Finite Group Databasebeta

← #13150023

Subgroups — #13150023

Order 1728.

938 conjugacy classes of subgroups.

The subgroup lattice has 448 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#11720863C214yesmaximal, complemented
5#11720862C214yesmaximal, complemented
6#11720861C214yesmaximal, complemented
7#10594780—31nomaximal
8#10589847—31nomaximal
9C22×C3×C4×C9C2211yescharacteristic
10C2×#6233C412yes—
11#88736C412yes—
12#88736C412yes—
13#88736C2212yes—
14#88736C2212yes—
15C2×C4⋊(C3⋊D9)C411yescharacteristic
16C2×C4×C3⋊D9C2212yes—
17C2×#6226C411yescharacteristic
18C9×#208C2212yes—
19C9×#208C2212yescomplemented
20C2×C9×C3⋊C8C2212yescomplemented
21C3×#3667C2212yes—
22C3×#3667C2212yescomplemented
23C2×C3×C9⋊C8C2212yescomplemented
24#88414C2218yescomplemented
25(C3⋊D9)⋊C8C2218yescomplemented
26C2×#3828—31no—
27C2×C3×#208—31no—
28C2×C3×#208D311yescharacteristic
29#63818—34no—
30#63817—34no—
31#63817—34no—
32C2×#3697—31no—
33C2×#3697—61no—
34C2×C9×M16—31no—
35C2×#3667D311yescharacteristic, complemented
36#63469—34no—
37M16×D9—34no—
38#63464—34no—
39C23×C3×C9C2×C411yescharacteristic
40C22×C3⋊D9C2×C412yes—
41C2×C3×C4×C9C2×C411yescharacteristic
42C2×C3×C4×C9C2×C411yescharacteristic
43C2×C3×C4×C9C2×C411yescharacteristic
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47#6233C2×C414yes—
48#6233C2×C414yes—
49C2×C3⋊Dic9C2×C412yes—
50C4⋊(C3⋊D9)C2×C412yes—
51C4⋊(C3⋊D9)C2×C412yes—
52C4×C3⋊D9C2×C414yes—
53C4×C3⋊D9C2314yes—
54#6226C2×C412yes—
55#6226C2×C412yes—
56C9×C3⋊C8C2314yes—
57C3×C9⋊C8C2314yes—
58#5745—91no—
59#5745—91no—
60C22×C32×C4C2×D311yescharacteristic
61C2×#638—32no—
62#3828—32no—
63#3828—32no—
64#3828—32no—
65#3828—32no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C4×C3⋊D3—32no—
68C2×#634—31no—
69C3×#208—32no—
70C3×#208—32no—
71C3×#208C2×D312yes—
72C3×#208C2×D312yes—
73C2×C3×C3⋊C8—32no—
74C2×C3×C3⋊C8C2×D312yes—
75#3712—38no—
76#3710—38no—
77C22×C4×C9—21no—
78C22×C4×C9C2×D311yescharacteristic
79C2×#611—32no—
80C2×#611—62no—
81#3697—32no—
82#3697—32no—
83#3697—32no—
84#3697—32no—
85#3697—62no—
86#3697—62no—
87#3697—62no—
88#3697—62no—
89C2×D36—31no—
90C2×D36—61no—
91C2×C4×D9—32no—
92C2×C4×D9—62no—
93C2×Dic18—31no—
94C2×Dic18—61no—
95C9×M16—32no—
96C9×M16—32no—
97C2×C8×C9—32no—
98#3667C2×D312yes—
99#3667C2×D312yescomplemented
100C2×C9⋊C8C2×D312yescomplemented
101#3663—38no—
102C8×D9—38no—
103C22×C3×C9—41no—
104C22×C3×C9C22×C411yescharacteristic
105C22×C3×C9C22×C411yescharacteristic
106C22×C3×C9C22×C411yescharacteristic
107C2×C3⋊D9—42no—
108C2×C3⋊D9C22×C414yes—
109C3×C4×C9C22×C411yescharacteristic
110C3×C4×C9C22×C411yescharacteristic
111C3×C4×C9C22×C411yescharacteristic
112C3×C4×C9C2411yescharacteristic
113C3⋊Dic9C22×C414yes—
114C2×#235—91no—
115C2×#235—91no—
116C2×#235—181no—
117C2×C3×M16—31no—
118C2×C3×M16—91no—
119C2×#208—31no—
120C2×#208D911yescharacteristic, complemented
121#907—94no—
122#907—94no—
123D3×M16—94no—
124D3×M16—94no—
125#901—94no—
126#901—94no—
127C23×C32C4×D311yescharacteristic
128C22×C3⋊D3—32no—
129C2×C32×C4C4×D311yescharacteristic
130C2×C32×C4C4×D311yescharacteristic
131C2×C32×C4C4×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135#638—34no—
136#638—34no—
137C2×C3⋊Dic3—32no—
138C4⋊(C3⋊D3)—32no—
139C4⋊(C3⋊D3)—32no—
140C4×C3⋊D3—34no—
141C4×C3⋊D3—34no—
142#634—32no—
143#634—32no—
144C23×C9—21no—
145C23×C9C4×D311yescharacteristic
146C22×D9—32no—
147C22×D9—62no—
148C3×C3⋊C8—34no—
149C3×C3⋊C8C22×D314yes—
150C2×C4×C9—21no—
151C2×C4×C9—21no—
152C2×C4×C9—21no—
153C2×C4×C9—21no—
154C2×C4×C9—21no—
155C2×C4×C9—21no—
156C2×C4×C9C4×D311yescharacteristic
157C2×C4×C9C4×D311yescharacteristic
158C2×C4×C9C4×D311yescharacteristic
159C2×C4×C9C22×D311yescharacteristic
160C2×C4×C9C22×D311yescharacteristic
161C2×C4×C9C22×D311yescharacteristic
162#611—34no—
163#611—34no—
164#611—64no—
165#611—64no—
166C2×Dic9—32no—
167C2×Dic9—62no—
168D36—32no—
169D36—32no—
170D36—62no—
171D36—62no—
172C4×D9—34no—
173C4×D9—34no—
174C4×D9—64no—
175C4×D9—64no—
176Dic18—32no—
177Dic18—32no—
178Dic18—62no—
179Dic18—62no—
180C8×C9—34no—
181C9⋊C8C22×D314yes—
182#568—271noSylow
183C2×C3×C9—21no—
184C2×C3×C9—21no—
185C2×C3×C9—21no—
186C2×C3×C9C23×C411yescharacteristic
187C3⋊D9—24no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D911yescharacteristic
190C22×C3×C4D3211yescharacteristic
191C2×#67—92no—
192C2×#67—92no—
193C2×#67—182no—
194#235—92no—
195#235—92no—
196#235—92no—
197#235—92no—
198#235—92no—
199#235—92no—
200#235—92no—
201#235—92no—
202#235—182no—
203#235—182no—
204#235—182no—
205#235—182no—
206C2×D12—91no—
207C2×D12—91no—
208C2×D12—181no—
209C2×C4×D3—92no—
210C2×C4×D3—92no—
211C2×C4×D3—182no—
212C2×Dic6—91no—
213C2×Dic6—91no—
214C2×Dic6—181no—
215C3×M16—32no—
216C3×M16—32no—
217C3×M16—92no—
218C3×M16—92no—
219C2×C3×C8—32no—
220C2×C3×C8—92no—
221#208—32no—
222#208—32no—
223#208C2×D912yes—
224#208C2×D912yescomplemented
225C2×C3⋊C8—32no—
226C2×C3⋊C8C2×D912yescomplemented
227ΓL(1,𝔽25)—98no—
228ΓL(1,𝔽25)—98no—
229D3×C8—98no—
230D3×C8—98no—
231C22×C32—41no—
232C22×C32C2×C4×D311yescharacteristic
233C22×C32C2×C4×D311yescharacteristic
234C22×C32C2×C4×D311yescharacteristic
235C2×C3⋊D3—34no—
236C2×C3⋊D3—122no—
237C32×C4C2×C4×D311yescharacteristic
238C32×C4C2×C4×D311yescharacteristic
239C32×C4C2×C4×D311yescharacteristic
240C32×C4C23×D311yescharacteristic
241C3⋊Dic3—34no—
242C22×C9—21no—
243C22×C9—21no—
244C22×C9—21no—
245C22×C9—41no—
246C22×C9—42no—
247C22×C9C2×C4×D311yescharacteristic
248C22×C9C2×C4×D311yescharacteristic
249C22×C9C2×C4×D311yescharacteristic
250C2×D9—34no—
251C2×D9—64no—
252C2×D9—122no—
253C2×D9—124no—
254C4×C9—21no—
255C4×C9—21no—
256C4×C9—21no—
257C4×C9—21no—
258C4×C9C2×C4×D311yescharacteristic
259C4×C9C2×C4×D311yescharacteristic
260C4×C9C2×C4×D311yescharacteristic
261C4×C9C23×D311yescharacteristic
262Dic9—34no—
263Dic9—64no—
264C2×#41—271no—
265#131—274no—
266#131—274no—
267C2×M16—31no—
268C2×M16—91no—
269C2×M16—274no—
270C3×C9#56811yescharacteristic, complemented, Sylow
271C23×C3—21no—
272C23×C3C4×D911yescharacteristic
273C23×C3#62411yescharacteristic
274C22×D3—92no—
275C22×D3—92no—
276C22×D3—182no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4C4×D911yescharacteristic
284C2×C3×C4C4×D911yescharacteristic
285C2×C3×C4C4×D911yescharacteristic
286C2×C3×C4C22×D911yescharacteristic
287C2×C3×C4C22×D911yescharacteristic
288C2×C3×C4C22×D911yescharacteristic
289C2×C3×C4#62411yescharacteristic
290C2×C3×C4#62411yescharacteristic
291C2×C3×C4#62411yescharacteristic
292C2×C3×C4C2×D3211yescharacteristic
293C2×C3×C4C2×D3211yescharacteristic
294C2×C3×C4C2×D3211yescharacteristic
295#67—94no—
296#67—94no—
297#67—94no—
298#67—94no—
299#67—184no—
300#67—184no—
301C2×Dic3—92no—
302C2×Dic3—92no—
303C2×Dic3—182no—
304D12—92no—
305D12—92no—
306D12—92no—
307D12—92no—
308D12—182no—
309D12—182no—
310C4×D3—94no—
311C4×D3—94no—
312C4×D3—94no—
313C4×D3—94no—
314C4×D3—184no—
315C4×D3—184no—
316Dic6—92no—
317Dic6—92no—
318Dic6—92no—
319Dic6—92no—
320Dic6—182no—
321Dic6—182no—
322C3×C8—34no—
323C3×C8—94no—
324C3⋊C8—34no—
325C3⋊C8C22×D914yes—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32C22×C4×D311yescharacteristic
330C3⋊D3—64no—
331C2×C9—22no—
332C2×C9—21no—
333C2×C9—21no—
334C2×C9—21no—
335C2×C9—21no—
336C2×C9—41no—
337C2×C9—41no—
338C2×C9C22×C4×D311yescharacteristic
339D9—64no—
340D9—124no—
341#41—272no—
342#41—272no—
343#41—272no—
344#41—272no—
345C2×Q8—271no—
346C2×D4—272no—
347C2×D4—271no—
348C22×C4—272no—
349C22×C4D3×D911yescharacteristic
350M16—32no—
351M16—32no—
352M16—92no—
353M16—92no—
354M16—278no—
355C2×C8—32no—
356C2×C8—92no—
357C2×C8—278no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—42no—
364C22×C3C2×C4×D911yescharacteristic
365C22×C3C2×C4×D911yescharacteristic
366C22×C3C2×C4×D911yescharacteristic
367C22×C3C2×#62411yescharacteristic
368C22×C3C2×#62411yescharacteristic
369C22×C3C2×#62411yescharacteristic
370C2×D3—94no—
371C2×D3—94no—
372C2×D3—184no—
373C2×D3—362no—
374C2×D3—362no—
375C2×D3—364no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4C2×C4×D911yescharacteristic
381C3×C4C2×C4×D911yescharacteristic
382C3×C4C2×C4×D911yescharacteristic
383C3×C4C23×D911yescharacteristic
384C3×C4C2×#62411yescharacteristic
385C3×C4C2×#62411yescharacteristic
386C3×C4C2×#62411yescharacteristic
387C3×C4C22×D3211yescharacteristic
388Dic3—94no—
389Dic3—94no—
390Dic3—184no—
391C32#574511yescharacteristic
392C9—21no—
393C9#574511yescharacteristic, complemented
394C23—272no—
395C23(C3⋊D9)⋊C411yescharacteristic
396Q8—272no—
397Q8—272no—
398D4—274no—
399D4—274no—
400D4—272no—
401D4—272no—
402C2×C4—272no—
403C2×C4—274no—
404C2×C4—274no—
405C2×C4(C3⋊D9)⋊C411yescharacteristic
406C2×C4(C3⋊D9)⋊C411yescharacteristic
407C2×C4(C3⋊D9)⋊C411yescharacteristic
408C2×C4C2×D3×D911yescharacteristic
409C2×C4C2×D3×D911yescharacteristic
410C2×C4C2×D3×D911yescharacteristic
411C8—34no—
412C8—94no—
413C2×C3—22no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—41no—
422C2×C3—41no—
423C2×C3C22×C4×D911yescharacteristic
424C2×C3C22×#62411yescharacteristic
425D3—184no—
426D3—184no—
427D3—364no—
428C22—41no—
429C22—274no—
430C22—1082no—
431C22C2×(C3⋊D9)⋊C411yescharacteristic
432C22C2×(C3⋊D9)⋊C411yescharacteristic
433C22C2×(C3⋊D9)⋊C411yescharacteristic
434C4—274no—
435C4C2×(C3⋊D9)⋊C411yescharacteristic
436C4C2×(C3⋊D9)⋊C411yescharacteristic
437C4C2×(C3⋊D9)⋊C411yescharacteristic
438C4C22×D3×D911yescharacteristic
439C3—21no—
440C3#1058984711yescharacteristic, complemented
441C3#1059478011yescharacteristic
442C2—21no—
443C2—21no—
444C2—21no—
445C2—544no—
446C2C22×(C3⋊D9)⋊C411yescharacteristic
subgroupG/H
centerC4C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC3×C4C22×D32
FittingC22×C3×C4×C9C22
socleC2×C32C22×C4×D3
prime radicalC2×#88736C2
cosocleC3×C4×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9#568
LCSG ▷ 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. #11720863C2semidirect
5. #11720862C2semidirect
6. #11720861C2semidirect
19. C9×#208C22semidirect
20. C2×C9×C3⋊C8C22semidirect
22. C3×#3667C22semidirect
23. C2×C3×C9⋊C8C22semidirect
24. #88414C22semidirect
25. (C3⋊D9)⋊C8C22semidirect
35. C2×#3667D3semidirect
99. #3667C2×D3semidirect
100. C2×C9⋊C8C2×D3semidirect
120. C2×#208D9semidirect
224. #208C2×D9semidirect
226. C2×C3⋊C8C2×D9semidirect
270. C3×C9#568semidirect
393. C9#5745semidirect
440. C3#10589847semidirect

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
#117208632
#117208622
#117208612
#105947803
#105898473