Finite Group Databasebeta

← C3×#29296 #10647903

Subgroups — C3×#29296 #10647903

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2857C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic
3C3×#2083C212yesmaximal, complemented
4C3×#2057C212yesmaximal, complemented
5C3×#1942C214yesmaximal, complemented
6C3×#1852C212yesmaximal
7C3×#1852C212yesmaximal
8C3×C4×C22⋊C8C211yesmaximal, characteristic
9#29296C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C22⋊Q8—21no—
12C22×C3×C4⋊C4—21no—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C2×C3×#107C2214yes—
17C2×C3×#107C2214yes—
18C2×C3×C22⋊C8C2212yes—
19C2×C3×C4×C8C2212yes—
20C3×#391—21no—
21C3×#389—21no—
22C3×#389—21no—
23C3×#389—21no—
24C3×#389C2212yes—
25C3×#389C2212yes—
26C3×#389C2212yes—
27C3×#384C2211yescharacteristic
28C3×#380—21no—
29C3×#380C2212yes—
30C2×C3×#95—21no—
31C3×#340C2214yes—
32C3×#340C2214yes—
33C3×#336C2212yes—
34#2857C2×C311yescharacteristic
35C2×#389C2×C311yescharacteristic
36#2083C2×C312yescomplemented
37#2057C2×C312yescomplemented
38#1942C2×C314yescomplemented
39#1852C2×C312yes—
40#1852C2×C312yes—
41C4×C22⋊C8C2×C311yescharacteristic
42C22×C3×Q8—21no—
43C23×C3×C4—21no—
44C23×C3×C4—21no—
45C23×C3×C4D411yescharacteristic
46C23×C3×C4D411yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C22×C3×C8C2314yes—
49C3×C4⋊Q8—42no—
50C3×C22⋊Q8—44no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—42no—
61C2×C3×C4⋊C4—44no—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C42—42no—
68C2×C3×C42D411yescharacteristic
69C2×C3×C42D411yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C3×#107—28no—
73C3×#107—44no—
74C3×C22⋊C8Q818yescomplemented
75C3×C4×C8—42no—
76C3×#95—22no—
77C3×#95—22no—
78C3×#95—22no—
79C3×#95—22no—
80C2×C4⋊Q8C22×C311yescharacteristic
81C2×C22⋊Q8—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4C22×C311yescharacteristic
84C22×C4⋊C4C22×C311yescharacteristic
85C22×C42C22×C311yescharacteristic
86C2×#107C22×C314yes—
87C2×#107C22×C314yes—
88C2×C22⋊C8C22×C312yes—
89C2×C4×C8C22×C312yes—
90#391—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389C22×C312yes—
95#389C22×C312yes—
96#389C22×C312yes—
97#384C22×C311yescharacteristic
98#380—21no—
99#380C22×C312yes—
100C2×#95—21no—
101#340C22×C314yes—
102#340C22×C314yes—
103#336C22×C312yes—
104C24×C3C2×D411yescharacteristic
105C2×C3×Q8—42no—
106C2×C3×Q8—42no—
107C2×C3×Q8—42no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—24no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—42no—
120C22×C3×C4—42no—
121C22×C3×C4—42no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4D814yes—
125C22×C3×C4Q1614yes—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×Q814yes—
132C22×C3×C4C2411yescharacteristic
133C2×C3×C8—24no—
134C2×C3×C8—24no—
135C2×C3×C8C2×Q818yes—
136C3×C4⋊C4—24no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—24no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—24no—
141C3×C4⋊C4—42no—
142C3×C4⋊C4—42no—
143C3×C4⋊C4—42no—
144C3×C4⋊C4—44no—
145C3×C4⋊C4—44no—
146C3×C4⋊C4—42no—
147C3×C4⋊C4—44no—
148C3×C4⋊C4#4114yes—
149C3×C22⋊C4—44no—
150C3×C42—22no—
151C3×C42—22no—
152C3×C42—41no—
153C3×C42—41no—
154C22×Q8—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C3×D411yescharacteristic
158C23×C4C3×D411yescharacteristic
159C23×C4C23×C311yescharacteristic
160C22×C8C23×C314yes—
161C4⋊Q8—42no—
162C22⋊Q8—44no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C23×C312yes—
175C2×C4⋊C4C23×C312yes—
176C2×C4⋊C4C23×C312yes—
177C2×C4⋊C4C23×C312yes—
178C2×C22⋊C4—22no—
179C2×C42—42no—
180C2×C42C3×D411yescharacteristic
181C2×C42C3×D411yescharacteristic
182C2×C42C23×C311yescharacteristic
183C2×C42C23×C311yescharacteristic
184#107—28no—
185#107—44no—
186C22⋊C8C3×Q818yescomplemented
187C4×C8—42no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3C4⋊Q812yes—
197C23×C3C2×D812yes—
198C23×C3C2×Q1612yes—
199C23×C3C22×D411yescharacteristic
200C3×Q8—84no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—28no—
212C2×C3×C4—24no—
213C2×C3×C4—44no—
214C2×C3×C4—44no—
215C2×C3×C4—48no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—84no—
228C2×C3×C4C4⋊Q812yes—
229C2×C3×C4C2×D812yes—
230C2×C3×C4C2×D812yes—
231C2×C3×C4C2×Q1612yes—
232C2×C3×C4C2×Q1612yes—
233C2×C3×C4C22×Q812yes—
234C2×C3×C4C2×#4112yes—
235C2×C3×C4H(5,𝔽2)12yes—
236C3×C8—28no—
237C24C2×C3×D411yescharacteristic
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C3×D814yes—
258C22×C4C3×Q1614yes—
259C22×C4C2×C3×D411yescharacteristic
260C22×C4C2×C3×D411yescharacteristic
261C22×C4C2×C3×D411yescharacteristic
262C22×C4C2×C3×D411yescharacteristic
263C22×C4C2×C3×D411yescharacteristic
264C22×C4C2×C3×Q814yes—
265C22×C4C24×C311yescharacteristic
266C2×C8—24no—
267C2×C8—24no—
268C2×C8C2×C3×Q818yes—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4C3×#4114yes—
282C22⋊C4—44no—
283C42—22no—
284C42—22no—
285C42—41no—
286C42—41no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3#50014yes—
294C22×C3#52511yescharacteristic
295C22×C3C2×C4⋊Q811yescharacteristic
296C22×C3#53411yescharacteristic
297C22×C3C22×D811yescharacteristic
298C22×C3C22×Q1611yescharacteristic
299C22×C3#57511yescharacteristic
300C22×C3#57511yescharacteristic
301C3×C4—22no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—22no—
306C3×C4—44no—
307C3×C4—82no—
308C3×C4—82no—
309C3×C4—82no—
310C3×C4#55414yes—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23C3×C4⋊Q812yes—
316C23C2×C3×D812yes—
317C23C2×C3×Q1612yes—
318C23C22×C3×D411yescharacteristic
319Q8—84no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—28no—
331C2×C4—24no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—84no—
347C2×C4C3×C4⋊Q812yes—
348C2×C4C2×C3×D812yes—
349C2×C4C2×C3×D812yes—
350C2×C4C2×C3×Q1612yes—
351C2×C4C2×C3×Q1612yes—
352C2×C4C22×C3×Q812yes—
353C2×C4C2×C3×#4112yes—
354C2×C4C3×H(5,𝔽2)12yes—
355C8—28no—
356C2×C3—24no—
357C2×C3#258311yescharacteristic
358C2×C3#310011yescharacteristic
359C2×C3#310411yescharacteristic
360C2×C3C2×#50011yescharacteristic
361C2×C3#316611yescharacteristic
362C2×C3#318611yescharacteristic
363C2×C3#318911yescharacteristic
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22C3×#50014yes—
371C22C3×#52511yescharacteristic
372C22C2×C3×C4⋊Q811yescharacteristic
373C22C3×#53411yescharacteristic
374C22C22×C3×D811yescharacteristic
375C22C22×C3×Q1611yescharacteristic
376C22C3×#57511yescharacteristic
377C22C3×#57511yescharacteristic
378C4—22no—
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—44no—
384C4—82no—
385C4—82no—
386C4—82no—
387C4C3×#55414yes—
388C3#2929611yescharacteristic, direct product factor, Sylow
389C2—24no—
390C2C3×#258311yescharacteristic
391C2C3×#310011yescharacteristic
392C2C3×#310411yescharacteristic
393C2C2×C3×#50011yescharacteristic
394C2C3×#316611yescharacteristic
395C2C3×#318611yescharacteristic
396C2C3×#318911yescharacteristic
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
3. C3×#2083C2semidirect
4. C3×#2057C2semidirect
5. C3×#1942C2semidirect
9. #29296C3direct (N × H)
36. #2083C2×C3semidirect
37. #2057C2×C3semidirect
38. #1942C2×C3semidirect
74. C3×C22⋊C8Q8semidirect
186. C22⋊C8C3×Q8semidirect
388. C3#29296direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28572
C2×C3×#3892
C3×#20832
C3×#20572
C3×#19422
C3×#18522
C3×C4×C22⋊C82
#292963