Finite Group Databasebeta

← #13140152

Subgroups — #13140152

Order 1728.

911 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×C4⋊#1084C211yesmaximal, characteristic, complemented
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4C2×#88450C212yesmaximal
5#11720334C212yesmaximal, complemented
6#10591325—31nomaximal
7#10591313—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—22no—
14C2×C4×(C3⋊D3)⋊C3—22no—
15C2×C4×(C3⋊D3)⋊C3C414yescomplemented
16C4⋊#1084—22no—
17C4⋊((C3⋊Dic3)⋊C3)—22no—
18#88450—24no—
19C22×C4×C3⋊D3D311yescharacteristic
20C22×C3×C4×D3—31no—
21C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
22C2×C3×C4⋊Dic3—31no—
23C2×C3×C4⋊Dic3—31no—
24C2×C3×C4⋊Dic3—31no—
25C2×C3×C4⋊Dic3—61no—
26C2×#3722—32no—
27C2×(C2×D3)⋊Dic3—32no—
28#63579—32no—
29#63579—32no—
30C23×H(3,𝔽3)C2311yescharacteristic
31C22×(C3⋊D3)⋊C3—21no—
32C22×(C3⋊D3)⋊C3—21no—
33C22×(C3⋊D3)⋊C3—21no—
34C22×(C3⋊D3)⋊C3—21no—
35C22×(C3⋊D3)⋊C3C2×C412yes—
36C22×(C3⋊D3)⋊C3D412yes—
37C22×(C3⋊D3)⋊C3Q812yes—
38C2×C4×H(3,𝔽3)C2×C412yes—
39C2×C4×H(3,𝔽3)D412yes—
40C2×C4×H(3,𝔽3)D412yes—
41C2×#1084—22no—
42C2×#1084—22no—
43C2×#1084—22no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—22no—
47C2×(C3⋊Dic3)⋊C3—22no—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3C2×C412yes—
50C4×(C3⋊D3)⋊C3—28no—
51#5212—91no—
52#4987—91no—
53#4987—91no—
54C23×C3⋊D3C2×D311yescharacteristic
55C23×C3×D3—31no—
56C22×C32×C4—21no—
57C22×C32×C4C2×D311yescharacteristic
58C22×C32×C4C2×D311yescharacteristic
59C22×C3⋊Dic3C2×D312yes—
60C22×C3⋊Dic3C2×D311yescharacteristic
61C2×C4×C3⋊D3—22no—
62C2×C4×C3⋊D3—22no—
63C2×C4×C3⋊D3Dic314yescomplemented
64C22×C3×Dic3—31no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—62no—
69C2×C3×C4×D3—34no—
70C2×C3×C4×D3—62no—
71C2×C3×C4×D3—62no—
72C4⋊(C3⋊Dic3)—22no—
73C3×C4⋊Dic3—64no—
74C3×C4⋊Dic3—62no—
75C3×C4⋊Dic3—62no—
76C3×C4⋊Dic3—62no—
77#3722—64no—
78(C2×D3)⋊Dic3—64no—
79C22×H(3,𝔽3)C22×C411yescharacteristic
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C22×H(3,𝔽3)C2×Q811yescharacteristic
84C22×H(3,𝔽3)#4111yescharacteristic
85C22×H(3,𝔽3)#4111yescharacteristic
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—22no—
88C2×(C3⋊D3)⋊C3—22no—
89C2×(C3⋊D3)⋊C3—22no—
90C2×(C3⋊D3)⋊C3—22no—
91C2×(C3⋊D3)⋊C3—22no—
92C2×(C3⋊D3)⋊C3C4⋊C414yescomplemented
93C4×H(3,𝔽3)C22⋊C414yescomplemented
94#1084—42no—
95(C3⋊Dic3)⋊C3—22no—
96(C3⋊Dic3)⋊C3—42no—
97C23×C3×C4—91no—
98C22×C4×D3—31no—
99C22×C4×D3—91no—
100C2×C3×C4⋊C4—91no—
101C2×C3×C4⋊C4—91no—
102C2×#217—92no—
103C2×(C2×D3)⋊C4—92no—
104C2×(C2×D3)⋊C4—92no—
105C2×C4⋊Dic3—31no—
106C2×C4⋊Dic3—31no—
107C2×C4⋊Dic3—31no—
108C2×C4⋊Dic3—61no—
109C2×C4⋊Dic3—91no—
110C3⋊#95—92no—
111C3⋊#95—92no—
112C3⋊#95—92no—
113C23×C32—21no—
114C23×C32C22×D311yescharacteristic
115C23×C32C22×D311yescharacteristic
116C22×C3⋊D3—21no—
117C22×C3⋊D3—21no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3—21no—
120C22×C3⋊D3Dic612yes—
121C22×C3⋊D3D1212yes—
122C22×C3⋊D3C2×Dic312yes—
123C22×C3×D3—32no—
124C22×C3×D3—32no—
125C22×C3×D3—32no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4C4×D312yes—
134C2×C32×C4D1212yes—
135C2×C32×C4C2×Dic312yes—
136C2×C32×C4#6712yes—
137C2×C32×C4#6712yes—
138C2×C32×C4#6712yes—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3C2×Dic312yes—
145C4×C3⋊D3—28no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—122no—
160C2×C3×Dic3—122no—
161C3×C4×D3—68no—
162#380—271noSylow
163C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
164C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
165C2×H(3,𝔽3)#11711yescharacteristic
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)#12111yescharacteristic
168C2×H(3,𝔽3)C22⋊Q811yescharacteristic
169C2×H(3,𝔽3)C22⋊Q811yescharacteristic
170(C3⋊D3)⋊C3—24no—
171C24×C3—91no—
172C23×D3—31no—
173C23×D3—91no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—61no—
177C22×C3×C4—94no—
178C22×C3×C4—91no—
179C22×C3×C4—92no—
180C22×C3×C4—92no—
181C22×C3×C4—182no—
182C22×C3×C4—182no—
183C22×C3×C4D3211yescharacteristic
184C22×Dic3—31no—
185C22×Dic3—32no—
186C22×Dic3—32no—
187C22×Dic3—32no—
188C22×Dic3—62no—
189C22×Dic3—91no—
190C22×Dic3—92no—
191C2×C4×D3—34no—
192C2×C4×D3—62no—
193C2×C4×D3—62no—
194C2×C4×D3—94no—
195C2×C4×D3—182no—
196C2×C4×D3—182no—
197C3×C4⋊C4—182no—
198C3×C4⋊C4—182no—
199#217—184no—
200(C2×D3)⋊C4—184no—
201(C2×D3)⋊C4—184no—
202C4⋊Dic3—62no—
203C4⋊Dic3—62no—
204C4⋊Dic3—64no—
205C4⋊Dic3—62no—
206C4⋊Dic3—182no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32C2×Dic611yescharacteristic
215C22×C32C2×C4×D311yescharacteristic
216C22×C32C2×D1211yescharacteristic
217C22×C32C2×D1211yescharacteristic
218C22×C32#23511yescharacteristic
219C22×C32#23511yescharacteristic
220C22×C32D3×D411yescharacteristic
221C22×C32#23711yescharacteristic
222C22×C32D3×Q811yescharacteristic
223C22×C32#23911yescharacteristic
224C22×C32C22×Dic311yescharacteristic
225C22×C32C2×#6711yescharacteristic
226C22×C32C2×#6711yescharacteristic
227C22×C32C2×#6711yescharacteristic
228C2×C3⋊D3—22no—
229C2×C3⋊D3—22no—
230C2×C3⋊D3—22no—
231C2×C3⋊D3—22no—
232C2×C3⋊D3—22no—
233C2×C3⋊D3—22no—
234C2×C3⋊D3C4⋊Dic314yescomplemented
235C2×C3×D3—34no—
236C2×C3×D3—62no—
237C2×C3×D3—62no—
238C2×C3×D3—62no—
239C2×C3×D3—62no—
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242C32×C4—24no—
243C32×C4(C2×D3)⋊C414yescomplemented
244C32×C4#21714yescomplemented
245C3⋊Dic3—22no—
246C3⋊Dic3—42no—
247C3×Dic3—62no—
248C3×Dic3—124no—
249C3×Dic3—122no—
250C3×Dic3—122no—
251C3×Dic3—122no—
252C23×C4—91no—
253C2×C4⋊C4—91no—
254C2×C4⋊C4—91no—
255C2×C22⋊C4—272no—
256#95—272no—
257H(3,𝔽3)#38011yescharacteristic, complemented, Sylow
258C23×C3—31no—
259C23×C3—31no—
260C23×C3—61no—
261C23×C3—92no—
262C23×C3—92no—
263C23×C3—92no—
264C23×C3—181no—
265C23×C3—181no—
266C23×C3—181no—
267C23×C3—181no—
268C23×C3C2×D3211yescharacteristic
269C22×D3—32no—
270C22×D3—32no—
271C22×D3—32no—
272C22×D3—61no—
273C22×D3—61no—
274C22×D3—61no—
275C22×D3—61no—
276C22×D3—92no—
277C22×D3—92no—
278C22×D3—92no—
279C22×D3—181no—
280C22×D3—181no—
281C22×D3—181no—
282C22×D3—181no—
283C2×C3×C4—32no—
284C2×C3×C4—32no—
285C2×C3×C4—32no—
286C2×C3×C4—32no—
287C2×C3×C4—32no—
288C2×C3×C4—32no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—92no—
293C2×C3×C4—188no—
294C2×C3×C4—181no—
295C2×C3×C4—181no—
296C2×C3×C4—182no—
297C2×C3×C4—182no—
298C2×C3×C4—182no—
299C2×C3×C4—182no—
300C2×C3×C4—182no—
301C2×C3×C4—182no—
302C2×C3×C4D3×Dic312yes—
303C2×C3×C4#62512yes—
304C2×C3×C4#62612yes—
305C2×Dic3—32no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—62no—
309C2×Dic3—62no—
310C2×Dic3—62no—
311C2×Dic3—62no—
312C2×Dic3—62no—
313C2×Dic3—62no—
314C2×Dic3—64no—
315C2×Dic3—62no—
316C2×Dic3—62no—
317C2×Dic3—62no—
318C2×Dic3—92no—
319C2×Dic3—122no—
320C2×Dic3—122no—
321C2×Dic3—181no—
322C2×Dic3—181no—
323C2×Dic3—182no—
324C2×Dic3—182no—
325C2×Dic3—182no—
326C4×D3—68no—
327C4×D3—188no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32D3×C4⋊C411yescharacteristic
336C2×C32#89211yescharacteristic
337C2×C32#89511yescharacteristic
338C2×C32#89711yescharacteristic
339C2×C32#92411yescharacteristic
340C2×C32#92411yescharacteristic
341C2×C32C2×C4⋊Dic311yescharacteristic
342C2×C32#92611yescharacteristic
343C2×C32C2×(C2×D3)⋊C411yescharacteristic
344C2×C32#93011yescharacteristic
345C2×C32#93011yescharacteristic
346C2×C32#93811yescharacteristic
347C2×C32#94611yescharacteristic
348C2×C32C2×#21711yescharacteristic
349C3⋊D3—24no—
350C3×D3—64no—
351C24—91no—
352C22×C4—91no—
353C22×C4—94no—
354C22×C4—92no—
355C22×C4—92no—
356C22×C4—182no—
357C22×C4—182no—
358C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
359C4⋊C4—182no—
360C4⋊C4—182no—
361C22⋊C4—544no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—31no—
370C22×C3—31no—
371C22×C3—31no—
372C22×C3—31no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—31no—
376C22×C3—61no—
377C22×C3—61no—
378C22×C3—61no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—61no—
383C22×C3—94no—
384C22×C3—182no—
385C22×C3—182no—
386C22×C3—182no—
387C22×C3—182no—
388C22×C3—182no—
389C22×C3—182no—
390C22×C3D3×Dic611yescharacteristic
391C22×C3#379511yescharacteristic
392C22×C3#379911yescharacteristic
393C22×C3D3×D1211yescharacteristic
394C22×C3C2×D3×Dic311yescharacteristic
395C22×C3C2×#62511yescharacteristic
396C22×C3C2×#62611yescharacteristic
397C2×D3—34no—
398C2×D3—62no—
399C2×D3—62no—
400C2×D3—62no—
401C2×D3—62no—
402C2×D3—62no—
403C2×D3—62no—
404C2×D3—94no—
405C2×D3—182no—
406C2×D3—182no—
407C2×D3—182no—
408C2×D3—182no—
409C2×D3—182no—
410C2×D3—182no—
411C3×C4—34no—
412C3×C4—34no—
413C3×C4—64no—
414C3×C4—182no—
415C3×C4—362no—
416C3×C4—362no—
417C3×C4(C2×D3)⋊Dic314yes—
418Dic3—62no—
419Dic3—122no—
420Dic3—122no—
421Dic3—124no—
422Dic3—122no—
423Dic3—182no—
424Dic3—362no—
425C32—21no—
426C32#498711yescharacteristic, complemented
427C32#521211yescharacteristic, complemented
428C23—92no—
429C23—92no—
430C23—92no—
431C23—181no—
432C23—181no—
433C23—181no—
434C23—181no—
435C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
436C2×C4—92no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4—188no—
440C2×C4—182no—
441C2×C4—182no—
442C2×C4—182no—
443C2×C4—182no—
444C2×C4—182no—
445C2×C4—182no—
446C2×C4((C3⋊D3)⋊C3)⋊C412yes—
447C2×C4#619812yes—
448C2×C4#620012yes—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—31no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—61no—
464C2×C3—61no—
465C2×C3—61no—
466C2×C3—61no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—61no—
470C2×C3—184no—
471C2×C3#6384111yescharacteristic
472C2×C3#6385611yescharacteristic
473C2×C3#6385711yescharacteristic
474C2×C3D3×C4⋊Dic311yescharacteristic
475C2×C3#6390811yescharacteristic
476C2×C3#6390911yescharacteristic
477C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
478D3—64no—
479D3—184no—
480C22—94no—
481C22—182no—
482C22—182no—
483C22—182no—
484C22—182no—
485C22—182no—
486C22—182no—
487C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
488C22#8864811yescharacteristic
489C22#8865211yescharacteristic
490C22#8865411yescharacteristic
491C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
492C22C2×#619811yescharacteristic
493C22C2×#620011yescharacteristic
494C4—182no—
495C4—362no—
496C4—362no—
497C4#8845014yescomplemented
498C3—31no—
499C3—31no—
500C3—61no—
501C3#1059131311yescharacteristic
502C2—184no—
503C2#1172108111yescharacteristic
504C2#1172110611yescharacteristic
505C2#1172110711yescharacteristic
506C2#1172112911yescharacteristic
507C2#1172114111yescharacteristic
508C2#1172115611yescharacteristic
509C2C2×#8845011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#380
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊#1084C2semidirect
3. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #11720334C2semidirect
15. C2×C4×(C3⋊D3)⋊C3C4semidirect
21. C2×C4⋊(C3⋊Dic3)D3semidirect
63. C2×C4×C3⋊D3Dic3semidirect
92. C2×(C3⋊D3)⋊C3C4⋊C4semidirect
93. C4×H(3,𝔽3)C22⋊C4semidirect
234. C2×C3⋊D3C4⋊Dic3semidirect
243. C32×C4(C2×D3)⋊C4semidirect
244. C32×C4#217semidirect
257. H(3,𝔽3)#380semidirect
426. C32#4987semidirect
427. C32#5212semidirect
497. C4#88450semidirect

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C2×C4⋊#10842
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884502
#117203342
#105913253
#105913133