Finite Group Databasebeta

← #13137843

Subgroups — #13137843

Order 1728.

530 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 89 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×#88639C211yesmaximal, characteristic
2#11721753C211yesmaximal, characteristic, complemented
3C2×#88424C211yesmaximal, characteristic, complemented
4C2×#88418C211yesmaximal, characteristic, complemented
5C9×#809C211yesmaximal, characteristic, complemented
6C3×#63368C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic, complemented
8#10590618—31nomaximal
9#10588463—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×#6195—21no—
15C2×D3×Dic9—21no—
16#88639—22no—
17#88639—22no—
18C3×C9×C4⋊C4C2211yescharacteristic
19#88536—21no—
20C4⋊(C3⋊Dic9)—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23#88424—22no—
24#88418—22no—
25C2×#3796—31no—
26#64106—31no—
27C2×#3719—31no—
28C2×#3716—31no—
29C2×#3699—31no—
30C3×#810—31no—
31C3×#809D311yescharacteristic
32#63557—31no—
33C2×#3672—31no—
34C2×Dic36—31no—
35#63459—31no—
36#63459—61no—
37C9×#102—31no—
38(C2×D9)⋊C8—31no—
39#63368D311yescharacteristic, complemented
40C22×C3⋊D9D411yescharacteristic
41C22×D3×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9D411yescharacteristic
46C4×C3⋊D9—21no—
47C4×C3⋊D9—21no—
48#6226—21no—
49#6226—21no—
50C2×C3×Dic9—21no—
51C9×D12—21no—
52C9×D12D412yes—
53C3×Dic18—21no—
54C3×Dic18D412yes—
55#6195—24no—
56D3×Dic9—24no—
57C9×C3⋊C8—21no—
58C3×C9⋊C8—21no—
59#4808—91no—
60#4783—91no—
61C2×C4×C3⋊D3—31no—
62C2×#634—31no—
63C2×C3×D12C2×D311yescharacteristic
64C2×C3×Dic6—31no—
65C2×#626—61no—
66C2×D3×Dic3—61no—
67#3796—62no—
68#3796—62no—
69C32×C4⋊C4C2×D311yescharacteristic
70#3752—61no—
71C4⋊(C3⋊Dic3)—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8C2×D311yescharacteristic
74#3719—62no—
75#3716—62no—
76C2×D4×C9—31no—
77C2×#611—61no—
78C22×Dic9—61no—
79#3699—62no—
80#3699—62no—
81C2×C4×D9—31no—
82C2×C4×D9—61no—
83C2×Dic18—31no—
84C2×Dic18—61no—
85C2×Dic18C2×D311yescharacteristic
86C2×C8×C9—31no—
87C9×C4⋊C4—21no—
88C9×C4⋊C4C2×D311yescharacteristic
89#3672—62no—
90(C2×D9)⋊C4—61no—
91(C2×D9)⋊C4—62no—
92C4⋊Dic9—61no—
93C4⋊Dic9—62no—
94C2×C9⋊C8C2×D311yescharacteristic
95Dic36—62no—
96C22×C3×C9C2×D411yescharacteristic
97C2×C3⋊D9—21no—
98C2×C3⋊D9—21no—
99C2×C3⋊D9—21no—
100C2×D3×C9—22no—
101C2×D3×C9—41no—
102C2×D3×C9—41no—
103C3×C4×C9—41no—
104C3×C4×C9C2×D411yescharacteristic
105C3×C4×C9C2×D411yescharacteristic
106C3⋊Dic9—21no—
107C3⋊Dic9—41no—
108C3×Dic9—22no—
109C2×#239—91no—
110C2×#237—91no—
111C2×#216—91no—
112C2×#214—91no—
113C2×Dic12—91no—
114C2×#204—91no—
115#897—91no—
116#897—91no—
117#897—181no—
118C3×#103—91no—
119C3×#102—31no—
120(C2×D3)⋊C8—91no—
121(C2×D3)⋊C8—91no—
122#810—31no—
123#809D911yescharacteristic, complemented
124C22×C3⋊D3—31no—
125C22×C3×D3—21no—
126C2×C32×C4—21no—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—31no—
129C2×C3⋊Dic3—61no—
130C4×C3⋊D3—61no—
131C4×C3⋊D3—61no—
132#634—61no—
133#634—61no—
134C2×C3×Dic3—61no—
135C3×D12—21no—
136C3×D12D1212yes—
137C3×Dic6—32no—
138C3×Dic6—61no—
139#626—64no—
140D3×Dic3—64no—
141C23×C9—61no—
142C22×D9—31no—
143C22×D9—61no—
144C3×C3⋊C8—21no—
145C3×C3⋊C8—61no—
146D4×C9—32no—
147D4×C9—61no—
148C2×C4×C9—22no—
149C2×C4×C9—21no—
150C2×C4×C9—21no—
151C2×C4×C9C22×D311yescharacteristic
152#611—64no—
153C2×Dic9—21no—
154C2×Dic9—31no—
155C2×Dic9—61no—
156C2×Dic9—64no—
157C2×Dic9—62no—
158C2×Dic9—61no—
159C4×D9—61no—
160C4×D9—61no—
161C4×D9—121no—
162C4×D9—121no—
163Dic18—21no—
164Dic18—61no—
165Dic18—61no—
166Dic18—121no—
167Dic18—121no—
168Dic18D1212yes—
169C8×C9—61no—
170C9⋊C8—21no—
171#452—271noSylow
172C2×C3×C9#12011yescharacteristic
173C2×C3×C9#13511yescharacteristic
174C2×C3×C9#13711yescharacteristic
175C3⋊D9—41no—
176D3×C9—42no—
177C2×C3×Q8—91no—
178C2×C3×D4—31no—
179C2×#67—181no—
180C22×Dic3—181no—
181#239—182no—
182#239—182no—
183#237—182no—
184#237—182no—
185C2×D12—181no—
186C2×D12C2×D911yescharacteristic
187C2×C4×D3—91no—
188C2×C4×D3—91no—
189C2×C4×D3—181no—
190C2×C4×D3—181no—
191C2×Dic6—31no—
192C2×Dic6—91no—
193C2×Dic6—91no—
194C2×Dic6—181no—
195C2×C3×C8—31no—
196C2×C3×C8—91no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4C2×D911yescharacteristic
199C3×C4⋊C4D3211yescharacteristic
200#216—182no—
201#214—182no—
202(C2×D3)⋊C4—181no—
203(C2×D3)⋊C4—181no—
204(C2×D3)⋊C4—182no—
205C4⋊Dic3—181no—
206C4⋊Dic3—181no—
207C4⋊Dic3—182no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8C2×D911yescharacteristic
210Dic12—182no—
211#204—182no—
212C22×C32D3×D411yescharacteristic
213C2×C3⋊D3—61no—
214C2×C3⋊D3—61no—
215C2×C3⋊D3—61no—
216C2×C3×D3—22no—
217C2×C3×D3—41no—
218C2×C3×D3—41no—
219C32×C4—41no—
220C32×C4C2×D1211yescharacteristic
221C32×C4D3×D411yescharacteristic
222C3⋊Dic3—61no—
223C3⋊Dic3—121no—
224C3×Dic3—62no—
225C22×C9—21no—
226C22×C9—62no—
227C22×C9—121no—
228C22×C9—121no—
229C22×C9D3×D411yescharacteristic
230C2×D9—62no—
231C2×D9—61no—
232C2×D9—61no—
233C2×D9—61no—
234C2×D9—121no—
235C2×D9—121no—
236C4×C9—21no—
237C4×C9—21no—
238C4×C9—42no—
239C4×C9—41no—
240C4×C9C2×D1211yescharacteristic
241C4×C9D3×D411yescharacteristic
242Dic9—22no—
243Dic9—61no—
244Dic9—121no—
245Dic9—122no—
246Dic9—121no—
247C2×#41—271no—
248C2×Q16—271no—
249C2×SD16—271no—
250C22⋊Q8—271no—
251#103—91no—
252#102—31no—
253C22⋊C8—271no—
254C3×C9#45211yescharacteristic, complemented, Sylow
255C23×C3—61no—
256C22×D3—21no—
257C22×D3—91no—
258C22×D3—91no—
259C22×D3—181no—
260C3×Q8—92no—
261C3×Q8—181no—
262C3×D4—32no—
263C3×D4—61no—
264C2×C3×C4—22no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—181no—
269C2×C3×C4C22×D911yescharacteristic
270C2×C3×C4C2×D3211yescharacteristic
271#67—184no—
272C2×Dic3—61no—
273C2×Dic3—91no—
274C2×Dic3—91no—
275C2×Dic3—181no—
276C2×Dic3—184no—
277C2×Dic3—182no—
278C2×Dic3—181no—
279C2×Dic3—181no—
280D12—21no—
281D12—184no—
282D12D3612yes—
283C4×D3—184no—
284C4×D3—181no—
285C4×D3—181no—
286C4×D3—181no—
287C4×D3—181no—
288C4×D3—361no—
289C4×D3—361no—
290Dic6—32no—
291Dic6—61no—
292Dic6—181no—
293Dic6—181no—
294Dic6—181no—
295Dic6—181no—
296Dic6—361no—
297Dic6—361no—
298C3×C8—61no—
299C3×C8—181no—
300C3⋊C8—21no—
301C3⋊C8—61no—
302C2×C32#88211yescharacteristic
303C2×C32#91211yescharacteristic
304C2×C32#91511yescharacteristic
305C3⋊D3—121no—
306C3×D3—42no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—21no—
310C2×C9—122no—
311C2×C9#88211yescharacteristic
312C2×C9#91611yescharacteristic
313C2×C9#91811yescharacteristic
314D9—122no—
315D9—121no—
316#41—542no—
317#41—542no—
318C2×Q8—91no—
319C2×Q8—271no—
320C2×D4—31no—
321C2×D4—541no—
322C22×C4—271no—
323C22×C4—541no—
324Q16—542no—
325SD16—542no—
326C2×C8—31no—
327C2×C8—91no—
328C4⋊C4—541no—
329C4⋊C4D3×D911yescharacteristic
330C22⋊C4—541no—
331C22×C3—21no—
332C22×C3—62no—
333C22×C3—121no—
334C22×C3—121no—
335C22×C3D4×D911yescharacteristic
336C22×C3#381111yescharacteristic
337C2×D3—22no—
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—182no—
341C2×D3—181no—
342C2×D3—181no—
343C2×D3—181no—
344C2×D3—181no—
345C2×D3—181no—
346C2×D3—181no—
347C2×D3—361no—
348C2×D3—361no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—42no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4—182no—
355C3×C4C2×D3611yescharacteristic
356C3×C4D4×D911yescharacteristic
357C3×C4D3×D1211yescharacteristic
358C3×C4D3×D1211yescharacteristic
359Dic3—62no—
360Dic3—181no—
361Dic3—181no—
362Dic3—361no—
363Dic3—362no—
364Dic3—361no—
365Dic3—361no—
366C32#478311yescharacteristic
367C9—21no—
368C9#480811yescharacteristic, complemented
369C23—61no—
370C23—271no—
371Q8—92no—
372Q8—181no—
373Q8—541no—
374Q8—541no—
375D4—32no—
376D4—61no—
377D4—544no—
378C2×C4—21no—
379C2×C4—181no—
380C2×C4—271no—
381C2×C4—544no—
382C2×C4—541no—
383C2×C4—541no—
384C2×C4—541no—
385C2×C4C2×D3×D911yescharacteristic
386C8—61no—
387C8—181no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—122no—
392C2×C3#6344411yescharacteristic
393C2×C3#6347411yescharacteristic
394C2×C3#6347711yescharacteristic
395C2×C3#6392311yescharacteristic
396C2×C3#6393611yescharacteristic
397C2×C3#6395111yescharacteristic
398D3—42no—
399D3—362no—
400D3—361no—
401D3—361no—
402C22—62no—
403C22—121no—
404C22—121no—
405C22—541no—
406C22—541no—
407C22—541no—
408C22#8866811yescharacteristic
409C4—41no—
410C4—182no—
411C4—541no—
412C4—1081no—
413C4D3×D3611yescharacteristic
414C4D9×D1211yescharacteristic
415C3—21no—
416C3#1058846311yescharacteristic, complemented
417C3#1059061811yescharacteristic
418C2—122no—
419C2—1081no—
420C2#1172105111yescharacteristic
421C2#1172117811yescharacteristic
422C2#1172121711yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88639C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#452
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721753C2semidirect
3. C2×#88424C2semidirect
4. C2×#88418C2semidirect
5. C9×#809C2semidirect
6. C3×#63368C2semidirect
7. #11720282C2semidirect
39. #63368D3semidirect
123. #809D9semidirect
254. C3×C9#452semidirect
368. C9#4808semidirect
416. C3#10588463semidirect

Maximal subgroups

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

subgroupindex
C2×#886392
#117217532
C2×#884242
C2×#884182
C9×#8092
C3×#633682
#117202822
#105906183
#105884633