Finite Group Databasebeta

← C3×#33778 #10652385

Subgroups — C3×#33778 #10652385

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3168C211yesmaximal, characteristic, complemented
2C3×#3147C211yesmaximal, characteristic, complemented
3C3×#2249C212yesmaximal, complemented
4C3×#2242C212yesmaximal, complemented
5C3×#2209C211yesmaximal, characteristic, complemented
6C3×#2208C211yesmaximal, characteristic, complemented
7C3×#2207C211yesmaximal, characteristic, complemented
8C3×#2206C211yesmaximal, characteristic, complemented
9C3×#2189C212yesmaximal, complemented
10C3×#2186C212yesmaximal, complemented
11C3×#2162C211yesmaximal, characteristic, complemented
12#33778C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#135—21no—
14C2×C3×#135C2212yes—
15C3×#532—21no—
16C3×#529—21no—
17C2×C3×Q32C2212yes—
18C2×C3×SD32C2212yes—
19C2×C3×SD32C2212yes—
20C2×C3×D16C2212yes—
21C3×#499C2211yescharacteristic
22C3×#495—21no—
23C3×#495C2211yescharacteristic
24C3×#495C2212yes—
25C3×#494C2211yescharacteristic
26C3×#493C2211yescharacteristic
27C3×#492—21no—
28C3×#488—21no—
29C3×#467C2211yescharacteristic
30C3×#466C2211yescharacteristic
31C3×#465—21no—
32C3×C4×Q16C2211yescharacteristic
33C3×C4×SD16—21no—
34C3×C4×D8C2211yescharacteristic
35C2×C3×C4×C8C2211yescharacteristic
36C3×C4⋊C16C2211yescharacteristic
37C3×C4⋊C16C2211yescharacteristic
38C3×#358C2212yes—
39C3×#358C2212yes—
40C3×#357C2212yes—
41C3×#357C2212yes—
42C3×C22⋊C16C2212yes—
43C3×C4×C16C2212yes—
44#3168C2×C311yescharacteristic, complemented
45#3147C2×C311yescharacteristic, complemented
46#2249C2×C312yescomplemented
47#2242C2×C312yescomplemented
48#2209C2×C311yescharacteristic, complemented
49#2208C2×C311yescharacteristic, complemented
50#2207C2×C311yescharacteristic, complemented
51#2206C2×C311yescharacteristic, complemented
52#2189C2×C312yescomplemented
53#2186C2×C312yescomplemented
54#2162C2×C311yescharacteristic, complemented
55C2×C3×#41—22no—
56C3×#135—42no—
57C3×#135—42no—
58C3×#135—44no—
59C2×C3×Q16—21no—
60C2×C3×Q16C2311yescharacteristic
61C2×C3×Q16C2312yes—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—21no—
65C2×C3×D8—21no—
66C2×C3×D8C2311yescharacteristic
67C2×C3×D8C2312yes—
68C22×C3×C8D411yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×C4⋊Q8—21no—
71C3×#127—21no—
72C3×#126—41no—
73C3×#125—21no—
74C3×#124—21no—
75C3×#124—22no—
76C3×#123—41no—
77C3×C22⋊Q8—21no—
78C3×#121—21no—
79C3×#121—41no—
80C3×#121—41no—
81C3×C4×Q8—21no—
82C3×C4×D4—21no—
83C3×C4×D4—41no—
84C3×C4×D4—41no—
85C3×C4×D4—41no—
86C3×#117—41no—
87C2×C3×C42D411yescharacteristic
88C3×Q32—42no—
89C3×SD32—42no—
90C3×SD32—42no—
91C3×D16—42no—
92C2×C3×C16C2312yes—
93C2×C3×C16C2312yes—
94C3×#107C2311yescharacteristic
95C3×#107C2311yescharacteristic
96C3×#106—21no—
97C3×#103—21no—
98C3×#103—21no—
99C3×#102—21no—
100C3×#102—21no—
101C3×C4×C8D411yescharacteristic
102C3×C4×C8D411yescharacteristic
103C3×C4×C8C2311yescharacteristic
104C3×C4×C8C2311yescharacteristic
105C2×#135—21no—
106C2×#135C22×C312yes—
107#532—21no—
108#529—21no—
109C2×Q32C22×C312yes—
110C2×SD32C22×C312yes—
111C2×SD32C22×C312yes—
112C2×D16C22×C312yes—
113#499C22×C311yescharacteristic
114#495—21no—
115#495C22×C312yes—
116#495C22×C311yescharacteristic
117#494C22×C311yescharacteristic
118#493C22×C311yescharacteristic
119#492—21no—
120#488—21no—
121#467C22×C311yescharacteristic
122#466C22×C311yescharacteristic
123#465—21no—
124C4×Q16C22×C311yescharacteristic
125C4×SD16—21no—
126C4×D8C22×C311yescharacteristic
127C2×C4×C8C22×C311yescharacteristic
128C4⋊C16C22×C311yescharacteristic
129C4⋊C16C22×C311yescharacteristic
130#358C22×C312yes—
131#358C22×C312yes—
132#357C22×C312yes—
133#357C22×C312yes—
134C22⋊C16C22×C312yes—
135C4×C16C22×C312yes—
136C3×#41—44no—
137C3×#41—44no—
138C2×C3×Q8—21no—
139C2×C3×Q8—22no—
140C2×C3×D4—21no—
141C2×C3×D4—22no—
142C2×C3×D4—41no—
143C2×C3×D4—42no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C22×C3×C4—41no—
147C22×C3×C4—42no—
148C22×C3×C4D811yescharacteristic
149C22×C3×C4D811yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C3×Q16—21no—
152C3×Q16—22no—
153C3×Q16—21no—
154C3×Q16—22no—
155C3×Q16—42no—
156C3×SD16—44no—
157C3×SD16—42no—
158C3×SD16—44no—
159C3×D8—21no—
160C3×D8—22no—
161C3×D8—21no—
162C3×D8—22no—
163C3×D8—42no—
164C2×C3×C8—22no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8C2×D411yescharacteristic
168C2×C3×C8C2×D411yescharacteristic
169C2×C3×C8C2×D411yescharacteristic
170C2×C3×C8C2411yescharacteristic
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—41no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4—41no—
176C3×C4⋊C4—41no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—41no—
179C3×C22⋊C4—41no—
180C3×C22⋊C4—41no—
181C3×C22⋊C4—41no—
182C3×C22⋊C4—41no—
183C3×C22⋊C4—42no—
184C3×C22⋊C4—42no—
185C3×C22⋊C4—41no—
186C3×C42—41no—
187C3×C42D811yescharacteristic
188C3×C42D811yescharacteristic
189C3×C42C2×D411yescharacteristic
190C3×C42C2×D411yescharacteristic
191C3×C16—22no—
192C3×C16—22no—
193C2×#41—22no—
194#135—42no—
195#135—42no—
196#135—44no—
197C2×Q16—21no—
198C2×Q16C23×C312yes—
199C2×Q16C23×C311yescharacteristic
200C2×SD16—22no—
201C2×SD16—22no—
202C2×SD16—21no—
203C2×D8—21no—
204C2×D8C23×C312yes—
205C2×D8C23×C311yescharacteristic
206C22×C8C3×D411yescharacteristic
207C22×C8C23×C311yescharacteristic
208C4⋊Q8—21no—
209#127—21no—
210#126—41no—
211#125—21no—
212#124—21no—
213#124—22no—
214#123—41no—
215C22⋊Q8—21no—
216#121—21no—
217#121—41no—
218#121—41no—
219C4×Q8—21no—
220C4×D4—21no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224#117—41no—
225C2×C42C3×D411yescharacteristic
226Q32—42no—
227SD32—42no—
228SD32—42no—
229D16—42no—
230C2×C16C23×C312yes—
231C2×C16C23×C312yes—
232#107C23×C311yescharacteristic
233#107C23×C311yescharacteristic
234#106—21no—
235#103—21no—
236#103—21no—
237#102—21no—
238#102—21no—
239C4×C8C3×D411yescharacteristic
240C4×C8C3×D411yescharacteristic
241C4×C8C23×C311yescharacteristic
242C4×C8C23×C311yescharacteristic
243C23×C3—41no—
244C23×C3—42no—
245C23×C3C2×D811yescharacteristic
246C3×Q8—42no—
247C3×Q8—41no—
248C3×Q8—42no—
249C3×Q8—41no—
250C3×D4—42no—
251C3×D4—41no—
252C3×D4—42no—
253C3×D4—41no—
254C3×D4—82no—
255C3×D4—82no—
256C3×D4—84no—
257C3×D4—82no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—22no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—84no—
267C2×C3×C4—82no—
268C2×C3×C4C2×D811yescharacteristic
269C2×C3×C4C2×D811yescharacteristic
270C2×C3×C4C2×D811yescharacteristic
271C2×C3×C4C2×D811yescharacteristic
272C2×C3×C4C2×D811yescharacteristic
273C2×C3×C4C22×D411yescharacteristic
274C3×C8—21no—
275C3×C8—21no—
276C3×C8AGL(1,ℤ/8)12yes—
277C3×C8H(5,𝔽2)11yescharacteristic
278C3×C8H(5,𝔽2)11yescharacteristic
279#41—44no—
280#41—44no—
281C2×Q8—21no—
282C2×Q8—22no—
283C2×D4—21no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4C3×D811yescharacteristic
292C22×C4C3×D811yescharacteristic
293C22×C4C2×C3×D411yescharacteristic
294Q16—21no—
295Q16—22no—
296Q16—21no—
297Q16—22no—
298Q16—42no—
299SD16—44no—
300SD16—44no—
301SD16—42no—
302D8—21no—
303D8—22no—
304D8—21no—
305D8—22no—
306D8—42no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8C2×C3×D411yescharacteristic
311C2×C8C2×C3×D411yescharacteristic
312C2×C8C2×C3×D411yescharacteristic
313C2×C8C24×C311yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C42—41no—
330C42C3×D811yescharacteristic
331C42C3×D811yescharacteristic
332C42C2×C3×D411yescharacteristic
333C42C2×C3×D411yescharacteristic
334C16—22no—
335C16—22no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—81no—
340C22×C3—82no—
341C22×C3—82no—
342C22×C3—81no—
343C22×C3—82no—
344C22×C3—81no—
345C22×C3C22×D811yescharacteristic
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—81no—
351C3×C4—81no—
352C3×C4—82no—
353C3×C4—81no—
354C3×C4#50812yes—
355C3×C4#53511yescharacteristic
356C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
357C23—41no—
358C23—42no—
359C23C2×C3×D811yescharacteristic
360Q8—42no—
361Q8—41no—
362Q8—42no—
363Q8—41no—
364D4—42no—
365D4—41no—
366D4—42no—
367D4—41no—
368D4—82no—
369D4—84no—
370D4—82no—
371D4—82no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4C2×C3×D811yescharacteristic
383C2×C4C2×C3×D811yescharacteristic
384C2×C4C2×C3×D811yescharacteristic
385C2×C4C2×C3×D811yescharacteristic
386C2×C4C2×C3×D811yescharacteristic
387C2×C4C22×C3×D411yescharacteristic
388C8—21no—
389C8—21no—
390C8C3×AGL(1,ℤ/8)12yes—
391C8C3×H(5,𝔽2)11yescharacteristic
392C8C3×H(5,𝔽2)11yescharacteristic
393C2×C3—41no—
394C2×C3—161no—
395C2×C3—162no—
396C2×C3#320511yescharacteristic
397C2×C3C2×#50811yescharacteristic
398C2×C3#341411yescharacteristic
399C22—21no—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—82no—
407C22—81no—
408C22C22×C3×D811yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—81no—
414C4—81no—
415C4—82no—
416C4—81no—
417C4C3×#50812yes—
418C4C3×#53511yescharacteristic
419C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
420C3#3377811yescharacteristic, direct product factor, Sylow
421C2—41no—
422C2—161no—
423C2—162no—
424C2C3×#320511yescharacteristic
425C2C2×C3×#50811yescharacteristic
426C2C3×#341411yescharacteristic
subgroupG/H
center, socleC22×C3C22×D8
commutator, Frattini, cosocleC2×C8C24×C3
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3168C2semidirect
2. C3×#3147C2semidirect
3. C3×#2249C2semidirect
4. C3×#2242C2semidirect
5. C3×#2209C2semidirect
6. C3×#2208C2semidirect
7. C3×#2207C2semidirect
8. C3×#2206C2semidirect
9. C3×#2189C2semidirect
10. C3×#2186C2semidirect
11. C3×#2162C2semidirect
12. #33778C3direct (N × H)
44. #3168C2×C3semidirect
45. #3147C2×C3semidirect
46. #2249C2×C3semidirect
47. #2242C2×C3semidirect
48. #2209C2×C3semidirect
49. #2208C2×C3semidirect
50. #2207C2×C3semidirect
51. #2206C2×C3semidirect
52. #2189C2×C3semidirect
53. #2186C2×C3semidirect
54. #2162C2×C3semidirect
420. C3#33778direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31682
C3×#31472
C3×#22492
C3×#22422
C3×#22092
C3×#22082
C3×#22072
C3×#22062
C3×#21892
C3×#21862
C3×#21622
#337783