Finite Group Databasebeta

← #13140866

Subgroups — #13140866

Order 1728.

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

normal subgroups only

This group has 638 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
2#11721856C211yesmaximal, characteristic, complemented
3C2×C9×#215C211yesmaximal, characteristic, complemented
4C2×C3×#3675C211yesmaximal, characteristic, complemented
5#11720295C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic, complemented
8#10591652—31nomaximal
9#10588849—31nomaximal
10C22×D3×D9—21no—
11C2×C3×Q8×C9C2211yescharacteristic, complemented
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×C9×D12C2211yescharacteristic
14C2×C3×D36C2211yescharacteristic
15C2×#6194—21no—
16#88646—22no—
17#88646—22no—
18#88536—21no—
19C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
20#88534—21no—
21C9×#215—22no—
22C3×#3675—22no—
23C2×C9×C3⋊C8C2211yescharacteristic, complemented
24C2×C3×C9⋊C8C2211yescharacteristic, complemented
25C2×#3802—31no—
26#64155—31no—
27C2×C3×#215—31no—
28C2×C3×#215D311yescharacteristic
29C2×D4×D9—31no—
30#63563—31no—
31#63563—31no—
32#63557—31no—
33C2×C9×SD16—31no—
34#63508—31no—
35#63508—61no—
36C2×#3675D311yescharacteristic, complemented
37#63392—31no—
38#63380—31no—
39(C2×D9)⋊C8—31no—
40C22×C3⋊D9D411yescharacteristic
41C22×D3×C9—21no—
42C22×C3×D9—21no—
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
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9D411yescharacteristic
52C4×C3⋊D9—21no—
53C4×C3⋊D9—21no—
54C9×D12—21no—
55C9×D12D412yes—
56C3×D36—21no—
57C3×D36D412yes—
58#6194—24no—
59C9×C3⋊C8—21no—
60C3×C9⋊C8—21no—
61#5169—91no—
62#5169—91no—
63C22×D32—61no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C4×C3⋊D3—31no—
66C2×C3×D12—31no—
67C2×C3×D12C2×D311yescharacteristic
68C2×#625—61no—
69#3802—62no—
70#3802—62no—
71C23×D9—61no—
72#3752—61no—
73C4⋊(C3⋊Dic3)—31no—
74#3750—61no—
75C3×#215—22no—
76C3×#215—62no—
77C2×C3×C3⋊C8—31no—
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×D36C2×D311yescharacteristic
86C2×C4×D9—31no—
87C2×C4×D9—61no—
88C9×SD16—62no—
89C2×C8×C9—31no—
90#3675—22no—
91(C2×D9)⋊C4—61no—
92(C2×D9)⋊C4—62no—
93C4⋊Dic9—31no—
94C4⋊Dic9—61no—
95Dic9⋊C4—61no—
96Dic9⋊C4—62no—
97C2×C9⋊C8C2×D311yescharacteristic, complemented
98C22×C3×C9C2×D411yescharacteristic
99C2×C3⋊D9—21no—
100C2×C3⋊D9—21no—
101C2×C3⋊D9SD1612yescomplemented
102C2×D3×C9—22no—
103C2×D3×C9—41no—
104C2×D3×C9—41no—
105C2×C3×D9—22no—
106C2×C3×D9—41no—
107C2×C3×D9—41no—
108D3×D9—48no—
109C3×C4×C9—41no—
110C3×C4×C9C2×D411yescharacteristic
111C3×C4×C9C2×D411yescharacteristic
112C3⋊Dic9—21no—
113C3⋊Dic9—41no—
114C2×D3×D4—91no—
115C2×D3×D4—91no—
116C2×C3×SD16—31no—
117C2×C3×SD16—91no—
118#946—91no—
119#946—91no—
120#946—181no—
121C2×#215—31no—
122C2×#215D911yescharacteristic, complemented
123#832—91no—
124#832—91no—
125#821—91no—
126#821—91no—
127(C2×D3)⋊C8—91no—
128(C2×D3)⋊C8—91no—
129C22×C3⋊D3—31no—
130C22×C3×D3—21no—
131C22×C3×D3—61no—
132C2×D32—64no—
133C2×D32—122no—
134C2×D32—122no—
135C32×Q8—21no—
136C32×Q8—21no—
137C2×C32×C4—21no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C3⋊Dic3—31no—
140C2×C3⋊Dic3—61no—
141C4×C3⋊D3—61no—
142C4×C3⋊D3—61no—
143C3×D12—21no—
144C3×D12—32no—
145C3×D12—61no—
146C3×D12#6712yes—
147#625—64no—
148C23×C9—61no—
149C22×D9—21no—
150C22×D9—31no—
151C22×D9—61no—
152C22×D9—64no—
153C22×D9—122no—
154C22×D9—122no—
155C3×C3⋊C8—21no—
156C3×C3⋊C8—61no—
157Q8×C9—21no—
158Q8×C9—21no—
159Q8×C9—41no—
160Q8×C9—41no—
161D4×C9—32no—
162D4×C9—61no—
163C2×C4×C9—22no—
164C2×C4×C9—21no—
165C2×C4×C9—21no—
166C2×C4×C9C22×D311yescharacteristic
167#611—64no—
168C2×Dic9—31no—
169C2×Dic9—61no—
170C2×Dic9—62no—
171C2×Dic9—61no—
172D36—21no—
173D36#6712yes—
174C4×D9—61no—
175C4×D9—61no—
176C4×D9—121no—
177C4×D9—121no—
178C8×C9—61no—
179C9⋊C8—21no—
180#450—271noSylow
181C2×C3×C9#12011yescharacteristic
182C2×C3×C9C2×SD1611yescharacteristic
183C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
184C3⋊D9—22no—
185D3×C9—42no—
186C3×D9—42no—
187C23×D3—181no—
188C23×D3—181no—
189C2×C3×Q8—21no—
190C2×C3×Q8C2×D911yescharacteristic, complemented
191C2×C3×Q8D3211yescharacteristic
192C2×C3×D4—31no—
193C2×C3×D4—91no—
194C2×#67—181no—
195C2×#67—181no—
196D3×D4—182no—
197D3×D4—182no—
198D3×D4—182no—
199D3×D4—182no—
200C2×D12—31no—
201C2×D12C2×D911yescharacteristic
202C2×C4×D3—91no—
203C2×C4×D3—91no—
204C2×C4×D3—181no—
205C3×SD16—62no—
206C3×SD16—182no—
207C2×C3×C8—31no—
208C2×C3×C8—91no—
209#215—22no—
210#215—62no—
211(C2×D3)⋊C4—181no—
212(C2×D3)⋊C4—181no—
213(C2×D3)⋊C4—182no—
214C4⋊Dic3—91no—
215C4⋊Dic3—91no—
216C4⋊Dic3—181no—
217Dic3⋊C4—181no—
218Dic3⋊C4—181no—
219Dic3⋊C4—182no—
220C2×C3⋊C8—31no—
221C2×C3⋊C8C2×D911yescharacteristic, complemented
222C22×C32D3×D411yescharacteristic
223C2×C3⋊D3—32no—
224C2×C3⋊D3—61no—
225C2×C3⋊D3—61no—
226C2×C3×D3—22no—
227C2×C3×D3—41no—
228C2×C3×D3—41no—
229C2×C3×D3—62no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232D32—128no—
233C32×C4—41no—
234C32×C4D3×D411yescharacteristic
235C32×C4C2×#6711yescharacteristic
236C3⋊Dic3—61no—
237C3⋊Dic3—121no—
238C22×C9—21no—
239C22×C9—62no—
240C22×C9—121no—
241C22×C9—121no—
242C22×C9D3×D411yescharacteristic
243C2×D9—22no—
244C2×D9—32no—
245C2×D9—41no—
246C2×D9—41no—
247C2×D9—62no—
248C2×D9—61no—
249C2×D9—61no—
250C2×D9—121no—
251C2×D9—121no—
252C2×D9—128no—
253C4×C9—21no—
254C4×C9—21no—
255C4×C9—42no—
256C4×C9—41no—
257C4×C9D3×D411yescharacteristic
258C4×C9C2×#6711yescharacteristic
259Dic9—61no—
260Dic9—121no—
261Dic9—122no—
262Dic9—121no—
263C22×D4—271no—
264C2×SD16—31no—
265C2×SD16—91no—
266C22⋊Q8—271no—
267#102—271no—
268#102—271no—
269C22⋊C8—271no—
270C3×C9#45011yescharacteristic, complemented, Sylow
271C23×C3—61no—
272C23×C3—181no—
273C22×D3—21no—
274C22×D3—61no—
275C22×D3—91no—
276C22×D3—91no—
277C22×D3—181no—
278C22×D3—184no—
279C22×D3—184no—
280C22×D3—362no—
281C22×D3—362no—
282C22×D3—362no—
283C22×D3—362no—
284C3×Q8—21no—
285C3×Q8—21no—
286C3×Q8—21no—
287C3×Q8—21no—
288C3×Q8—41no—
289C3×Q8—41no—
290C3×D4—32no—
291C3×D4—61no—
292C3×D4—92no—
293C3×D4—181no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4C22×D911yescharacteristic
299C2×C3×C4C2×D3211yescharacteristic
300#67—184no—
301#67—184no—
302C2×Dic3—91no—
303C2×Dic3—91no—
304C2×Dic3—181no—
305C2×Dic3—182no—
306C2×Dic3—181no—
307C2×Dic3—181no—
308D12—21no—
309D12—32no—
310D12—61no—
311D12#61112yes—
312C4×D3—181no—
313C4×D3—181no—
314C4×D3—181no—
315C4×D3—181no—
316C4×D3—361no—
317C4×D3—361no—
318C3×C8—61no—
319C3×C8—181no—
320C3⋊C8—21no—
321C3⋊C8—61no—
322C2×C32D3×SD1611yescharacteristic
323C2×C32#91411yescharacteristic
324C2×C32#93711yescharacteristic
325C3⋊D3—62no—
326C3×D3—42no—
327C3×D3—122no—
328C2×C9—21no—
329C2×C9—21no—
330C2×C9—21no—
331C2×C9—122no—
332C2×C9D3×SD1611yescharacteristic
333C2×C9#91411yescharacteristic
334C2×C9#93711yescharacteristic
335D9—42no—
336D9—62no—
337D9—122no—
338C24—541no—
339C2×Q8D3×D911yescharacteristic, complemented
340C2×D4—31no—
341C2×D4—91no—
342C2×D4—541no—
343C2×D4—542no—
344C2×D4—542no—
345C22×C4—271no—
346SD16—62no—
347SD16—182no—
348C2×C8—31no—
349C2×C8—91no—
350C4⋊C4—271no—
351C4⋊C4—541no—
352C22⋊C4—541no—
353C22×C3—21no—
354C22×C3—62no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—182no—
358C22×C3—361no—
359C22×C3—361no—
360C22×C3D4×D911yescharacteristic
361C22×C3#381111yescharacteristic
362C2×D3—22no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—62no—
366C2×D3—92no—
367C2×D3—92no—
368C2×D3—121no—
369C2×D3—121no—
370C2×D3—182no—
371C2×D3—181no—
372C2×D3—181no—
373C2×D3—181no—
374C2×D3—181no—
375C2×D3—361no—
376C2×D3—361no—
377C2×D3—368no—
378C2×D3—368no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—42no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4D4×D911yescharacteristic
385C3×C4C2×#61111yescharacteristic
386C3×C4D3×#6711yescharacteristic
387C3×C4D3×#6711yescharacteristic
388Dic3—181no—
389Dic3—181no—
390Dic3—361no—
391Dic3—362no—
392Dic3—361no—
393Dic3—361no—
394C32#516911yescharacteristic
395C9—21no—
396C9#516911yescharacteristic, complemented
397C23—61no—
398C23—181no—
399C23—271no—
400C23—544no—
401C23—1082no—
402C23—1082no—
403Q8—21no—
404Q8—21no—
405D4—32no—
406D4—61no—
407D4—92no—
408D4—181no—
409D4—544no—
410C2×C4—21no—
411C2×C4—271no—
412C2×C4—541no—
413C2×C4—541no—
414C2×C4—541no—
415C2×C4C2×D3×D911yescharacteristic
416C8—61no—
417C8—181no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—122no—
422C2×C3—362no—
423C2×C3SD16×D911yescharacteristic
424C2×C3#6347611yescharacteristic
425C2×C3#6349911yescharacteristic
426C2×C3#6394011yescharacteristic
427C2×C3#6394111yescharacteristic
428C2×C3#6398311yescharacteristic
429D3—42no—
430D3—122no—
431D3—182no—
432D3—182no—
433D3—362no—
434C22—62no—
435C22—121no—
436C22—121no—
437C22—182no—
438C22—272no—
439C22—361no—
440C22—361no—
441C22—541no—
442C22—541no—
443C22—1088no—
444C22#8866811yescharacteristic
445C4—41no—
446C4—541no—
447C4—1081no—
448C4D3×#61111yescharacteristic
449C4D9×#6711yescharacteristic
450C3—21no—
451C3#1058884911yescharacteristic, complemented
452C3#1059165211yescharacteristic
453C2—122no—
454C2—362no—
455C2—542no—
456C2#1172121211yescharacteristic
457C2#1172121311yescharacteristic
458C2#1172129211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88646C2
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. #11721856C2semidirect
3. C2×C9×#215C2semidirect
4. C2×C3×#3675C2semidirect
5. #11720295C2semidirect
6. #11720283C2semidirect
7. #11720282C2semidirect
11. C2×C3×Q8×C9C22semidirect
19. C4⋊(C3⋊Dic9)C22semidirect
23. C2×C9×C3⋊C8C22semidirect
24. C2×C3×C9⋊C8C22semidirect
36. C2×#3675D3semidirect
80. C2×Q8×C9C2×D3semidirect
97. C2×C9⋊C8C2×D3semidirect
101. C2×C3⋊D9SD16semidirect
122. C2×#215D9semidirect
190. C2×C3×Q8C2×D9semidirect
221. C2×C3⋊C8C2×D9semidirect
270. C3×C9#450semidirect
339. C2×Q8D3×D9semidirect
396. C9#5169semidirect
451. C3#10588849semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
#117218562
C2×C9×#2152
C2×C3×#36752
#117202952
#117202832
#117202822
#105916523
#105888493