Finite Group Databasebeta

← #13138673

Subgroups — #13138673

Order 1728.

614 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 93 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×D9×D12C211yesmaximal, characteristic
2C2×C9×#204C211yesmaximal, characteristic, complemented
3#11720944C211yesmaximal, characteristic, complemented
4C2×#88426C211yesmaximal, characteristic, complemented
5#11720543C211yesmaximal, characteristic, complemented
6C3×(C2×D9)⋊C8C211yesmaximal, characteristic, complemented
7#11720295C211yesmaximal, characteristic, complemented
8#10590942—31nomaximal
9#10588849—31nomaximal
10C22×D3×D9—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×C4×D9C2211yescharacteristic
14C2×C9×Dic6C2211yescharacteristic, complemented
15C2×#6195—21no—
16D9×D12—22no—
17D9×D12—22no—
18C2×C3×C8×C9C2211yescharacteristic, complemented
19C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
20C2×C3×C9⋊C8C2211yescharacteristic, complemented
21C9×#204—22no—
22(C2×D9)⋊Dic3—21no—
23#88442—21no—
24#88426—22no—
25C2×D3×D12—31no—
26C2×C3×#204D311yescharacteristic
27#63856—31no—
28C2×#3717—31no—
29C2×D4×D9—31no—
30#63648—31no—
31C3×(C2×D3)⋊C8—31no—
32#63563—31no—
33C2×C9×SD16—31no—
34#63508—31no—
35C2×#3675—31no—
36#63392—31no—
37#63380—31no—
38#63380—61no—
39(C2×D9)⋊C8D311yescharacteristic, complemented
40C22×C3⋊D9—21no—
41C22×D3×C9—21no—
42C22×C3×D9D411yescharacteristic
43C2×D3×D9—24no—
44C2×D3×D9—42no—
45C2×D3×D9—42no—
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3⋊Dic9—21no—
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)D412yes—
50C2×C9×Dic3—21no—
51C2×C3×Dic9D411yescharacteristic
52C9×D12—21no—
53C9×D12D412yes—
54C3×C4×D9—21no—
55C3×C4×D9—21no—
56C9×Dic6—21no—
57C9×Dic6—21no—
58#6195—24no—
59C3×C8×C9—21no—
60C3×C9⋊C8—21no—
61#5169—91no—
62#4731—91no—
63C22×D32—61no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C3×D12C2×D311yescharacteristic
66C2×C3×C4×D3—31no—
67C2×C3×Dic6C2×D311yescharacteristic
68C2×#626—61no—
69D3×D12—62no—
70D3×D12—62no—
71C23×D9—61no—
72C2×C32×C8C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)—31no—
74C2×C3×C3⋊C8—31no—
75C3×#204—22no—
76#3724—61no—
77(C2×D3)⋊Dic3—61no—
78#3717—62no—
79C2×Q8×C9—31no—
80C2×D4×C9—31no—
81C2×#611—61no—
82D4×D9—62no—
83D4×D9—62no—
84C2×D36—31no—
85C2×D36—61no—
86C2×C4×D9C2×D311yescharacteristic
87C9×SD16—62no—
88C2×C8×C9—21no—
89C2×C8×C9C2×D311yescharacteristic, complemented
90#3675—62no—
91(C2×D9)⋊C4—61no—
92C4⋊Dic9—31no—
93C4⋊Dic9—61no—
94Dic9⋊C4—61no—
95C2×C9⋊C8C2×D311yescharacteristic, complemented
96C22×C3×C9C2×D411yescharacteristic
97C2×C3⋊D9—22no—
98C2×C3⋊D9—41no—
99C2×C3⋊D9—41no—
100C2×D3×C9—22no—
101C2×D3×C9—41no—
102C2×D3×C9—41no—
103C2×C3×D9—21no—
104C2×C3×D9—21no—
105C2×C3×D9SD1612yescomplemented
106D3×D9—48no—
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3⋊Dic9—41no—
110C9×Dic3—41no—
111C3×Dic9—21no—
112C2×D3×D4—91no—
113C22×D12—91no—
114C2×C3×SD16—31no—
115#946—91no—
116C2×#215—91no—
117#924—91no—
118C2×#204—91no—
119C2×#204D911yescharacteristic, complemented
120C3×C22⋊C8—91no—
121#832—91no—
122#821—91no—
123#821—91no—
124#821—91no—
125#821—181no—
126(C2×D3)⋊C8—31no—
127C22×C3⋊D3—61no—
128C22×C3×D3—21no—
129C22×C3×D3—31no—
130C2×D32—64no—
131C2×D32—122no—
132C2×D32—122no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—61no—
135C4⋊(C3⋊D3)—32no—
136C4⋊(C3⋊D3)—61no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—31no—
139C3×D12—21no—
140C3×D12#6712yes—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×Dic6—21no—
144C3×Dic6—21no—
145#626—64no—
146C23×C9—61no—
147C22×D9—64no—
148C22×D9—61no—
149C22×D9—121no—
150C22×D9—122no—
151C22×D9—122no—
152C22×D9D1211yescharacteristic
153C32×C8—21no—
154C3×C3⋊C8—61no—
155Q8×C9—61no—
156Q8×C9—61no—
157D4×C9—32no—
158D4×C9—61no—
159C2×C4×C9—21no—
160C2×C4×C9—61no—
161C2×C4×C9C22×D311yescharacteristic
162#611—64no—
163C2×Dic9—61no—
164C2×Dic9—121no—
165C2×Dic9D1211yescharacteristic
166D36—32no—
167D36—62no—
168D36—61no—
169D36—121no—
170C4×D9—21no—
171C4×D9—21no—
172C8×C9—21no—
173C8×C9—41no—
174C9⋊C8—21no—
175#450—271noSylow
176C2×C3×C9#12011yescharacteristic
177C2×C3×C9C2×SD1611yescharacteristic
178C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
179C3⋊D9—42no—
180D3×C9—42no—
181C3×D9—22no—
182C23×D3—181no—
183C23×D3—181no—
184C2×C3×Q8—31no—
185C2×C3×D4—31no—
186C22×C3×C4—91no—
187C2×#67—181no—
188D3×D4—182no—
189D3×D4—182no—
190C2×D12—91no—
191C2×D12—91no—
192C2×D12—181no—
193C2×D12—182no—
194C2×D12—181no—
195C2×D12—182no—
196C2×D12C2×D911yescharacteristic
197C2×C4×D3—31no—
198C2×Dic6C2×D911yescharacteristic, complemented
199C3×SD16—62no—
200C2×C3×C8—21no—
201C2×C3×C8—91no—
202C2×C3×C8C2×D911yescharacteristic, complemented
203C2×C3×C8D3211yescharacteristic
204#217—181no—
205#215—182no—
206(C2×D3)⋊C4—181no—
207C4⋊Dic3—91no—
208C4⋊Dic3—91no—
209C4⋊Dic3—181no—
210Dic3⋊C4—181no—
211Dic3⋊C4—181no—
212C2×C3⋊C8—31no—
213#204—22no—
214#204—182no—
215C22×C32D3×D411yescharacteristic
216C2×C3⋊D3—62no—
217C2×C3⋊D3—121no—
218C2×C3⋊D3—121no—
219C2×C3×D3—22no—
220C2×C3×D3—32no—
221C2×C3×D3—41no—
222C2×C3×D3—41no—
223C2×C3×D3—61no—
224C2×C3×D3—61no—
225D32—128no—
226C32×C4D3×D411yescharacteristic
227C32×C4C2×#6711yescharacteristic
228C3⋊Dic3—121no—
229C3×Dic3—41no—
230C3×Dic3—61no—
231C22×C9—21no—
232C22×C9—62no—
233C22×C9—121no—
234C22×C9—121no—
235C22×C9C2×D1211yescharacteristic
236C2×D9—21no—
237C2×D9—21no—
238C2×D9—62no—
239C2×D9—122no—
240C2×D9—128no—
241C2×D9—121no—
242C2×D9—121no—
243C2×D9—241no—
244C2×D9—241no—
245C2×D9#20412yescomplemented
246C4×C9—21no—
247C4×C9—21no—
248C4×C9—121no—
249C4×C9D3×D411yescharacteristic
250C4×C9D3×D411yescharacteristic
251Dic9—21no—
252Dic9—121no—
253Dic9—241no—
254C22×D4—271no—
255C2×SD16—31no—
256C2×SD16—271no—
257C22⋊Q8—271no—
258#102—271no—
259#102—271no—
260C22⋊C8—91no—
261C3×C9#45011yescharacteristic, complemented, Sylow
262C23×C3—61no—
263C23×C3—91no—
264C22×D3—21no—
265C22×D3—31no—
266C22×D3—184no—
267C22×D3—184no—
268C22×D3—181no—
269C22×D3—181no—
270C22×D3—361no—
271C22×D3—362no—
272C22×D3—362no—
273C22×D3—362no—
274C22×D3—362no—
275C3×Q8—61no—
276C3×Q8—61no—
277C3×D4—32no—
278C3×D4—61no—
279C2×C3×C4—21no—
280C2×C3×C4—61no—
281C2×C3×C4—91no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4C22×D911yescharacteristic
285C2×C3×C4C2×D3211yescharacteristic
286#67—184no—
287C2×Dic3—21no—
288C2×Dic3—31no—
289C2×Dic3—181no—
290C2×Dic3—181no—
291C2×Dic3—361no—
292D12—21no—
293D12—92no—
294D12—92no—
295D12—182no—
296D12—184no—
297D12—181no—
298D12—181no—
299D12—361no—
300D12#61112yes—
301C4×D3—61no—
302C4×D3—61no—
303Dic6—21no—
304Dic6—21no—
305C3×C8—21no—
306C3×C8—21no—
307C3×C8—41no—
308C3×C8—181no—
309C3⋊C8—61no—
310C2×C32D3×SD1611yescharacteristic
311C2×C32#91411yescharacteristic
312C2×C32#93711yescharacteristic
313C3⋊D3—122no—
314C3×D3—42no—
315C3×D3—62no—
316C2×C9—21no—
317C2×C9—21no—
318C2×C9—21no—
319C2×C9—122no—
320C2×C9#88211yescharacteristic
321C2×C9C2×#20411yescharacteristic
322C2×C9#90811yescharacteristic
323D9—22no—
324D9—122no—
325D9—242no—
326C24—541no—
327C2×Q8—31no—
328C2×D4—31no—
329C2×D4—271no—
330C2×D4—541no—
331C2×D4—542no—
332C2×D4—542no—
333C22×C4—91no—
334SD16—62no—
335SD16—542no—
336C2×C8—91no—
337C2×C8D3×D911yescharacteristic, complemented
338C4⋊C4—271no—
339C4⋊C4—541no—
340C22⋊C4—541no—
341C22×C3—21no—
342C22×C3—62no—
343C22×C3—92no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—181no—
347C22×C3—181no—
348C22×C3D4×D911yescharacteristic
349C22×C3D3×D1211yescharacteristic
350C2×D3—22no—
351C2×D3—32no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—61no—
355C2×D3—61no—
356C2×D3—182no—
357C2×D3—182no—
358C2×D3—362no—
359C2×D3—368no—
360C2×D3—368no—
361C2×D3—361no—
362C2×D3—361no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—721no—
366C2×D3—721no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—121no—
370C3×C4—181no—
371C3×C4D4×D911yescharacteristic
372C3×C4C2×#61111yescharacteristic
373C3×C4#380211yescharacteristic
374C3×C4D3×#6711yescharacteristic
375Dic3—41no—
376Dic3—61no—
377Dic3—361no—
378Dic3—361no—
379Dic3—721no—
380C32#516911yescharacteristic
381C9—21no—
382C9#473111yescharacteristic, complemented
383C23—61no—
384C23—91no—
385C23—544no—
386C23—541no—
387C23—1082no—
388C23—1082no—
389Q8—61no—
390Q8—61no—
391D4—32no—
392D4—61no—
393D4—272no—
394D4—544no—
395D4—541no—
396C2×C4—61no—
397C2×C4—91no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—541no—
401C2×C4C2×D3×D911yescharacteristic
402C8—21no—
403C8—181no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—122no—
408C2×C3—182no—
409C2×C3SD16×D911yescharacteristic
410C2×C3#6347611yescharacteristic
411C2×C3#6349911yescharacteristic
412C2×C3D3×#20411yescharacteristic
413C2×C3#6379411yescharacteristic
414C2×C3#6392211yescharacteristic
415D3—42no—
416D3—62no—
417D3—362no—
418D3—362no—
419D3—722no—
420C22—62no—
421C22—92no—
422C22—121no—
423C22—121no—
424C22—181no—
425C22—181no—
426C22—542no—
427C22—1088no—
428C22—1081no—
429C22—1081no—
430C22D9×D1211yescharacteristic
431C4—121no—
432C4—181no—
433C4—1081no—
434C4#8864611yescharacteristic
435C4D3×#61111yescharacteristic
436C3—21no—
437C3#1058884911yescharacteristic, complemented
438C3#1059094211yescharacteristic
439C2—122no—
440C2—182no—
441C2—1082no—
442C2D9×#20411yescharacteristic
443C2#1172081511yescharacteristic
444C2#1172105011yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×D9×D12C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#450
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C9×#204C2semidirect
3. #11720944C2semidirect
4. C2×#88426C2semidirect
5. #11720543C2semidirect
6. C3×(C2×D9)⋊C8C2semidirect
7. #11720295C2semidirect
14. C2×C9×Dic6C22semidirect
18. C2×C3×C8×C9C22semidirect
19. C4⋊(C3⋊Dic9)C22semidirect
20. C2×C3×C9⋊C8C22semidirect
39. (C2×D9)⋊C8D3semidirect
89. C2×C8×C9C2×D3semidirect
95. C2×C9⋊C8C2×D3semidirect
105. C2×C3×D9SD16semidirect
119. C2×#204D9semidirect
198. C2×Dic6C2×D9semidirect
202. C2×C3×C8C2×D9semidirect
245. C2×D9#204semidirect
261. C3×C9#450semidirect
337. C2×C8D3×D9semidirect
382. C9#4731semidirect
437. C3#10588849semidirect

Maximal subgroups

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

subgroupindex
C2×D9×D122
C2×C9×#2042
#117209442
C2×#884262
#117205432
C3×(C2×D9)⋊C82
#117202952
#105909423
#105888493