Finite Group Databasebeta

← #13136402

Subgroups — #13136402

Order 1728.

476 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C27×#123C211yesmaximal, characteristic, complemented
2#11720188C211yesmaximal, characteristic, complemented
3#11720187C211yesmaximal, characteristic, complemented
4C4×#6171C211yesmaximal, characteristic, complemented
5C4×#6171C211yesmaximal, characteristic, complemented
6#11720147C211yesmaximal, characteristic, complemented
7#11720142C211yesmaximal, characteristic, complemented
8C4⋊C4×D27C211yesmaximal, characteristic, complemented
9#11720138C211yesmaximal, characteristic, complemented
10#11720133C211yesmaximal, characteristic, complemented
11#11720133C211yesmaximal, characteristic, complemented
12#11720132C211yesmaximal, characteristic, complemented
13#6171⋊C4C211yesmaximal, characteristic, complemented
14#11720128C211yesmaximal, characteristic, complemented
15#11720126C211yesmaximal, characteristic, complemented
16#10589763—31nomaximal
17C2×D4×C27C2211yescharacteristic
18C22×C4×C27C2211yescharacteristic
19C2×#6171C2211yescharacteristic
20C2×#6171C2211yescharacteristic
21C2×#6171C2211yescharacteristic
22C2×#6171C2211yescharacteristic
23C22×Dic27C2211yescharacteristic
24C2×D108C2211yescharacteristic
25C2×C4×D27C2211yescharacteristic
26C2×C4×D27C2211yescharacteristic
27C2×C4×D27C2211yescharacteristic
28C2×C4×D27C2211yescharacteristic
29C4⋊C4×C27C2211yescharacteristic
30C4⋊C4×C27C2211yescharacteristic
31C22⋊C4×C27C2211yescharacteristic
32C22⋊C4×C27C2211yescharacteristic
33C22⋊C4×C27C2211yescharacteristic
34#88372C2211yescharacteristic
35#88372C2211yescharacteristic
36#88372C2211yescharacteristic
37(C2×D27)⋊C4C2211yescharacteristic
38(C2×D27)⋊C4C2211yescharacteristic
39(C2×D27)⋊C4C2211yescharacteristic
40(C2×D27)⋊C4C2211yescharacteristic
41C4⋊Dic27C2211yescharacteristic
42C4⋊Dic27C2211yescharacteristic
43Dic27⋊C4C2211yescharacteristic
44Dic27⋊C4C2211yescharacteristic
45Dic27⋊C4C2211yescharacteristic
46Dic27⋊C4C2211yescharacteristic
47Dic27⋊C4C2211yescharacteristic
48Dic27⋊C4C2211yescharacteristic
49C4×Dic27C2211yescharacteristic
50C4×Dic27C2211yescharacteristic
51C4×Dic27C2211yescharacteristic
52C9×#123D311yescharacteristic
53#63501—31no—
54#63500—31no—
55C4×#611—31no—
56C4×#611—31no—
57#63460—31no—
58#63455—31no—
59C4⋊C4×D9—31no—
60#63451—31no—
61#63446—31no—
62#63446—31no—
63#63445—31no—
64#611⋊C4—31no—
65#63441—31no—
66#63439—31no—
67C23×C27C2311yescharacteristic
68C23×C27C2311yescharacteristic
69C22×D27C2311yescharacteristic
70C22×D27C2311yescharacteristic
71D4×C27—41no—
72C2×C4×C27—41no—
73C2×C4×C27C2311yescharacteristic
74C2×C4×C27C2311yescharacteristic
75C2×C4×C27C2311yescharacteristic
76C2×C4×C27C2311yescharacteristic
77C2×C4×C27C2311yescharacteristic
78#6171—22no—
79#6171—22no—
80#6171—22no—
81#6171—41no—
82C2×Dic27—22no—
83C2×Dic27C2311yescharacteristic
84C2×Dic27C2311yescharacteristic
85C2×Dic27C2311yescharacteristic
86C2×Dic27C2311yescharacteristic
87C2×Dic27C2311yescharacteristic
88C2×Dic27C2311yescharacteristic
89D108—22no—
90C4×D27—22no—
91C4×D27—41no—
92C4×D27—41no—
93C4×D27—41no—
94#5661—91no—
95C2×D4×C9C2×D311yescharacteristic
96C22×C4×C9C2×D311yescharacteristic
97C2×#611—31no—
98C2×#611—31no—
99C2×#611—31no—
100C2×#611—31no—
101C22×Dic9—31no—
102C2×D36—31no—
103C2×C4×D9—31no—
104C2×C4×D9—31no—
105C2×C4×D9—31no—
106C2×C4×D9—31no—
107C9×C4⋊C4C2×D311yescharacteristic
108C9×C4⋊C4C2×D311yescharacteristic
109C9×C22⋊C4C2×D311yescharacteristic
110C9×C22⋊C4C2×D311yescharacteristic
111C9×C22⋊C4C2×D311yescharacteristic
112#3676—31no—
113#3676—31no—
114#3676—31no—
115(C2×D9)⋊C4—31no—
116(C2×D9)⋊C4—31no—
117(C2×D9)⋊C4—31no—
118(C2×D9)⋊C4—31no—
119C4⋊Dic9—31no—
120C4⋊Dic9—31no—
121Dic9⋊C4—31no—
122Dic9⋊C4—31no—
123Dic9⋊C4—31no—
124Dic9⋊C4—31no—
125Dic9⋊C4—31no—
126Dic9⋊C4—31no—
127C4×Dic9—31no—
128C4×Dic9—31no—
129C4×Dic9—31no—
130C22×C27—21no—
131C22×C27—21no—
132C22×C27—21no—
133C22×C27—21no—
134C22×C27—21no—
135C22×C27—21no—
136C22×C27C2411yescharacteristic
137C2×D27—21no—
138C2×D27—21no—
139C2×D27—21no—
140C2×D27—21no—
141C2×D27—21no—
142C2×D27#4112yes—
143C4×C27—21no—
144C4×C27—21no—
145C4×C27—21no—
146C4×C27—21no—
147C4×C27—21no—
148Dic27—21no—
149Dic27—21no—
150Dic27—21no—
151Dic27—21no—
152Dic27—21no—
153Dic27#4112yes—
154C3×#123D911yescharacteristic
155#939—91no—
156#938—91no—
157C4×#67—91no—
158C4×#67—91no—
159#898—91no—
160#893—91no—
161D3×C4⋊C4—91no—
162#889—91no—
163#884—91no—
164#884—91no—
165#883—91no—
166#67⋊C4—91no—
167#879—91no—
168#877—91no—
169C23×C9C22×D311yescharacteristic
170C23×C9C22×D311yescharacteristic
171C22×D9—31no—
172C22×D9—31no—
173D4×C9—41no—
174C2×C4×C9—41no—
175C2×C4×C9C22×D311yescharacteristic
176C2×C4×C9C22×D311yescharacteristic
177C2×C4×C9C22×D311yescharacteristic
178C2×C4×C9C22×D311yescharacteristic
179C2×C4×C9C22×D311yescharacteristic
180#611—62no—
181#611—62no—
182#611—62no—
183#611—121no—
184C2×Dic9—31no—
185C2×Dic9—31no—
186C2×Dic9—31no—
187C2×Dic9—31no—
188C2×Dic9—31no—
189C2×Dic9—31no—
190C2×Dic9—62no—
191D36—62no—
192C4×D9—62no—
193C4×D9—121no—
194C4×D9—121no—
195C4×D9—121no—
196#553—271noSylow
197C2×C27—41no—
198C2×C27—41no—
199C2×C27C2×#4111yescharacteristic
200C2×C27C2×#4111yescharacteristic
201C2×C27H(5,𝔽2)11yescharacteristic
202D27—22no—
203D27—41no—
204C2×C3×D4C2×D911yescharacteristic
205C22×C3×C4C2×D911yescharacteristic
206C2×#67—91no—
207C2×#67—91no—
208C2×#67—91no—
209C2×#67—91no—
210C22×Dic3—91no—
211C2×D12—91no—
212C2×C4×D3—91no—
213C2×C4×D3—91no—
214C2×C4×D3—91no—
215C2×C4×D3—91no—
216C3×C4⋊C4C2×D911yescharacteristic
217C3×C4⋊C4C2×D911yescharacteristic
218C3×C22⋊C4C2×D911yescharacteristic
219C3×C22⋊C4C2×D911yescharacteristic
220C3×C22⋊C4C2×D911yescharacteristic
221#217—91no—
222#217—91no—
223#217—91no—
224(C2×D3)⋊C4—91no—
225(C2×D3)⋊C4—91no—
226(C2×D3)⋊C4—91no—
227(C2×D3)⋊C4—91no—
228C4⋊Dic3—91no—
229C4⋊Dic3—91no—
230Dic3⋊C4—91no—
231Dic3⋊C4—91no—
232Dic3⋊C4—91no—
233Dic3⋊C4—91no—
234Dic3⋊C4—91no—
235Dic3⋊C4—91no—
236C4×Dic3—91no—
237C4×Dic3—91no—
238C4×Dic3—91no—
239C22×C9—21no—
240C22×C9—21no—
241C22×C9—21no—
242C22×C9—21no—
243C22×C9—21no—
244C22×C9—21no—
245C22×C9C23×D311yescharacteristic
246C2×D9—32no—
247C2×D9—61no—
248C2×D9—61no—
249C2×D9—61no—
250C2×D9—61no—
251C2×D9—61no—
252C4×C9—21no—
253C4×C9—21no—
254C4×C9—21no—
255C4×C9—21no—
256C4×C9—21no—
257Dic9—32no—
258Dic9—61no—
259Dic9—61no—
260Dic9—61no—
261Dic9—61no—
262Dic9—61no—
263#126—271no—
264#126—271no—
265#125—271no—
266#123—271no—
267#123D2711yescharacteristic, complemented
268#121—271no—
269#121—271no—
270#121—271no—
271#121—271no—
272C4×D4—271no—
273C4×D4—271no—
274C4×D4—271no—
275C4×D4—271no—
276#117—271no—
277C2×C4⋊C4—271no—
278C27#55311yescharacteristic, complemented, Sylow
279C23×C3C22×D911yescharacteristic
280C23×C3C22×D911yescharacteristic
281C22×D3—91no—
282C22×D3—91no—
283C3×D4—41no—
284C2×C3×C4—41no—
285C2×C3×C4C22×D911yescharacteristic
286C2×C3×C4C22×D911yescharacteristic
287C2×C3×C4C22×D911yescharacteristic
288C2×C3×C4C22×D911yescharacteristic
289C2×C3×C4C22×D911yescharacteristic
290#67—182no—
291#67—182no—
292#67—182no—
293#67—361no—
294C2×Dic3—91no—
295C2×Dic3—91no—
296C2×Dic3—91no—
297C2×Dic3—91no—
298C2×Dic3—91no—
299C2×Dic3—91no—
300C2×Dic3—182no—
301D12—182no—
302C4×D3—182no—
303C4×D3—361no—
304C4×D3—361no—
305C4×D3—361no—
306C2×C9—41no—
307C2×C9—41no—
308C2×C9#100411yescharacteristic
309C2×C9D3×#4111yescharacteristic
310C2×C9D3×#4111yescharacteristic
311D9—62no—
312D9—121no—
313C2×D4—271no—
314C2×D4—271no—
315C2×D4—271no—
316C2×D4—271no—
317C2×D4—271no—
318C2×D4C2×D2711yescharacteristic
319C22×C4—271no—
320C22×C4—271no—
321C22×C4—271no—
322C22×C4—271no—
323C22×C4—271no—
324C22×C4C2×D2711yescharacteristic
325C4⋊C4—271no—
326C4⋊C4—271no—
327C4⋊C4—271no—
328C4⋊C4—271no—
329C4⋊C4—271no—
330C4⋊C4—271no—
331C4⋊C4—271no—
332C4⋊C4—271no—
333C4⋊C4C2×D2711yescharacteristic
334C4⋊C4C2×D2711yescharacteristic
335C22⋊C4—271no—
336C22⋊C4—271no—
337C22⋊C4—271no—
338C22⋊C4—271no—
339C22⋊C4—271no—
340C22⋊C4—271no—
341C22⋊C4—271no—
342C22⋊C4C2×D2711yescharacteristic
343C22⋊C4C2×D2711yescharacteristic
344C22⋊C4C2×D2711yescharacteristic
345C42—271no—
346C42—271no—
347C42—271no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3C23×D911yescharacteristic
355C2×D3—92no—
356C2×D3—181no—
357C2×D3—181no—
358C2×D3—181no—
359C2×D3—181no—
360C2×D3—181no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366Dic3—92no—
367Dic3—181no—
368Dic3—181no—
369Dic3—181no—
370Dic3—181no—
371Dic3—181no—
372C9#566111yescharacteristic
373C23—271no—
374C23—271no—
375C23C22×D2711yescharacteristic
376C23C22×D2711yescharacteristic
377D4—41no—
378D4—542no—
379D4—542no—
380D4—542no—
381D4—542no—
382D4—1081no—
383C2×C4—41no—
384C2×C4—271no—
385C2×C4—271no—
386C2×C4—271no—
387C2×C4—271no—
388C2×C4—271no—
389C2×C4—271no—
390C2×C4—542no—
391C2×C4—542no—
392C2×C4—1081no—
393C2×C4—1081no—
394C2×C4—1081no—
395C2×C4C22×D2711yescharacteristic
396C2×C4C22×D2711yescharacteristic
397C2×C4C22×D2711yescharacteristic
398C2×C4C22×D2711yescharacteristic
399C2×C4C22×D2711yescharacteristic
400C2×C3—41no—
401C2×C3—41no—
402C2×C3#6371011yescharacteristic
403C2×C3#41×D911yescharacteristic
404C2×C3#41×D911yescharacteristic
405D3—182no—
406D3—361no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—272no—
414C22—541no—
415C22—541no—
416C22—541no—
417C22—541no—
418C22—541no—
419C22C23×D2711yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—272no—
426C4—541no—
427C4—541no—
428C4—541no—
429C4—541no—
430C4—541no—
431C3#1058976311yescharacteristic
432C2—41no—
433C2—41no—
434C2—542no—
435C2—1081no—
436C2#1172067011yescharacteristic
437C2#41×D2711yescharacteristic
438C2#41×D2711yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#123C2
socleC22×C3C23×D9
hypercenter#123D27
hypocenterC27#553
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C27×#123C2semidirect
2. #11720188C2semidirect
3. #11720187C2semidirect
4. C4×#6171C2semidirect
5. C4×#6171C2semidirect
6. #11720147C2semidirect
7. #11720142C2semidirect
8. C4⋊C4×D27C2semidirect
9. #11720138C2semidirect
10. #11720133C2semidirect
11. #11720133C2semidirect
12. #11720132C2semidirect
13. #6171⋊C4C2semidirect
14. #11720128C2semidirect
15. #11720126C2semidirect
267. #123D27semidirect
278. C27#553semidirect

Maximal subgroups

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

subgroupindex
C27×#1232
#117201882
#117201872
C4×#61712
#117201472
#117201422
C4⋊C4×D272
#117201382
#117201332
#117201322
#6171⋊C42
#117201282
#117201262
#105897633