Finite Group Databasebeta

← #13139415

Subgroups — #13139415

Order 1728.

614 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic
2C2×C9×#208C211yesmaximal, characteristic, complemented
3C2×C3×#3667C211yesmaximal, characteristic, complemented
4#11720290C212yesmaximal, complemented
5#11720288C212yesmaximal, complemented
6#10591078—31nomaximal
7#10588814—31nomaximal
8C22×C3×C4×C9C2211yescharacteristic
9C2×#6233C412yes—
10#88736—21no—
11#88736—21no—
12#88736—21no—
13#88736—21no—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C412yes—
16C2×#6226C2211yescharacteristic
17C9×#208—21no—
18C9×#208C2212yes—
19C2×C9×C3⋊C8—21no—
20C3×#3667—21no—
21C3×#3667C2212yes—
22C2×C3×C9⋊C8—21no—
23C2×#3828—31no—
24C2×C3×#208—31no—
25C2×C3×#208D311yescharacteristic
26C2×#3697—31no—
27C2×#3697—61no—
28#63561—32no—
29#63559—32no—
30C2×C9×M16—31no—
31C2×#3667D311yescharacteristic, complemented
32#63383—32no—
33#63382—32no—
34C23×C3×C9C2×C411yescharacteristic
35C22×C3⋊D9C2×C412yes—
36C2×C3×C4×C9C2×C411yescharacteristic
37C2×C3×C4×C9D411yescharacteristic
38C2×C3×C4×C9D411yescharacteristic
39C2×C3×C4×C9D411yescharacteristic
40C2×C3×C4×C9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42#6233—22no—
43#6233—22no—
44C2×C3⋊Dic9C2×C412yes—
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—21no—
47C4×C3⋊D9—22no—
48C4×C3⋊D9—22no—
49#6226—21no—
50#6226—21no—
51C9×C3⋊C8—22no—
52C3×C9⋊C8—22no—
53#5132—91no—
54#5132—91no—
55C22×C32×C4C2×D311yescharacteristic
56C2×#638—32no—
57#3828—61no—
58#3828—61no—
59#3828—61no—
60#3828—61no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×C4×C3⋊D3—32no—
63C2×#634—31no—
64C3×#208—21no—
65C3×#208—32no—
66C3×#208—61no—
67C3×#208C2×D312yes—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8—61no—
70C22×C4×C9—21no—
71C22×C4×C9C2×D311yescharacteristic
72C2×#611—32no—
73C2×#611—62no—
74#3697—62no—
75#3697—62no—
76#3697—62no—
77#3697—62no—
78#3697—61no—
79#3697—61no—
80#3697—61no—
81#3697—61no—
82C2×D36—31no—
83C2×D36—61no—
84C2×C4×D9—32no—
85C2×C4×D9—62no—
86C2×Dic18—31no—
87C2×Dic18—61no—
88C9×M16—32no—
89C9×M16—61no—
90C2×C8×C9—61no—
91#3667—21no—
92#3667C2×D312yes—
93C2×C9⋊C8—21no—
94C22×C3×C9—41no—
95C22×C3×C9C22⋊C411yescharacteristic
96C22×C3×C9C22⋊C411yescharacteristic
97C22×C3×C9C22×C411yescharacteristic
98C2×C3⋊D9—22no—
99C2×C3⋊D9—42no—
100C3×C4×C9C22⋊C411yescharacteristic
101C3×C4×C9C22⋊C411yescharacteristic
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—22no—
105C2×#235—91no—
106C2×#235—91no—
107C2×#235—181no—
108C2×C3×M16—31no—
109C2×C3×M16—91no—
110C2×#208—31no—
111C2×#208D911yescharacteristic, complemented
112#824—92no—
113#824—92no—
114#823—92no—
115#823—92no—
116C23×C32C4×D311yescharacteristic
117C22×C3⋊D3—32no—
118C2×C32×C4C4×D311yescharacteristic
119C2×C32×C4D1211yescharacteristic
120C2×C32×C4D1211yescharacteristic
121C2×C32×C4#6711yescharacteristic
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124#638—62no—
125#638—62no—
126C2×C3⋊Dic3—32no—
127C4⋊(C3⋊D3)—61no—
128C4⋊(C3⋊D3)—61no—
129C4×C3⋊D3—62no—
130C4×C3⋊D3—62no—
131#634—61no—
132#634—61no—
133C23×C9—21no—
134C23×C9C4×D311yescharacteristic
135C22×D9—32no—
136C22×D9—62no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—62no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9C4×D311yescharacteristic
146C2×C4×C9D1211yescharacteristic
147C2×C4×C9D1211yescharacteristic
148C2×C4×C9#6711yescharacteristic
149C2×C4×C9#6711yescharacteristic
150C2×C4×C9C22×D311yescharacteristic
151#611—64no—
152#611—64no—
153#611—62no—
154#611—62no—
155C2×Dic9—32no—
156C2×Dic9—62no—
157D36—62no—
158D36—62no—
159D36—61no—
160D36—61no—
161C4×D9—64no—
162C4×D9—64no—
163C4×D9—62no—
164C4×D9—62no—
165Dic18—62no—
166Dic18—62no—
167Dic18—61no—
168Dic18—61no—
169C8×C9—62no—
170C9⋊C8—22no—
171#413—271noSylow
172C2×C3×C9—21no—
173C2×C3×C9—21no—
174C2×C3×C9—21no—
175C2×C3×C9C2×C22⋊C411yescharacteristic
176C3⋊D9—42no—
177C22×C3×C4—21no—
178C22×C3×C4C2×D911yescharacteristic
179C22×C3×C4D3211yescharacteristic
180C2×#67—92no—
181C2×#67—92no—
182C2×#67—182no—
183#235—182no—
184#235—182no—
185#235—182no—
186#235—182no—
187#235—181no—
188#235—181no—
189#235—181no—
190#235—181no—
191#235—181no—
192#235—181no—
193#235—181no—
194#235—181no—
195C2×D12—91no—
196C2×D12—91no—
197C2×D12—181no—
198C2×C4×D3—92no—
199C2×C4×D3—92no—
200C2×C4×D3—182no—
201C2×Dic6—91no—
202C2×Dic6—91no—
203C2×Dic6—181no—
204C3×M16—32no—
205C3×M16—61no—
206C3×M16—92no—
207C3×M16—181no—
208C2×C3×C8—61no—
209C2×C3×C8—181no—
210#208—21no—
211#208—32no—
212#208—61no—
213#208C2×D912yes—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—61no—
216C22×C32—41no—
217C22×C32(C2×D3)⋊C411yescharacteristic
218C22×C32(C2×D3)⋊C411yescharacteristic
219C22×C32C2×C4×D311yescharacteristic
220C2×C3⋊D3—62no—
221C2×C3⋊D3—122no—
222C32×C4(C2×D3)⋊C411yescharacteristic
223C32×C4(C2×D3)⋊C411yescharacteristic
224C32×C4C2×D1211yescharacteristic
225C32×C4C2×#6711yescharacteristic
226C3⋊Dic3—62no—
227C22×C9—21no—
228C22×C9—21no—
229C22×C9—21no—
230C22×C9—41no—
231C22×C9—42no—
232C22×C9(C2×D3)⋊C411yescharacteristic
233C22×C9(C2×D3)⋊C411yescharacteristic
234C22×C9C2×C4×D311yescharacteristic
235C2×D9—64no—
236C2×D9—62no—
237C2×D9—124no—
238C2×D9—122no—
239C4×C9—21no—
240C4×C9—21no—
241C4×C9—21no—
242C4×C9—21no—
243C4×C9(C2×D3)⋊C411yescharacteristic
244C4×C9(C2×D3)⋊C411yescharacteristic
245C4×C9C2×D1211yescharacteristic
246C4×C9C2×#6711yescharacteristic
247Dic9—64no—
248Dic9—62no—
249C2×#41—271no—
250C2×M16—31no—
251C2×M16—91no—
252#101—272no—
253#100—272no—
254C3×C9#41311yescharacteristic, complemented, Sylow
255C23×C3—21no—
256C23×C3C4×D911yescharacteristic
257C23×C3#62411yescharacteristic
258C22×D3—92no—
259C22×D3—92no—
260C22×D3—182no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4C4×D911yescharacteristic
268C2×C3×C4D3611yescharacteristic
269C2×C3×C4D3611yescharacteristic
270C2×C3×C4#61111yescharacteristic
271C2×C3×C4#61111yescharacteristic
272C2×C3×C4C22×D911yescharacteristic
273C2×C3×C4#62411yescharacteristic
274C2×C3×C4#62611yescharacteristic
275C2×C3×C4#62611yescharacteristic
276C2×C3×C4#62611yescharacteristic
277C2×C3×C4#62611yescharacteristic
278C2×C3×C4C2×D3211yescharacteristic
279#67—184no—
280#67—184no—
281#67—182no—
282#67—182no—
283#67—182no—
284#67—182no—
285C2×Dic3—92no—
286C2×Dic3—92no—
287C2×Dic3—182no—
288D12—182no—
289D12—182no—
290D12—181no—
291D12—181no—
292D12—181no—
293D12—181no—
294C4×D3—184no—
295C4×D3—184no—
296C4×D3—182no—
297C4×D3—182no—
298C4×D3—182no—
299C4×D3—182no—
300Dic6—182no—
301Dic6—182no—
302Dic6—181no—
303Dic6—181no—
304Dic6—181no—
305Dic6—181no—
306C3×C8—62no—
307C3×C8—182no—
308C3⋊C8—22no—
309C3⋊C8—62no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32C2×(C2×D3)⋊C411yescharacteristic
314C3⋊D3—122no—
315C2×C9—22no—
316C2×C9—21no—
317C2×C9—21no—
318C2×C9—21no—
319C2×C9—21no—
320C2×C9—41no—
321C2×C9—41no—
322C2×C9C2×(C2×D3)⋊C411yescharacteristic
323D9—124no—
324D9—122no—
325#41—541no—
326#41—541no—
327#41—541no—
328#41—541no—
329C2×Q8—271no—
330C2×D4—272no—
331C2×D4—271no—
332C22×C4—272no—
333C22×C4D3×D911yescharacteristic
334M16—32no—
335M16—61no—
336M16—92no—
337M16—181no—
338C2×C8—61no—
339C2×C8—181no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—42no—
346C22×C3(C2×D9)⋊C411yescharacteristic
347C22×C3(C2×D9)⋊C411yescharacteristic
348C22×C3C2×C4×D911yescharacteristic
349C22×C3#372211yescharacteristic
350C22×C3#372211yescharacteristic
351C22×C3C2×#62411yescharacteristic
352C2×D3—184no—
353C2×D3—182no—
354C2×D3—182no—
355C2×D3—364no—
356C2×D3—362no—
357C2×D3—362no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4(C2×D9)⋊C411yescharacteristic
363C3×C4(C2×D9)⋊C411yescharacteristic
364C3×C4C2×D3611yescharacteristic
365C3×C4C2×#61111yescharacteristic
366C3×C4#372211yescharacteristic
367C3×C4#372211yescharacteristic
368C3×C4C2×#62611yescharacteristic
369C3×C4C2×#62611yescharacteristic
370Dic3—184no—
371Dic3—182no—
372Dic3—182no—
373C32#513211yescharacteristic
374C9—21no—
375C9#513211yescharacteristic, complemented
376C23—272no—
377C23(C3⋊D9)⋊C411yescharacteristic
378Q8—541no—
379Q8—541no—
380D4—542no—
381D4—542no—
382D4—541no—
383D4—541no—
384C2×C4—272no—
385C2×C4—542no—
386C2×C4—542no—
387C2×C4(C3⋊D9)⋊C411yescharacteristic
388C2×C4#619211yescharacteristic
389C2×C4#619211yescharacteristic
390C2×C4#619511yescharacteristic
391C2×C4#619511yescharacteristic
392C2×C4C2×D3×D911yescharacteristic
393C8—62no—
394C8—182no—
395C2×C3—22no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3C2×(C2×D9)⋊C411yescharacteristic
406C2×C3C2×#372211yescharacteristic
407D3—364no—
408D3—362no—
409D3—362no—
410C22—41no—
411C22—542no—
412C22—1082no—
413C22#8844511yescharacteristic
414C22#8844511yescharacteristic
415C22C2×(C3⋊D9)⋊C411yescharacteristic
416C4—542no—
417C4#8844511yescharacteristic
418C4#8844511yescharacteristic
419C4C2×#619211yescharacteristic
420C4C2×#619511yescharacteristic
421C3—21no—
422C3#1058881411yescharacteristic, complemented
423C3#1059107811yescharacteristic
424C2—21no—
425C2—21no—
426C2—21no—
427C2—1082no—
428C2C2×#8844511yescharacteristic
subgroupG/H
centerC4#88445
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC2×C32C2×(C2×D3)⋊C4
prime radicalC2×#88736C2
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#413
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C9×#208C2semidirect
3. C2×C3×#3667C2semidirect
4. #11720290C2semidirect
5. #11720288C2semidirect
31. C2×#3667D3semidirect
111. C2×#208D9semidirect
254. C3×C9#413semidirect
375. C9#5132semidirect
422. C3#10588814semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×C9×#2082
C2×C3×#36672
#117202902
#117202882
#105910783
#105888143