Finite Group Databasebeta

← #13155536

Subgroups — #13155536

Order 1728.

538 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63874C211yesmaximal, characteristic
2#11722072C211yesmaximal, characteristic
3Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4#11722023C211yesmaximal, characteristic
5Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590647—31nomaximal
9#10590265—31nomaximal
10#10590253—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4×Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C3×#3724C411yescharacteristic
19C3×#3724C411yescharacteristic
20C3×#3722C411yescharacteristic
21C3×#3722C411yescharacteristic
22C3×#872—31no—
23C3×#872—31no—
24#63874D311yescharacteristic
25#63647—31no—
26C4⋊(C3⋊(C3⋊C8))—31no—
27C4×C3⋊(C3⋊C8)—31no—
28C3×#815—31no—
29C3×Dic3⋊C8—31no—
30C3×C4⋊(C3⋊C8)—31no—
31C3×#803—31no—
32Dic3⋊(C3⋊C8)—31no—
33Dic3⋊(C3⋊C8)—31no—
34Dic3⋊(C3⋊C8)—61no—
35Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
36#63557—31no—
37(C2×D3)⋊(C3⋊C8)—31no—
38(C2×D3)⋊(C3⋊C8)—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40(C2×D3)⋊(C3⋊C8)—31no—
41(C2×D3)⋊(C3⋊C8)—61no—
42#63554D311yescharacteristic, complemented
43Dic3×C3⋊C8—31no—
44Dic3×C3⋊C8—31no—
45Dic3×C3⋊C8—61no—
46C22×C3×C3⋊D3C2×C411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3C2×C411yescharacteristic
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C2×C32×Dic3C2×C411yescharacteristic
52C2×C32×Dic3C2×C411yescharacteristic
53C2×C32×Dic3C2×C411yescharacteristic
54C2×C32×Dic3C2×C411yescharacteristic
55C3⋊(C3⋊(C3⋊C8))—21no—
56C3×C3⋊(C3⋊C8)—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5000—91no—
60#4840—91no—
61#4709—91no—
62C2×C4×C3⋊D3C2×D311yescharacteristic
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—61no—
66C2×C32×C8—31no—
67C32×C42—31no—
68C32×C42—31no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—61no—
73C2×C3⋊(C3⋊C8)—61no—
74C2×C3⋊(C3⋊C8)—61no—
75C2×C3⋊(C3⋊C8)—121no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×(C2×D3)⋊C4—31no—
79C3×(C2×D3)⋊C4—31no—
80C3×(C2×D3)⋊C4—31no—
81C3×(C2×D3)⋊C4—31no—
82C3×Dic3⋊C4—31no—
83C3×Dic3⋊C4—31no—
84C3×Dic3⋊C4—31no—
85C3×Dic3⋊C4—31no—
86C3×C4×Dic3—21no—
87C3×C4×Dic3—21no—
88C3×C4×Dic3C2×D311yescharacteristic
89C3×C4×Dic3C2×D311yescharacteristic
90C3×C4×Dic3C2×D311yescharacteristic
91C3×C4×Dic3C2×D311yescharacteristic
92C2×C3×C3⋊C8—21no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—61no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101#3724Dic311yescharacteristic
102#3724Dic311yescharacteristic
103#3722Dic311yescharacteristic
104#3722Dic311yescharacteristic
105C22×C33C22×C411yescharacteristic
106C2×C3×C3⋊D3—21no—
107C2×C3×C3⋊D3—21no—
108C2×C3×C3⋊D3—21no—
109C33×C4#4111yescharacteristic
110C33×C4#4111yescharacteristic
111C3×C3⋊Dic3—21no—
112C32×Dic3—21no—
113C32×Dic3—21no—
114C32×Dic3—21no—
115C32×Dic3—21no—
116C3×#117—91no—
117#872—31no—
118#872—31no—
119C3×C4⋊C8—91no—
120C3×#97—91no—
121#830—91no—
122#830—91no—
123(C2×D3)⋊C8—91no—
124(C2×D3)⋊C8—91no—
125(C2×D3)⋊C8—91no—
126(C2×D3)⋊C8—91no—
127(C2×D3)⋊C8—181no—
128#815—31no—
129Dic3⋊C8—31no—
130Dic3⋊C8—91no—
131C8×Dic3—91no—
132C4⋊(C3⋊C8)—31no—
133C4⋊(C3⋊C8)—91no—
134C4⋊(C3⋊C8)—91no—
135C4⋊(C3⋊C8)—181no—
136#803—31no—
137C4×C3⋊C8—91no—
138C4×C3⋊C8—91no—
139C4×C3⋊C8—181no—
140C22×C3⋊D3C2×Dic311yescharacteristic
141C22×C3×D3—31no—
142C22×C3×D3—31no—
143C22×C3×D3—61no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—31no—
148C2×C32×C4—31no—
149C2×C32×C4—31no—
150C2×C32×C4—31no—
151C2×C32×C4—41no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3C2×Dic311yescharacteristic
156C4×C3⋊D3—21no—
157C4×C3⋊D3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3C4×D311yescharacteristic
166C2×C3×Dic3C4×D311yescharacteristic
167C2×C3×Dic3C4×D311yescharacteristic
168C2×C3×Dic3C4×D311yescharacteristic
169C2×C3×Dic3C2×Dic311yescharacteristic
170C2×C3×Dic3C2×Dic311yescharacteristic
171C2×C3×Dic3C2×Dic311yescharacteristic
172C2×C3×Dic3C2×Dic311yescharacteristic
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—121no—
178C3×C4×D3—121no—
179C32×C8—61no—
180C3⋊(C3⋊C8)—21no—
181C3⋊(C3⋊C8)—21no—
182C3⋊(C3⋊C8)—61no—
183C3⋊(C3⋊C8)—62no—
184C3⋊(C3⋊C8)—61no—
185C3⋊(C3⋊C8)—61no—
186C3⋊(C3⋊C8)—121no—
187C3⋊(C3⋊C8)—121no—
188C3⋊(C3⋊C8)—122no—
189C3×C3⋊C8—21no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—41no—
192C3×C3⋊C8—62no—
193C3×C3⋊C8—61no—
194C3×C3⋊C8—61no—
195C3×C3⋊C8—61no—
196C3×C3⋊C8—61no—
197C3×C3⋊C8—121no—
198#433—271noSylow
199C2×C33#11711yescharacteristic
200C2×C33#13111yescharacteristic
201C2×C33#13111yescharacteristic
202C3×C3⋊D3—41no—
203C22×C3×C4—91no—
204C2×C4×D3—31no—
205C2×C4×D3—31no—
206C2×C4×D3—61no—
207C2×C3×C8—31no—
208C2×C3×C8—31no—
209C2×C3×C8—61no—
210C2×C3×C8—91no—
211C2×C3×C8—91no—
212C3×C4⋊C4—91no—
213C3×C4⋊C4—91no—
214C3×C22⋊C4—91no—
215C3×C22⋊C4—91no—
216C3×C42—31no—
217C3×C42—31no—
218C3×C42—31no—
219C3×C42—31no—
220C3×C42—61no—
221C3×C42—61no—
222(C2×D3)⋊C4—31no—
223(C2×D3)⋊C4—31no—
224(C2×D3)⋊C4—31no—
225(C2×D3)⋊C4—31no—
226Dic3⋊C4—31no—
227Dic3⋊C4—31no—
228Dic3⋊C4—31no—
229Dic3⋊C4—31no—
230C4×Dic3D3211yescharacteristic
231C4×Dic3D3211yescharacteristic
232C2×C3⋊C8—31no—
233C2×C3⋊C8—31no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—31no—
236C2×C3⋊C8—61no—
237C2×C3⋊C8—61no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—91no—
240C2×C3⋊C8—91no—
241C2×C3⋊C8—181no—
242C2×C3⋊C8—181no—
243C2×C3⋊C8—181no—
244C2×C3⋊C8—361no—
245C2×C3⋊C8D3211yescharacteristic
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32C2×C4×D311yescharacteristic
251C22×C32C2×C4×D311yescharacteristic
252C22×C32C22×Dic311yescharacteristic
253C2×C3⋊D3—21no—
254C2×C3⋊D3—21no—
255C2×C3⋊D3—21no—
256C2×C3×D3—62no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—61no—
274C32×C4—61no—
275C32×C4—61no—
276C32×C4—61no—
277C32×C4#23511yescharacteristic
278C32×C4#23511yescharacteristic
279C32×C4#23511yescharacteristic
280C32×C4#23511yescharacteristic
281C32×C4#23711yescharacteristic
282C32×C4#23911yescharacteristic
283C3⋊Dic3—21no—
284C3×Dic3—22no—
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3—21no—
288C3×Dic3—21no—
289C3×Dic3—21no—
290C3×Dic3—21no—
291C3×Dic3—21no—
292C3×Dic3—21no—
293C3×Dic3—21no—
294C3×Dic3—21no—
295C3×Dic3—41no—
296C3×Dic3—61no—
297C3×Dic3—61no—
298C3×Dic3—121no—
299#117—91no—
300C4⋊C8—91no—
301C4⋊C8—271no—
302C22⋊C8—271no—
303C22⋊C8—271no—
304#97—91no—
305C4×C8—271no—
306C33#43311yescharacteristic, complemented, Sylow
307C23×C3—91no—
308C22×D3—31no—
309C22×D3—31no—
310C22×D3—61no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—41no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—91no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4C2×D3211yescharacteristic
331C2×C3×C4C2×D3211yescharacteristic
332C2×C3×C4C2×D3211yescharacteristic
333C2×Dic3—31no—
334C2×Dic3—31no—
335C2×Dic3—61no—
336C2×Dic3D3×Dic311yescharacteristic
337C2×Dic3D3×Dic311yescharacteristic
338C2×Dic3D3×Dic311yescharacteristic
339C2×Dic3D3×Dic311yescharacteristic
340C4×D3—61no—
341C4×D3—61no—
342C4×D3—61no—
343C4×D3—61no—
344C4×D3—121no—
345C4×D3—121no—
346C3×C8—61no—
347C3×C8—61no—
348C3×C8—121no—
349C3×C8—181no—
350C3×C8—181no—
351C3⋊C8—21no—
352C3⋊C8—62no—
353C3⋊C8—61no—
354C3⋊C8—61no—
355C3⋊C8—61no—
356C3⋊C8—61no—
357C3⋊C8—121no—
358C3⋊C8—182no—
359C3⋊C8—181no—
360C3⋊C8—181no—
361C3⋊C8—181no—
362C3⋊C8—361no—
363C3⋊C8—361no—
364C3⋊C8—362no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32#87211yescharacteristic
378C2×C32#89211yescharacteristic
379C2×C32#90111yescharacteristic
380C2×C32#90711yescharacteristic
381C2×C32#90711yescharacteristic
382C2×C32#90711yescharacteristic
383C2×C32#92611yescharacteristic
384C2×C32#94811yescharacteristic
385C2×C32#94811yescharacteristic
386C3⋊D3—41no—
387C3×D3—122no—
388C3×D3—121no—
389C3×D3—121no—
390C22×C4—91no—
391C2×C8—31no—
392C2×C8—91no—
393C2×C8—91no—
394C2×C8—271no—
395C4⋊C4—91no—
396C4⋊C4—91no—
397C22⋊C4—91no—
398C22⋊C4—91no—
399C42—31no—
400C42—31no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3—181no—
406C22×C3—181no—
407C22×C3—181no—
408C22×C3C4×D3211yescharacteristic
409C22×C3C2×D3×Dic311yescharacteristic
410C22×C3C2×D3×Dic311yescharacteristic
411C2×D3—62no—
412C2×D3—61no—
413C2×D3—61no—
414C2×D3—61no—
415C2×D3—61no—
416C2×D3—61no—
417C2×D3—61no—
418C2×D3—121no—
419C2×D3—121no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—62no—
429C3×C4—62no—
430C3×C4—62no—
431C3×C4—61no—
432C3×C4—61no—
433C3×C4—61no—
434C3×C4—61no—
435C3×C4—61no—
436C3×C4—61no—
437C3×C4—61no—
438C3×C4—61no—
439C3×C4—121no—
440C3×C4—181no—
441C3×C4#379511yescharacteristic
442C3×C4#379811yescharacteristic
443C3×C4#379811yescharacteristic
444C3×C4#379911yescharacteristic
445C3×C4#379911yescharacteristic
446C3×C4#380411yescharacteristic
447Dic3—21no—
448Dic3—21no—
449Dic3—21no—
450Dic3—21no—
451Dic3—61no—
452Dic3—61no—
453Dic3—121no—
454C32—21no—
455C32—21no—
456C32—21no—
457C32—41no—
458C32#470911yescharacteristic, complemented
459C32#484011yescharacteristic, complemented
460C32#500011yescharacteristic, complemented
461C23—91no—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—31no—
465C2×C4—31no—
466C2×C4—91no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4D3311yescharacteristic
470C8—61no—
471C8—181no—
472C8—181no—
473C8—541no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—361no—
487C2×C3#6380411yescharacteristic
488C2×C3#6380511yescharacteristic
489C2×C3#6381411yescharacteristic
490C2×C3#6381511yescharacteristic
491C2×C3#6381511yescharacteristic
492C2×C3#6381511yescharacteristic
493C2×C3#6383611yescharacteristic
494C2×C3#6384111yescharacteristic
495C2×C3#6385511yescharacteristic
496D3—122no—
497D3—121no—
498D3—121no—
499C22—181no—
500C22—181no—
501C22—181no—
502C22D32×Dic311yescharacteristic
503C4—61no—
504C4—61no—
505C4—61no—
506C4—61no—
507C4—181no—
508C4#8896011yescharacteristic
509C4#8896511yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059025311yescharacteristic, complemented
515C3#1059026511yescharacteristic, complemented
516C3#1059064711yescharacteristic, complemented
517C2—361no—
518C2#1172282211yescharacteristic
519C2#1172282311yescharacteristic
520C2#1172291511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C2×C4
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
5. Dic3×C3⋊(C3⋊C8)C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×#63554C2semidirect
35. Dic3⋊(C3⋊C8)D3semidirect
42. #63554D3semidirect
306. C33#433semidirect
458. C32#4709semidirect
459. C32#4840semidirect
460. C32#5000semidirect
514. C3#10590253semidirect
515. C3#10590265semidirect
516. C3#10590647semidirect

Maximal subgroups

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

subgroupindex
C3×#638742
#117220722
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Dic3×C3⋊(C3⋊C8)2
C3×Dic3⋊(C3⋊C8)2
C3×#635542
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