Finite Group Databasebeta

← #13155530

Subgroups — #13155530

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6294C211yesmaximal, characteristic
2#11722051C211yesmaximal, characteristic, complemented
3#11722023C211yesmaximal, characteristic
4#11722020C211yesmaximal, characteristic, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6C3×#63557C211yesmaximal, characteristic
7C3×#63555C211yesmaximal, characteristic, complemented
8#10590665—31nomaximal
9#10590661—31nomaximal
10#10590660—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×#6294C411yescharacteristic
13C2×#6294C411yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88813C411yescharacteristic
21#88813C411yescharacteristic
22C4×#624—31no—
23C4×D3×Dic3—31no—
24C4×D3×Dic3—31no—
25#63647—31no—
26#63642—31no—
27#63641—31no—
28C3×(C2×D3)⋊C8—31no—
29C3×(C2×D3)⋊C8—31no—
30C3×#815—31no—
31C3×#815—31no—
32C3×Dic3⋊C8—31no—
33C3×Dic3⋊C8—31no—
34#63572D311yescharacteristic, complemented
35Dic3⋊(C3⋊C8)—31no—
36Dic3⋊(C3⋊C8)—31no—
37Dic3⋊(C3⋊C8)—61no—
38#63557D311yescharacteristic
39(C2×D3)⋊(C3⋊C8)—31no—
40(C2×D3)⋊(C3⋊C8)—31no—
41(C2×D3)⋊(C3⋊C8)—61no—
42#63555D311yescharacteristic, complemented
43#63554—31no—
44#63554—31no—
45#63554—61no—
46C22×C3×C3⋊D3C2×C411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3C2×C411yescharacteristic
49C2×C3×C3⋊Dic3C2×C411yescharacteristic
50C2×C3×C3⋊Dic3C2×C411yescharacteristic
51C2×C3×C3⋊Dic3C2×C411yescharacteristic
52C2×C3×C3⋊Dic3C2×C411yescharacteristic
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55#6294—22no—
56#6294—22no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59C32×C3⋊C8—21no—
60C32×C3⋊C8—21no—
61#4835—91no—
62#4719—91no—
63#4705—91no—
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—61no—
68C2×#624—31no—
69C2×#624—31no—
70C2×D3×Dic3—31no—
71C2×D3×Dic3—31no—
72C2×D3×Dic3—31no—
73C2×D3×Dic3—31no—
74C2×C32×C8—31no—
75C2×C32×C8—31no—
76C2×C32×C8—31no—
77C4×C3⋊Dic3C2×D311yescharacteristic
78C4×C3⋊Dic3C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—61no—
85C2×C3⋊(C3⋊C8)—121no—
86C3×C4×Dic3—31no—
87C3×C4×Dic3—31no—
88C3×C4×Dic3—31no—
89C3×C4×Dic3—31no—
90C3×C4×Dic3—61no—
91C3×C4×Dic3—61no—
92C2×C3×C3⋊C8—21no—
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101Dic32—31no—
102Dic32—31no—
103Dic32—31no—
104Dic32—31no—
105Dic32—31no—
106Dic32—31no—
107C22×C33C22×C411yescharacteristic
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3M1612yes—
111C33×C4#4111yescharacteristic
112C33×C4#4111yescharacteristic
113C3×C3⋊Dic3—21no—
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3M1612yes—
118C2×C4×Dic3—91no—
119C42×D3—91no—
120C42×D3—91no—
121C3×C4⋊C8—91no—
122C3×C22⋊C8—91no—
123C3×#97—91no—
124#830—91no—
125(C2×D3)⋊C8—31no—
126(C2×D3)⋊C8—31no—
127(C2×D3)⋊C8—91no—
128(C2×D3)⋊C8—91no—
129(C2×D3)⋊C8—181no—
130#815—31no—
131#815—31no—
132#815—91no—
133#815—91no—
134#815—181no—
135Dic3⋊C8—31no—
136Dic3⋊C8—31no—
137Dic3⋊C8—91no—
138Dic3⋊C8—91no—
139Dic3⋊C8—181no—
140C4⋊(C3⋊C8)—91no—
141#803—91no—
142C22×C3⋊D3C4×D311yescharacteristic
143C22×C3×D3—31no—
144C22×C3×D3—31no—
145C22×C3×D3—61no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—41no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C3⋊Dic3C4×D311yescharacteristic
154C2×C3⋊Dic3C4×D311yescharacteristic
155C2×C3⋊Dic3C4×D311yescharacteristic
156C2×C3⋊Dic3C4×D311yescharacteristic
157C2×C3⋊Dic3C4×D311yescharacteristic
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—31no—
168C2×C3×Dic3—31no—
169C2×C3×Dic3—31no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—61no—
178C3×C4×D3—61no—
179C3×C4×D3—121no—
180C3×C4×D3—121no—
181#624—62no—
182#624—62no—
183D3×Dic3—62no—
184D3×Dic3—62no—
185D3×Dic3—62no—
186D3×Dic3—62no—
187C32×C8—61no—
188C32×C8—61no—
189C32×C8—61no—
190C3⋊(C3⋊C8)—61no—
191C3⋊(C3⋊C8)—62no—
192C3⋊(C3⋊C8)—61no—
193C3⋊(C3⋊C8)—61no—
194C3⋊(C3⋊C8)—121no—
195C3⋊(C3⋊C8)—121no—
196C3⋊(C3⋊C8)—122no—
197C3×C3⋊C8—22no—
198C3×C3⋊C8—21no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—41no—
205C3×C3⋊C8—41no—
206#432—271noSylow
207C2×C33#11711yescharacteristic
208C2×C33C2×M1611yescharacteristic
209C2×C33C2×M1611yescharacteristic
210C3×C3⋊D3—22no—
211C22×C3×C4—91no—
212C22×Dic3—91no—
213C22×Dic3—91no—
214C2×C4×D3—31no—
215C2×C4×D3—31no—
216C2×C4×D3—61no—
217C2×C4×D3—91no—
218C2×C4×D3—91no—
219C2×C4×D3—91no—
220C2×C4×D3—91no—
221C2×C3×C8—31no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—61no—
228C2×C3×C8—61no—
229C2×C3×C8—61no—
230C3×C42—91no—
231C3×C42—91no—
232C4×Dic3—31no—
233C4×Dic3—31no—
234C4×Dic3—31no—
235C4×Dic3—31no—
236C4×Dic3—61no—
237C4×Dic3—61no—
238C4×Dic3—91no—
239C4×Dic3—91no—
240C4×Dic3—91no—
241C4×Dic3—91no—
242C4×Dic3—91no—
243C4×Dic3—91no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—91no—
246C2×C3⋊C8—91no—
247C2×C3⋊C8—181no—
248C2×C3⋊C8—181no—
249C2×C3⋊C8—181no—
250C2×C3⋊C8—361no—
251C2×C3⋊C8D3211yescharacteristic
252C2×C3⋊C8D3211yescharacteristic
253C2×C3⋊C8D3211yescharacteristic
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—41no—
258C22×C32C2×C4×D311yescharacteristic
259C22×C32C2×C4×D311yescharacteristic
260C22×C32C2×C4×D311yescharacteristic
261C2×C3⋊D3—21no—
262C2×C3⋊D3—21no—
263C2×C3⋊D3ΓL(1,𝔽25)12yes—
264C2×C3×D3—32no—
265C2×C3×D3—32no—
266C2×C3×D3—62no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4#23511yescharacteristic
282C32×C4#23511yescharacteristic
283C32×C4#23711yescharacteristic
284C32×C4#23711yescharacteristic
285C32×C4#23711yescharacteristic
286C32×C4#23911yescharacteristic
287C3⋊Dic3—21no—
288C3⋊Dic3—21no—
289C3⋊Dic3—21no—
290C3⋊Dic3—21no—
291C3⋊Dic3ΓL(1,𝔽25)12yes—
292C3×Dic3—32no—
293C3×Dic3—32no—
294C3×Dic3—62no—
295C3×Dic3—61no—
296C3×Dic3—61no—
297C3×Dic3—61no—
298C3×Dic3—61no—
299C3×Dic3—61no—
300C3×Dic3—61no—
301C3×Dic3—61no—
302C3×Dic3—61no—
303C3×Dic3—121no—
304C3×Dic3—121no—
305C3×Dic3—121no—
306C3×Dic3—121no—
307C2×C42—271no—
308C4⋊C8—91no—
309C4⋊C8—271no—
310C22⋊C8—91no—
311C22⋊C8—271no—
312#97—91no—
313#97—271no—
314C33#43211yescharacteristic, complemented, Sylow
315C23×C3—91no—
316C22×D3—31no—
317C22×D3—31no—
318C22×D3—61no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—41no—
323C2×C3×C4—91no—
324C2×C3×C4—91no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
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—31no—
336C2×Dic3—31no—
337C2×Dic3—31no—
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—31no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—182no—
349C2×Dic3—182no—
350C4×D3—61no—
351C4×D3—61no—
352C4×D3—61no—
353C4×D3—61no—
354C4×D3—121no—
355C4×D3—121no—
356C4×D3—182no—
357C4×D3—182no—
358C4×D3—182no—
359C4×D3—182no—
360C3×C8—62no—
361C3×C8—61no—
362C3×C8—61no—
363C3×C8—61no—
364C3×C8—61no—
365C3×C8—61no—
366C3×C8—61no—
367C3×C8—121no—
368C3×C8—121no—
369C3⋊C8—21no—
370C3⋊C8—21no—
371C3⋊C8—21no—
372C3⋊C8—182no—
373C3⋊C8—181no—
374C3⋊C8—181no—
375C3⋊C8—181no—
376C3⋊C8—361no—
377C3⋊C8—361no—
378C3⋊C8—362no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32#87211yescharacteristic
392C2×C32#87711yescharacteristic
393C2×C32#89211yescharacteristic
394C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
395C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
396C2×C32D3×M1611yescharacteristic
397C2×C32D3×M1611yescharacteristic
398C2×C32D3×M1611yescharacteristic
399C2×C32D3×M1611yescharacteristic
400C3⋊D3—22no—
401C3×D3—62no—
402C3×D3—62no—
403C3×D3—122no—
404C22×C4—91no—
405C22×C4—271no—
406C22×C4—271no—
407C2×C8—31no—
408C2×C8—31no—
409C2×C8—31no—
410C2×C8—271no—
411C42—91no—
412C42—91no—
413C42—271no—
414C42—271no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—41no—
419C22×C3—92no—
420C22×C3—181no—
421C22×C3—181no—
422C22×C3C4×D3211yescharacteristic
423C22×C3C4×D3211yescharacteristic
424C22×C3C4×D3211yescharacteristic
425C2×D3—32no—
426C2×D3—32no—
427C2×D3—62no—
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—61no—
431C2×D3—61no—
432C2×D3—121no—
433C2×D3—121no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—41no—
441C3×C4—41no—
442C3×C4—92no—
443C3×C4—181no—
444C3×C4—181no—
445C3×C4—181no—
446C3×C4—181no—
447C3×C4#379511yescharacteristic
448C3×C4#379611yescharacteristic
449C3×C4#379911yescharacteristic
450C3×C4#380411yescharacteristic
451C3×C4#380411yescharacteristic
452C3×C4#380511yescharacteristic
453Dic3—32no—
454Dic3—32no—
455Dic3—62no—
456Dic3—61no—
457Dic3—61no—
458Dic3—61no—
459Dic3—61no—
460Dic3—61no—
461Dic3—61no—
462Dic3—61no—
463Dic3—61no—
464Dic3—121no—
465Dic3—121no—
466Dic3—121no—
467Dic3—121no—
468C32—21no—
469C32—21no—
470C32—21no—
471C32—41no—
472C32#470511yescharacteristic, complemented
473C32#471911yescharacteristic, complemented
474C32#483511yescharacteristic, complemented
475C23—91no—
476C2×C4—91no—
477C2×C4—91no—
478C2×C4—91no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4—542no—
484C2×C4—542no—
485C2×C4D3311yescharacteristic
486C8—61no—
487C8—61no—
488C8—61no—
489C8—541no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—182no—
503C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
504C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
505C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
506C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
507C2×C3#6379111yescharacteristic
508C2×C3#6379111yescharacteristic
509C2×C3#6383611yescharacteristic
510C2×C3#6387711yescharacteristic
511C2×C3#6387811yescharacteristic
512D3—62no—
513D3—62no—
514D3—122no—
515C22—92no—
516C22—181no—
517C22—181no—
518C22#8894911yescharacteristic
519C4—92no—
520C4—181no—
521C4—181no—
522C4—181no—
523C4—181no—
524C4#8896111yescharacteristic
525C4#8896211yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3#1059066011yescharacteristic, complemented
531C3#1059066111yescharacteristic, complemented
532C3#1059066511yescharacteristic, complemented
533C2—182no—
534C2#1172282011yescharacteristic
535C2#1172282011yescharacteristic
536C2#1172292211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C2×C4
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11722051C2semidirect
4. #11722020C2semidirect
5. C3×#63572C2semidirect
7. C3×#63555C2semidirect
34. #63572D3semidirect
42. #63555D3semidirect
314. C33#432semidirect
472. C32#4705semidirect
473. C32#4719semidirect
474. C32#4835semidirect
530. C3#10590660semidirect
531. C3#10590661semidirect
532. C3#10590665semidirect

Maximal subgroups

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

subgroupindex
C4×#62942
#117220512
#117220232
#117220202
C3×#635722
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