Finite Group Databasebeta

← #13176866

Subgroups — #13176866

Order 1728.

2204 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×#89076C211yesmaximal, characteristic
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C2×C32×#208C211yesmaximal, characteristic, complemented
4#11723358C214yesmaximal, complemented
5#11723357C214yesmaximal, complemented
6#11723356C214yesmaximal, complemented
7#10596216—31nomaximal
8#10594780—34nomaximal
9C22×C33×C4C2211yescharacteristic
10C2×#6312C412yes—
11#89076C412yes—
12#89076C412yes—
13#89076C2212yes—
14#89076C2212yes—
15C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2212yes—
17C2×#6308C411yescharacteristic
18C3×#3748C2212yes—
19C3×#3748C2212yescomplemented
20C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
21C32×#208C2212yes—
22C32×#208C2212yescomplemented
23C2×C32×C3⋊C8C2212yescomplemented
24#88788C2218yescomplemented
25#88785C2218yescomplemented
26C2×#3828—31no—
27C2×#3828—34no—
28C2×#3828—64no—
29C2×C32×M16—31no—
30C2×#3748D311yescharacteristic, complemented
31#64116—34no—
32M16×C3⋊D3—34no—
33#64110—34no—
34C2×C3×#208—34no—
35C2×C3×#208D314yescomplemented
36#63818—316no—
37#63817—316no—
38#63817—316no—
39C23×C33C2×C411yescharacteristic
40C22×C3⋊(C3⋊D3)C2×C412yes—
41C2×C33×C4C2×C411yescharacteristic
42C2×C33×C4C2×C411yescharacteristic
43C2×C33×C4C2×C411yescharacteristic
44C2×C33×C4C2311yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47#6312C2×C414yes—
48#6312C2×C414yes—
49C2×C3⋊(C3⋊Dic3)C2×C412yes—
50C4⋊(C3⋊(C3⋊D3))C2×C412yes—
51C4⋊(C3⋊(C3⋊D3))C2×C412yes—
52C4×C3⋊(C3⋊D3)C2×C414yes—
53C4×C3⋊(C3⋊D3)C2314yes—
54#6308C2×C412yes—
55#6308C2×C412yes—
56C3×C3⋊(C3⋊C8)C2314yes—
57C32×C3⋊C8C2314yes—
58#5745—94no—
59#5745—91no—
60C22×C32×C4—24no—
61C22×C32×C4C2×D314yes—
62C22×C32×C4C2×D311yescharacteristic
63C2×#638—32no—
64C2×#638—38no—
65C2×#638—68no—
66#3828—32no—
67#3828—32no—
68#3828—38no—
69#3828—38no—
70#3828—32no—
71#3828—32no—
72#3828—38no—
73#3828—38no—
74#3828—68no—
75#3828—68no—
76#3828—68no—
77#3828—68no—
78C2×C4⋊(C3⋊D3)—31no—
79C2×C4⋊(C3⋊D3)—34no—
80C2×C4⋊(C3⋊D3)—64no—
81C2×C4×C3⋊D3—32no—
82C2×C4×C3⋊D3—38no—
83C2×C4×C3⋊D3—68no—
84C2×#634—31no—
85C2×#634—34no—
86C2×#634—64no—
87C32×M16—32no—
88C32×M16—32no—
89C2×C32×C8—32no—
90#3748C2×D312yes—
91#3748C2×D312yescomplemented
92C2×C3⋊(C3⋊C8)C2×D312yescomplemented
93#3743—38no—
94C8×C3⋊D3—38no—
95C3×#208—38no—
96C3×#208—38no—
97C3×#208C2×D318yes—
98C3×#208C2×D318yescomplemented
99C2×C3×C3⋊C8—38no—
100C2×C3×C3⋊C8C2×D318yescomplemented
101#3712—332no—
102#3710—332no—
103C22×C33—41no—
104C22×C33C22×C411yescharacteristic
105C22×C33C22×C411yescharacteristic
106C22×C33C22×C411yescharacteristic
107C2×C3⋊(C3⋊D3)—42no—
108C2×C3⋊(C3⋊D3)C22×C414yes—
109C33×C4C22×C411yescharacteristic
110C33×C4C22×C411yescharacteristic
111C33×C4C22×C411yescharacteristic
112C33×C4C2411yescharacteristic
113C3⋊(C3⋊Dic3)C22×C414yes—
114C2×#235—94no—
115C2×#235—91no—
116C2×#235—184no—
117C2×C3×M16—34no—
118C2×C3×M16—91no—
119C2×#208—34no—
120C2×#208C3⋊D311yescharacteristic, complemented
121#907—916no—
122#907—94no—
123D3×M16—916no—
124D3×M16—94no—
125#901—916no—
126#901—94no—
127C23×C32—24no—
128C23×C32C4×D314yes—
129C23×C32C4×D311yescharacteristic
130C22×C3⋊D3—32no—
131C22×C3⋊D3—38no—
132C22×C3⋊D3—68no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4C4×D314yes—
140C2×C32×C4C4×D314yes—
141C2×C32×C4C4×D314yes—
142C2×C32×C4C4×D311yescharacteristic
143C2×C32×C4C4×D311yescharacteristic
144C2×C32×C4C4×D311yescharacteristic
145C2×C32×C4C22×D314yes—
146C2×C32×C4C22×D314yes—
147C2×C32×C4C22×D314yes—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151#638—34no—
152#638—34no—
153#638—316no—
154#638—316no—
155#638—616no—
156#638—616no—
157C2×C3⋊Dic3—32no—
158C2×C3⋊Dic3—38no—
159C2×C3⋊Dic3—68no—
160C4⋊(C3⋊D3)—32no—
161C4⋊(C3⋊D3)—32no—
162C4⋊(C3⋊D3)—38no—
163C4⋊(C3⋊D3)—38no—
164C4⋊(C3⋊D3)—68no—
165C4⋊(C3⋊D3)—68no—
166C4×C3⋊D3—34no—
167C4×C3⋊D3—34no—
168C4×C3⋊D3—316no—
169C4×C3⋊D3—316no—
170C4×C3⋊D3—616no—
171C4×C3⋊D3—616no—
172#634—32no—
173#634—32no—
174#634—38no—
175#634—38no—
176#634—68no—
177#634—68no—
178C32×C8—34no—
179C3⋊(C3⋊C8)C22×D314yes—
180C3×C3⋊C8—316no—
181C3×C3⋊C8C22×D3116yes—
182#568—271noSylow
183C2×C33—21no—
184C2×C33—21no—
185C2×C33—21no—
186C2×C33C23×C411yescharacteristic
187C3⋊(C3⋊D3)—24no—
188C22×C3×C4—24no—
189C22×C3×C4D3214yes—
190C22×C3×C4C2×C3⋊D311yescharacteristic
191C2×#67—98no—
192C2×#67—92no—
193C2×#67—188no—
194#235—98no—
195#235—98no—
196#235—92no—
197#235—92no—
198#235—98no—
199#235—98no—
200#235—92no—
201#235—92no—
202#235—188no—
203#235—188no—
204#235—188no—
205#235—188no—
206C2×D12—94no—
207C2×D12—91no—
208C2×D12—184no—
209C2×C4×D3—98no—
210C2×C4×D3—92no—
211C2×C4×D3—188no—
212C2×Dic6—94no—
213C2×Dic6—91no—
214C2×Dic6—184no—
215C3×M16—38no—
216C3×M16—38no—
217C3×M16—92no—
218C3×M16—92no—
219C2×C3×C8—38no—
220C2×C3×C8—92no—
221#208—38no—
222#208—38no—
223#208C2×C3⋊D312yes—
224#208C2×C3⋊D312yescomplemented
225C2×C3⋊C8—38no—
226C2×C3⋊C8C2×C3⋊D312yescomplemented
227ΓL(1,𝔽25)—932no—
228ΓL(1,𝔽25)—98no—
229D3×C8—932no—
230D3×C8—98no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—24no—
234C22×C32—41no—
235C22×C32—44no—
236C22×C32—48no—
237C22×C32C2×C4×D314yes—
238C22×C32C2×C4×D314yes—
239C22×C32C2×C4×D314yes—
240C22×C32C2×C4×D311yescharacteristic
241C22×C32C2×C4×D311yescharacteristic
242C22×C32C2×C4×D311yescharacteristic
243C2×C3⋊D3—34no—
244C2×C3⋊D3—316no—
245C2×C3⋊D3—616no—
246C2×C3⋊D3—122no—
247C2×C3⋊D3—128no—
248C2×C3⋊D3—1216no—
249C32×C4—24no—
250C32×C4—24no—
251C32×C4—24no—
252C32×C4—24no—
253C32×C4C2×C4×D314yes—
254C32×C4C2×C4×D314yes—
255C32×C4C2×C4×D314yes—
256C32×C4C2×C4×D311yescharacteristic
257C32×C4C2×C4×D311yescharacteristic
258C32×C4C2×C4×D311yescharacteristic
259C32×C4C23×D314yes—
260C32×C4C23×D311yescharacteristic
261C3⋊Dic3—34no—
262C3⋊Dic3—316no—
263C3⋊Dic3—616no—
264C2×#41—271no—
265#131—274no—
266#131—274no—
267C2×M16—31no—
268C2×M16—91no—
269C2×M16—274no—
270C33#56811yescharacteristic, complemented, Sylow
271C23×C3—24no—
272C23×C3#62414yes—
273C23×C3C4×C3⋊D311yescharacteristic
274C22×D3—98no—
275C22×D3—92no—
276C22×D3—188no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4#62414yes—
284C2×C3×C4#62414yes—
285C2×C3×C4#62414yes—
286C2×C3×C4C4×C3⋊D311yescharacteristic
287C2×C3×C4C4×C3⋊D311yescharacteristic
288C2×C3×C4C4×C3⋊D311yescharacteristic
289C2×C3×C4C2×D3214yes—
290C2×C3×C4C2×D3214yes—
291C2×C3×C4C2×D3214yes—
292C2×C3×C4C22×C3⋊D311yescharacteristic
293C2×C3×C4C22×C3⋊D311yescharacteristic
294C2×C3×C4C22×C3⋊D311yescharacteristic
295#67—916no—
296#67—916no—
297#67—94no—
298#67—94no—
299#67—1816no—
300#67—1816no—
301C2×Dic3—98no—
302C2×Dic3—92no—
303C2×Dic3—188no—
304D12—98no—
305D12—98no—
306D12—92no—
307D12—92no—
308D12—188no—
309D12—188no—
310C4×D3—916no—
311C4×D3—94no—
312C4×D3—916no—
313C4×D3—94no—
314C4×D3—1816no—
315C4×D3—1816no—
316Dic6—98no—
317Dic6—98no—
318Dic6—92no—
319Dic6—92no—
320Dic6—188no—
321Dic6—188no—
322C3×C8—316no—
323C3×C8—94no—
324C3⋊C8—316no—
325C3⋊C8C22×C3⋊D314yes—
326C2×C32—28no—
327C2×C32—24no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—24no—
332C2×C32—24no—
333C2×C32—24no—
334C2×C32—44no—
335C2×C32—44no—
336C2×C32C22×C4×D314yes—
337C2×C32C22×C4×D311yescharacteristic
338C3⋊D3—64no—
339C3⋊D3—616no—
340C3⋊D3—1216no—
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×C3⋊D311yescharacteristic
350M16—32no—
351M16—32no—
352M16—92no—
353M16—92no—
354M16—278no—
355C2×C8—32no—
356C2×C8—92no—
357C2×C8—278no—
358C22×C3—24no—
359C22×C3—24no—
360C22×C3—24no—
361C22×C3—44no—
362C22×C3—41no—
363C22×C3—48no—
364C22×C3C2×#62414yes—
365C22×C3C2×#62414yes—
366C22×C3C2×#62414yes—
367C22×C3C2×C4×C3⋊D311yescharacteristic
368C22×C3C2×C4×C3⋊D311yescharacteristic
369C22×C3C2×C4×C3⋊D311yescharacteristic
370C2×D3—916no—
371C2×D3—94no—
372C2×D3—1816no—
373C2×D3—362no—
374C2×D3—368no—
375C2×D3—3616no—
376C3×C4—24no—
377C3×C4—24no—
378C3×C4—24no—
379C3×C4—24no—
380C3×C4C2×#62414yes—
381C3×C4C2×#62414yes—
382C3×C4C2×#62414yes—
383C3×C4C2×C4×C3⋊D311yescharacteristic
384C3×C4C2×C4×C3⋊D311yescharacteristic
385C3×C4C2×C4×C3⋊D311yescharacteristic
386C3×C4C22×D3214yes—
387C3×C4C23×C3⋊D311yescharacteristic
388Dic3—916no—
389Dic3—94no—
390Dic3—1816no—
391C32—24no—
392C32#574514yescomplemented
393C32#574511yescharacteristic, complemented
394C23—272no—
395C23#628911yescharacteristic
396Q8—272no—
397Q8—272no—
398D4—274no—
399D4—274no—
400D4—272no—
401D4—272no—
402C2×C4—272no—
403C2×C4—274no—
404C2×C4—274no—
405C2×C4#628911yescharacteristic
406C2×C4#628911yescharacteristic
407C2×C4#628911yescharacteristic
408C2×C4C2×D3×C3⋊D311yescharacteristic
409C2×C4C2×D3×C3⋊D311yescharacteristic
410C2×C4C2×D3×C3⋊D311yescharacteristic
411C8—34no—
412C8—94no—
413C2×C3—28no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—24no—
418C2×C3—24no—
419C2×C3—24no—
420C2×C3—24no—
421C2×C3—44no—
422C2×C3—44no—
423C2×C3C22×#62414yes—
424C2×C3C22×C4×C3⋊D311yescharacteristic
425D3—184no—
426D3—1816no—
427D3—3616no—
428C22—41no—
429C22—274no—
430C22—1082no—
431C22C2×#628911yescharacteristic
432C22C2×#628911yescharacteristic
433C22C2×#628911yescharacteristic
434C4—274no—
435C4C2×#628911yescharacteristic
436C4C2×#628911yescharacteristic
437C4C2×#628911yescharacteristic
438C4C22×D3×C3⋊D311yescharacteristic
439C3—24no—
440C3#1059478014yescomplemented
441C3#1059621611yescharacteristic, complemented
442C2—21no—
443C2—21no—
444C2—21no—
445C2—544no—
446C2C22×#628911yescharacteristic
subgroupG/H
center, FrattiniC4C22×D3×C3⋊D3
commutator, socleC2×C33C23×C4
FittingC22×C33×C4C22
prime radicalC2×#89076C2
cosocleC33×C4C24
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. C2×C32×#208C2semidirect
4. #11723358C2semidirect
5. #11723357C2semidirect
6. #11723356C2semidirect
19. C3×#3748C22semidirect
20. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. C32×#208C22semidirect
23. C2×C32×C3⋊C8C22semidirect
24. #88788C22semidirect
25. #88785C22semidirect
30. C2×#3748D3semidirect
35. C2×C3×#208D3semidirect
91. #3748C2×D3semidirect
92. C2×C3⋊(C3⋊C8)C2×D3semidirect
98. C3×#208C2×D3semidirect
100. C2×C3×C3⋊C8C2×D3semidirect
120. C2×#208C3⋊D3semidirect
224. #208C2×C3⋊D3semidirect
226. C2×C3⋊C8C2×C3⋊D3semidirect
270. C33#568semidirect
392. C32#5745semidirect
393. C32#5745semidirect
440. C3#10594780semidirect
441. C3#10596216semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C2×C3×#37482
C2×C32×#2082
#117233582
#117233572
#117233562
#105962163
#105947803