Finite Group Databasebeta

← C3×#29706 #10648313

Subgroups — C3×#29706 #10648313

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3461C211yesmaximal, characteristic
2C3×#3079C212yesmaximal, complemented
3C3×#2567C211yesmaximal, characteristic, complemented
4C3×#2060C212yesmaximal, complemented
5C3×#2039C212yesmaximal
6C3×#2038C214yesmaximal, complemented
7C3×#1806C212yesmaximal, complemented
8C3×#1801C211yesmaximal, characteristic, complemented
9#29706C311yesmaximal, characteristic, direct product factor, Sylow
10C3×#544C2211yescharacteristic
11C3×#542—22no—
12C3×#541—22no—
13C3×#539—22no—
14C3×#537C2211yescharacteristic
15C3×#536C2212yes—
16C3×#533—22no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C3×#529—22no—
19C3×#529—22no—
20C3×C4×#41C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C3×#477—22no—
23C3×#476—22no—
24C3×#475—22no—
25C3×#474—22no—
26C3×#428C2214yes—
27C3×#423C2212yes—
28C2×C3×#104C2214yes—
29C3×#417C2214yes—
30C2×C3×#101C2212yes—
31C3×#395—22no—
32C3×#391C2212yes—
33C3×#389—22no—
34C3×#384C2212yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#343C2212yes—
37C3×#339C2214yes—
38#3461C2×C311yescharacteristic
39#3079C2×C312yescomplemented
40#2567C2×C311yescharacteristic, complemented
41#2060C2×C312yescomplemented
42#2039C2×C312yes—
43#2038C2×C314yescomplemented
44#1806C2×C312yescomplemented
45#1801C2×C311yescharacteristic, complemented
46C2×C3×#41C2312yes—
47C22×C3×Q8—21no—
48C22×C3×Q8C2311yescharacteristic
49C2×C3×M16C2314yes—
50C3×C4⋊Q8—22no—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—22no—
53C3×C4⋊Q8—22no—
54C3×#126—24no—
55C3×#126—24no—
56C3×#125—22no—
57C3×#125—24no—
58C3×#125—22no—
59C3×#124—22no—
60C3×#124—22no—
61C3×#123—24no—
62C3×#123—24no—
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—22no—
68C3×#121—22no—
69C3×#121—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8D412yes—
74C3×C4×Q8D412yes—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C3×C4×D4D412yes—
79C3×C4×D4D412yes—
80C3×#117—24no—
81C3×#117—22no—
82C3×#117C2312yes—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4D412yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—22no—
90C2×C3×C42D412yes—
91C2×C3×C42C2312yes—
92C2×C3×C42C2311yescharacteristic
93C3×#107—24no—
94C3×#106—24no—
95C3×C4⋊C8—22no—
96C3×C4⋊C8—22no—
97C3×#104—44no—
98C3×#103—24no—
99C3×#102—24no—
100C3×#101—42no—
101C3×#95—21no—
102C3×#95—24no—
103#544C22×C311yescharacteristic
104#542—22no—
105#541—22no—
106#539—22no—
107#537C22×C311yescharacteristic
108#536C22×C312yes—
109#533—22no—
110C2×C4⋊Q8C22×C311yescharacteristic
111#529—22no—
112#529—22no—
113C4×#41C22×C312yes—
114C2×C4×Q8C22×C311yescharacteristic
115#477—22no—
116#476—22no—
117#475—22no—
118#474—22no—
119#428C22×C314yes—
120#423C22×C312yes—
121C2×#104C22×C314yes—
122#417C22×C314yes—
123C2×#101C22×C312yes—
124#395—22no—
125#391C22×C312yes—
126#389—22no—
127#384C22×C312yes—
128C4×C4⋊C4C22×C311yescharacteristic
129#343C22×C312yes—
130#339C22×C314yes—
131C3×#41—22no—
132C3×#41—22no—
133C3×#41—22no—
134C3×#41—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—21no—
140C2×C3×Q8—42no—
141C2×C3×Q8—82no—
142C2×C3×Q8C2×D412yes—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4C2×D412yes—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C3×M16—24no—
157C3×M16—24no—
158C2×C3×C8—24no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—24no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—84no—
173C3×C4⋊C4C2×D412yes—
174C3×C4⋊C4C2×D412yes—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—22no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190C2×#41C23×C312yes—
191C22×Q8—21no—
192C22×Q8C23×C311yescharacteristic
193C2×M16C23×C314yes—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198#126—24no—
199#126—24no—
200#125—22no—
201#125—24no—
202#125—22no—
203#124—22no—
204#124—22no—
205#123—24no—
206#123—24no—
207C22⋊Q8—24no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212#121—22no—
213#121—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8C3×D412yes—
218C4×Q8C3×D412yes—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4C3×D412yes—
223C4×D4C3×D412yes—
224#117—24no—
225#117—22no—
226#117C23×C312yes—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4C3×D412yes—
231C2×C4⋊C4C23×C312yes—
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C42—22no—
234C2×C42C3×D412yes—
235C2×C42C23×C312yes—
236C2×C42C23×C311yescharacteristic
237#107—24no—
238#106—24no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241#104—44no—
242#103—24no—
243#102—24no—
244#101—42no—
245#95—21no—
246#95—24no—
247C23×C3—22no—
248C23×C3C22×D411yescharacteristic
249C3×Q8—22no—
250C3×Q8—22no—
251C3×Q8—42no—
252C3×Q8—42no—
253C3×Q8—84no—
254C3×Q8—82no—
255C3×D4—22no—
256C3×D4—22no—
257C3×D4—42no—
258C3×D4—42no—
259C3×D4—82no—
260C2×C3×C4—22no—
261C2×C3×C4—24no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—44no—
275C2×C3×C4—42no—
276C2×C3×C4—44no—
277C2×C3×C4—42no—
278C2×C3×C4—82no—
279C2×C3×C4#12011yescharacteristic
280C2×C3×C4#12012yes—
281C2×C3×C4#12011yescharacteristic
282C2×C3×C4C22×D412yes—
283C2×C3×C4H(5,𝔽2)11yescharacteristic
284C2×C3×C42−1+411yescharacteristic
285C2×C3×C42−1+412yes—
286C3×C8—44no—
287#41—22no—
288#41—22no—
289#41—22no—
290#41—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—21no—
294C2×Q8—22no—
295C2×Q8—21no—
296C2×Q8—42no—
297C2×Q8—82no—
298C2×Q8C2×C3×D412yes—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4C2×C3×D412yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4C2×C3×D412yes—
308C22×C4C2×C3×D411yescharacteristic
309C22×C4C2×C3×D412yes—
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C24×C311yescharacteristic
312M16—24no—
313M16—24no—
314C2×C8—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—84no—
329C4⋊C4C2×C3×D412yes—
330C4⋊C4C2×C3×D412yes—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42C2×C3×D412yes—
345C42C2×C3×D412yes—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3C2×#12011yescharacteristic
352C22×C3#53611yescharacteristic
353C22×C3#53711yescharacteristic
354C3×C4—21no—
355C3×C4—22no—
356C3×C4—44no—
357C3×C4—42no—
358C3×C4—42no—
359C3×C4—42no—
360C3×C4—42no—
361C3×C4—42no—
362C3×C4—82no—
363C3×C4#45612yes—
364C3×C4#53612yes—
365C3×C4#53712yes—
366C23—22no—
367C23C22×C3×D411yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—42no—
371Q8—42no—
372Q8—84no—
373Q8—82no—
374D4—22no—
375D4—22no—
376D4—42no—
377D4—42no—
378D4—82no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—82no—
398C2×C4C3×#12011yescharacteristic
399C2×C4C3×#12012yes—
400C2×C4C3×#12011yescharacteristic
401C2×C4C22×C3×D412yes—
402C2×C4C3×H(5,𝔽2)11yescharacteristic
403C2×C4C3×2−1+411yescharacteristic
404C2×C4C3×2−1+412yes—
405C8—44no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—82no—
409C2×C3#261611yescharacteristic
410C2×C3C2×#45611yescharacteristic
411C2×C3#302111yescharacteristic
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22C2×C3×#12011yescharacteristic
418C22C3×#53611yescharacteristic
419C22C3×#53711yescharacteristic
420C4—21no—
421C4—22no—
422C4—44no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—82no—
429C4C3×#45612yes—
430C4C3×#53612yes—
431C4C3×#53712yes—
432C3#2970611yescharacteristic, direct product factor, Sylow
433C2—21no—
434C2—21no—
435C2—82no—
436C2C3×#261611yescharacteristic
437C2C2×C3×#45611yescharacteristic
438C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3079C2semidirect
3. C3×#2567C2semidirect
4. C3×#2060C2semidirect
6. C3×#2038C2semidirect
7. C3×#1806C2semidirect
8. C3×#1801C2semidirect
9. #29706C3direct (N × H)
39. #3079C2×C3semidirect
40. #2567C2×C3semidirect
41. #2060C2×C3semidirect
43. #2038C2×C3semidirect
44. #1806C2×C3semidirect
45. #1801C2×C3semidirect
432. C3#29706direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34612
C3×#30792
C3×#25672
C3×#20602
C3×#20392
C3×#20382
C3×#18062
C3×#18012
#297063