Finite Group Databasebeta

← C3×#10593518 #13166884

Subgroups — C3×#10593518 #13166884

Order 1728.

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

normal subgroups only

This group has 652 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×#64330C211yesmaximal, characteristic
2C3×#64233C212yesmaximal, complemented
3C3×#63975C211yesmaximal, characteristic, complemented
4C3×#63786C211yesmaximal, characteristic, complemented
5C3×#63738C212yesmaximal, complemented
6#10593518C311yesmaximal, characteristic, direct product factor
7C2×C3×#638C2211yescharacteristic
8C2×C32×#67—21no—
9C3×#3811C2212yes—
10C3×D3×#67—22no—
11C2×C3×#624C2211yescharacteristic, complemented
12C3×#3805—21no—
13C3×#3804—22no—
14C3×C22⋊#157C2212yescomplemented
15C3×#3773—22no—
16C3×D32⋊C4—22no—
17C32×#217—21no—
18C3×Dic3⋊Dic3—21no—
19C3×#3722C2211yescharacteristic, complemented
20C3×#1004—61no—
21#64330C2×C311yescharacteristic
22#64233C2×C312yescomplemented
23C3×#883—61no—
24#63975C2×C311yescharacteristic, complemented
25#63786C2×C311yescharacteristic, complemented
26#63738C2×C312yescomplemented
27C23×C33D411yescharacteristic
28C22×C3×C3⋊D3C2311yescharacteristic
29C22×C32×D3—21no—
30C2×C3×D32D412yes—
31C2×C3×#157—22no—
32C3×#638—22no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C32×#67—22no—
35C32×#67—22no—
36C2×C32×Dic3—21no—
37C2×C32×Dic3—21no—
38C2×C32×Dic3—21no—
39C3×#627—21no—
40C3×#626—22no—
41C3×#625—21no—
42C3×#624—22no—
43C3×#624D412yes—
44C3×D3×Dic3—22no—
45C3×#458—91nomaximal
46C2×C32×D4—61no—
47C2×#638C22×C311yescharacteristic
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C2×C3×#67—41no—
51C2×C3×#67—62no—
52C2×C3×#67—61no—
53C2×C3×#67—61no—
54C3×#237—62no—
55C3×#237—62no—
56C3×D3×D4—62no—
57C3×D3×D4—62no—
58C3×#235—62no—
59C2×C3×C4×D3—61no—
60#3811C22×C312yes—
61D3×#67—22no—
62C2×#624C22×C311yescharacteristic, complemented
63#3805—21no—
64#3804—22no—
65C22⋊#157C22×C312yescomplemented
66#3773—22no—
67D32⋊C4—22no—
68C32×C22⋊C4—61no—
69C3×#217—21no—
70C3×#217—21no—
71C3×#217—41no—
72C3×(C2×D3)⋊C4—61no—
73C3×C4⋊Dic3—61no—
74C3×Dic3⋊C4—61no—
75Dic3⋊Dic3—21no—
76#3722C22×C311yescharacteristic, complemented
77C22×C33—21no—
78C22×C33—41no—
79C22×C33C2×D411yescharacteristic
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3C2×D412yes—
82C2×C32×D3—22no—
83C2×C32×D3—81no—
84C3×D32—42no—
85C3×#157—42no—
86C3×C3⋊Dic3—21no—
87C32×Dic3—22no—
88C32×Dic3—41no—
89C32×Dic3—41no—
90C3×H(5,𝔽2)—91no—
91#1004—61no—
92C3×#123—91no—
93C3×#123—92no—
94#883—61no—
95C3×#99—91no—
96C3×#99—92no—
97C23×C32—21no—
98C23×C32—21no—
99C23×C32—41no—
100C23×C32—41no—
101C23×C32—61no—
102C23×C32C3×D411yescharacteristic
103C22×C3⋊D3C23×C311yescharacteristic
104C22×C3×D3—21no—
105C22×C3×D3—21no—
106C22×C3×D3—41no—
107C22×C3×D3—62no—
108C22×C3×D3—61no—
109C22×C3×D3—61no—
110C2×D32C3×D412yes—
111C2×#157—22no—
112C32×D4—62no—
113C32×D4—62no—
114C2×C32×C4—61no—
115C2×C32×C4—61no—
116C2×C32×C4—61no—
117#638—22no—
118C2×C3⋊Dic3C3×D411yescharacteristic
119C3×#67—22no—
120C3×#67—22no—
121C3×#67—22no—
122C3×#67—22no—
123C3×#67—42no—
124C3×#67—42no—
125C3×#67—62no—
126C3×#67—62no—
127C3×#67—62no—
128C3×#67—62no—
129C3×#67—122no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—62no—
142C3×D12—62no—
143C3×C4×D3—62no—
144C3×C4×D3—62no—
145C3×C4×D3—122no—
146C3×Dic6—62no—
147#627—21no—
148#626—22no—
149#625—21no—
150#624—22no—
151#624C3×D412yes—
152D3×Dic3—22no—
153#458—91noSylow
154C2×C33—21no—
155C2×C33—41no—
156C2×C33#12011yescharacteristic
157C3×C3⋊D3—22no—
158C32×D3—42no—
159C3×#41—182no—
160C3×#41—181no—
161C2×C3×D4—61no—
162C2×C3×D4—61no—
163C2×C3×D4—92no—
164C2×C3×D4—91no—
165C2×C3×D4—121no—
166C2×C3×D4—182no—
167C22×C3×C4—91no—
168C2×#67—21no—
169C2×#67—61no—
170C2×#67—61no—
171C2×#67—62no—
172#237—62no—
173#237—62no—
174D3×D4—62no—
175D3×D4—62no—
176#235—62no—
177C2×C4×D3—61no—
178C3×C4⋊C4—181no—
179C3×C4⋊C4—182no—
180C3×C22⋊C4—61no—
181C3×C22⋊C4—61no—
182C3×C22⋊C4—91no—
183C3×C22⋊C4—92no—
184C3×C22⋊C4—121no—
185C3×C22⋊C4—182no—
186#217—21no—
187(C2×D3)⋊C4—61no—
188C4⋊Dic3—61no—
189Dic3⋊C4—61no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—42no—
204C22×C32—62no—
205C22×C32—81no—
206C22×C32—81no—
207C22×C32—81no—
208C22×C32—82no—
209C22×C32—241no—
210C22×C32C2×C3×D411yescharacteristic
211C2×C3⋊D3—41no—
212C2×C3⋊D3C2×C3×D412yes—
213C2×C3×D3—22no—
214C2×C3×D3—22no—
215C2×C3×D3—42no—
216C2×C3×D3—62no—
217C2×C3×D3—62no—
218C2×C3×D3—81no—
219C2×C3×D3—81no—
220C2×C3×D3—82no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—242no—
224C2×C3×D3—241no—
225D32—42no—
226#157—42no—
227C32×C4—62no—
228C32×C4—121no—
229C32×C4—121no—
230C3⋊Dic3—21no—
231C3×Dic3—22no—
232C3×Dic3—22no—
233C3×Dic3—42no—
234C3×Dic3—41no—
235C3×Dic3—41no—
236C3×Dic3—41no—
237C3×Dic3—41no—
238C3×Dic3—62no—
239C3×Dic3—81no—
240C3×Dic3—81no—
241C3×Dic3—121no—
242H(5,𝔽2)—91no—
243#123—91no—
244#123—92no—
245#99—91no—
246#99—92no—
247C33#45811yescharacteristic, complemented, Sylow
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—61no—
253C23×C3—61no—
254C23×C3—92no—
255C23×C3—91no—
256C23×C3—121no—
257C23×C3GO+(4,𝔽2)11yescharacteristic
258C22×D3—21no—
259C22×D3—62no—
260C22×D3—61no—
261C22×D3—61no—
262C3×Q8—181no—
263C3×D4—62no—
264C3×D4—62no—
265C3×D4—62no—
266C3×D4—62no—
267C3×D4—122no—
268C3×D4—122no—
269C3×D4—182no—
270C3×D4—182no—
271C3×D4—181no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—92no—
279C2×C3×C4—91no—
280C2×C3×C4—121no—
281C2×C3×C4—121no—
282C2×C3×C4—121no—
283C2×C3×C4—182no—
284C2×C3×C4—182no—
285C2×C3×C4—182no—
286#67—22no—
287#67—22no—
288#67—62no—
289#67—62no—
290#67—62no—
291#67—62no—
292#67—122no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—61no—
297C2×Dic3—61no—
298C2×Dic3—62no—
299D12—62no—
300C4×D3—62no—
301C4×D3—62no—
302C4×D3—122no—
303Dic6—62no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—42no—
317C2×C32—41no—
318C2×C32—81no—
319C2×C32—81no—
320C2×C32—81no—
321C2×C32—82no—
322C2×C32—122no—
323C2×C32C3×#12011yescharacteristic
324C3⋊D3—22no—
325C3×D3—42no—
326C3×D3—42no—
327C3×D3—82no—
328C3×D3—122no—
329C3×D3—122no—
330#41—182no—
331#41—181no—
332C2×D4—61no—
333C2×D4—92no—
334C2×D4—91no—
335C2×D4—182no—
336C22×C4—91no—
337C4⋊C4—181no—
338C4⋊C4—182no—
339C22⋊C4—61no—
340C22⋊C4—91no—
341C22⋊C4—92no—
342C22⋊C4—182no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—42no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—62no—
358C22×C3—62no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—82no—
363C22×C3—92no—
364C22×C3—122no—
365C22×C3—241no—
366C22×C3—241no—
367C22×C3—242no—
368C22×C3—361no—
369C22×C3—362no—
370C22×C3C2×GO+(4,𝔽2)11yescharacteristic
371C2×D3—22no—
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—81no—
375C2×D3—121no—
376C2×D3—121no—
377C2×D3—242no—
378C2×D3—241no—
379C3×C4—62no—
380C3×C4—62no—
381C3×C4—122no—
382C3×C4—121no—
383C3×C4—121no—
384C3×C4—121no—
385C3×C4—121no—
386C3×C4—181no—
387C3×C4—241no—
388C3×C4—241no—
389C3×C4—362no—
390Dic3—22no—
391Dic3—41no—
392Dic3—41no—
393Dic3—62no—
394Dic3—121no—
395C32—21no—
396C32—21no—
397C32—41no—
398C32—41no—
399C32C3×#45811yescharacteristic, complemented
400C23—61no—
401C23—92no—
402C23—91no—
403C23C3×GO+(4,𝔽2)11yescharacteristic
404Q8—181no—
405D4—62no—
406D4—62no—
407D4—182no—
408D4—182no—
409D4—181no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—92no—
414C2×C4—91no—
415C2×C4—182no—
416C2×C4—182no—
417C2×C4—182no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—42no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—81no—
433C2×C3—81no—
434C2×C3—81no—
435C2×C3—82no—
436C2×C3—122no—
437C2×C3—122no—
438C2×C3—182no—
439C2×C3—242no—
440C2×C3#6424111yescharacteristic
441D3—42no—
442D3—122no—
443D3—122no—
444C22—21no—
445C22—41no—
446C22—62no—
447C22—92no—
448C22—241no—
449C22—361no—
450C22—362no—
451C22C2×C3×GO+(4,𝔽2)11yescharacteristic
452C4—62no—
453C4—121no—
454C4—121no—
455C4—181no—
456C4—362no—
457C3—21no—
458C3—21no—
459C3—41no—
460C3—41no—
461C3#1059351811yescharacteristic, direct product factor
462C2—21no—
463C2—41no—
464C2—122no—
465C2—182no—
466C2C3×#6424111yescharacteristic
subgroupG/H
centerC2×C3#64241
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC2×C33#120
prime radicalC3×#64330C2
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#458
LCSG ▷ C22×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C3×#64233C2semidirect
3. C3×#63975C2semidirect
4. C3×#63786C2semidirect
5. C3×#63738C2semidirect
6. #10593518C3direct (N × H)
11. C2×C3×#624C22semidirect
14. C3×C22⋊#157C22semidirect
19. C3×#3722C22semidirect
22. #64233C2×C3semidirect
24. #63975C2×C3semidirect
25. #63786C2×C3semidirect
26. #63738C2×C3semidirect
62. C2×#624C22×C3semidirect
65. C22⋊#157C22×C3semidirect
76. #3722C22×C3semidirect
247. C33#458semidirect
399. C32C3×#458semidirect
461. C3#10593518direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643302
C3×#642332
C3×#639752
C3×#637862
C3×#637382
#105935183
C3×#4589