Finite Group Databasebeta

← #13160064

Subgroups — #13160064

Order 1728.

1148 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89020C211yesmaximal, characteristic
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×#88787C211yesmaximal, characteristic, complemented
4#11722190C211yesmaximal, characteristic, complemented
5C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6#11722024C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592426—31nomaximal
9#10590953—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6293C411yescharacteristic
14C2×#6292C411yescharacteristic
15C2×#6291C411yescharacteristic
16C2×#6290C411yescharacteristic
17#89020—22no—
18#89020—22no—
19C2×C33×C8C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×C8×C3⋊D3—22no—
23C2×C32×C3⋊C8C2211yescharacteristic
24#88787—22no—
25C2×#3828—31no—
26C2×#3798—34no—
27C2×#3743—31no—
28C2×C8×C3⋊D3D311yescharacteristic, complemented
29C2×C3×D3×C8—34no—
30C2×#3711—34no—
31C32×C22⋊C8—31no—
32#63657—31no—
33#63647—31no—
34#63647—31no—
35#63647—34no—
36#63647—64no—
37C3×(C2×D3)⋊C8D314yescomplemented
38#63557—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
41C22×C3×C3⋊D3C2×C411yescharacteristic
42C22×C32×D3C2×C411yescharacteristic
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
45C4×C3⋊(C3⋊D3)—21no—
46C4×C3⋊(C3⋊D3)—21no—
47C2×C3×C3⋊Dic3C2×C411yescharacteristic
48C3×C4×C3⋊D3D412yes—
49C3×C4×C3⋊D3D412yes—
50C2×C32×Dic3C2×C411yescharacteristic
51C32×C4×D3—21no—
52C32×C4×D3—21no—
53#6293—22no—
54#6292—22no—
55#6291—22no—
56#6290—22no—
57C33×C8—21no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61#5110—91no—
62#4727—94no—
63C22×C32×C4—31no—
64C2×#638—31no—
65C2×#638—31no—
66#3828—62no—
67#3828—62no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4×C3⋊D3—31no—
70C2×C4×C3⋊D3—34no—
71C2×C4×C3⋊D3—64no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×#634—31no—
74C2×C3×C4×D3—34no—
75C2×C3×C4×D3C2×D314yes—
76C2×#627—34no—
77C2×#626—34no—
78C2×#626—34no—
79C2×#625—34no—
80#3798—68no—
81#3798—68no—
82C2×C32×C8—24no—
83C2×C32×C8—31no—
84C2×C32×C8C2×D314yes—
85C2×C32×C8C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90#3743—62no—
91C8×C3⋊D3—22no—
92C2×C3×C3⋊C8—34no—
93C2×C3×C3⋊C8C2×D314yes—
94C3×D3×C8—68no—
95#3711—68no—
96C22×C33C22×C411yescharacteristic
97C2×C3⋊(C3⋊D3)—21no—
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3⋊(C3⋊D3)—21no—
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3C22⋊C412yes—
103C2×C32×D3—21no—
104C2×C32×D3—21no—
105C2×C32×D3—21no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3⋊(C3⋊Dic3)—21no—
109C3×C3⋊Dic3C22⋊C412yes—
110C32×Dic3—21no—
111C2×#235—94no—
112C2×#235—91no—
113C22×C3×C8—91no—
114C2×#208—91no—
115C2×ΓL(1,𝔽25)—94no—
116C2×D3×C8—34no—
117C3×C22⋊C8—34no—
118#830—94no—
119(C2×D3)⋊C8—94no—
120(C2×D3)⋊C8—91no—
121(C2×D3)⋊C8—94no—
122(C2×D3)⋊C8—91no—
123(C2×D3)⋊C8—91no—
124(C2×D3)⋊C8—184no—
125(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
126C23×C32—31no—
127C22×C3⋊D3—31no—
128C22×C3⋊D3—34no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3C4×D311yescharacteristic
131C22×C3×D3—34no—
132C22×C3×D3C4×D314yes—
133C2×C32×C4—24no—
134C2×C32×C4—31no—
135C2×C32×C4—61no—
136C2×C32×C4—61no—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D311yescharacteristic
139#638—62no—
140#638—62no—
141C2×C3⋊Dic3—31no—
142C2×C3⋊Dic3—34no—
143C2×C3⋊Dic3—64no—
144C2×C3⋊Dic3C4×D311yescharacteristic
145C4⋊(C3⋊D3)—62no—
146C4×C3⋊D3—61no—
147C4×C3⋊D3—61no—
148C4×C3⋊D3—64no—
149C4×C3⋊D3—64no—
150C4×C3⋊D3—124no—
151C4×C3⋊D3—124no—
152C4×C3⋊D3D1212yes—
153C4×C3⋊D3#6712yes—
154#634—62no—
155C2×C3×Dic3—34no—
156C2×C3×Dic3C4×D314yes—
157C3×C4×D3—24no—
158C3×C4×D3—24no—
159C3×C4×D3—38no—
160C3×C4×D3—38no—
161#627—68no—
162#626—68no—
163#626—68no—
164#625—68no—
165C32×C8—21no—
166C32×C8—24no—
167C32×C8—44no—
168C32×C8—61no—
169C3⋊(C3⋊C8)—21no—
170C3⋊(C3⋊C8)—61no—
171C3⋊(C3⋊C8)—64no—
172C3⋊(C3⋊C8)—124no—
173C3×C3⋊C8—24no—
174C3×C3⋊C8—64no—
175#408—271noSylow
176C2×C33C2×C22⋊C411yescharacteristic
177C2×C33#13111yescharacteristic
178C2×C33#13111yescharacteristic
179C3⋊(C3⋊D3)—41no—
180C3×C3⋊D3—22no—
181C32×D3—41no—
182C22×C3×C4—34no—
183C22×C3×C4—91no—
184C2×#67—94no—
185C2×#67—94no—
186C2×#67—91no—
187C2×#67—91no—
188#235—182no—
189#235—182no—
190#235—188no—
191#235—188no—
192C2×D12—94no—
193C2×D12—91no—
194C2×C4×D3—34no—
195C2×C4×D3—94no—
196C2×C4×D3—91no—
197C2×C4×D3—184no—
198C2×C4×D3C2×C3⋊D311yescharacteristic
199C2×Dic6—94no—
200C2×Dic6—91no—
201C2×C3×C8—24no—
202C2×C3×C8—34no—
203C2×C3×C8—91no—
204C2×C3×C8—182no—
205C2×C3×C8D3214yes—
206C2×C3×C8C2×C3⋊D311yescharacteristic
207#208—182no—
208C2×C3⋊C8—34no—
209C2×C3⋊C8—94no—
210C2×C3⋊C8—91no—
211C2×C3⋊C8—184no—
212C2×C3⋊C8C2×C3⋊D311yescharacteristic
213ΓL(1,𝔽25)—188no—
214D3×C8—68no—
215C22×C32—24no—
216C22×C32—61no—
217C22×C32—61no—
218C22×C32—61no—
219C22×C32C2×C4×D314yes—
220C22×C32C2×C4×D311yescharacteristic
221C2×C3⋊D3—21no—
222C2×C3⋊D3—21no—
223C2×C3⋊D3—68no—
224C2×C3⋊D3—64no—
225C2×C3⋊D3—61no—
226C2×C3⋊D3—61no—
227C2×C3⋊D3—61no—
228C2×C3⋊D3—64no—
229C2×C3⋊D3—64no—
230C2×C3⋊D3—124no—
231C2×C3⋊D3—124no—
232C2×C3⋊D3(C2×D3)⋊C412yes—
233C2×C3×D3—24no—
234C2×C3×D3—24no—
235C2×C3×D3—24no—
236C2×C3×D3—38no—
237C2×C3×D3—64no—
238C2×C3×D3—64no—
239C32×C4—24no—
240C32×C4—24no—
241C32×C4—61no—
242C32×C4C2×D1211yescharacteristic
243C32×C4D3×D414yes—
244C32×C4D3×D414yes—
245C32×C4C2×#6711yescharacteristic
246C3⋊Dic3—61no—
247C3⋊Dic3—64no—
248C3⋊Dic3—124no—
249C3⋊Dic3(C2×D3)⋊C412yes—
250C3×Dic3—24no—
251C3×Dic3—38no—
252C2×#41—271no—
253C2×M16—271no—
254C22×C8—91no—
255C22⋊C8—31no—
256C22⋊C8—271no—
257C22⋊C8—271no—
258C22⋊C8—271no—
259C33#40811yescharacteristic, complemented, Sylow
260C23×C3—34no—
261C23×C3—91no—
262C22×D3—34no—
263C22×D3—94no—
264C22×D3—91no—
265C22×D3—184no—
266C22×D3C4×C3⋊D311yescharacteristic
267C2×C3×C4—24no—
268C2×C3×C4—34no—
269C2×C3×C4—64no—
270C2×C3×C4—64no—
271C2×C3×C4—91no—
272C2×C3×C4—92no—
273C2×C3×C4—92no—
274C2×C3×C4C2×D3214yes—
275C2×C3×C4C22×C3⋊D311yescharacteristic
276#67—182no—
277#67—188no—
278#67—182no—
279#67—188no—
280C2×Dic3—34no—
281C2×Dic3—94no—
282C2×Dic3—91no—
283C2×Dic3—184no—
284C2×Dic3C4×C3⋊D311yescharacteristic
285D12—188no—
286D12—182no—
287C4×D3—21no—
288C4×D3—21no—
289C4×D3—38no—
290C4×D3—38no—
291C4×D3—184no—
292C4×D3—184no—
293C4×D3—181no—
294C4×D3—181no—
295C4×D3—364no—
296C4×D3—364no—
297Dic6—182no—
298Dic6—188no—
299C3×C8—24no—
300C3×C8—21no—
301C3×C8—44no—
302C3×C8—64no—
303C3×C8—181no—
304C3⋊C8—21no—
305C3⋊C8—64no—
306C3⋊C8—184no—
307C3⋊C8—181no—
308C3⋊C8—364no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—121no—
313C2×C32D3×C22⋊C414yes—
314C2×C32#90114yes—
315C2×C32#90111yescharacteristic
316C2×C32#90111yescharacteristic
317C2×C32#90714yes—
318C2×C32C2×(C2×D3)⋊C411yescharacteristic
319C3⋊D3—22no—
320C3⋊D3—128no—
321C3⋊D3—124no—
322C3⋊D3—121no—
323C3×D3—44no—
324C3×D3—68no—
325#41—542no—
326#41—542no—
327C2×Q8—271no—
328C2×D4—271no—
329C2×D4—271no—
330C2×D4—271no—
331C22×C4—31no—
332C22×C4—91no—
333C22×C4—271no—
334M16—542no—
335C2×C8—31no—
336C2×C8—91no—
337C2×C8—182no—
338C2×C8—271no—
339C2×C8D3×C3⋊D311yescharacteristic
340C22×C3—24no—
341C22×C3—64no—
342C22×C3—64no—
343C22×C3—64no—
344C22×C3—92no—
345C22×C3—181no—
346C22×C3—181no—
347C22×C3C4×D3214yes—
348C22×C3C2×C4×C3⋊D311yescharacteristic
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—38no—
353C2×D3—64no—
354C2×D3—64no—
355C2×D3—188no—
356C2×D3—184no—
357C2×D3—184no—
358C2×D3—181no—
359C2×D3—181no—
360C2×D3—181no—
361C2×D3—184no—
362C2×D3—364no—
363C2×D3—364no—
364C3×C4—24no—
365C3×C4—24no—
366C3×C4—64no—
367C3×C4—92no—
368C3×C4D3×D1214yes—
369C3×C4D3×#6714yes—
370C3×C4D4×C3⋊D311yescharacteristic
371C3×C4D4×C3⋊D311yescharacteristic
372Dic3—21no—
373Dic3—38no—
374Dic3—181no—
375Dic3—184no—
376Dic3—364no—
377C32—24no—
378C32#472714yescomplemented
379C32#511011yescharacteristic, complemented
380C23—31no—
381C23—91no—
382C23—271no—
383Q8—542no—
384D4—542no—
385D4—542no—
386D4—542no—
387C2×C4—31no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—91no—
391C2×C4—92no—
392C2×C4—92no—
393C2×C4—271no—
394C2×C4—541no—
395C2×C4—541no—
396C2×C4C2×D3×C3⋊D311yescharacteristic
397C8—21no—
398C8—61no—
399C8—181no—
400C8—541no—
401C2×C3—24no—
402C2×C3—24no—
403C2×C3—24no—
404C2×C3—124no—
405C2×C3—182no—
406C2×C3#6380314yes—
407C2×C3#6380414yes—
408C2×C3D3×(C2×D3)⋊C414yes—
409C2×C3C22⋊C4×C3⋊D311yescharacteristic
410C2×C3#6411011yescharacteristic
411C2×C3#6411611yescharacteristic
412D3—41no—
413D3—68no—
414D3—368no—
415D3—361no—
416D3—364no—
417C22—61no—
418C22—61no—
419C22—61no—
420C22—92no—
421C22—181no—
422C22—181no—
423C22—541no—
424C22—541no—
425C22—541no—
426C22C4×D3×C3⋊D311yescharacteristic
427C4—61no—
428C4—92no—
429C4—541no—
430C4C3⋊D3×D1211yescharacteristic
431C4C3⋊D3×#6711yescharacteristic
432C3—24no—
433C3#1059095314yescomplemented
434C3#1059242611yescharacteristic, complemented
435C2—121no—
436C2—182no—
437C2—1081no—
438C2#1172333111yescharacteristic
439C2#1172333311yescharacteristic
440C2C3⋊D3×(C2×D3)⋊C411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC2×#89020C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#408
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×C3⋊D3C2semidirect
3. C2×#88787C2semidirect
4. #11722190C2semidirect
5. C32×(C2×D3)⋊C8C2semidirect
6. #11722024C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
28. C2×C8×C3⋊D3D3semidirect
37. C3×(C2×D3)⋊C8D3semidirect
125. (C2×D3)⋊C8C3⋊D3semidirect
259. C33#408semidirect
378. C32#4727semidirect
379. C32#5110semidirect
433. C3#10590953semidirect
434. C3#10592426semidirect

Maximal subgroups

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

subgroupindex
C2×#890202
C2×C3×C8×C3⋊D32
C2×#887872
#117221902
C32×(C2×D3)⋊C82
#117220242
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105924263
#105909533