Finite Group Databasebeta

← #10592482⋊C3 #13141800

Subgroups — #10592482⋊C3 #13141800

Order 1728.

582 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#3830⋊C3C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic, complemented
3C2×#3746⋊C3C211yesmaximal, characteristic, complemented
4#64106⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592482C311yesmaximal, characteristic, complemented
9C3×#4783—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C22×(C3⋊Dic3)⋊C3—21no—
13#3830⋊C3—22no—
14#3830⋊C3—22no—
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#3754⋊C3—22no—
21#3752⋊C3—21no—
22C4⋊((C3⋊Dic3)⋊C3)—21no—
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
24#3746⋊C3—22no—
25C2×#3830C2×C311yescharacteristic
26C2×C3×#237—31no—
27C2×#3754C2×C311yescharacteristic, complemented
28C2×#3746C2×C311yescharacteristic, complemented
29#64106C2×C311yescharacteristic, complemented
30C2×C3×#214—31no—
31C2×C3×Dic12—31no—
32C3×#897—31no—
33C32×#102—21no—
34C32×#102D311yescharacteristic, complemented
35C32×#102C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63637C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—31no—
39C3×#810—31no—
40C23×H(3,𝔽3)—21no—
41C22×(C3⋊D3)⋊C3D411yescharacteristic
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)D412yes—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46#638⋊C3—24no—
47C2×(C3⋊Dic3)⋊C3—24no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4×(C3⋊D3)⋊C3—21no—
52C4×(C3⋊D3)⋊C3—21no—
53#634⋊C3—21no—
54#634⋊C3—21no—
55#634⋊C3—21no—
56#634⋊C3D412yes—
57C8×H(3,𝔽3)—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59C3×#452—91no—
60#4783—31no—
61#4783—91no—
62C2×C32×D4—21no—
63C2×C32×D4C2×D311yescharacteristic
64C2×C32×D4C22×C311yescharacteristic
65C2×#638—21no—
66C22×C3⋊Dic3—21no—
67#3830—22no—
68#3830—22no—
69C2×C4×C3⋊D3C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C2×#634C22×C311yescharacteristic
72C2×C3×#67—61no—
73C22×C3×Dic3—61no—
74C3×#237—62no—
75C3×#237—62no—
76C2×C3×C4×D3—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—31no—
79C2×C32×C8—21no—
80C2×C32×C8C2×D311yescharacteristic
81C2×C32×C8C22×C311yescharacteristic
82C32×C4⋊C4—21no—
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C22×C311yescharacteristic
85#3754—22no—
86#3752—21no—
87C4⋊(C3⋊Dic3)—21no—
88C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
89#3746—22no—
90C3×#214—62no—
91C3×(C2×D3)⋊C4—61no—
92C3×C4⋊Dic3—61no—
93C2×C3×C3⋊C8—31no—
94C3×Dic12—62no—
95C22×H(3,𝔽3)—22no—
96C22×H(3,𝔽3)—41no—
97C22×H(3,𝔽3)—41no—
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C2×(C3⋊D3)⋊C3—21no—
100C2×(C3⋊D3)⋊C3—21no—
101C2×(C3⋊D3)⋊C3—21no—
102C4×H(3,𝔽3)—41no—
103C4×H(3,𝔽3)C2×D411yescharacteristic
104C4×H(3,𝔽3)C2×D411yescharacteristic
105(C3⋊Dic3)⋊C3—22no—
106(C3⋊Dic3)⋊C3—21no—
107(C3⋊Dic3)⋊C3—41no—
108C2×C3×#41—91no—
109C2×#237—31no—
110C2×#237—91no—
111C2×C3×Q16—91no—
112C2×C3×SD16—91no—
113C3×C22⋊Q8—91no—
114C2×#214—31no—
115C2×#214—91no—
116C2×Dic12—31no—
117C2×Dic12—91no—
118#897—31no—
119#897—91no—
120C3×#103—91no—
121C3×#102—31no—
122C3×#102—31no—
123C3×#102—61no—
124C3×#102C3×D311yescharacteristic
125C3×C22⋊C8—91no—
126(C2×D3)⋊C8—31no—
127(C2×D3)⋊C8—91no—
128#810—31no—
129#810—91no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—41no—
133C22×C3⋊D3C3×D411yescharacteristic
134C22×C3×D3—31no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—22no—
138C32×D4—41no—
139C32×D4D1212yes—
140C32×D4C3×D412yes—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—41no—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C23×C311yescharacteristic
147#638—24no—
148C2×C3⋊Dic3—24no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3C3×D411yescharacteristic
152C4×C3⋊D3—21no—
153C4×C3⋊D3—21no—
154#634—21no—
155#634—21no—
156#634—21no—
157#634C3×D412yes—
158C3×#67—64no—
159C2×C3×Dic3—31no—
160C2×C3×Dic3—64no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C3×C4×D3—61no—
164C3×C4×D3—61no—
165C3×Dic6—32no—
166C3×Dic6—61no—
167C3×Dic6—61no—
168C3×Dic6—61no—
169C32×C8—21no—
170C32×C8—21no—
171C32×C8—41no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—61no—
174#452—91noSylow
175C2×H(3,𝔽3)—42no—
176C2×H(3,𝔽3)#12011yescharacteristic
177C2×H(3,𝔽3)#13511yescharacteristic
178C2×H(3,𝔽3)#13711yescharacteristic
179(C3⋊D3)⋊C3—41no—
180C3×#41—182no—
181C3×#41—182no—
182C2×C3×Q8—91no—
183C2×C3×Q8—91no—
184C2×C3×D4—31no—
185C2×C3×D4—31no—
186C2×C3×D4—61no—
187C2×C3×D4—181no—
188C2×C3×D4C2×C3×D311yescharacteristic
189C22×C3×C4—91no—
190C22×C3×C4—181no—
191C2×#67—61no—
192C2×#67—181no—
193C22×Dic3—61no—
194C22×Dic3—181no—
195#237—62no—
196#237—62no—
197#237—182no—
198#237—182no—
199C2×C4×D3—31no—
200C2×C4×D3—91no—
201C2×Dic6—31no—
202C2×Dic6—31no—
203C2×Dic6—91no—
204C2×Dic6—91no—
205C3×Q16—182no—
206C3×SD16—182no—
207C2×C3×C8—31no—
208C2×C3×C8—31no—
209C2×C3×C8—61no—
210C2×C3×C8—91no—
211C2×C3×C8C2×C3×D311yescharacteristic
212C3×C4⋊C4—31no—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—61no—
215C3×C4⋊C4—181no—
216C3×C4⋊C4C2×C3×D311yescharacteristic
217C3×C22⋊C4—181no—
218#214—62no—
219#214—182no—
220(C2×D3)⋊C4—61no—
221(C2×D3)⋊C4—181no—
222C4⋊Dic3—61no—
223C4⋊Dic3—181no—
224C2×C3⋊C8—31no—
225C2×C3⋊C8—91no—
226Dic12—62no—
227Dic12—182no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—21no—
231C22×C32—42no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—81no—
237C22×C32—81no—
238C22×C32D3×D411yescharacteristic
239C22×C32C2×C3×D411yescharacteristic
240C2×C3⋊D3—21no—
241C2×C3⋊D3—21no—
242C2×C3⋊D3—21no—
243C2×C3×D3—61no—
244C2×C3×D3—61no—
245C2×C3×D3—61no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—41no—
249C32×C4—41no—
250C32×C4—81no—
251C32×C4C2×D1211yescharacteristic
252C32×C4D3×D411yescharacteristic
253C32×C4C2×C3×D411yescharacteristic
254C32×C4C2×C3×D411yescharacteristic
255C3⋊Dic3—22no—
256C3⋊Dic3—21no—
257C3⋊Dic3—41no—
258C3×Dic3—62no—
259C3×Dic3—61no—
260C3×Dic3—121no—
261C2×#41—91no—
262C2×Q16—91no—
263C2×SD16—91no—
264C22⋊Q8—91no—
265#103—91no—
266#102(C3⋊D3)⋊C311yescharacteristic, complemented
267C22⋊C8—91no—
268H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
269C23×C3—21no—
270C23×C3—61no—
271C23×C3—61no—
272C23×C3—91no—
273C23×C3—121no—
274C22×D3—31no—
275C22×D3—91no—
276C3×Q8—92no—
277C3×Q8—181no—
278C3×Q8—181no—
279C3×Q8—181no—
280C3×D4—21no—
281C3×D4—32no—
282C3×D4—32no—
283C3×D4—61no—
284C3×D4—61no—
285C3×D4—62no—
286C3×D4—121no—
287C3×D4—184no—
288C3×D4C3×D1212yes—
289C2×C3×C4—21no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—91no—
296C2×C3×C4—121no—
297C2×C3×C4—181no—
298C2×C3×C4—181no—
299C2×C3×C4—181no—
300C2×C3×C4—184no—
301C2×C3×C4—181no—
302C2×C3×C4C22×C3×D311yescharacteristic
303#67—64no—
304#67—184no—
305C2×Dic3—31no—
306C2×Dic3—64no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—91no—
310C2×Dic3—181no—
311C2×Dic3—184no—
312C2×Dic3—181no—
313C4×D3—61no—
314C4×D3—61no—
315C4×D3—181no—
316C4×D3—181no—
317Dic6—32no—
318Dic6—61no—
319Dic6—61no—
320Dic6—61no—
321Dic6—92no—
322Dic6—181no—
323Dic6—181no—
324Dic6—181no—
325C3×C8—21no—
326C3×C8—61no—
327C3×C8—61no—
328C3×C8—121no—
329C3×C8—181no—
330C3⋊C8—61no—
331C3⋊C8—181no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—42no—
336C2×C32—42no—
337C2×C32—82no—
338C2×C32#88211yescharacteristic
339C2×C32#91211yescharacteristic
340C2×C32#91511yescharacteristic
341C2×C32C3×#12011yescharacteristic
342C2×C32C3×#13511yescharacteristic
343C2×C32C3×#13711yescharacteristic
344C3⋊D3—41no—
345C3×D3—121no—
346#41—182no—
347#41—182no—
348C2×Q8—91no—
349C2×Q8—91no—
350C2×D4—181no—
351C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
352C22×C4—91no—
353C22×C4—181no—
354Q16—182no—
355SD16—182no—
356C2×C8—91no—
357C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
358C4⋊C4—181no—
359C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
360C22⋊C4—181no—
361C22×C3—22no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—62no—
367C22×C3—62no—
368C22×C3—61no—
369C22×C3—122no—
370C22×C3—121no—
371C22×C3—121no—
372C22×C3—121no—
373C22×C3—121no—
374C22×C3—181no—
375C22×C3—181no—
376C22×C3—181no—
377C22×C3—241no—
378C22×C3—241no—
379C22×C3C3×D3×D411yescharacteristic
380C2×D3—61no—
381C2×D3—61no—
382C2×D3—61no—
383C2×D3—181no—
384C2×D3—181no—
385C2×D3—181no—
386C3×C4—31no—
387C3×C4—31no—
388C3×C4—31no—
389C3×C4—31no—
390C3×C4—41no—
391C3×C4—61no—
392C3×C4—61no—
393C3×C4—121no—
394C3×C4—121no—
395C3×C4—181no—
396C3×C4—182no—
397C3×C4—241no—
398C3×C4—361no—
399C3×C4C2×C3×D1211yescharacteristic
400C3×C4C3×D3×D411yescharacteristic
401Dic3—62no—
402Dic3—61no—
403Dic3—121no—
404Dic3—182no—
405Dic3—181no—
406Dic3—361no—
407C32—21no—
408C32#478311yescharacteristic, complemented
409C32C3×#45211yescharacteristic, complemented
410C23—21no—
411C23—91no—
412Q8—92no—
413Q8—181no—
414Q8—181no—
415Q8—181no—
416D4—21no—
417D4—184no—
418D4C4⋊((C3⋊D3)⋊C3)12yes—
419C2×C4—21no—
420C2×C4—91no—
421C2×C4—184no—
422C2×C4—181no—
423C2×C4—181no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
427C8—21no—
428C8—181no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—42no—
436C2×C3—61no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—122no—
440C2×C3—122no—
441C2×C3—242no—
442C2×C3—361no—
443C2×C3C3×#88211yescharacteristic
444C2×C3C3×#91211yescharacteristic
445C2×C3C3×#91511yescharacteristic
446D3—121no—
447D3—361no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—181no—
452C22—181no—
453C22—181no—
454C22D4×(C3⋊D3)⋊C311yescharacteristic
455C4—41no—
456C4—181no—
457C4—182no—
458C4—361no—
459C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
460C4D4×(C3⋊D3)⋊C311yescharacteristic
461C3—31no—
462C3—31no—
463C3—61no—
464C3C3×#478311yescharacteristic
465C2—42no—
466C2—361no—
467C2#64091⋊C311yescharacteristic
468C2#64121⋊C311yescharacteristic
469C2#64124⋊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
prime radicalC2×#3830⋊C3C2
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#452
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
2. C2×#3754⋊C3C2semidirect
3. C2×#3746⋊C3C2semidirect
4. #64106⋊C3C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #63647⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592482C3semidirect
27. C2×#3754C2×C3semidirect
28. C2×#3746C2×C3semidirect
29. #64106C2×C3semidirect
34. C32×#102D3semidirect
35. C32×#102C2×C3semidirect
36. #63647C2×C3semidirect
37. #63637C2×C3semidirect
266. #102(C3⋊D3)⋊C3semidirect
268. H(3,𝔽3)#452semidirect
408. C32#4783semidirect
409. C32C3×#452semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
C2×#3754⋊C32
C2×#3746⋊C32
#64106⋊C32
H(3,𝔽3)×#1022
#63647⋊C32
#63637⋊C32
#105924823
C3×#47833