C3×C8—24no—
155C2×C3×C8—24no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4—42no—
162C3×C4⋊C4—42no—
163C3×C4⋊C4—44no—
164C3×C4⋊C4—42no—
165C3×C22⋊C4—44no—
166C3×C42—22no—
167C3×C42—22no—
168C3×C42—41no—
169C3×C42—41no—
170C22×Q8—21no—
171C22×Q8—22no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C3×D414yes—
175C22×Q8C23×C312yes—
176C22×Q8C23×C312yes—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4C3×D411yescharacteristic
180C23×C4C3×D411yescharacteristic
181C23×C4C23×C311yescharacteristic
182C2×Q16—28no—
183C2×Q16—44no—
184C2×Q16—44no—
185C22×C8C23×C314yes—
186C4⋊Q8—42no—
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4C23×C312yes—
197C2×C4⋊C4C23×C312yes—
198C2×C22⋊C4—22no—
199C2×C42—42no—
200C2×C42C3×D411yescharacteristic
201C2×C42C3×D411yescharacteristic
202C2×C42C23×C311yescharacteristic
203C2×C42C23×C311yescharacteristic
204C4⋊C8—42no—
205#103—28no—
206#103—44no—
207C22⋊C8—24no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3#12112yes—
219C23×C3C2×Q1612yes—
220C23×C3#13712yes—
221C23×C3C22×D411yescharacteristic
222C3×Q8—28no—
223C3×Q8—48no—
224C3×Q8—44no—
225C3×Q8—44no—
226C3×Q8—44no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—44no—
242C2×C3×C4—44no—
243C2×C3×C4—48no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—84no—
256C2×C3×C4#12112yes—
257C2×C3×C4C2×Q1612yes—
258C2×C3×C4C2×Q1612yes—
259C2×C3×C4#13712yes—
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C2×#4111yescharacteristic
262C2×C3×C4H(5,𝔽2)12yes—
263C2×C3×C4H(5,𝔽2)12yes—
264C3×C8—44no—
265C24C2×C3×D411yescharacteristic
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—48no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—48no—
278C2×Q8C2×C3×D414yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C3×Q1614yescomplemented
297C22×C4C2×C3×D411yescharacteristic
298C22×C4C2×C3×D411yescharacteristic
299C22×C4C2×C3×D412yes—
300C22×C4C2×C3×D411yescharacteristic
301C22×C4C2×C3×D411yescharacteristic
302C22×C4C2×C3×D411yescharacteristic
303C22×C4C3×#4112yes—
304C22×C4C24×C311yescharacteristic
305Q16—416no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C22⋊C4—44no—
319C42—22no—
320C42—22no—
321C42—41no—
322C42—41no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3#46214yes—
330C22×C3C2×#12111yescharacteristic
331C22×C3#53411yescharacteristic
332C22×C3#53511yescharacteristic
333C22×C3C22×Q1611yescharacteristic
334C22×C3C2×#13711yescharacteristic
335C22×C3C2×#13711yescharacteristic
336C22×C3#57511yescharacteristic
337C3×C4—22no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—44no—
342C3×C4—41no—
343C3×C4—82no—
344C3×C4—82no—
345C3×C4—82no—
346C3×C4#53412yes—
347C3×C4#53812yes—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23C3×#12112yes—
353C23C2×C3×Q1612yes—
354C23C3×#13712yes—
355C23C22×C3×D411yescharacteristic
356Q8—28no—
357Q8—48no—
358Q8—44no—
359Q8—44no—
360Q8—44no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—84no—
390C2×C4C3×#12112yes—
391C2×C4C2×C3×Q1612yes—
392C2×C4C2×C3×Q1612yes—
393C2×C4C3×#13712yes—
394C2×C4C22×C3×D411yescharacteristic
395C2×C4C2×C3×#4111yescharacteristic
396C2×C4C3×H(5,𝔽2)12yes—
397C2×C4C3×H(5,𝔽2)12yes—
398C8—44no—
399C2×C3—24no—
400C2×C3#261711yescharacteristic
401C2×C3C2×#46211yescharacteristic
402C2×C3#303711yescharacteristic
403C2×C3#318911yescharacteristic
404C2×C3#319511yescharacteristic
405C2×C3#320911yescharacteristic
406C2×C3#321511yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22C3×#46214yes—
414C22C2×C3×#12111yescharacteristic
415C22C3×#53411yescharacteristic
416C22C3×#53511yescharacteristic
417C22C22×C3×Q1611yescharacteristic
418C22C2×C3×#13711yescharacteristic
419C22C2×C3×#13711yescharacteristic
420C22C3×#57511yescharacteristic
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—44no—
426C4—41no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4C3×#53412yes—
431C4C3×#53812yes—
432C3#2971511yescharacteristic, direct product factor, Sylow
433C2—24no—
434C2C3×#261711yescharacteristic
435C2C2×C3×#46211yescharacteristic
436C2C3×#303711yescharacteristic
437C2C3×#318911yescharacteristic
438C2C3×#319511yescharacteristic
439C2C3×#320911yescharacteristic
440C2C3×#321511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#2072C2semidirect
5. C3×#2056C2semidirect
6. C3×#2016C2semidirect
9. #29715C3direct (N × H)
38. #2072C2×C3semidirect
39. #2056C2×C3semidirect
40. #2016C2×C3semidirect
143. C22×C3×C4Q16semidirect
296. C22×C4C3×Q16semidirect
432. C3#29715direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4512
C3×#28572
C2×C3×#3912
C3×#20722
C3×#20562
C3×#20162
C3×#18722
C3×#17972
#297153