Finite Group Databasebeta

← #13140502

Subgroups — #13140502

Order 1728.

688 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88646C211yesmaximal, characteristic, complemented
2#11721838C211yesmaximal, characteristic, complemented
3C2×C9×#213C211yesmaximal, characteristic, complemented
4C2×C3×#3673C211yesmaximal, characteristic, complemented
5#11720295C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic, complemented
8#10591516—31nomaximal
9#10588836—31nomaximal
10C22×D3×D9—21no—
11C2×C3×D4×C9C2211yescharacteristic, complemented
12C2×#6233—21no—
13C2×C4×C3⋊D9C2211yescharacteristic
14C2×C9×D12C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×#6194—21no—
17#88646—22no—
18#88646—22no—
19#88551—21no—
20C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
21C9×#213—22no—
22C3×#3673—22no—
23C2×C9×C3⋊C8C2211yescharacteristic, complemented
24C2×C3×C9⋊C8C2211yescharacteristic, complemented
25C2×#3802—31no—
26#64147—31no—
27C2×C3×#213—31no—
28C2×C3×#213D311yescharacteristic
29C2×D4×D9—31no—
30#63563—31no—
31#63563—31no—
32#63557—31no—
33C2×C9×D8—31no—
34#63500—31no—
35#63500—61no—
36C2×#3673D311yescharacteristic, complemented
37#63392—31no—
38#63380—31no—
39(C2×D9)⋊C8—31no—
40C23×C3×C9—21no—
41C22×C3⋊D9D411yescharacteristic
42C22×D3×C9—21no—
43C22×C3×D9—21no—
44C2×D3×D9—24no—
45C2×D3×D9—42no—
46C2×D3×D9—42no—
47C3×D4×C9—21no—
48C3×D4×C9—21no—
49C2×C3×C4×C9C2311yescharacteristic
50#6233—42no—
51C2×C3⋊Dic9—21no—
52C2×C3⋊Dic9D411yescharacteristic
53C4×C3⋊D9—21no—
54C4×C3⋊D9—21no—
55C9×D12—21no—
56C9×D12D412yescomplemented
57C3×D36—21no—
58C3×D36D412yescomplemented
59#6194—24no—
60C9×C3⋊C8—21no—
61C3×C9⋊C8—21no—
62#5156—91no—
63#5156—91no—
64C22×D32—61no—
65C2×C32×D4C2×D311yescharacteristic
66C2×#638—61no—
67C2×C4×C3⋊D3—31no—
68C2×C3×D12—31no—
69C2×C3×D12C2×D311yescharacteristic
70C2×#625—61no—
71#3802—62no—
72#3802—62no—
73C23×D9—61no—
74#3757—61no—
75C4⋊(C3⋊Dic3)—31no—
76C3×#213—22no—
77C3×#213—62no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8C2×D311yescharacteristic
80C2×D4×C9—21no—
81C2×D4×C9—31no—
82C2×D4×C9C2×D311yescharacteristic, complemented
83C2×#611—61no—
84C2×#611—61no—
85C2×#611—62no—
86D4×D9—62no—
87D4×D9—62no—
88C2×D36C2×D311yescharacteristic
89C2×C4×D9—31no—
90C2×C4×D9—61no—
91C9×D8—62no—
92C2×C8×C9—31no—
93#3676—61no—
94#3676—62no—
95#3673—22no—
96C4⋊Dic9—31no—
97C4⋊Dic9—61no—
98C2×C9⋊C8C2×D311yescharacteristic, complemented
99C22×C3×C9—41no—
100C22×C3×C9—41no—
101C22×C3×C9—41no—
102C22×C3×C9C2×D411yescharacteristic
103C2×C3⋊D9—21no—
104C2×C3⋊D9—21no—
105C2×C3⋊D9D812yescomplemented
106C2×D3×C9—22no—
107C2×D3×C9—41no—
108C2×D3×C9—41no—
109C2×C3×D9—22no—
110C2×C3×D9—41no—
111C2×C3×D9—41no—
112D3×D9—48no—
113C3×C4×C9C2×D411yescharacteristic
114C3×C4×C9C2×D411yescharacteristic
115C3⋊Dic9—21no—
116C3⋊Dic9—41no—
117C2×D3×D4—91no—
118C2×D3×D4—91no—
119C2×C3×D8—31no—
120C2×C3×D8—91no—
121#938—91no—
122#938—91no—
123#938—181no—
124C2×#213—31no—
125C2×#213D911yescharacteristic, complemented
126#832—91no—
127#832—91no—
128#821—91no—
129#821—91no—
130(C2×D3)⋊C8—91no—
131(C2×D3)⋊C8—91no—
132C23×C32—21no—
133C22×C3⋊D3—31no—
134C22×C3×D3—21no—
135C22×C3×D3—61no—
136C2×D32—64no—
137C2×D32—122no—
138C2×D32—122no—
139C32×D4—21no—
140C32×D4—21no—
141C2×C32×C4C22×D311yescharacteristic
142#638—122no—
143C2×C3⋊Dic3—31no—
144C2×C3⋊Dic3—61no—
145C4×C3⋊D3—61no—
146C4×C3⋊D3—61no—
147C3×D12—21no—
148C3×D12—32no—
149C3×D12—61no—
150C3×D12#6712yes—
151#625—64no—
152C23×C9—22no—
153C23×C9—21no—
154C23×C9—61no—
155C22×D9—21no—
156C22×D9—31no—
157C22×D9—61no—
158C22×D9—64no—
159C22×D9—122no—
160C22×D9—122no—
161C3×C3⋊C8—21no—
162C3×C3⋊C8—61no—
163D4×C9—21no—
164D4×C9—21no—
165D4×C9—32no—
166D4×C9—41no—
167D4×C9—41no—
168D4×C9—61no—
169C2×C4×C9—21no—
170C2×C4×C9C22×D311yescharacteristic
171#611—64no—
172#611—122no—
173#611—124no—
174C2×Dic9—31no—
175C2×Dic9—61no—
176C2×Dic9—62no—
177C2×Dic9—61no—
178D36—21no—
179D36#6712yescomplemented
180C4×D9—61no—
181C4×D9—61no—
182C4×D9—121no—
183C4×D9—121no—
184C8×C9—61no—
185C9⋊C8—21no—
186#447—271noSylow
187C2×C3×C9—81no—
188C2×C3×C9#12011yescharacteristic
189C2×C3×C9C2×D811yescharacteristic
190C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
191C3⋊D9—22no—
192D3×C9—42no—
193C3×D9—42no—
194C23×D3—181no—
195C23×D3—181no—
196C2×C3×D4—21no—
197C2×C3×D4—31no—
198C2×C3×D4—91no—
199C2×C3×D4C2×D911yescharacteristic, complemented
200C2×C3×D4D3211yescharacteristic
201C2×#67—181no—
202C2×#67—181no—
203C2×#67—181no—
204C2×#67—181no—
205C2×#67—182no—
206D3×D4—182no—
207D3×D4—182no—
208D3×D4—182no—
209D3×D4—182no—
210C2×D12—31no—
211C2×D12C2×D911yescharacteristic
212C2×C4×D3—91no—
213C2×C4×D3—91no—
214C2×C4×D3—181no—
215C3×D8—62no—
216C3×D8—182no—
217C2×C3×C8—31no—
218C2×C3×C8—91no—
219#217—181no—
220#217—181no—
221#217—182no—
222#213—22no—
223#213—62no—
224C4⋊Dic3—91no—
225C4⋊Dic3—91no—
226C4⋊Dic3—181no—
227C2×C3⋊C8—31no—
228C2×C3⋊C8C2×D911yescharacteristic, complemented
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32D3×D411yescharacteristic
233C2×C3⋊D3—32no—
234C2×C3⋊D3—61no—
235C2×C3⋊D3—61no—
236C2×C3×D3—22no—
237C2×C3×D3—41no—
238C2×C3×D3—41no—
239C2×C3×D3—62no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242D32—128no—
243C32×C4D3×D411yescharacteristic
244C32×C4C2×#6711yescharacteristic
245C3⋊Dic3—61no—
246C3⋊Dic3—121no—
247C22×C9—21no—
248C22×C9—42no—
249C22×C9—42no—
250C22×C9—42no—
251C22×C9—41no—
252C22×C9—41no—
253C22×C9—41no—
254C22×C9—62no—
255C22×C9—121no—
256C22×C9—121no—
257C22×C9D3×D411yescharacteristic
258C2×D9—22no—
259C2×D9—32no—
260C2×D9—41no—
261C2×D9—41no—
262C2×D9—62no—
263C2×D9—61no—
264C2×D9—61no—
265C2×D9—121no—
266C2×D9—121no—
267C2×D9—128no—
268C4×C9—21no—
269C4×C9—21no—
270C4×C9D3×D411yescharacteristic
271C4×C9C2×#6711yescharacteristic
272Dic9—61no—
273Dic9—121no—
274Dic9—122no—
275Dic9—121no—
276C22×D4—271no—
277C2×D8—31no—
278C2×D8—91no—
279#121—271no—
280#102—271no—
281#102—271no—
282C22⋊C8—271no—
283C3×C9#44711yescharacteristic, complemented, Sylow
284C23×C3—22no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—61no—
288C23×C3—181no—
289C22×D3—21no—
290C22×D3—61no—
291C22×D3—91no—
292C22×D3—91no—
293C22×D3—181no—
294C22×D3—184no—
295C22×D3—184no—
296C22×D3—362no—
297C22×D3—362no—
298C22×D3—362no—
299C22×D3—362no—
300C3×D4—21no—
301C3×D4—21no—
302C3×D4—21no—
303C3×D4—21no—
304C3×D4—32no—
305C3×D4—41no—
306C3×D4—41no—
307C3×D4—61no—
308C3×D4—92no—
309C3×D4—181no—
310C2×C3×C4—21no—
311C2×C3×C4C22×D911yescharacteristic
312C2×C3×C4C2×D3211yescharacteristic
313#67—184no—
314#67—184no—
315#67—362no—
316#67—362no—
317#67—364no—
318C2×Dic3—91no—
319C2×Dic3—91no—
320C2×Dic3—181no—
321C2×Dic3—182no—
322C2×Dic3—181no—
323C2×Dic3—181no—
324D12—21no—
325D12—32no—
326D12—61no—
327D12#61112yescomplemented
328C4×D3—181no—
329C4×D3—181no—
330C4×D3—181no—
331C4×D3—181no—
332C4×D3—361no—
333C4×D3—361no—
334C3×C8—61no—
335C3×C8—181no—
336C3⋊C8—21no—
337C3⋊C8—61no—
338C2×C32—81no—
339C2×C32D3×D811yescharacteristic
340C2×C32#91111yescharacteristic
341C2×C32#93711yescharacteristic
342C3⋊D3—62no—
343C3×D3—42no—
344C3×D3—122no—
345C2×C9—21no—
346C2×C9—21no—
347C2×C9—21no—
348C2×C9—82no—
349C2×C9—81no—
350C2×C9—122no—
351C2×C9D3×D811yescharacteristic
352C2×C9#91111yescharacteristic
353C2×C9#93711yescharacteristic
354D9—42no—
355D9—62no—
356D9—122no—
357C24—541no—
358C2×D4—31no—
359C2×D4—91no—
360C2×D4—541no—
361C2×D4—542no—
362C2×D4—542no—
363C2×D4—541no—
364C2×D4D3×D911yescharacteristic, complemented
365C22×C4—271no—
366D8—62no—
367D8—182no—
368C2×C8—31no—
369C2×C8—91no—
370C4⋊C4—271no—
371C22⋊C4—541no—
372C22×C3—21no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—62no—
383C22×C3—121no—
384C22×C3—121no—
385C22×C3—182no—
386C22×C3—361no—
387C22×C3—361no—
388C22×C3D4×D911yescharacteristic
389C22×C3#381111yescharacteristic
390C2×D3—22no—
391C2×D3—41no—
392C2×D3—41no—
393C2×D3—62no—
394C2×D3—92no—
395C2×D3—92no—
396C2×D3—121no—
397C2×D3—121no—
398C2×D3—182no—
399C2×D3—181no—
400C2×D3—181no—
401C2×D3—181no—
402C2×D3—181no—
403C2×D3—361no—
404C2×D3—361no—
405C2×D3—368no—
406C2×D3—368no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4D4×D911yescharacteristic
410C3×C4C2×#61111yescharacteristic
411C3×C4D3×#6711yescharacteristic
412C3×C4D3×#6711yescharacteristic
413Dic3—181no—
414Dic3—181no—
415Dic3—361no—
416Dic3—362no—
417Dic3—361no—
418Dic3—361no—
419C32#515611yescharacteristic
420C9—21no—
421C9#515611yescharacteristic, complemented
422C23—21no—
423C23—61no—
424C23—181no—
425C23—271no—
426C23—544no—
427C23—1082no—
428C23—1082no—
429D4—21no—
430D4—21no—
431D4—32no—
432D4—61no—
433D4—92no—
434D4—181no—
435D4—544no—
436D4—1082no—
437C2×C4—271no—
438C2×C4—541no—
439C2×C4—541no—
440C2×C4—541no—
441C2×C4C2×D3×D911yescharacteristic
442C8—61no—
443C8—181no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—82no—
448C2×C3—81no—
449C2×C3—81no—
450C2×C3—122no—
451C2×C3—362no—
452C2×C3D8×D911yescharacteristic
453C2×C3#6347311yescharacteristic
454C2×C3#6349911yescharacteristic
455C2×C3#6392611yescharacteristic
456C2×C3#6392711yescharacteristic
457C2×C3#6398311yescharacteristic
458D3—42no—
459D3—122no—
460D3—182no—
461D3—182no—
462D3—362no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—62no—
467C22—121no—
468C22—121no—
469C22—182no—
470C22—272no—
471C22—361no—
472C22—361no—
473C22—541no—
474C22—541no—
475C22—1088no—
476C22#8866811yescharacteristic
477C4—541no—
478C4—1081no—
479C4D3×#61111yescharacteristic
480C4D9×#6711yescharacteristic
481C3—21no—
482C3#1058883611yescharacteristic, complemented
483C3#1059151611yescharacteristic
484C2—81no—
485C2—122no—
486C2—362no—
487C2—542no—
488C2#1172117411yescharacteristic
489C2#1172117511yescharacteristic
490C2#1172129211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
hypercenterC2×D4D3×D9
hypocenterC3×C9#447
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88646C2semidirect
2. #11721838C2semidirect
3. C2×C9×#213C2semidirect
4. C2×C3×#3673C2semidirect
5. #11720295C2semidirect
6. #11720283C2semidirect
7. #11720282C2semidirect
11. C2×C3×D4×C9C22semidirect
20. C4⋊(C3⋊Dic9)C22semidirect
23. C2×C9×C3⋊C8C22semidirect
24. C2×C3×C9⋊C8C22semidirect
36. C2×#3673D3semidirect
56. C9×D12D4semidirect
58. C3×D36D4semidirect
82. C2×D4×C9C2×D3semidirect
98. C2×C9⋊C8C2×D3semidirect
105. C2×C3⋊D9D8semidirect
125. C2×#213D9semidirect
179. D36#67semidirect
199. C2×C3×D4C2×D9semidirect
228. C2×C3⋊C8C2×D9semidirect
283. C3×C9#447semidirect
327. D12#611semidirect
364. C2×D4D3×D9semidirect
421. C9#5156semidirect
482. C3#10588836semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
#117218382
C2×C9×#2132
C2×C3×#36732
#117202952
#117202832
#117202822
#105915163
#105888363