Finite Group Databasebeta

← C3×#29311 #10647918

Subgroups — C3×#29311 #10647918

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2628C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic
3C3×#2085C212yesmaximal, complemented
4C3×#2058C212yesmaximal, complemented
5C3×#1943C212yesmaximal, complemented
6C3×#1941C212yesmaximal, complemented
7C3×#1854C212yesmaximal
8C3×#1853C212yesmaximal
9C3×#1750C211yesmaximal, characteristic
10#29311C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×C3×#107C2212yes—
16C2×C3×#106C2212yes—
17C2×C3×#106C2212yes—
18C2×C3×#106C2212yes—
19C2×C3×C22⋊C8C2212yes—
20C2×C3×#97C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C3×#400—21no—
23C3×#399—21no—
24C3×#395—21no—
25C3×#393—21no—
26C3×#389C2212yes—
27C3×#389C2212yes—
28C3×#389C2212yes—
29C3×#385—21no—
30C3×#385—21no—
31C3×#384—21no—
32C3×#382—21no—
33C3×#382C2211yescharacteristic
34C3×#380C2212yes—
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#336C2212yes—
40#2628C2×C311yescharacteristic
41#2317C2×C311yescharacteristic
42#2085C2×C312yescomplemented
43#2058C2×C312yescomplemented
44#1943C2×C312yescomplemented
45#1941C2×C312yescomplemented
46#1854C2×C312yes—
47#1853C2×C312yes—
48#1750C2×C311yescharacteristic
49C22×C3×Q8—21no—
50C23×C3×C4—21no—
51C23×C3×C4C2311yescharacteristic
52C22×C3×C8C2312yes—
53C22×C3×C8C2312yes—
54C3×C4⋊Q8—41no—
55C3×C4⋊Q8—41no—
56C3×#117—24no—
57C3×#117—41no—
58C3×#117—41no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—41no—
65C2×C3×C4⋊C4—41no—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4D411yescharacteristic
75C2×C3×C22⋊C4D411yescharacteristic
76C2×C3×C42—21no—
77C2×C3×C42C2311yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C3×#107—24no—
80C3×#106—24no—
81C3×#106—42no—
82C3×#106—42no—
83C3×C22⋊C8Q818yescomplemented
84C3×#97—41no—
85C3×C4×C8—41no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—22no—
89C3×#95—21no—
90C3×#95—21no—
91C3×#95—22no—
92C2×C4⋊Q8C22×C311yescharacteristic
93C2×#117C22×C311yescharacteristic
94C2×#117C22×C311yescharacteristic
95C22×C4⋊C4C22×C311yescharacteristic
96C2×#107C22×C312yes—
97C2×#106C22×C312yes—
98C2×#106C22×C312yes—
99C2×#106C22×C312yes—
100C2×C22⋊C8C22×C312yes—
101C2×#97C22×C311yescharacteristic
102C2×C4×C8C22×C311yescharacteristic
103#400—21no—
104#399—21no—
105#395—21no—
106#393—21no—
107#389C22×C312yes—
108#389C22×C312yes—
109#389C22×C312yes—
110#385—21no—
111#385—21no—
112#384—21no—
113#382—21no—
114#382C22×C311yescharacteristic
115#380C22×C312yes—
116#340C22×C312yes—
117#340C22×C312yes—
118#340C22×C312yes—
119#340C22×C312yes—
120#336C22×C312yes—
121C24×C3C2×D411yescharacteristic
122C2×C3×Q8—42no—
123C2×C3×Q8—82no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—42no—
132C22×C3×C4—82no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×Q814yes—
139C22×C3×C4C2411yescharacteristic
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C2×C3×C8C2×Q814yes—
145C2×C3×C8C2×Q814yes—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4—42no—
157C3×C4⋊C4—82no—
158C3×C4⋊C4—82no—
159C3×C4⋊C4—82no—
160C3×C4⋊C4#4114yes—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—82no—
165C3×C42—24no—
166C3×C42—21no—
167C3×C42—21no—
168C3×C42—21no—
169C3×C42—21no—
170C22×Q8—21no—
171C23×C4—21no—
172C23×C4C23×C311yescharacteristic
173C22×C8C23×C312yes—
174C22×C8C23×C312yes—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177#117—24no—
178#117—41no—
179#117—41no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C3×D411yescharacteristic
188C2×C4⋊C4C3×D411yescharacteristic
189C2×C4⋊C4C23×C312yes—
190C2×C4⋊C4C23×C312yes—
191C2×C4⋊C4C23×C312yes—
192C2×C4⋊C4C23×C312yes—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C3×D411yescharacteristic
196C2×C22⋊C4C3×D411yescharacteristic
197C2×C42—21no—
198C2×C42C23×C311yescharacteristic
199C2×C42C23×C311yescharacteristic
200#107—24no—
201#106—24no—
202#106—42no—
203#106—42no—
204C22⋊C8C3×Q818yescomplemented
205#97—41no—
206C4×C8—41no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—21no—
212#95—22no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3C4⋊Q812yes—
218C23×C3C22×D411yescharacteristic
219C3×Q8—84no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—84no—
245C2×C3×C4C4⋊Q812yes—
246C2×C3×C4C22×Q812yes—
247C2×C3×C4C2×#4112yes—
248C2×C3×C4H(5,𝔽2)11yescharacteristic
249C2×C3×C4H(5,𝔽2)11yescharacteristic
250C3×C8—24no—
251C3×C8—24no—
252C24C2×C3×D411yescharacteristic
253C2×Q8—42no—
254C2×Q8—82no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4—82no—
264C22×C4C2×C3×D411yescharacteristic
265C22×C4C2×C3×D411yescharacteristic
266C22×C4C2×C3×D411yescharacteristic
267C22×C4C2×C3×D411yescharacteristic
268C22×C4C2×C3×D411yescharacteristic
269C22×C4C2×C3×Q814yes—
270C22×C4C24×C311yescharacteristic
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8C2×C3×Q814yes—
276C2×C8C2×C3×Q814yes—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4C3×#4114yes—
292C22⋊C4—24no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—82no—
296C42—24no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C22×C3—22no—
302C22×C3—24no—
303C22×C3—42no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3#52511yescharacteristic
307C22×C3C2×C4⋊Q811yescharacteristic
308C22×C3#53411yescharacteristic
309C22×C3#57712yes—
310C22×C3#57712yes—
311C3×C4—22no—
312C3×C4—44no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—82no—
318C3×C4—82no—
319C3×C4—82no—
320C3×C4#55412yes—
321C3×C4#55412yes—
322C23—22no—
323C23—22no—
324C23—21no—
325C23—21no—
326C23C3×C4⋊Q812yes—
327C23C22×C3×D411yescharacteristic
328Q8—84no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—84no—
354C2×C4C3×C4⋊Q812yes—
355C2×C4C22×C3×Q812yes—
356C2×C4C2×C3×#4112yes—
357C2×C4C3×H(5,𝔽2)11yescharacteristic
358C2×C4C3×H(5,𝔽2)11yescharacteristic
359C8—24no—
360C8—24no—
361C2×C3—42no—
362C2×C3#258311yescharacteristic
363C2×C3#312512yes—
364C2×C3#316312yes—
365C2×C3#320011yescharacteristic
366C2×C3#320011yescharacteristic
367C22—24no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22C3×#52511yescharacteristic
373C22C2×C3×C4⋊Q811yescharacteristic
374C22C3×#53411yescharacteristic
375C22C3×#57712yes—
376C22C3×#57712yes—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—44no—
383C4—82no—
384C4—82no—
385C4—82no—
386C4C3×#55412yes—
387C4C3×#55412yes—
388C3#2931111yescharacteristic, direct product factor, Sylow
389C2—42no—
390C2C3×#258311yescharacteristic
391C2C3×#312512yes—
392C2C3×#316312yes—
393C2C3×#320011yescharacteristic
394C2C3×#320011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2085C2semidirect
4. C3×#2058C2semidirect
5. C3×#1943C2semidirect
6. C3×#1941C2semidirect
10. #29311C3direct (N × H)
42. #2085C2×C3semidirect
43. #2058C2×C3semidirect
44. #1943C2×C3semidirect
45. #1941C2×C3semidirect
83. C3×C22⋊C8Q8semidirect
204. C22⋊C8C3×Q8semidirect
388. C3#29311direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26282
C3×#23172
C3×#20852
C3×#20582
C3×#19432
C3×#19412
C3×#18542
C3×#18532
C3×#17502
#293113