Finite Group Databasebeta

← #13165471

Subgroups — #13165471

Order 1728.

776 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 68 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×D3×#625C211yesmaximal, characteristic
2C2×#88925C211yesmaximal, characteristic, complemented
3#11722917C211yesmaximal, characteristic, complemented
4#11722793C211yesmaximal, characteristic, complemented
5#11722773C211yesmaximal, characteristic, complemented
6#11722772C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593493—31nomaximal
9C22×C3×D32—21no—
10C2×#6295C2211yescharacteristic
11C2×#6290—21no—
12C2×D3×C3⋊Dic3C2211yescharacteristic
13C2×C3×#625C2211yescharacteristic
14C2×C3⋊#622C2211yescharacteristic, complemented
15C2×C3×#622C2211yescharacteristic, complemented
16D3×#625—22no—
17D3×#625C414yescomplemented
18#88925—22no—
19#88815C2211yescharacteristic, complemented
20Dic3×C3⋊Dic3—21no—
21C3×#3724C2211yescharacteristic, complemented
22C3×(C2×D3)⋊Dic3—21no—
23C2×D3×#67—61no—
24C22×#625—31no—
25C2×#3792—31no—
26#63957—31no—
27#63878—61no—
28#63739—31no—
29#63739—31no—
30#63739—31no—
31#63739D311yescharacteristic, complemented
32C22×C3×C3⋊D3—21no—
33C22×C32×D3—21no—
34C22×C32×D3D411yescharacteristic
35C2×C3×D32—24no—
36C2×C3×D32—42no—
37C2×C3×D32—42no—
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3C2311yescharacteristic
41C2×C32×Dic3—21no—
42#6295—21no—
43#6295C2×C412yes—
44#6290—24no—
45D3×C3⋊Dic3C2×C412yes—
46D3×C3⋊Dic3D412yes—
47C3×#625—21no—
48C3×#625C2×C412yes—
49C3⋊#622—21no—
50C3×#622—21no—
51#4770—91nomaximal
52C23×C3×D3—61no—
53C23×C3×D3—61no—
54C22×D32—21no—
55C2×#638—61no—
56C22×C3⋊Dic3—31no—
57C2×C3×#67—61no—
58D3×#67—64no—
59D3×#67—64no—
60C2×#626—61no—
61C2×#625—31no—
62C2×#625—34no—
63C2×#625—62no—
64C2×#625—61no—
65C2×#625—61no—
66C2×#625—121no—
67C2×#625C2×D311yescharacteristic
68C2×D3×Dic3—61no—
69C2×D3×Dic3—61no—
70#3792—62no—
71C2×#622—31no—
72C2×#622C2×D311yescharacteristic, complemented
73C4×C3⋊Dic3—61no—
74C3×#217—61no—
75C3×(C2×D3)⋊C4—61no—
76C3×Dic3⋊C4—61no—
77#3724—31no—
78#3724C2×D311yescharacteristic, complemented
79Dic3⋊Dic3—61no—
80(C2×D3)⋊Dic3—21no—
81Dic32—61no—
82Dic32—61no—
83Dic32—121no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—22no—
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C2×C32×D3—21no—
89C2×C32×D3—21no—
90C2×C32×D3—22no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C2×C32×D3C22⋊C412yes—
94C3×D32—48no—
95C3⋊(C3⋊Dic3)C22⋊C412yes—
96C3×C3⋊Dic3—41no—
97C3×C3⋊Dic3C22×C411yescharacteristic
98C3×C3⋊Dic3C2×D411yescharacteristic
99C32×Dic3—41no—
100C22×#67—181no—
101C2×D3×D4—91no—
102C2×ΓL(1,𝔽25)—91no—
103#892—91no—
104#877—181no—
105C3×#102—91no—
106#832—91no—
107#821—91no—
108#809—91no—
109C23×C32—31no—
110C23×C32—61no—
111C22×C3⋊D3—21no—
112C22×C3×D3—21no—
113C22×C3×D3—21no—
114C22×C3×D3—21no—
115C22×C3×D3—21no—
116C22×C3×D3—41no—
117C22×C3×D3—64no—
118C22×C3×D3—64no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C22×C3×D3—121no—
122C22×C3×D3—122no—
123C22×C3×D3—122no—
124C22×C3×D3—122no—
125C22×C3×D3—122no—
126C2×D32—24no—
127C2×D32—42no—
128C2×D32—42no—
129C2×C32×C4—61no—
130#638—64no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—31no—
133C2×C3⋊Dic3—32no—
134C2×C3⋊Dic3—32no—
135C2×C3⋊Dic3—61no—
136C2×C3⋊Dic3—61no—
137C2×C3⋊Dic3—121no—
138C2×C3⋊Dic3—121no—
139C2×C3⋊Dic3C22×D311yescharacteristic
140C3×#67—62no—
141C3×#67—62no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—121no—
150#626—62no—
151#626—62no—
152#625—21no—
153#625—32no—
154#625—64no—
155#625—64no—
156#625—61no—
157#625—124no—
158#625C4×D312yes—
159D3×Dic3—62no—
160D3×Dic3—62no—
161D3×Dic3—62no—
162D3×Dic3—62no—
163#622—21no—
164#622—61no—
165#418—271noSylow
166C2×C33C2×C22⋊C411yescharacteristic
167C2×C33AGL(1,ℤ/8)11yescharacteristic
168C2×C33AGL(1,ℤ/8)11yescharacteristic
169C3×C3⋊D3—42no—
170C32×D3—22no—
171C32×D3—42no—
172C24×C3—181no—
173C23×D3—61no—
174C23×D3—61no—
175C2×C3×D4—91no—
176C2×#67—61no—
177C2×#67—181no—
178C2×#67—184no—
179C2×#67—181no—
180C2×#67—184no—
181C2×#67—181no—
182C2×#67—181no—
183C2×#67—361no—
184C22×Dic3—181no—
185C22×Dic3—181no—
186D3×D4—94no—
187D3×D4—182no—
188C2×D12—91no—
189C2×C4×D3—91no—
190C2×C3×C8—91no—
191C3×C4⋊C4—91no—
192C3×C22⋊C4—181no—
193#217—61no—
194(C2×D3)⋊C4—61no—
195C4⋊Dic3—91no—
196Dic3⋊C4—61no—
197Dic3⋊C4—181no—
198C4×Dic3—181no—
199C4×Dic3—181no—
200C4×Dic3—181no—
201C4×Dic3—361no—
202C2×C3⋊C8—91no—
203ΓL(1,𝔽25)—182no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—32no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—62no—
210C22×C32—61no—
211C22×C32—61no—
212C22×C32—121no—
213C22×C32—121no—
214C22×C32D3×D411yescharacteristic
215C2×C3⋊D3—22no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3×D3—22no—
219C2×C3×D3—22no—
220C2×C3×D3—22no—
221C2×C3×D3—22no—
222C2×C3×D3—22no—
223C2×C3×D3—22no—
224C2×C3×D3—41no—
225C2×C3×D3—41no—
226C2×C3×D3—42no—
227C2×C3×D3—41no—
228C2×C3×D3—41no—
229C2×C3×D3—41no—
230C2×C3×D3—41no—
231C2×C3×D3—62no—
232C2×C3×D3—62no—
233C2×C3×D3—81no—
234C2×C3×D3—81no—
235C2×C3×D3—122no—
236C2×C3×D3—128no—
237C2×C3×D3—128no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242C2×C3×D3—241no—
243C2×C3×D3—241no—
244D32—48no—
245C32×C4—121no—
246C3⋊Dic3—32no—
247C3⋊Dic3—41no—
248C3⋊Dic3—62no—
249C3⋊Dic3—62no—
250C3⋊Dic3—122no—
251C3⋊Dic3—122no—
252C3⋊Dic3C2×C4×D311yescharacteristic
253C3⋊Dic3D3×D411yescharacteristic
254C3×Dic3—41no—
255C3×Dic3—42no—
256C3×Dic3—41no—
257C3×Dic3—61no—
258C3×Dic3—61no—
259C3×Dic3—61no—
260C3×Dic3—61no—
261C3×Dic3—121no—
262C3×Dic3—122no—
263C3×Dic3—121no—
264C22×D4—271no—
265C2×M16—271no—
266#117—271no—
267#102—91no—
268#102—271no—
269#102—271no—
270#102—271no—
271C33#41811yescharacteristic, complemented, Sylow
272C23×C3—61no—
273C23×C3—61no—
274C23×C3—61no—
275C23×C3—61no—
276C23×C3—121no—
277C23×C3—181no—
278C23×C3—184no—
279C23×C3—362no—
280C23×C3—362no—
281C22×D3—21no—
282C22×D3—64no—
283C22×D3—61no—
284C22×D3—61no—
285C22×D3—64no—
286C22×D3—121no—
287C22×D3—122no—
288C22×D3—122no—
289C22×D3—122no—
290C22×D3—122no—
291C22×D3GO+(4,𝔽2)11yescharacteristic
292C3×D4—92no—
293C3×D4—181no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—91no—
297C2×C3×C4—121no—
298C2×C3×C4—181no—
299#67—62no—
300#67—62no—
301#67—182no—
302#67—182no—
303#67—184no—
304#67—184no—
305#67—184no—
306#67—364no—
307C2×Dic3—21no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—91no—
313C2×Dic3—121no—
314C2×Dic3—182no—
315C2×Dic3—182no—
316C2×Dic3—182no—
317C2×Dic3—182no—
318C2×Dic3—181no—
319C2×Dic3—181no—
320C2×Dic3—361no—
321C2×Dic3—361no—
322D12—92no—
323D12—181no—
324C4×D3—92no—
325C4×D3—92no—
326C3×C8—181no—
327C3⋊C8—181no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—62no—
341C2×C32—122no—
342C2×C32D3×C22⋊C411yescharacteristic
343C2×C32#91111yescharacteristic
344C2×C32#91411yescharacteristic
345C3⋊D3—42no—
346C3×D3—24no—
347C3×D3—42no—
348C3×D3—42no—
349C3×D3—42no—
350C3×D3—82no—
351C3×D3—122no—
352C3×D3—122no—
353C3×D3—242no—
354C24—181no—
355C2×D4—91no—
356C2×D4—271no—
357C2×D4—274no—
358C2×D4—542no—
359C2×D4—541no—
360C22×C4—271no—
361M16—542no—
362C2×C8—91no—
363C2×C8—271no—
364C4⋊C4—91no—
365C4⋊C4—271no—
366C22⋊C4—181no—
367C42—541no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—62no—
373C22×C3—62no—
374C22×C3—62no—
375C22×C3—62no—
376C22×C3—62no—
377C22×C3—62no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3—122no—
381C22×C3—121no—
382C22×C3—121no—
383C22×C3—121no—
384C22×C3—121no—
385C22×C3—182no—
386C22×C3—241no—
387C22×C3—241no—
388C22×C3—361no—
389C22×C3—361no—
390C22×C3—368no—
391C22×C3C2×GO+(4,𝔽2)11yescharacteristic
392C2×D3—21no—
393C2×D3—21no—
394C2×D3—22no—
395C2×D3—41no—
396C2×D3—41no—
397C2×D3—62no—
398C2×D3—62no—
399C2×D3—122no—
400C2×D3—128no—
401C2×D3—128no—
402C2×D3—121no—
403C2×D3—121no—
404C2×D3—121no—
405C2×D3—121no—
406C2×D3—241no—
407C2×D3—241no—
408C2×D3D32⋊C412yes—
409C3×C4—91no—
410C3×C4—91no—
411C3×C4—121no—
412C3×C4—122no—
413C3×C4—121no—
414C3×C4—361no—
415Dic3—41no—
416Dic3—61no—
417Dic3—61no—
418Dic3—61no—
419Dic3—61no—
420Dic3—92no—
421Dic3—121no—
422Dic3—122no—
423Dic3—121no—
424Dic3—182no—
425Dic3—182no—
426Dic3—362no—
427Dic3—362no—
428C32—21no—
429C32—21no—
430C32—41no—
431C32—41no—
432C32#477011yescharacteristic, complemented
433C23—31no—
434C23—61no—
435C23—184no—
436C23—181no—
437C23—362no—
438C23—362no—
439D4—92no—
440D4—181no—
441D4—272no—
442D4—541no—
443D4—544no—
444C2×C4—61no—
445C2×C4—91no—
446C2×C4—181no—
447C2×C4—272no—
448C2×C4—272no—
449C2×C4—271no—
450C8—181no—
451C8—541no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—64no—
465C2×C3—122no—
466C2×C3—122no—
467C2×C3—122no—
468C2×C3—242no—
469C2×C3—362no—
470C2×C3C2×D32⋊C411yescharacteristic
471C2×C3#6423611yescharacteristic
472C2×C3#6423611yescharacteristic
473D3—22no—
474D3—42no—
475D3—122no—
476D3—122no—
477D3—242no—
478C22—32no—
479C22—61no—
480C22—61no—
481C22—62no—
482C22—121no—
483C22—121no—
484C22—182no—
485C22—368no—
486C22—361no—
487C22—361no—
488C22D3×GO+(4,𝔽2)11yescharacteristic
489C4—91no—
490C4—91no—
491C4—121no—
492C4—272no—
493C4—361no—
494C3—21no—
495C3—21no—
496C3—41no—
497C3—41no—
498C3#1059349311yescharacteristic, complemented
499C2—62no—
500C2—122no—
501C2—362no—
502C2D3×D32⋊C411yescharacteristic
503C2#1172413811yescharacteristic
504C2#1172413911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC2×D3×#625C2
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33#418
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#88925C2semidirect
3. #11722917C2semidirect
4. #11722793C2semidirect
5. #11722773C2semidirect
6. #11722772C2semidirect
7. C3×#63739C2semidirect
14. C2×C3⋊#622C22semidirect
15. C2×C3×#622C22semidirect
17. D3×#625C4semidirect
19. #88815C22semidirect
21. C3×#3724C22semidirect
31. #63739D3semidirect
72. C2×#622C2×D3semidirect
78. #3724C2×D3semidirect
271. C33#418semidirect
432. C32#4770semidirect
498. C3#10593493semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6252
C2×#889252
#117229172
#117227932
#117227732
#117227722
C3×#637392
#105934933
#47709