Finite Group Databasebeta

← C3×#29786 #10648393

Subgroups — C3×#29786 #10648393

Order 768.

754 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#470C211yesmaximal, characteristic
2C2×C3×#424C211yesmaximal, characteristic
3C3×#2625C212yesmaximal
4C3×#2088C212yesmaximal, complemented
5C3×#2068C212yesmaximal, complemented
6C3×#2066C212yesmaximal, complemented
7C3×#2056C212yesmaximal, complemented
8C3×#1892C212yesmaximal
9C3×#1809C211yesmaximal, characteristic
10#29786C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×Q16C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×#117C2211yescharacteristic
16C3×#470—24no—
17C2×C3×#107C2211yescharacteristic
18C2×C3×#106C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C3×#424—24no—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C3×#400—22no—
27C3×#398—22no—
28C3×#397—22no—
29C3×#397—22no—
30C3×#391C2212yes—
31C3×#391C2212yes—
32C3×#389C2212yes—
33C3×#389C2212yes—
34C3×#380C2212yes—
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#340C2212yes—
38C2×#470C2×C311yescharacteristic
39C2×#424C2×C311yescharacteristic
40#2625C2×C312yes—
41#2088C2×C312yescomplemented
42#2068C2×C312yescomplemented
43#2066C2×C312yescomplemented
44#2056C2×C312yescomplemented
45#1892C2×C312yes—
46#1809C2×C311yescharacteristic
47C22×C3×Q8C2312yes—
48C23×C3×C4C2311yescharacteristic
49C2×C3×Q16—22no—
50C2×C3×Q16—41no—
51C2×C3×Q16—41no—
52C2×C3×M16—41no—
53C2×C3×M16—41no—
54C2×C3×M16D414yes—
55C22×C3×C8—22no—
56C22×C3×C8—41no—
57C22×C3×C8—41no—
58C22×C3×C8C2312yes—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C22⋊Q8—24no—
62C3×C22⋊Q8—24no—
63C3×#117—22no—
64C3×#117—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4D412yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4D411yescharacteristic
75C2×C3×C22⋊C4D411yescharacteristic
76C2×C3×C42C2312yes—
77C3×#107—41no—
78C3×#106—22no—
79C3×#106—41no—
80C3×C4⋊C8—24no—
81C3×C4⋊C8—24no—
82C3×#103—24no—
83C3×#103—42no—
84C3×#103—42no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—22no—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—22no—
92C22×Q16C22×C311yescharacteristic
93C22×M16C22×C311yescharacteristic
94C23×C8C22×C311yescharacteristic
95C2×C22⋊Q8C22×C312yes—
96C2×#117C22×C311yescharacteristic
97#470—24no—
98C2×#107C22×C311yescharacteristic
99C2×#106C22×C311yescharacteristic
100C2×#106C22×C311yescharacteristic
101#424—24no—
102C2×C4⋊C8C22×C312yes—
103C2×C4⋊C8C22×C312yes—
104C2×#103C22×C312yes—
105C2×#103C22×C312yes—
106C2×#103C22×C312yes—
107#400—22no—
108#398—22no—
109#397—22no—
110#397—22no—
111#391C22×C312yes—
112#391C22×C312yes—
113#389C22×C312yes—
114#389C22×C312yes—
115#380C22×C312yes—
116#340C22×C312yes—
117#340C22×C312yes—
118#340C22×C312yes—
119C24×C3C2×D411yescharacteristic
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D412yes—
138C22×C3×C4#4112yes—
139C22×C3×C4C2411yescharacteristic
140C3×Q16—44no—
141C3×M16—44no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8—21no—
145C2×C3×C8—21no—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8—44no—
150C2×C3×C8C2×D412yes—
151C2×C3×C8C2×D412yes—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—44no—
160C3×C4⋊C4—82no—
161C3×C4⋊C4—82no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C22⋊C4—44no—
165C3×C22⋊C4—82no—
166C3×C42—22no—
167C3×C42—22no—
168C22×Q8C23×C312yes—
169C23×C4C23×C311yescharacteristic
170C2×Q16—22no—
171C2×Q16—41no—
172C2×Q16—41no—
173C2×M16—41no—
174C2×M16—41no—
175C2×M16C3×D414yes—
176C22×C8—22no—
177C22×C8—41no—
178C22×C8—41no—
179C22×C8C23×C312yes—
180C22×C8C23×C311yescharacteristic
181C22×C8C23×C311yescharacteristic
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184#117—22no—
185#117—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C3×D412yes—
190C2×C4⋊C4C23×C312yes—
191C2×C4⋊C4C23×C312yes—
192C2×C4⋊C4C23×C312yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C3×D411yescharacteristic
196C2×C22⋊C4C3×D411yescharacteristic
197C2×C42C23×C312yes—
198#107—41no—
199#106—22no—
200#106—41no—
201C4⋊C8—24no—
202C4⋊C8—24no—
203#103—24no—
204#103—42no—
205#103—42no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3#12112yes—
218C23×C3C22×D411yescharacteristic
219C3×Q8—42no—
220C3×Q8—42no—
221C3×Q8—42no—
222C3×Q8—42no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
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—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4#12112yes—
248C2×C3×C4C22×D411yescharacteristic
249C2×C3×C4C2×#4111yescharacteristic
250C2×C3×C4H(5,𝔽2)11yescharacteristic
251C2×C3×C4H(5,𝔽2)11yescharacteristic
252C2×C3×C42−1+411yescharacteristic
253C2×C3×C42−1+411yescharacteristic
254C3×C8—22no—
255C3×C8—22no—
256C3×C8—42no—
257C24C2×C3×D411yescharacteristic
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4C2×C3×D412yes—
273C22×C4C2×C3×D412yes—
274C22×C4C2×C3×D411yescharacteristic
275C22×C4C2×C3×D412yes—
276C22×C4C3×#4112yes—
277C22×C4C24×C311yescharacteristic
278Q16—44no—
279M16—44no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—44no—
288C2×C8C2×C3×D412yes—
289C2×C8C2×C3×D412yes—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—44no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—44no—
303C22⋊C4—82no—
304C42—22no—
305C42—22no—
306C22×C3—22no—
307C22×C3—24no—
308C22×C3—42no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3C2×#12111yescharacteristic
312C22×C3#53711yescharacteristic
313C22×C3#53711yescharacteristic
314C22×C3#57712yes—
315C22×C3#57812yes—
316C3×C4—22no—
317C3×C4—42no—
318C3×C4—42no—
319C3×C4—82no—
320C3×C4—82no—
321C3×C4—82no—
322C3×C4—82no—
323C3×C4#53411yescharacteristic
324C3×C4#53611yescharacteristic
325C3×C4#53911yescharacteristic
326C3×C4#53911yescharacteristic
327C23—22no—
328C23—22no—
329C23—21no—
330C23—21no—
331C23C3×#12112yes—
332C23C22×C3×D411yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—42no—
336Q8—42no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4C3×#12112yes—
362C2×C4C22×C3×D411yescharacteristic
363C2×C4C2×C3×#4111yescharacteristic
364C2×C4C3×H(5,𝔽2)11yescharacteristic
365C2×C4C3×H(5,𝔽2)11yescharacteristic
366C2×C4C3×2−1+411yescharacteristic
367C2×C4C3×2−1+411yescharacteristic
368C8—22no—
369C8—22no—
370C8—42no—
371C2×C3—42no—
372C2×C3#261811yescharacteristic
373C2×C3#306312yes—
374C2×C3#325812yes—
375C2×C3#326111yescharacteristic
376C2×C3#326311yescharacteristic
377C22—22no—
378C22—24no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22C2×C3×#12111yescharacteristic
383C22C3×#53711yescharacteristic
384C22C3×#53711yescharacteristic
385C22C3×#57712yes—
386C22C3×#57812yes—
387C4—22no—
388C4—42no—
389C4—42no—
390C4—82no—
391C4—82no—
392C4—82no—
393C4—82no—
394C4C3×#53411yescharacteristic
395C4C3×#53611yescharacteristic
396C4C3×#53911yescharacteristic
397C4C3×#53911yescharacteristic
398C3#2978611yescharacteristic, direct product factor, Sylow
399C2—42no—
400C2C3×#261811yescharacteristic
401C2C3×#306312yes—
402C2C3×#325812yes—
403C2C3×#326111yescharacteristic
404C2C3×#326311yescharacteristic
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
4. C3×#2088C2semidirect
5. C3×#2068C2semidirect
6. C3×#2066C2semidirect
7. C3×#2056C2semidirect
10. #29786C3direct (N × H)
41. #2088C2×C3semidirect
42. #2068C2×C3semidirect
43. #2066C2×C3semidirect
44. #2056C2×C3semidirect
398. C3#29786direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4702
C2×C3×#4242
C3×#26252
C3×#20882
C3×#20682
C3×#20662
C3×#20562
C3×#18922
C3×#18092
#297863