Finite Group Databasebeta

← #10592474⋊C3 #13141792

Subgroups — #10592474⋊C3 #13141792

Order 1728.

519 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 94 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
1#64147⋊C3C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#64104⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592474C311yesmaximal, characteristic, complemented
9C3×#4775—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4⋊((C3⋊D3)⋊C3)—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17#3757⋊C3—21no—
18#3753⋊C3—22no—
19#3752⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C8×(C3⋊D3)⋊C3—22no—
23#64147C2×C311yescharacteristic, complemented
24C2×#3753C2×C311yescharacteristic, complemented
25C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
26#64104C2×C311yescharacteristic, complemented
27C3×#938—31no—
28C2×C3×#213—31no—
29C2×C3×D3×C8—31no—
30C3×#895—31no—
31C32×#102—21no—
32C32×#102D311yescharacteristic, complemented
33C32×#102C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic, complemented
35#63634C2×C311yescharacteristic, complemented
36C3×#821—31no—
37C3×#807—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#638⋊C3—42no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—42no—
51C4×(C3⋊D3)⋊C3—21no—
52C4×(C3⋊D3)⋊C3—21no—
53C8×H(3,𝔽3)—21no—
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55C3×#466—91no—
56#4775—31no—
57#4775—91no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C22×C311yescharacteristic
61C2×#638—21no—
62C2×C4⋊(C3⋊D3)—21no—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×C4×C3⋊D3C22×C311yescharacteristic
65C2×C3×#67—61no—
66C2×C3×D12—31no—
67C2×C3×D12—61no—
68C2×C3×C4×D3—31no—
69C2×C32×C8—21no—
70C2×C32×C8C2×D311yescharacteristic
71C2×C32×C8C22×C311yescharacteristic
72C32×C4⋊C4—21no—
73C32×C4⋊C4C2×D311yescharacteristic
74C32×C4⋊C4C22×C311yescharacteristic
75#3757—21no—
76#3753—22no—
77#3752—21no—
78C4⋊(C3⋊Dic3)C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C8×C3⋊D3—22no—
81C3×#217—61no—
82C3×#213—62no—
83C3×(C2×D3)⋊C4—61no—
84C3×C4⋊Dic3—31no—
85C2×C3×C3⋊C8—31no—
86C3×D3×C8—62no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3D812yescomplemented
97C4×H(3,𝔽3)—41no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)#4111yescharacteristic
100(C3⋊Dic3)⋊C3—21no—
101(C3⋊Dic3)⋊C3—41no—
102C2×C3×D8—91no—
103C22×C3×C8—91no—
104C3×#121—91no—
105C3×#121—91no—
106#938—31no—
107#938—91no—
108C2×#213—31no—
109C2×#213—91no—
110C2×D3×C8—31no—
111C2×D3×C8—91no—
112#895—31no—
113#895—91no—
114C3×#107—91no—
115C3×#102—31no—
116C3×#102—31no—
117C3×#102—61no—
118C3×#102—91no—
119C3×#102C3×D311yescharacteristic
120#821—31no—
121#821—91no—
122#807—31no—
123#807—91no—
124C23×C32—21no—
125C23×C32—21no—
126C23×C32—41no—
127C22×C3⋊D3—21no—
128C22×C3⋊D3C3×D411yescharacteristic
129C22×C3×D3—31no—
130C22×C3×D3—61no—
131C32×D4—22no—
132C32×D4—21no—
133C32×D4—21no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—41no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—41no—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C23×C311yescharacteristic
143#638—42no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3C3×D411yescharacteristic
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)—21no—
148C4⋊(C3⋊D3)—42no—
149C4×C3⋊D3—21no—
150C4×C3⋊D3—21no—
151C3×#67—122no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—61no—
154C3×D12—61no—
155C3×D12—61no—
156C3×D12—122no—
157C3×C4×D3—61no—
158C3×C4×D3—61no—
159C32×C8—21no—
160C32×C8—21no—
161C32×C8—41no—
162C3⋊(C3⋊C8)C3×D412yescomplemented
163C3×C3⋊C8—32no—
164#466—91noSylow
165C2×H(3,𝔽3)—81no—
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)C2×D811yescharacteristic
168C2×H(3,𝔽3)#13511yescharacteristic
169(C3⋊D3)⋊C3—22no—
170(C3⋊D3)⋊C3—81no—
171C2×C3×D4—31no—
172C2×C3×D4—31no—
173C2×C3×D4—61no—
174C2×C3×D4—91no—
175C2×C3×D4—181no—
176C2×C3×D4—181no—
177C2×C3×D4C2×C3×D311yescharacteristic
178C22×C3×C4—91no—
179C2×#67—61no—
180C2×#67—181no—
181C2×D12—31no—
182C2×D12—61no—
183C2×D12—91no—
184C2×D12—181no—
185C2×C4×D3—31no—
186C2×C4×D3—91no—
187C3×D8—182no—
188C2×C3×C8—31no—
189C2×C3×C8—31no—
190C2×C3×C8—61no—
191C2×C3×C8—91no—
192C2×C3×C8—182no—
193C2×C3×C8C2×C3×D311yescharacteristic
194C3×C4⋊C4—31no—
195C3×C4⋊C4—31no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—91no—
198C3×C4⋊C4C2×C3×D311yescharacteristic
199C3×C22⋊C4—181no—
200C3×C22⋊C4—181no—
201#217—61no—
202#217—181no—
203#213—62no—
204#213—182no—
205(C2×D3)⋊C4—61no—
206(C2×D3)⋊C4—181no—
207C4⋊Dic3—31no—
208C4⋊Dic3—91no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—91no—
211D3×C8—62no—
212D3×C8—182no—
213C22×C32—21no—
214C22×C32—42no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—81no—
222C22×C32—81no—
223C22×C32D3×D411yescharacteristic
224C22×C32C2×C3×D411yescharacteristic
225C2×C3⋊D3—21no—
226C2×C3⋊D3—21no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3C3×D812yescomplemented
231C2×C3×D3—32no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4—81no—
242C32×C4#23511yescharacteristic
243C32×C4D3×D411yescharacteristic
244C32×C4C2×C3×D411yescharacteristic
245C32×C4C3×#4111yescharacteristic
246C3⋊Dic3—21no—
247C3⋊Dic3—41no—
248C3×Dic3—61no—
249C3×Dic3—121no—
250C2×D8—91no—
251C22×C8—91no—
252#121—91no—
253#121—91no—
254#107—91no—
255#102—91no—
256#102(C3⋊D3)⋊C311yescharacteristic, complemented
257H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
258C23×C3—21no—
259C23×C3—61no—
260C23×C3—61no—
261C23×C3—91no—
262C23×C3—121no—
263C23×C3—181no—
264C22×D3—31no—
265C22×D3—61no—
266C22×D3—91no—
267C22×D3—181no—
268C3×D4—21no—
269C3×D4—21no—
270C3×D4—61no—
271C3×D4—61no—
272C3×D4—61no—
273C3×D4—61no—
274C3×D4—62no—
275C3×D4—121no—
276C3×D4—181no—
277C3×D4—181no—
278C3×D4—362no—
279C3×D4—362no—
280C2×C3×C4—21no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—91no—
287C2×C3×C4—121no—
288C2×C3×C4—181no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4C22×C3×D311yescharacteristic
292#67—122no—
293#67—362no—
294C2×Dic3—31no—
295C2×Dic3—61no—
296C2×Dic3—91no—
297C2×Dic3—181no—
298D12—61no—
299D12—61no—
300D12—122no—
301D12—181no—
302D12—181no—
303D12—362no—
304C4×D3—61no—
305C4×D3—61no—
306C4×D3—181no—
307C4×D3—181no—
308C3×C8—21no—
309C3×C8—61no—
310C3×C8—61no—
311C3×C8—92no—
312C3×C8—121no—
313C3⋊C8—32no—
314C3⋊C8—92no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—82no—
319C2×C32—81no—
320C2×C32—81no—
321C2×C32#88411yescharacteristic
322C2×C32D3×D811yescharacteristic
323C2×C32#91611yescharacteristic
324C2×C32C3×#12111yescharacteristic
325C2×C32C2×C3×D811yescharacteristic
326C2×C32C3×#13511yescharacteristic
327C3⋊D3—22no—
328C3⋊D3—81no—
329C3×D3—62no—
330C3×D3—241no—
331C2×D4—91no—
332C2×D4—181no—
333C2×D4—181no—
334C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
335C22×C4—91no—
336D8—182no—
337C2×C8—91no—
338C2×C8—182no—
339C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
340C4⋊C4—91no—
341C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
342C22⋊C4—181no—
343C22⋊C4—181no—
344C22×C3—31no—
345C22×C3—31no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—61no—
350C22×C3—92no—
351C22×C3—121no—
352C22×C3—121no—
353C22×C3—121no—
354C22×C3—121no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—122no—
358C22×C3—181no—
359C22×C3—181no—
360C22×C3—241no—
361C22×C3—241no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3C3×D3×D411yescharacteristic
366C2×D3—32no—
367C2×D3—61no—
368C2×D3—61no—
369C2×D3—92no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—121no—
373C2×D3—181no—
374C2×D3—181no—
375C2×D3—361no—
376C2×D3—361no—
377C2×D3—361no—
378C3×C4—31no—
379C3×C4—31no—
380C3×C4—31no—
381C3×C4—31no—
382C3×C4—41no—
383C3×C4—61no—
384C3×C4—61no—
385C3×C4—121no—
386C3×C4—121no—
387C3×C4—181no—
388C3×C4—241no—
389C3×C4—361no—
390C3×C4C3×#23511yescharacteristic
391C3×C4C3×D3×D411yescharacteristic
392Dic3—61no—
393Dic3—121no—
394Dic3—181no—
395Dic3—361no—
396C32—21no—
397C32#477511yescharacteristic, complemented
398C32C3×#46611yescharacteristic, complemented
399C23—21no—
400C23—91no—
401C23—181no—
402D4—21no—
403D4—21no—
404D4—181no—
405D4—181no—
406D4—362no—
407D4—362no—
408C2×C4—21no—
409C2×C4—91no—
410C2×C4—181no—
411C2×C4—181no—
412C2×C4—181no—
413C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
414C8—21no—
415C8—92no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—31no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—81no—
426C2×C3—182no—
427C2×C3—241no—
428C2×C3—241no—
429C2×C3—242no—
430C2×C3—721no—
431C2×C3C3×#88411yescharacteristic
432C2×C3C3×D3×D811yescharacteristic
433C2×C3C3×#91611yescharacteristic
434D3—62no—
435D3—182no—
436D3—241no—
437D3—721no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—92no—
442C22—181no—
443C22—181no—
444C22—361no—
445C22—361no—
446C22—361no—
447C22D4×(C3⋊D3)⋊C311yescharacteristic
448C4—41no—
449C4—181no—
450C4—361no—
451C4#3828⋊C311yescharacteristic
452C4D4×(C3⋊D3)⋊C311yescharacteristic
453C3—31no—
454C3—31no—
455C3—61no—
456C3C3×#477511yescharacteristic
457C2—81no—
458C2—182no—
459C2—721no—
460C2#64093⋊C311yescharacteristic
461C2D8×(C3⋊D3)⋊C311yescharacteristic
462C2#64125⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#466
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. #64147⋊C3C2semidirect
2. C2×#3753⋊C3C2semidirect
3. C2×C8×(C3⋊D3)⋊C3C2semidirect
4. #64104⋊C3C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #63648⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592474C3semidirect
23. #64147C2×C3semidirect
24. C2×#3753C2×C3semidirect
25. C2×C8×C3⋊D3C2×C3semidirect
26. #64104C2×C3semidirect
32. C32×#102D3semidirect
33. C32×#102C2×C3semidirect
34. #63648C2×C3semidirect
35. #63634C2×C3semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
96. C2×(C3⋊D3)⋊C3D8semidirect
162. C3⋊(C3⋊C8)C3×D4semidirect
230. C2×C3⋊D3C3×D8semidirect
256. #102(C3⋊D3)⋊C3semidirect
257. H(3,𝔽3)#466semidirect
397. C32#4775semidirect
398. C32C3×#466semidirect

Maximal subgroups

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

subgroupindex
#64147⋊C32
C2×#3753⋊C32
C2×C8×(C3⋊D3)⋊C32
#64104⋊C32
H(3,𝔽3)×#1022
#63648⋊C32
#63634⋊C32
#105924743
C3×#47753