Finite Group Databasebeta

← C3×#10593415 #13166779

Subgroups — C3×#10593415 #13166779

Order 1728.

630 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64308C211yesmaximal, characteristic
2C3×#64233C212yesmaximal, complemented
3C3×#63897C211yesmaximal, characteristic, complemented
4C3×#63774C211yesmaximal, characteristic, complemented
5C3×#63738C212yesmaximal, complemented
6#10593415C311yesmaximal, characteristic, direct product factor
7C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
8C32×#235—21no—
9C3×#3811—21no—
10C3×#3811C2212yes—
11C2×C3×#624C2211yescharacteristic, complemented
12C3×D3×D12—22no—
13C3×#3799—22no—
14C3×C22⋊#157C2212yescomplemented
15C3×#3773—22no—
16C3×D32⋊C4—22no—
17C32×Dic3⋊C4—21no—
18C3×#3722—21no—
19C3×#3722C2211yescharacteristic, complemented
20C3×#1009—61no—
21#64308C2×C311yescharacteristic
22#64233C2×C312yescomplemented
23C3×#894—61no—
24#63897C2×C311yescharacteristic, complemented
25#63774C2×C311yescharacteristic, complemented
26#63738C2×C312yescomplemented
27C22×C3×C3⋊D3D411yescharacteristic
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3×D32—21no—
30C2×C3×D32D412yes—
31C2×C3×#157—22no—
32C2×C33×C4D411yescharacteristic
33C3×C4⋊(C3⋊D3)—22no—
34C32×#67—22no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3—21no—
37C32×D12—21no—
38C32×C4×D3—22no—
39C32×Dic6—21no—
40C3×#626—22no—
41C3×#626—22no—
42C3×#624—21no—
43C3×#624—22no—
44C3×#624D412yes—
45C3×#458—91nomaximal
46C32×#41—61no—
47C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
48C3×#239—62no—
49C3×D3×D4—62no—
50C3×D3×D4—62no—
51C3×D3×D4—62no—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—41no—
55C3×#235—62no—
56C2×C3×D12—62no—
57C2×C3×D12—61no—
58C2×C3×D12—61no—
59C2×C3×C4×D3—61no—
60#3811—21no—
61#3811C22×C312yes—
62C2×#624C22×C311yescharacteristic, complemented
63D3×D12—22no—
64#3799—22no—
65C22⋊#157C22×C312yescomplemented
66#3773—22no—
67D32⋊C4—22no—
68C32×C4⋊C4—61no—
69C3×(C2×D3)⋊C4—61no—
70C3×(C2×D3)⋊C4—61no—
71C3×(C2×D3)⋊C4—61no—
72C3×Dic3⋊C4—21no—
73C3×Dic3⋊C4—21no—
74C3×Dic3⋊C4—41no—
75#3722—21no—
76#3722C22×C311yescharacteristic, complemented
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3C2×D412yes—
82C2×C32×D3—22no—
83C3×D32—42no—
84C3×D32—81no—
85C3×#157—42no—
86C33×C4—21no—
87C32×Dic3—22no—
88C32×Dic3—41no—
89C32×Dic3—41no—
90C3×H(5,𝔽2)—91no—
91#1009—61no—
92C3×#123—91no—
93C3×#123—92no—
94#894—61no—
95C3×#99—91no—
96C3×#99—92no—
97C22×C3⋊D3C3×D411yescharacteristic
98C22×C3⋊D3C23×C311yescharacteristic
99C22×C3×D3—62no—
100C22×C3×D3—61no—
101C22×C3×D3—61no—
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—62no—
105C2×D32—21no—
106C2×D32C3×D412yes—
107C2×#157—22no—
108C32×Q8—61no—
109C32×D4—61no—
110C32×D4—62no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4—62no—
116C2×C32×C4—61no—
117C2×C32×C4—61no—
118C2×C32×C4C3×D411yescharacteristic
119C4⋊(C3⋊D3)—22no—
120C3×#67—22no—
121C3×#67—22no—
122C3×#67—42no—
123C3×#67—62no—
124C3×#67—62no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—41no—
130C2×C3×Dic3—41no—
131C3×D12—21no—
132C3×D12—21no—
133C3×D12—41no—
134C3×D12—62no—
135C3×D12—62no—
136C3×D12—62no—
137C3×D12—62no—
138C3×D12—122no—
139C3×C4×D3—22no—
140C3×C4×D3—22no—
141C3×C4×D3—42no—
142C3×C4×D3—62no—
143C3×C4×D3—62no—
144C3×C4×D3—122no—
145C3×Dic6—21no—
146C3×Dic6—21no—
147C3×Dic6—41no—
148#626—22no—
149#626—22no—
150#624—21no—
151#624—22no—
152#624C3×D412yes—
153#458—91noSylow
154C2×C33—21no—
155C2×C33#12011yescharacteristic
156C3×C3⋊D3—22no—
157C3×C3⋊D3—41no—
158C32×D3—42no—
159C3×#41—61no—
160C3×#41—61no—
161C3×#41—121no—
162C3×#41—182no—
163C2×C3×D4—92no—
164C2×C3×D4—91no—
165C2×C3×D4—182no—
166C2×C3×D4—181no—
167C22×C3×C4—91no—
168#239—62no—
169D3×D4—62no—
170D3×D4—62no—
171D3×D4—62no—
172#235—21no—
173#235—62no—
174C2×D12—61no—
175C2×D12—61no—
176C2×D12—62no—
177C2×C4×D3—61no—
178C3×C4⋊C4—61no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4—121no—
181C3×C4⋊C4—182no—
182C3×C22⋊C4—91no—
183C3×C22⋊C4—92no—
184C3×C22⋊C4—181no—
185C3×C22⋊C4—182no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—61no—
189Dic3⋊C4—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—62no—
195C22×C32C2×C3×D411yescharacteristic
196C2×C3⋊D3—21no—
197C2×C3⋊D3—41no—
198C2×C3⋊D3—41no—
199C2×C3⋊D3C2×C3×D412yes—
200C2×C3×D3—22no—
201C2×C3×D3—22no—
202C2×C3×D3—42no—
203C2×C3×D3—62no—
204C2×C3×D3—62no—
205C2×C3×D3—62no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C2×C3×D3—242no—
212C2×C3×D3—241no—
213C2×C3×D3—241no—
214C2×C3×D3—242no—
215D32—42no—
216D32—81no—
217#157—42no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—62no—
225C32×C4—81no—
226C32×C4—121no—
227C32×C4—121no—
228C3×Dic3—22no—
229C3×Dic3—22no—
230C3×Dic3—42no—
231C3×Dic3—41no—
232C3×Dic3—41no—
233C3×Dic3—41no—
234C3×Dic3—42no—
235C3×Dic3—41no—
236C3×Dic3—81no—
237H(5,𝔽2)—91no—
238#123—91no—
239#123—92no—
240#99—91no—
241#99—92no—
242C33#45811yescharacteristic, complemented, Sylow
243C23×C3—92no—
244C23×C3—91no—
245C23×C3—91no—
246C23×C3—181no—
247C22×D3—62no—
248C22×D3—61no—
249C22×D3—61no—
250C22×D3—61no—
251C22×D3—61no—
252C22×D3—62no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—121no—
256C3×D4—61no—
257C3×D4—61no—
258C3×D4—62no—
259C3×D4—62no—
260C3×D4—121no—
261C3×D4—122no—
262C3×D4—182no—
263C3×D4—182no—
264C3×D4—182no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—92no—
276C2×C3×C4—122no—
277C2×C3×C4—121no—
278C2×C3×C4—121no—
279C2×C3×C4—181no—
280C2×C3×C4—182no—
281C2×C3×C4—182no—
282C2×C3×C4GO+(4,𝔽2)11yescharacteristic
283#67—22no—
284#67—62no—
285#67—62no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288D12—21no—
289D12—62no—
290D12—62no—
291D12—62no—
292D12—62no—
293D12—122no—
294C4×D3—22no—
295C4×D3—62no—
296C4×D3—62no—
297C4×D3—122no—
298Dic6—21no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32—81no—
310C2×C32—122no—
311C2×C32C3×#12011yescharacteristic
312C3⋊D3—22no—
313C3⋊D3—41no—
314C3×D3—42no—
315C3×D3—42no—
316C3×D3—82no—
317C3×D3—122no—
318C3×D3—122no—
319C3×D3—122no—
320C3×D3—241no—
321#41—61no—
322#41—182no—
323C2×D4—92no—
324C2×D4—91no—
325C2×D4—182no—
326C2×D4—181no—
327C22×C4—91no—
328C4⋊C4—61no—
329C4⋊C4—182no—
330C22⋊C4—91no—
331C22⋊C4—92no—
332C22⋊C4—181no—
333C22⋊C4—182no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—62no—
339C22×C3—62no—
340C22×C3—92no—
341C22×C3—122no—
342C22×C3—181no—
343C22×C3—361no—
344C22×C3—361no—
345C22×C3—362no—
346C22×C3—721no—
347C22×C3C2×GO+(4,𝔽2)11yescharacteristic
348C2×D3—22no—
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—62no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—242no—
358C2×D3—241no—
359C2×D3—241no—
360C2×D3—242no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—62no—
368C3×C4—62no—
369C3×C4—81no—
370C3×C4—122no—
371C3×C4—121no—
372C3×C4—121no—
373C3×C4—121no—
374C3×C4—122no—
375C3×C4—121no—
376C3×C4—241no—
377C3×C4—362no—
378Dic3—22no—
379Dic3—41no—
380Dic3—41no—
381C32—21no—
382C32—21no—
383C32—41no—
384C32—41no—
385C32C3×#45811yescharacteristic, complemented
386C23—92no—
387C23—91no—
388C23—91no—
389C23—181no—
390Q8—61no—
391D4—61no—
392D4—62no—
393D4—182no—
394D4—182no—
395D4—182no—
396C2×C4—62no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—92no—
400C2×C4—181no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4C3×GO+(4,𝔽2)11yescharacteristic
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—81no—
415C2×C3—122no—
416C2×C3—122no—
417C2×C3—182no—
418C2×C3—242no—
419C2×C3—361no—
420C2×C3#6424111yescharacteristic
421D3—42no—
422D3—122no—
423D3—122no—
424D3—122no—
425D3—241no—
426C22—62no—
427C22—92no—
428C22—181no—
429C22—361no—
430C22—361no—
431C22—362no—
432C22—721no—
433C22C2×C3×GO+(4,𝔽2)11yescharacteristic
434C4—21no—
435C4—62no—
436C4—121no—
437C4—121no—
438C4—362no—
439C3—21no—
440C3—21no—
441C3—41no—
442C3—41no—
443C3#1059341511yescharacteristic, direct product factor
444C2—21no—
445C2—122no—
446C2—182no—
447C2—361no—
448C2C3×#6424111yescharacteristic
subgroupG/H
centerC2×C3#64241
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33#120
prime radicalC3×#64308C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#458
LCSG ▷ C22×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64233C2semidirect
3. C3×#63897C2semidirect
4. C3×#63774C2semidirect
5. C3×#63738C2semidirect
6. #10593415C3direct (N × H)
11. C2×C3×#624C22semidirect
14. C3×C22⋊#157C22semidirect
19. C3×#3722C22semidirect
22. #64233C2×C3semidirect
24. #63897C2×C3semidirect
25. #63774C2×C3semidirect
26. #63738C2×C3semidirect
62. C2×#624C22×C3semidirect
65. C22⋊#157C22×C3semidirect
76. #3722C22×C3semidirect
242. C33#458semidirect
385. C32C3×#458semidirect
443. C3#10593415direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643082
C3×#642332
C3×#638972
C3×#637742
C3×#637382
#105934153
C3×#4589