Finite Group Databasebeta

← #13150056

Subgroups — #13150056

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721813C211yesmaximal, characteristic
2C9×#926C211yesmaximal, characteristic
3C3×#63486C211yesmaximal, characteristic
4#11721247C212yesmaximal
5#11721246C212yesmaximal
6#11720932C211yesmaximal, characteristic, complemented
7#11720927C211yesmaximal, characteristic, complemented
8#11720918C212yesmaximal, complemented
9#11720917C212yesmaximal, complemented
10#11720912C211yesmaximal, characteristic, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10594824—31nomaximal
13#10589620—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17#88551C2212yes—
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19#88534C2212yes—
20C9×#217C2211yescharacteristic
21C9×#217C2211yescharacteristic
22C3×#3676C2212yes—
23C9×C4⋊Dic3C2211yescharacteristic
24C9×C4⋊Dic3C2211yescharacteristic
25C4×C9×Dic3C2211yescharacteristic
26C4×C9×Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2212yes—
29Dic3⋊Dic9C2212yes—
30Dic3⋊Dic9C2212yes—
31#88442C2212yes—
32#88442C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37#64133—31no—
38C3×#926D311yescharacteristic
39C3×#924—31no—
40#63956—32no—
41#63956—32no—
42#63869—31no—
43#63866—31no—
44#63847—32no—
45#63846—32no—
46#63844—31no—
47Dic3×Dic6—31no—
48C9×#117—31no—
49#63486—31no—
50#63486—61no—
51#63486D311yescharacteristic
52#63451—32no—
53#63451—32no—
54#63441—32no—
55#63441—32no—
56#63432—31no—
57#63431—31no—
58C4×Dic18—31no—
59C4×Dic18—31no—
60C23×C3×C9C2311yescharacteristic
61C2×C3×C4×C9—22no—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2312yes—
66#6226—22no—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2312yes—
73C3×Dic18—22no—
74#5625—91no—
75#5518—91no—
76C22×C32×C4C2×D311yescharacteristic
77C2×#634—31no—
78C2×C3×Dic6—31no—
79#3757—32no—
80C4⋊(C3⋊Dic3)—31no—
81#3750—32no—
82C3×#217—32no—
83C3×#217C2×D311yescharacteristic
84C3×#217C2×D311yescharacteristic
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×Dic3⋊C4—32no—
89C3×C4×Dic3C2×D311yescharacteristic
90C3×C4×Dic3C2×D311yescharacteristic
91#3724—32no—
92#3724—32no—
93Dic3⋊Dic3—32no—
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97Dic32—32no—
98Dic32—32no—
99C22×C4×C9—21no—
100C22×C4×C9C2×D311yescharacteristic
101C2×Dic18—31no—
102C2×Dic18—61no—
103C2×Dic18C2×D311yescharacteristic
104C9×C4⋊C4—31no—
105C9×C4⋊C4—31no—
106C9×C22⋊C4—31no—
107C9×C22⋊C4—31no—
108C42×C9—31no—
109C42×C9—31no—
110#3676—32no—
111#3676—62no—
112#3676C2×D312yes—
113C4⋊Dic9—31no—
114C4⋊Dic9—32no—
115C4⋊Dic9—32no—
116C4⋊Dic9—61no—
117C4⋊Dic9C2×D311yescharacteristic
118Dic9⋊C4—32no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—62no—
124Dic9⋊C4C2×D312yes—
125C4×Dic9—32no—
126C4×Dic9—32no—
127C22×C3×C9—21no—
128C22×C3×C9—21no—
129C22×C3×C9—21no—
130C22×C3×C9C2411yescharacteristic
131C3×C4×C9—21no—
132C3×C4×C9#4112yes—
133C3⋊Dic9—22no—
134C3⋊Dic9—22no—
135C9×Dic3—21no—
136C9×Dic3—21no—
137C9×Dic3—21no—
138C9×Dic3—21no—
139C3×Dic9—22no—
140C3×Dic9—22no—
141C3×C22⋊Q8—91no—
142C3×#117—31no—
143Q8×Dic3—91no—
144#944—91no—
145#926D911yescharacteristic
146#924—31no—
147#924—91no—
148#924—91no—
149#924—181no—
150#890—92no—
151#890—91no—
152#889—92no—
153#889—92no—
154#889—92no—
155#887—91no—
156#879—92no—
157#879—92no—
158#879—92no—
159#879—92no—
160#870—91no—
161#869—91no—
162C4×Dic6—91no—
163C4×Dic6—91no—
164C23×C32C22×D311yescharacteristic
165C2×C32×C4—22no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—32no—
170#634—62no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3C22×D311yescharacteristic
174C2×C3×Dic3C22×D311yescharacteristic
175C2×C3×Dic3C22×D311yescharacteristic
176C2×C3×Dic3C22×D311yescharacteristic
177C3×Dic6—62no—
178C23×C9—21no—
179C23×C9C22×D311yescharacteristic
180C2×C4×C9—22no—
181C2×C4×C9—21no—
182C2×C4×C9—21no—
183C2×C4×C9—31no—
184C2×C4×C9—31no—
185C2×C4×C9—31no—
186C2×C4×C9—31no—
187C2×C4×C9—42no—
188C2×C4×C9C22×D311yescharacteristic
189C2×C4×C9C22×D311yescharacteristic
190C2×Dic9—32no—
191C2×Dic9—32no—
192C2×Dic9—62no—
193C2×Dic9—62no—
194C2×Dic9C22×D312yes—
195C2×Dic9C22×D312yes—
196Dic18—22no—
197Dic18—62no—
198Dic18—122no—
199#541—271noSylow
200C2×C3×C9—41no—
201C2×C3×C9C2×#4111yescharacteristic
202C2×C3×C92−1+411yescharacteristic
203C2×C3×C92−1+411yescharacteristic
204C2×C3×Q8—91no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D911yescharacteristic
207C22×C3×C4D3211yescharacteristic
208C2×Dic6—31no—
209C2×Dic6—91no—
210C2×Dic6—91no—
211C2×Dic6—181no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—91no—
215C3×C4⋊C4—92no—
216C3×C22⋊C4—31no—
217C3×C22⋊C4—31no—
218C3×C22⋊C4—92no—
219C3×C42—31no—
220C3×C42—31no—
221#217—32no—
222#217—92no—
223#217—92no—
224#217—182no—
225#217C2×D911yescharacteristic
226#217C2×D911yescharacteristic
227C4⋊Dic3—31no—
228C4⋊Dic3—92no—
229C4⋊Dic3—92no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—92no—
233C4⋊Dic3—92no—
234C4⋊Dic3—181no—
235C4⋊Dic3C2×D911yescharacteristic
236C4⋊Dic3C2×D911yescharacteristic
237Dic3⋊C4—32no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—182no—
249C4×Dic3—92no—
250C4×Dic3—92no—
251C4×Dic3—92no—
252C4×Dic3—92no—
253C4×Dic3C2×D911yescharacteristic
254C4×Dic3C2×D911yescharacteristic
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32C23×D311yescharacteristic
259C32×C4—21no—
260C32×C4#23512yes—
261C3⋊Dic3—62no—
262C3⋊Dic3—62no—
263C3×Dic3—21no—
264C3×Dic3—21no—
265C3×Dic3—21no—
266C3×Dic3—21no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C22×C9—22no—
270C22×C9—21no—
271C22×C9—21no—
272C22×C9—21no—
273C22×C9—21no—
274C22×C9—41no—
275C22×C9—41no—
276C22×C9C23×D311yescharacteristic
277C4×C9—21no—
278C4×C9—22no—
279C4×C9—41no—
280C4×C9—61no—
281C4×C9—61no—
282C4×C9—61no—
283C4×C9—61no—
284C4×C9#23712yes—
285Dic9—22no—
286Dic9—22no—
287Dic9—62no—
288Dic9—62no—
289Dic9—122no—
290Dic9—122no—
291C4⋊Q8—271no—
292#126—272no—
293#126—272no—
294#125—271no—
295#125—272no—
296#125—272no—
297C22⋊Q8—91no—
298C22⋊Q8—271no—
299C4×Q8—271no—
300C4×Q8—271no—
301#117—31no—
302C3×C9#54111yescharacteristic, complemented, Sylow
303C23×C3—21no—
304C23×C3C22×D911yescharacteristic
305C23×C3C2×D3211yescharacteristic
306C3×Q8—182no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—31no—
315C2×C3×C4—42no—
316C2×C3×C4—92no—
317C2×C3×C4—92no—
318C2×C3×C4C22×D911yescharacteristic
319C2×C3×C4C22×D911yescharacteristic
320C2×C3×C4C2×D3211yescharacteristic
321C2×C3×C4C2×D3211yescharacteristic
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—92no—
325C2×Dic3—92no—
326C2×Dic3—92no—
327C2×Dic3—92no—
328C2×Dic3—182no—
329C2×Dic3—182no—
330C2×Dic3C22×D911yescharacteristic
331C2×Dic3C22×D911yescharacteristic
332C2×Dic3C22×D911yescharacteristic
333C2×Dic3C22×D911yescharacteristic
334Dic6—62no—
335Dic6—182no—
336Dic6—182no—
337Dic6—362no—
338C2×C32—41no—
339C2×C32C2×#23511yescharacteristic
340C2×C32#101011yescharacteristic
341C2×C32#101011yescharacteristic
342C2×C9—21no—
343C2×C9—21no—
344C2×C9—21no—
345C2×C9—42no—
346C2×C9—41no—
347C2×C9C2×#23711yescharacteristic
348C2×C9#100711yescharacteristic
349C2×C9#101011yescharacteristic
350C2×Q8—91no—
351C2×Q8—271no—
352C22×C4D3×D911yescharacteristic
353C4⋊C4—31no—
354C4⋊C4—31no—
355C4⋊C4—91no—
356C4⋊C4—92no—
357C4⋊C4—272no—
358C4⋊C4—272no—
359C4⋊C4—272no—
360C4⋊C4—272no—
361C4⋊C4—272no—
362C4⋊C4—272no—
363C4⋊C4—271no—
364C4⋊C4—272no—
365C22⋊C4—31no—
366C22⋊C4—31no—
367C22⋊C4—92no—
368C22⋊C4—272no—
369C42—31no—
370C42—31no—
371C42—272no—
372C42—272no—
373C22×C3—22no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3C23×D911yescharacteristic
384C22×C3C22×D3211yescharacteristic
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—22no—
388C3×C4—41no—
389C3×C4—61no—
390C3×C4—61no—
391C3×C4—61no—
392C3×C4—61no—
393C3×C4—182no—
394C3×C4—182no—
395C3×C4#369712yes—
396C3×C4#380412yes—
397Dic3—21no—
398Dic3—21no—
399Dic3—21no—
400Dic3—21no—
401Dic3—62no—
402Dic3—62no—
403Dic3—182no—
404Dic3—182no—
405Dic3—182no—
406Dic3—182no—
407Dic3—362no—
408Dic3—362no—
409C32#551811yescharacteristic
410C9—21no—
411C9#562511yescharacteristic, complemented
412C23C2×D3×D911yescharacteristic
413Q8—182no—
414Q8—542no—
415C2×C4—22no—
416C2×C4—31no—
417C2×C4—31no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—92no—
421C2×C4—92no—
422C2×C4—272no—
423C2×C4—272no—
424C2×C4C2×D3×D911yescharacteristic
425C2×C4C2×D3×D911yescharacteristic
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—42no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3C2×#369711yescharacteristic
433C2×C3#6371611yescharacteristic
434C2×C3#6371611yescharacteristic
435C2×C3#6429711yescharacteristic
436C2×C3#6429811yescharacteristic
437C2×C3C2×#380411yescharacteristic
438C22—21no—
439C22—21no—
440C22—21no—
441C22C22×D3×D911yescharacteristic
442C4—21no—
443C4—61no—
444C4—61no—
445C4—61no—
446C4—61no—
447C4—182no—
448C4—182no—
449C4—542no—
450C4—542no—
451C4#8865812yes—
452C3—21no—
453C3#1058962011yescharacteristic, complemented
454C3#1059482411yescharacteristic
455C2—41no—
456C2#1172246411yescharacteristic
457C2#1172246511yescharacteristic
458C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC23×C32C22×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#541
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11720932C2semidirect
7. #11720927C2semidirect
8. #11720918C2semidirect
9. #11720917C2semidirect
10. #11720912C2semidirect
11. Dic3×Dic18C2semidirect
302. C3×C9#541semidirect
411. C9#5625semidirect
453. C3#10589620semidirect

Maximal subgroups

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

subgroupindex
#117218132
C9×#9262
C3×#634862
#117212472
#117212462
#117209322
#117209272
#117209182
#117209172
#117209122
Dic3×Dic182
#105948243
#105896203