Finite Group Databasebeta

← #13140865

Subgroups — #13140865

Order 1728.

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

normal subgroups only

This group has 632 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×#88548C211yesmaximal, characteristic, complemented
3C3×#63508C211yesmaximal, characteristic, complemented
4C2×C9×#215C211yesmaximal, characteristic, complemented
5#11720294C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic, complemented
7(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591651—31nomaximal
9#10588849—31nomaximal
10C22×D3×D9—21no—
11C2×C3×Q8×C9C2211yescharacteristic, complemented
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C9×D12C2211yescharacteristic
14C2×C3×C4×D9C2211yescharacteristic
15C2×#6195—21no—
16D9×D12—22no—
17D9×D12—22no—
18#88548—22no—
19C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
20C9×#215—22no—
21C3×(C2×D9)⋊C4—21no—
22C3×C4⋊Dic9C2211yescharacteristic, complemented
23C3×Dic9⋊C4—21no—
24C2×C9×C3⋊C8C2211yescharacteristic, complemented
25C2×D3×D12—31no—
26C2×#3755—31no—
27C3×#946—31no—
28C2×C3×#215D311yescharacteristic
29C2×D4×D9—31no—
30#63564—31no—
31#63562—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×C9×SD16—31no—
34#63508D311yescharacteristic, complemented
35C2×#3675—31no—
36C2×#3675—61no—
37#63392—31no—
38#63380—31no—
39(C2×D9)⋊C8—31no—
40C22×C3⋊D9—21no—
41C22×D3×C9—21no—
42C22×C3×D9D411yescharacteristic
43C2×D3×D9—24no—
44C2×D3×D9—42no—
45C2×D3×D9—42no—
46C3×Q8×C9—21no—
47C3×Q8×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9C2311yescharacteristic
50C4⋊(C3⋊D9)—21no—
51C4⋊(C3⋊D9)D412yes—
52C2×C3×Dic9—21no—
53C2×C3×Dic9D411yescharacteristic
54C9×D12—21no—
55C9×D12D412yes—
56C3×C4×D9—21no—
57C3×C4×D9—21no—
58#6195—24no—
59C3⋊(C9⋊C8)—21no—
60C9×C3⋊C8—21no—
61#5169—91no—
62#5046—91no—
63C22×D32—61no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C4⋊(C3⋊D3)—31no—
66C2×C3×D12C2×D311yescharacteristic
67C2×C3×C4×D3—31no—
68C2×#626—61no—
69D3×D12—62no—
70D3×D12—62no—
71C23×D9—61no—
72#3755—62no—
73C2×C3⋊(C3⋊C8)—31no—
74C3×#215—22no—
75C3×(C2×D3)⋊C4—61no—
76C3×C4⋊Dic3—31no—
77C3×Dic3⋊C4—61no—
78C2×C3×C3⋊C8C2×D311yescharacteristic
79C2×Q8×C9—21no—
80C2×Q8×C9C2×D311yescharacteristic, complemented
81C2×D4×C9—31no—
82C2×#611—61no—
83D4×D9—62no—
84D4×D9—62no—
85C2×D36—31no—
86C2×D36—61no—
87C2×C4×D9C2×D311yescharacteristic
88C9×SD16—62no—
89C2×C8×C9—31no—
90#3675—62no—
91#3675—64no—
92(C2×D9)⋊C4—21no—
93C4⋊Dic9C2×D311yescharacteristic, complemented
94Dic9⋊C4—21no—
95C2×C9⋊C8—31no—
96C2×C9⋊C8—61no—
97C22×C3×C9C2×D411yescharacteristic
98C2×C3⋊D9—22no—
99C2×C3⋊D9—41no—
100C2×C3⋊D9—41no—
101C2×D3×C9—22no—
102C2×D3×C9—41no—
103C2×D3×C9—41no—
104C2×C3×D9—21no—
105C2×C3×D9—21no—
106C2×C3×D9SD1612yescomplemented
107D3×D9—48no—
108C3×C4×C9—41no—
109C3×C4×C9C2×D411yescharacteristic
110C3×C4×C9C2×D411yescharacteristic
111C3×Dic9—21no—
112C3×Dic9—41no—
113C2×D3×D4—91no—
114C22×D12—91no—
115C2×C3×SD16—31no—
116C3×C22⋊Q8—91no—
117#946—31no—
118C2×#215—91no—
119C2×#215—91no—
120C2×#215—181no—
121C2×#215D911yescharacteristic, complemented
122#832—91no—
123#830—91no—
124#821—91no—
125(C2×D3)⋊C8—91no—
126#809—91no—
127#809—91no—
128C22×C3⋊D3—61no—
129C22×C3×D3—21no—
130C22×C3×D3—31no—
131C2×D32—64no—
132C2×D32—122no—
133C2×D32—122no—
134C32×Q8—21no—
135C32×Q8—21no—
136C2×C32×C4—21no—
137C2×C32×C4C22×D311yescharacteristic
138C4⋊(C3⋊D3)—32no—
139C4⋊(C3⋊D3)—61no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—61no—
142C3×D12—21no—
143C3×D12#6712yes—
144C3×C4×D3—61no—
145C3×C4×D3—61no—
146#626—64no—
147C23×C9—61no—
148C22×D9—64no—
149C22×D9—61no—
150C22×D9—121no—
151C22×D9—122no—
152C22×D9—122no—
153C22×D9#6711yescharacteristic
154C3⋊(C3⋊C8)—61no—
155C3×C3⋊C8—21no—
156Q8×C9—22no—
157Q8×C9—21no—
158Q8×C9—21no—
159Q8×C9—41no—
160D4×C9—32no—
161D4×C9—61no—
162C2×C4×C9—21no—
163C2×C4×C9—21no—
164C2×C4×C9—41no—
165C2×C4×C9C22×D311yescharacteristic
166#611—64no—
167C2×Dic9—21no—
168C2×Dic9#6711yescharacteristic
169D36—32no—
170D36—62no—
171D36—61no—
172D36—121no—
173C4×D9—21no—
174C4×D9—21no—
175C8×C9—61no—
176C9⋊C8—62no—
177C9⋊C8—61no—
178#450—271noSylow
179C2×C3×C9#12011yescharacteristic
180C2×C3×C9C2×SD1611yescharacteristic
181C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
182C3⋊D9—42no—
183D3×C9—42no—
184C3×D9—22no—
185C23×D3—181no—
186C23×D3—181no—
187C2×C3×Q8—21no—
188C2×C3×Q8C2×D911yescharacteristic, complemented
189C2×C3×Q8D3211yescharacteristic
190C2×C3×D4—31no—
191C22×C3×C4—91no—
192C2×#67—181no—
193D3×D4—182no—
194D3×D4—182no—
195C2×D12—91no—
196C2×D12—91no—
197C2×D12—181no—
198C2×D12—182no—
199C2×D12—181no—
200C2×D12—182no—
201C2×D12C2×D911yescharacteristic
202C2×C4×D3—31no—
203C3×SD16—62no—
204C2×C3×C8—31no—
205C3×C4⋊C4—91no—
206C3×C4⋊C4—181no—
207C3×C22⋊C4—181no—
208#215—22no—
209#215—182no—
210#215—182no—
211#215—184no—
212(C2×D3)⋊C4—61no—
213C4⋊Dic3—31no—
214Dic3⋊C4—61no—
215C2×C3⋊C8—91no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—181no—
218C2×C3⋊C8C2×D911yescharacteristic, complemented
219C22×C32D3×D411yescharacteristic
220C2×C3⋊D3—62no—
221C2×C3⋊D3—121no—
222C2×C3⋊D3—121no—
223C2×C3×D3—22no—
224C2×C3×D3—32no—
225C2×C3×D3—41no—
226C2×C3×D3—41no—
227C2×C3×D3—61no—
228C2×C3×D3—61no—
229D32—128no—
230C32×C4—41no—
231C32×C4D3×D411yescharacteristic
232C32×C4C2×#6711yescharacteristic
233C3×Dic3—61no—
234C3×Dic3—121no—
235C22×C9—21no—
236C22×C9—62no—
237C22×C9—121no—
238C22×C9—121no—
239C22×C9C2×#6711yescharacteristic
240C2×D9—21no—
241C2×D9—21no—
242C2×D9—62no—
243C2×D9—122no—
244C2×D9—128no—
245C2×D9—121no—
246C2×D9—121no—
247C2×D9—241no—
248C2×D9—241no—
249C2×D9#21512yescomplemented
250C4×C9—21no—
251C4×C9—21no—
252C4×C9—42no—
253C4×C9—41no—
254C4×C9D3×D411yescharacteristic
255C4×C9D3×D411yescharacteristic
256Dic9—21no—
257Dic9—41no—
258C22×D4—271no—
259C2×SD16—31no—
260C2×SD16—271no—
261C22⋊Q8—91no—
262#102—271no—
263#102—271no—
264C22⋊C8—271no—
265C3×C9#45011yescharacteristic, complemented, Sylow
266C23×C3—61no—
267C23×C3—91no—
268C22×D3—21no—
269C22×D3—31no—
270C22×D3—184no—
271C22×D3—184no—
272C22×D3—181no—
273C22×D3—181no—
274C22×D3—361no—
275C22×D3—362no—
276C22×D3—362no—
277C22×D3—362no—
278C22×D3—362no—
279C3×Q8—22no—
280C3×Q8—21no—
281C3×Q8—21no—
282C3×Q8—21no—
283C3×Q8—21no—
284C3×Q8—41no—
285C3×D4—32no—
286C3×D4—61no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—41no—
291C2×C3×C4—91no—
292C2×C3×C4—181no—
293C2×C3×C4—181no—
294C2×C3×C4—181no—
295C2×C3×C4C22×D911yescharacteristic
296C2×C3×C4C2×D3211yescharacteristic
297#67—184no—
298C2×Dic3—31no—
299C2×Dic3—61no—
300D12—21no—
301D12—92no—
302D12—92no—
303D12—182no—
304D12—184no—
305D12—181no—
306D12—181no—
307D12—361no—
308D12#61112yes—
309C4×D3—61no—
310C4×D3—61no—
311C3×C8—61no—
312C3⋊C8—21no—
313C3⋊C8—182no—
314C3⋊C8—181no—
315C3⋊C8—181no—
316C2×C32D3×SD1611yescharacteristic
317C2×C32#91411yescharacteristic
318C2×C32#93711yescharacteristic
319C3⋊D3—122no—
320C3×D3—42no—
321C3×D3—62no—
322C2×C9—21no—
323C2×C9—21no—
324C2×C9—21no—
325C2×C9—122no—
326C2×C9#93711yescharacteristic
327C2×C9C2×#21511yescharacteristic
328C2×C9#94911yescharacteristic
329D9—22no—
330D9—122no—
331D9—242no—
332C24—541no—
333C2×Q8D3×D911yescharacteristic, complemented
334C2×D4—31no—
335C2×D4—271no—
336C2×D4—541no—
337C2×D4—542no—
338C2×D4—542no—
339C22×C4—91no—
340SD16—62no—
341SD16—542no—
342C2×C8—31no—
343C2×C8—271no—
344C4⋊C4—91no—
345C4⋊C4—181no—
346C22⋊C4—181no—
347C22×C3—21no—
348C22×C3—62no—
349C22×C3—92no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3D4×D911yescharacteristic
355C22×C3D3×#6711yescharacteristic
356C2×D3—22no—
357C2×D3—32no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—182no—
363C2×D3—182no—
364C2×D3—362no—
365C2×D3—368no—
366C2×D3—368no—
367C2×D3—361no—
368C2×D3—361no—
369C2×D3—361no—
370C2×D3—361no—
371C2×D3—721no—
372C2×D3—721no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—42no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—181no—
379C3×C4—361no—
380C3×C4D4×D911yescharacteristic
381C3×C4C2×#61111yescharacteristic
382C3×C4D3×#6711yescharacteristic
383C3×C4#381111yescharacteristic
384Dic3—61no—
385Dic3—121no—
386C32#516911yescharacteristic
387C9—21no—
388C9#504611yescharacteristic, complemented
389C23—61no—
390C23—91no—
391C23—544no—
392C23—541no—
393C23—1082no—
394C23—1082no—
395Q8—21no—
396Q8—21no—
397D4—32no—
398D4—61no—
399D4—272no—
400D4—544no—
401D4—541no—
402C2×C4—21no—
403C2×C4—91no—
404C2×C4—181no—
405C2×C4—181no—
406C2×C4—181no—
407C2×C4C2×D3×D911yescharacteristic
408C8—61no—
409C8—541no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—122no—
414C2×C3—182no—
415C2×C3SD16×D911yescharacteristic
416C2×C3#6347611yescharacteristic
417C2×C3#6349911yescharacteristic
418C2×C3D3×#21511yescharacteristic
419C2×C3#6393911yescharacteristic
420C2×C3#6398311yescharacteristic
421D3—42no—
422D3—62no—
423D3—362no—
424D3—362no—
425D3—722no—
426C22—62no—
427C22—92no—
428C22—121no—
429C22—121no—
430C22—181no—
431C22—181no—
432C22—542no—
433C22—1088no—
434C22—1081no—
435C22—1081no—
436C22D9×#6711yescharacteristic
437C4—41no—
438C4—181no—
439C4—361no—
440C4D3×#61111yescharacteristic
441C4#8866811yescharacteristic
442C3—21no—
443C3#1058884911yescharacteristic, complemented
444C3#1059165111yescharacteristic
445C2—122no—
446C2—182no—
447C2—1082no—
448C2D9×#21511yescharacteristic
449C2#1172121111yescharacteristic
450C2#1172129211yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×D9×D12C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#450
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#88548C2semidirect
3. C3×#63508C2semidirect
4. C2×C9×#215C2semidirect
5. #11720294C2semidirect
6. #11720284C2semidirect
7. (C2×D9)⋊(C3⋊C8)C2semidirect
11. C2×C3×Q8×C9C22semidirect
19. C2×C3⋊(C9⋊C8)C22semidirect
22. C3×C4⋊Dic9C22semidirect
24. C2×C9×C3⋊C8C22semidirect
34. #63508D3semidirect
80. C2×Q8×C9C2×D3semidirect
93. C4⋊Dic9C2×D3semidirect
106. C2×C3×D9SD16semidirect
121. C2×#215D9semidirect
188. C2×C3×Q8C2×D9semidirect
218. C2×C3⋊C8C2×D9semidirect
249. C2×D9#215semidirect
265. C3×C9#450semidirect
333. C2×Q8D3×D9semidirect
388. C9#5046semidirect
443. 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×#885482
C3×#635082
C2×C9×#2152
#117202942
#117202842
(C2×D9)⋊(C3⋊C8)2
#105916513
#105888493