Finite Group Databasebeta

← #13139498

Subgroups — #13139498

Order 1728.

806 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C22×C4×C3⋊D9C211yesmaximal, characteristic
2C2×#88414C212yesmaximal, complemented
3C9×#830C211yesmaximal, characteristic, complemented
4C3×#63389C211yesmaximal, characteristic, complemented
5#11720282C212yesmaximal, complemented
6#10591159—31nomaximal
7#10588408—31nomaximal
8C23×C3⋊D9C411yescharacteristic
9C22×C3×C4×C9C2211yescharacteristic
10C22×C3⋊Dic9C411yescharacteristic
11C2×C4×C3⋊D9—24no—
12C2×C4×C3⋊D9C411yescharacteristic
13C2×C4×C3⋊D9C411yescharacteristic
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C9×C3⋊C8C2212yescomplemented
17C2×C3×C9⋊C8C2212yescomplemented
18#88414—24no—
19C22×C4×C3⋊D3—31no—
20C2×#3712—32no—
21C22×C4×D9—31no—
22C22×C4×D9—61no—
23C3×#830—31no—
24C3×#830D311yescharacteristic
25#63557—32no—
26C2×#3663—32no—
27C9×C22⋊C8—31no—
28#63389D311yescharacteristic, complemented
29(C2×D9)⋊C8—32no—
30C23×C3×C9C2×C411yescharacteristic
31C22×C3⋊D9—22no—
32C22×C3⋊D9—22no—
33C22×C3⋊D9—22no—
34C22×C3⋊D9C2×C411yescharacteristic
35C22×C3⋊D9C2×C411yescharacteristic
36C2×C3×C4×C9—22no—
37C2×C3×C4×C9C2×C411yescharacteristic
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—22no—
40C2×C3⋊Dic9C2×C411yescharacteristic
41C2×C3⋊Dic9C2×C411yescharacteristic
42C4×C3⋊D9—22no—
43C4×C3⋊D9—22no—
44C4×C3⋊D9—22no—
45C4×C3⋊D9D414yes—
46C9×C3⋊C8—22no—
47C3×C9⋊C8—22no—
48#4726—91no—
49#4726—91no—
50C23×C3⋊D3—31no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C3⋊Dic3—31no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—31no—
57C2×C4×C3⋊D3—64no—
58C23×D9—31no—
59C23×D9—61no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8C2×D312yes—
62#3712—64no—
63C22×C4×C9—21no—
64C22×C4×C9C2×D311yescharacteristic
65C22×Dic9—31no—
66C22×Dic9—61no—
67C2×C4×D9—31no—
68C2×C4×D9—31no—
69C2×C4×D9—31no—
70C2×C4×D9—31no—
71C2×C4×D9—64no—
72C2×C4×D9—61no—
73C2×C4×D9—61no—
74C2×C4×D9—61no—
75C2×C4×D9—61no—
76C2×C4×D9—68no—
77C2×C8×C9—32no—
78C2×C9⋊C8C2×D312yescomplemented
79#3663—64no—
80C22×C3×C9—21no—
81C22×C3×C9—21no—
82C22×C3×C9M1612yescomplemented
83C22×C3×C9C22×C411yescharacteristic
84C2×C3⋊D9—28no—
85C2×C3⋊D9—21no—
86C2×C3⋊D9—21no—
87C2×C3⋊D9—21no—
88C2×C3⋊D9—21no—
89C2×C3⋊D9C22⋊C412yes—
90C2×C3⋊D9M1612yescomplemented
91C3×C4×C9—21no—
92C3×C4×C9C2×D412yes—
93C3⋊Dic9—21no—
94C3⋊Dic9C22⋊C412yes—
95C22×C4×D3—91no—
96C22×C4×D3—91no—
97C22×C4×D3—181no—
98C2×ΓL(1,𝔽25)—92no—
99C2×ΓL(1,𝔽25)—92no—
100C3×C22⋊C8—31no—
101C3×C22⋊C8—91no—
102#830—31no—
103#830D911yescharacteristic, complemented
104(C2×D3)⋊C8—92no—
105(C2×D3)⋊C8—92no—
106C23×C32C4×D311yescharacteristic
107C22×C3⋊D3—31no—
108C22×C3⋊D3—31no—
109C22×C3⋊D3—62no—
110C22×C3⋊D3—62no—
111C22×C3⋊D3—62no—
112C2×C32×C4—22no—
113C2×C32×C4C4×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—62no—
118C4×C3⋊D3—34no—
119C4×C3⋊D3—62no—
120C4×C3⋊D3—62no—
121C4×C3⋊D3—62no—
122C23×C9—21no—
123C23×C9C4×D311yescharacteristic
124C22×D9—31no—
125C22×D9—31no—
126C22×D9—61no—
127C22×D9—61no—
128C22×D9—62no—
129C22×D9—62no—
130C22×D9—64no—
131C22×D9—64no—
132C22×D9—64no—
133C22×D9—62no—
134C3×C3⋊C8—22no—
135C3×C3⋊C8—62no—
136C2×C4×C9—24no—
137C2×C4×C9—22no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9C4×D311yescharacteristic
141C2×C4×C9C22×D311yescharacteristic
142C2×Dic9—31no—
143C2×Dic9—31no—
144C2×Dic9—61no—
145C2×Dic9—61no—
146C2×Dic9—62no—
147C2×Dic9—64no—
148C4×D9—34no—
149C4×D9—64no—
150C4×D9—62no—
151C4×D9—62no—
152C4×D9—64no—
153C4×D9—64no—
154C4×D9—64no—
155C4×D9—62no—
156C8×C9—62no—
157C9⋊C8—22no—
158#407—271noSylow
159C2×C3×C9—22no—
160C2×C3×C9C2×C22⋊C411yescharacteristic
161C2×C3×C9C2×M1611yescharacteristic
162C2×C3×C9C2×M1611yescharacteristic
163C3⋊D9—22no—
164C3⋊D9—22no—
165C23×D3—91no—
166C23×D3—91no—
167C23×D3—181no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D911yescharacteristic
170C22×C3×C4D3211yescharacteristic
171C22×Dic3—91no—
172C22×Dic3—91no—
173C22×Dic3—181no—
174C2×C4×D3—91no—
175C2×C4×D3—91no—
176C2×C4×D3—91no—
177C2×C4×D3—91no—
178C2×C4×D3—91no—
179C2×C4×D3—91no—
180C2×C4×D3—91no—
181C2×C4×D3—91no—
182C2×C4×D3—181no—
183C2×C4×D3—181no—
184C2×C4×D3—181no—
185C2×C4×D3—181no—
186C2×C4×D3—184no—
187C2×C4×D3—184no—
188C2×C4×D3—188no—
189C2×C3×C8—32no—
190C2×C3×C8—92no—
191C2×C3⋊C8—32no—
192C2×C3⋊C8C2×D912yescomplemented
193ΓL(1,𝔽25)—184no—
194ΓL(1,𝔽25)—184no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32ΓL(1,𝔽25)12yes—
198C22×C32C2×C4×D311yescharacteristic
199C2×C3⋊D3—32no—
200C2×C3⋊D3—32no—
201C2×C3⋊D3—68no—
202C2×C3⋊D3—61no—
203C2×C3⋊D3—61no—
204C2×C3⋊D3—61no—
205C2×C3⋊D3—61no—
206C32×C4—21no—
207C32×C4D3×D412yes—
208C3⋊Dic3—32no—
209C3⋊Dic3—61no—
210C22×C9—22no—
211C22×C9—22no—
212C22×C9—21no—
213C22×C9—21no—
214C22×C9—22no—
215C22×C9—21no—
216C22×C9ΓL(1,𝔽25)12yescomplemented
217C22×C9C2×C4×D311yescharacteristic
218C2×D9—32no—
219C2×D9—32no—
220C2×D9—62no—
221C2×D9—62no—
222C2×D9—62no—
223C2×D9—616no—
224C2×D9—68no—
225C2×D9—62no—
226C2×D9—62no—
227C2×D9—62no—
228C2×D9—61no—
229C2×D9—61no—
230C2×D9—61no—
231C2×D9—61no—
232C4×C9—22no—
233C4×C9—21no—
234C4×C9—22no—
235C4×C9D3×D412yes—
236Dic9—32no—
237Dic9—62no—
238Dic9—62no—
239Dic9—61no—
240C23×C4—271no—
241C2×M16—272no—
242C22⋊C8—31no—
243C22⋊C8—91no—
244C22⋊C8—272no—
245C3×C9#40711yescharacteristic, complemented, Sylow
246C23×C3—21no—
247C23×C3C4×D911yescharacteristic
248C23×C3#62411yescharacteristic
249C22×D3—91no—
250C22×D3—91no—
251C22×D3—91no—
252C22×D3—91no—
253C22×D3—181no—
254C22×D3—181no—
255C22×D3—182no—
256C22×D3—182no—
257C22×D3—182no—
258C22×D3—182no—
259C22×D3—184no—
260C22×D3—184no—
261C22×D3—184no—
262C22×D3—182no—
263C22×D3—182no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4C4×D911yescharacteristic
270C2×C3×C4C22×D911yescharacteristic
271C2×C3×C4#62411yescharacteristic
272C2×C3×C4C2×D3211yescharacteristic
273C2×Dic3—91no—
274C2×Dic3—91no—
275C2×Dic3—91no—
276C2×Dic3—91no—
277C2×Dic3—181no—
278C2×Dic3—181no—
279C2×Dic3—182no—
280C2×Dic3—182no—
281C2×Dic3—184no—
282C4×D3—94no—
283C4×D3—94no—
284C4×D3—184no—
285C4×D3—182no—
286C4×D3—182no—
287C4×D3—182no—
288C4×D3—182no—
289C4×D3—184no—
290C4×D3—184no—
291C4×D3—184no—
292C4×D3—182no—
293C4×D3—182no—
294C3×C8—62no—
295C3×C8—182no—
296C3⋊C8—22no—
297C3⋊C8—62no—
298C2×C32—22no—
299C2×C32D3×C22⋊C411yescharacteristic
300C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
301C2×C32D3×M1611yescharacteristic
302C3⋊D3—62no—
303C3⋊D3—62no—
304C2×C9—24no—
305C2×C9—22no—
306C2×C9—21no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9D3×C22⋊C411yescharacteristic
310C2×C9C2×ΓL(1,𝔽25)11yescharacteristic
311C2×C9D3×M1611yescharacteristic
312D9—64no—
313D9—64no—
314D9—62no—
315D9—62no—
316C24—271no—
317C22×C4—271no—
318C22×C4—271no—
319C22×C4—271no—
320C22×C4—271no—
321C22×C4—271no—
322C22×C4—544no—
323C22×C4D3×D911yescharacteristic
324M16—544no—
325C2×C8—32no—
326C2×C8—92no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—22no—
335C22×C3#366312yescomplemented
336C22×C3C2×C4×D911yescharacteristic
337C22×C3#371212yes—
338C22×C3C2×#62411yescharacteristic
339C2×D3—92no—
340C2×D3—92no—
341C2×D3—92no—
342C2×D3—92no—
343C2×D3—182no—
344C2×D3—182no—
345C2×D3—182no—
346C2×D3—188no—
347C2×D3—1816no—
348C2×D3—188no—
349C2×D3—182no—
350C2×D3—182no—
351C2×D3—182no—
352C2×D3—181no—
353C2×D3—181no—
354C2×D3—181no—
355C2×D3—181no—
356C2×D3—181no—
357C2×D3—181no—
358C2×D3—181no—
359C2×D3—181no—
360C3×C4—22no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—22no—
364C3×C4D4×D912yes—
365C3×C4#381112yes—
366Dic3—92no—
367Dic3—92no—
368Dic3—182no—
369Dic3—182no—
370Dic3—181no—
371Dic3—181no—
372C32#472611yescharacteristic
373C9—21no—
374C9#472611yescharacteristic, complemented
375C23—271no—
376C23—271no—
377C23—542no—
378C23—542no—
379C23—542no—
380C23(C3⋊D9)⋊C411yescharacteristic
381C2×C4—22no—
382C2×C4—271no—
383C2×C4—274no—
384C2×C4—271no—
385C2×C4—542no—
386C2×C4—542no—
387C2×C4—542no—
388C2×C4—542no—
389C2×C4(C3⋊D9)⋊C411yescharacteristic
390C2×C4C2×D3×D911yescharacteristic
391C8—62no—
392C8—182no—
393C2×C3—24no—
394C2×C3—22no—
395C2×C3—22no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3C22⋊C4×D911yescharacteristic
400C2×C3C2×#366311yescharacteristic
401C2×C3M16×D911yescharacteristic
402C2×C3#6381811yescharacteristic
403C2×C3C2×#371211yescharacteristic
404C2×C3#6397411yescharacteristic
405D3—184no—
406D3—184no—
407D3—182no—
408D3—182no—
409D3—182no—
410D3—182no—
411C22—21no—
412C22—21no—
413C22—272no—
414C22—272no—
415C22—548no—
416C22—541no—
417C22—541no—
418C22—541no—
419C22—541no—
420C22#8841412yescomplemented
421C22C2×(C3⋊D9)⋊C411yescharacteristic
422C4—21no—
423C4—272no—
424C4—541no—
425C4#8866812yes—
426C3—21no—
427C3#1058840811yescharacteristic, complemented
428C3#1059115911yescharacteristic
429C2—22no—
430C2—542no—
431C2—542no—
432C2#1172086311yescharacteristic
433C2C2×#8841411yescharacteristic
434C2#1172127411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×C4×D3
prime radicalC23×C3⋊D9C4
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#407
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88414C2semidirect
3. C9×#830C2semidirect
4. C3×#63389C2semidirect
5. #11720282C2semidirect
16. C2×C9×C3⋊C8C22semidirect
17. C2×C3×C9⋊C8C22semidirect
28. #63389D3semidirect
78. C2×C9⋊C8C2×D3semidirect
82. C22×C3×C9M16semidirect
90. C2×C3⋊D9M16semidirect
103. #830D9semidirect
192. C2×C3⋊C8C2×D9semidirect
216. C22×C9ΓL(1,𝔽25)semidirect
245. C3×C9#407semidirect
335. C22×C3#3663semidirect
374. C9#4726semidirect
420. C22#88414semidirect
427. C3#10588408semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C2×#884142
C9×#8302
C3×#633892
#117202822
#105911593
#105884083