Finite Group Databasebeta

← #13175198

Subgroups — #13175198

Order 1728.

1924 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×#89019C211yesmaximal, characteristic, complemented
2C2×#89015C211yesmaximal, characteristic
3#11723913C211yesmaximal, characteristic, complemented
4C3×#64078C211yesmaximal, characteristic, complemented
5C32×#875C211yesmaximal, characteristic, complemented
6#11723500C214yesmaximal, complemented
7#11723451C211yesmaximal, characteristic, complemented
8#11723434C211yesmaximal, characteristic, complemented
9#11723422C214yesmaximal, complemented
10#10596146—31nomaximal
11#10593908—34nomaximal
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C2×C32×D12C2211yescharacteristic
16C2×C32×Dic6C2211yescharacteristic
17C2×#6293C2212yes—
18C2×#6291C2212yes—
19C2×#6289C2212yes—
20C2×D3×C3⋊Dic3C2212yes—
21#89019—24no—
22#89015—24no—
23C33×C42C2211yescharacteristic
24#88859C2212yes—
25#88857C2212yes—
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
27C3×C4⋊(C3⋊Dic3)C2214yes—
28C32×(C2×D3)⋊C4C2214yes—
29#88806C2214yes—
30(C2×D3)⋊(C3⋊Dic3)C2214yes—
31C2×Q8×C3⋊D3—31no—
32C2×#3830—31no—
33C2×#3797—34no—
34C2×#3796—34no—
35C32×#124—31no—
36#64153—31no—
37#64145—31no—
38#64092—34no—
39#64087—34no—
40#64081—31no—
41#64081—34no—
42#64081—64no—
43#64078D311yescharacteristic, complemented
44C3×#875D314yescomplemented
45C3×#869—34no—
46#63899—316no—
47#63873—34no—
48#63863—34no—
49#63854—316no—
50C22×C3⋊(C3⋊D3)C2311yescharacteristic
51C22×C32×D3C2312yes—
52C2×C33×C4C2312yes—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)C2312yes—
55C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
56C4×C3⋊(C3⋊D3)D414yes—
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2312yes—
59C3×#634—22no—
60C3×#634—22no—
61C2×C32×Dic3C2312yes—
62C32×D12—22no—
63C32×Dic6—22no—
64#6293—24no—
65#6291—24no—
66#6289—24no—
67D3×C3⋊Dic3—24no—
68#5724—91no—
69#5675—94no—
70C2×C32×Q8—31no—
71C2×C32×D4—31no—
72C2×#638—32no—
73C22×C3⋊Dic3—32no—
74Q8×C3⋊D3—64no—
75#3830—64no—
76C2×C4×C3⋊D3—31no—
77C2×C4×C3⋊D3—34no—
78C2×C4×C3⋊D3—32no—
79C2×C4×C3⋊D3—64no—
80C2×#634—32no—
81C2×#634C2×D311yescharacteristic
82C2×#634C2×D311yescharacteristic
83C2×C3×D12C2×D314yes—
84C2×C3×Dic6—34no—
85C2×C3×Dic6—34no—
86C2×C3×Dic6C2×D314yes—
87C2×#627—38no—
88C2×#626—38no—
89C2×#624—38no—
90C2×D3×Dic3—38no—
91#3797—616no—
92#3796—616no—
93C32×C22⋊C4—34no—
94C32×C42—24no—
95C32×C42C2×D314yes—
96C32×C42C2×D311yescharacteristic
97#3757—34no—
98#3752—32no—
99#3752—38no—
100#3752—68no—
101C4⋊(C3⋊Dic3)C2×D314yes—
102#3750—32no—
103#3750—38no—
104#3750—34no—
105#3750—68no—
106C4×C3⋊Dic3—31no—
107C4×C3⋊Dic3—34no—
108C4×C3⋊Dic3—64no—
109C3×(C2×D3)⋊C4C2×D3116yes—
110C3×C4⋊Dic3—316no—
111#3724—316no—
112(C2×D3)⋊Dic3—316no—
113C22×C33C2411yescharacteristic
114C2×C3⋊(C3⋊D3)—22no—
115C2×C3⋊(C3⋊D3)C2×D412yes—
116C2×C32×D3—22no—
117C2×C32×D3—24no—
118C33×C4—22no—
119C33×C4C2×D412yes—
120C3⋊(C3⋊Dic3)—22no—
121C3⋊(C3⋊Dic3)C2×D412yes—
122C3×C3⋊Dic3—22no—
123C3×C3⋊Dic3—22no—
124C32×Dic3—22no—
125C2×#239—91no—
126C2×D3×Q8—94no—
127C2×D3×Q8—91no—
128C2×#237—94no—
129C3×C4⋊Q8—91no—
130C3×#124—34no—
131#947—91no—
132#944—94no—
133#944—91no—
134#936—94no—
135#896—94no—
136#894—94no—
137#883—916no—
138#878—916no—
139#875C3⋊D311yescharacteristic, complemented
140#872—94no—
141#872—91no—
142#872—184no—
143#869—34no—
144C23×C32—32no—
145C22×C3⋊D3—31no—
146C22×C3⋊D3—34no—
147C22×C3⋊D3—64no—
148C22×C3×D3C22×D318yes—
149C32×Q8—62no—
150C32×D4—62no—
151C2×C32×C4—24no—
152C2×C32×C4—28no—
153C2×C32×C4—32no—
154C2×C32×C4C22×D318yes—
155C2×C32×C4C22×D314yes—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D311yescharacteristic
158#638—64no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—32no—
161C2×C3⋊Dic3—34no—
162C2×C3⋊Dic3—38no—
163C2×C3⋊Dic3—64no—
164C2×C3⋊Dic3—68no—
165C2×C3⋊Dic3—64no—
166C2×C3⋊Dic3C22×D312yes—
167C2×C3⋊Dic3C22×D312yes—
168C4×C3⋊D3—34no—
169C4×C3⋊D3—316no—
170C4×C3⋊D3—616no—
171C4×C3⋊D3—64no—
172#634—22no—
173#634—22no—
174#634—64no—
175C2×C3×Dic3—38no—
176C2×C3×Dic3—38no—
177C2×C3×Dic3C22×D318yes—
178C3×D12—28no—
179C3×Dic6—28no—
180C3×Dic6—68no—
181C3×Dic6—68no—
182#627—616no—
183#626—616no—
184#624—616no—
185D3×Dic3—616no—
186#536—271noSylow
187C2×C33C22×D411yescharacteristic
188C2×C332−1+412yes—
189C3⋊(C3⋊D3)—22no—
190C32×D3—42no—
191C2×C3×Q8—34no—
192C2×C3×Q8—91no—
193C2×C3×Q8—91no—
194C2×C3×D4—34no—
195C2×#67—98no—
196C22×Dic3—98no—
197#239—184no—
198D3×Q8—184no—
199D3×Q8—1816no—
200#237—1816no—
201C2×D12—92no—
202C2×D12C2×C3⋊D311yescharacteristic
203C2×C4×D3—94no—
204C2×C4×D3—91no—
205C2×C4×D3—98no—
206C2×C4×D3—92no—
207C2×C4×D3—92no—
208C2×C4×D3—184no—
209C2×Dic6—34no—
210C2×Dic6—34no—
211C2×Dic6—98no—
212C2×Dic6—92no—
213C2×Dic6C2×C3⋊D311yescharacteristic
214C3×C4⋊C4—94no—
215C3×C22⋊C4—316no—
216C3×C42—24no—
217C3×C42D3214yes—
218C3×C42C2×C3⋊D311yescharacteristic
219#217—916no—
220(C2×D3)⋊C4—94no—
221(C2×D3)⋊C4—98no—
222(C2×D3)⋊C4—92no—
223(C2×D3)⋊C4—188no—
224(C2×D3)⋊C4C2×C3⋊D314yes—
225C4⋊Dic3—316no—
226Dic3⋊C4—916no—
227Dic3⋊C4—98no—
228Dic3⋊C4—92no—
229Dic3⋊C4—94no—
230Dic3⋊C4—188no—
231C4×Dic3—94no—
232C4×Dic3—91no—
233C4×Dic3—184no—
234C22×C32—24no—
235C22×C32—62no—
236C22×C32—64no—
237C22×C32C23×D314yes—
238C22×C32C23×D311yescharacteristic
239C2×C3⋊D3—32no—
240C2×C3⋊D3—38no—
241C2×C3⋊D3—68no—
242C2×C3⋊D3—62no—
243C2×C3⋊D3—68no—
244C2×C3⋊D3—128no—
245C2×C3×D3—28no—
246C2×C3×D3—216no—
247C32×C4—22no—
248C32×C4—28no—
249C32×C4—28no—
250C32×C4—48no—
251C32×C4—62no—
252C32×C4D3×D418yes—
253C32×C4D3×D412yes—
254C3⋊Dic3—22no—
255C3⋊Dic3—22no—
256C3⋊Dic3—32no—
257C3⋊Dic3—38no—
258C3⋊Dic3—68no—
259C3⋊Dic3—62no—
260C3⋊Dic3—68no—
261C3⋊Dic3—128no—
262C3×Dic3—28no—
263C3×Dic3—68no—
264C3×Dic3—68no—
265C2×#41—271no—
266C22×Q8—271no—
267C4⋊Q8—91no—
268C4⋊Q8—271no—
269#124—31no—
270#124—271no—
271#123—274no—
272C22⋊Q8—274no—
273#117—271no—
274C33#53611yescharacteristic, complemented, Sylow
275C23×C3—38no—
276C22×D3—94no—
277C22×D3—91no—
278C22×D3—184no—
279C22×D3C22×C3⋊D312yes—
280C3×Q8—68no—
281C3×Q8—182no—
282C3×Q8—182no—
283C3×D4—68no—
284C2×C3×C4—24no—
285C2×C3×C4—28no—
286C2×C3×C4—38no—
287C2×C3×C4—92no—
288C2×C3×C4—92no—
289C2×C3×C4C2×D3218yes—
290C2×C3×C4C2×D3214yes—
291C2×C3×C4C22×C3⋊D312yes—
292C2×C3×C4C22×C3⋊D311yescharacteristic
293#67—1816no—
294C2×Dic3—38no—
295C2×Dic3—38no—
296C2×Dic3—94no—
297C2×Dic3—91no—
298C2×Dic3—98no—
299C2×Dic3—92no—
300C2×Dic3—184no—
301C2×Dic3—188no—
302C2×Dic3—1816no—
303C2×Dic3C22×C3⋊D312yes—
304D12—22no—
305D12—184no—
306C4×D3—916no—
307C4×D3—94no—
308C4×D3—1816no—
309C4×D3—184no—
310C4×D3—1816no—
311C4×D3—184no—
312Dic6—22no—
313Dic6—68no—
314Dic6—68no—
315Dic6—184no—
316Dic6—1816no—
317C2×C32—28no—
318C2×C32—24no—
319C2×C32—122no—
320C2×C32C2×D3×D414yes—
321C2×C32C2×D3×D411yescharacteristic
322C2×C32#100712yes—
323C2×C32#101018yes—
324C3⋊D3—62no—
325C3⋊D3—68no—
326C3⋊D3—128no—
327C3×D3—48no—
328#41—544no—
329C2×Q8—31no—
330C2×Q8—91no—
331C2×Q8—91no—
332C2×Q8—272no—
333C2×Q8—544no—
334C2×D4—31no—
335C2×D4—272no—
336C22×C4—272no—
337C22×C4—272no—
338C22×C4—271no—
339C4⋊C4—94no—
340C4⋊C4—274no—
341C4⋊C4—272no—
342C22⋊C4—34no—
343C22⋊C4—274no—
344C22⋊C4—272no—
345C42—271no—
346C42D3×C3⋊D311yescharacteristic
347C22×C3—24no—
348C22×C3—616no—
349C22×C3—68no—
350C22×C3C22×D3214yes—
351C22×C3C23×C3⋊D311yescharacteristic
352C2×D3—22no—
353C2×D3—24no—
354C2×D3—98no—
355C2×D3—92no—
356C2×D3—188no—
357C2×D3—188no—
358C2×D3—182no—
359C2×D3—368no—
360C3×C4—28no—
361C3×C4—22no—
362C3×C4—28no—
363C3×C4—48no—
364C3×C4—68no—
365C3×C4—182no—
366C3×C4—182no—
367C3×C4#380218yes—
368C3×C4D4×C3⋊D312yes—
369Dic3—22no—
370Dic3—68no—
371Dic3—68no—
372Dic3—98no—
373Dic3—92no—
374Dic3—188no—
375Dic3—188no—
376Dic3—182no—
377Dic3—368no—
378C32—24no—
379C32#567514yescomplemented
380C32#572411yescharacteristic, complemented
381C23—32no—
382C23—271no—
383Q8—62no—
384Q8—182no—
385Q8—182no—
386Q8—544no—
387D4—62no—
388D4—544no—
389C2×C4—32no—
390C2×C4—92no—
391C2×C4—92no—
392C2×C4—271no—
393C2×C4—274no—
394C2×C4—272no—
395C2×C4—544no—
396C2×C4—544no—
397C2×C4C2×D3×C3⋊D312yes—
398C2×C4C2×D3×C3⋊D311yescharacteristic
399C2×C3—28no—
400C2×C3—24no—
401C2×C3—128no—
402C2×C3#6429718yes—
403C2×C3C2×#380214yes—
404C2×C3C2×D4×C3⋊D311yescharacteristic
405C2×C3#6436712yes—
406D3—42no—
407D3—182no—
408D3—188no—
409D3—368no—
410C22—64no—
411C22—62no—
412C22—272no—
413C22—542no—
414C22C22×D3×C3⋊D311yescharacteristic
415C4—22no—
416C4—62no—
417C4—182no—
418C4—182no—
419C4—272no—
420C4—542no—
421C4#8902612yes—
422C3—24no—
423C3#1059390814yescomplemented
424C3#1059614611yescharacteristic, complemented
425C2—122no—
426C2—542no—
427C2#1172456212yes—
428C2C2×#8902611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC2×#89015C2
hypercenterC42D3×C3⋊D3
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#89019C2semidirect
3. #11723913C2semidirect
4. C3×#64078C2semidirect
5. C32×#875C2semidirect
6. #11723500C2semidirect
7. #11723451C2semidirect
8. #11723434C2semidirect
9. #11723422C2semidirect
43. #64078D3semidirect
44. C3×#875D3semidirect
139. #875C3⋊D3semidirect
274. C33#536semidirect
379. C32#5675semidirect
380. C32#5724semidirect
423. C3#10593908semidirect
424. C3#10596146semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
C2×#890152
#117239132
C3×#640782
C32×#8752
#117235002
#117234512
#117234342
#117234222
#105961463
#105939083