Finite Group Databasebeta

← #13148987

Subgroups — #13148987

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
2C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
3C9×#887C211yesmaximal, characteristic, complemented
4C4×#6192C211yesmaximal, characteristic
5C4×#6192C212yesmaximal
6#11720970C211yesmaximal, characteristic, complemented
7#11720944C211yesmaximal, characteristic, complemented
8#11720939C212yesmaximal, complemented
9#11720929C211yesmaximal, characteristic, complemented
10#11720925C212yesmaximal, complemented
11#11720923C212yesmaximal, complemented
12#10594536—31nomaximal
13#10589681—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×C9×Dic6C2211yescharacteristic
19C2×#6192C2211yescharacteristic
20C2×#6189C2212yes—
21C3×C9×C4⋊C4C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2212yes—
24C9×Dic3⋊C4C2212yes—
25C4×C9×Dic3C2211yescharacteristic
26C4×C9×Dic3C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2212yes—
29#88445C2211yescharacteristic
30#88445C2212yes—
31(C2×D9)⋊Dic3C2211yescharacteristic
32(C2×D9)⋊Dic3C2212yes—
33Dic3⋊Dic9C2212yes—
34Dic3⋊Dic9C2211yescharacteristic
35#88442C2212yes—
36Dic9⋊Dic3C2212yes—
37C4⋊C4×C3⋊D3—31no—
38C3×D3×C4⋊C4—31no—
39C3×#887D311yescharacteristic
40#63918—31no—
41C4×#626—31no—
42C4×#626—32no—
43#63871—31no—
44#63867—32no—
45#63865—32no—
46#63856—31no—
47#63851—32no—
48C4×Q8×C9—31no—
49#63508—31no—
50#63508—31no—
51#63458—32no—
52#63458—32no—
53C4⋊C4×D9—31no—
54C4⋊C4×D9—61no—
55C4⋊C4×D9D311yescharacteristic, complemented
56C4×D36—32no—
57C4×D36—31no—
58#63432—32no—
59#63431—31no—
60C22×C3⋊D9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C4×C3⋊D9—22no—
67C4×C3⋊D9—42no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C3×C4×D9—22no—
74C3×C4×D9—42no—
75C9×Dic6—22no—
76#6192Q814yescomplemented
77#6189—42no—
78#5579—91no—
79#5511—91no—
80C2×C4×C3⋊D3—31no—
81C2×C4×C3⋊D3—32no—
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—32no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×#627—32no—
86C2×#626—31no—
87C32×C4⋊C4C2×D311yescharacteristic
88C4⋊(C3⋊Dic3)—31no—
89#3750—32no—
90C3×C4⋊Dic3—31no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4C2×D312yes—
93C3×C4×Dic3C2×D312yes—
94C3×C4×Dic3C2×D311yescharacteristic
95#3724—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—31no—
98Dic3⋊Dic3—32no—
99#3722—31no—
100#3722—32no—
101(C2×D3)⋊Dic3—31no—
102(C2×D3)⋊Dic3—32no—
103C2×Q8×C9—31no—
104C2×D36—31no—
105C2×C4×D9—31no—
106C2×C4×D9—32no—
107C2×C4×D9—61no—
108C2×C4×D9—62no—
109C2×C4×D9C2×D311yescharacteristic
110C2×C4×D9C2×D312yes—
111C2×Dic18—32no—
112C9×C4⋊C4—21no—
113C9×C4⋊C4—32no—
114C9×C4⋊C4C2×D311yescharacteristic
115C42×C9—31no—
116C42×C9—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—31no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—31no—
121C4⋊Dic9—31no—
122C4⋊Dic9—32no—
123C4⋊Dic9—31no—
124C4⋊Dic9—61no—
125C4⋊Dic9C2×D311yescharacteristic
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—32no—
129Dic9⋊C4—62no—
130Dic9⋊C4C2×D312yes—
131C22×C3×C9C2411yescharacteristic
132C2×C3⋊D9—21no—
133C2×C3⋊D9—21no—
134C2×C3⋊D9C2×Q812yes—
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C2×C3×D9C2×Q812yes—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—21no—
141C3⋊Dic9—22no—
142C9×Dic3—22no—
143C9×Dic3—21no—
144C9×Dic3C2×Q812yes—
145C3×Dic9—21no—
146C3×Dic9—22no—
147C3×C4×Q8—31no—
148C2×C3×C4⋊C4—91no—
149#946—91no—
150#946—91no—
151C4×#67—91no—
152C4×#67—92no—
153#924—91no—
154#924—92no—
155#897—91no—
156#896—92no—
157#896—92no—
158#896—92no—
159D3×C4⋊C4—31no—
160D3×C4⋊C4—91no—
161D3×C4⋊C4—91no—
162D3×C4⋊C4—181no—
163#889—92no—
164#888—91no—
165#887D911yescharacteristic, complemented
166C4×D12—92no—
167C4×D12—91no—
168#870—92no—
169#869—91no—
170C22×C3⋊D3—31no—
171C22×C3×D3—31no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—32no—
176C4×C3⋊D3—62no—
177C4×C3⋊D3—122no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3C22×D312yes—
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D311yescharacteristic
183C3×C4×D3—62no—
184C3×C4×D3—122no—
185C3×Dic6—22no—
186#627—122no—
187#626—34no—
188C22×D9—31no—
189C22×D9—61no—
190C22×D9C22×D311yescharacteristic
191Q8×C9—62no—
192C2×C4×C9—21no—
193C2×C4×C9—22no—
194C2×C4×C9—31no—
195C2×C4×C9—32no—
196C2×C4×C9—31no—
197C2×C4×C9C22×D312yes—
198C2×C4×C9C22×D311yescharacteristic
199C2×Dic9—31no—
200C2×Dic9—32no—
201C2×Dic9—61no—
202C2×Dic9—62no—
203C2×Dic9C22×D311yescharacteristic
204C2×Dic9C22×D312yes—
205D36—34no—
206C4×D9—22no—
207C4×D9—42no—
208C4×D9—64no—
209C4×D9—62no—
210C4×D9—122no—
211C4×D9—124no—
212Dic18—122no—
213#554—271noSylow
214C2×C3×C9C22×Q811yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9H(5,𝔽2)11yescharacteristic
217C3⋊D9—22no—
218C3×D9—22no—
219C2×C3×Q8—31no—
220C22×C3×C4—91no—
221C22×C3×C4—92no—
222C2×#67—91no—
223C2×D12—91no—
224C2×C4×D3—31no—
225C2×C4×D3—32no—
226C2×C4×D3—91no—
227C2×C4×D3—91no—
228C2×C4×D3—92no—
229C2×C4×D3—92no—
230C2×C4×D3—181no—
231C2×C4×D3—182no—
232C2×Dic6—92no—
233C2×Dic6—92no—
234C2×Dic6C2×D911yescharacteristic
235C3×C4⋊C4—21no—
236C3×C4⋊C4—32no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4C2×D911yescharacteristic
240C3×C4⋊C4D3211yescharacteristic
241C3×C42—31no—
242C3×C42—32no—
243#217—91no—
244#217—92no—
245(C2×D3)⋊C4—91no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—91no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—92no—
251C4⋊Dic3—31no—
252C4⋊Dic3—92no—
253C4⋊Dic3—91no—
254C4⋊Dic3—91no—
255C4⋊Dic3—91no—
256C4⋊Dic3—92no—
257C4⋊Dic3—181no—
258Dic3⋊C4—32no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—91no—
265Dic3⋊C4—92no—
266Dic3⋊C4—182no—
267Dic3⋊C4C2×D912yes—
268C4×Dic3C2×D911yescharacteristic
269C4×Dic3C2×D912yes—
270C22×C32C23×D311yescharacteristic
271C2×C3⋊D3—32no—
272C2×C3⋊D3—61no—
273C2×C3⋊D3—61no—
274C2×C3×D3—32no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C32×C4—22no—
278C32×C4#23512yes—
279C3⋊Dic3—61no—
280C3⋊Dic3—62no—
281C3×Dic3—22no—
282C3×Dic3—21no—
283C3×Dic3—61no—
284C3×Dic3—62no—
285C3×Dic3D3×Q812yes—
286C22×C9—21no—
287C22×C9C23×D311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9—32no—
291C2×D9—62no—
292C2×D9—62no—
293C2×D9—62no—
294C2×D9—61no—
295C2×D9—61no—
296C2×D9D3×Q812yes—
297C4×C9—22no—
298C4×C9—22no—
299C4×C9—32no—
300C4×C9—42no—
301C4×C9—62no—
302C4×C9—61no—
303C4×C9#23512yes—
304Dic9—21no—
305Dic9—22no—
306Dic9—62no—
307Dic9—61no—
308Dic9—62no—
309Dic9—122no—
310C4⋊Q8—271no—
311#125—272no—
312C22⋊Q8—272no—
313C22⋊Q8—272no—
314C22⋊Q8—271no—
315C22⋊Q8—271no—
316C4×Q8—31no—
317C4×D4—272no—
318C4×D4—271no—
319C2×C4⋊C4—91no—
320C2×C4⋊C4—271no—
321C3×C9#55411yescharacteristic, complemented, Sylow
322C23×C3—91no—
323C22×D3—31no—
324C22×D3—91no—
325C22×D3—91no—
326C22×D3—181no—
327C3×Q8—62no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—31no—
331C2×C3×C4—32no—
332C2×C3×C4—31no—
333C2×C3×C4—91no—
334C2×C3×C4—92no—
335C2×C3×C4—182no—
336C2×C3×C4—362no—
337C2×C3×C4C22×D912yes—
338C2×C3×C4C22×D911yescharacteristic
339C2×C3×C4C2×D3212yes—
340C2×C3×C4C2×D3211yescharacteristic
341#67—94no—
342C2×Dic3—31no—
343C2×Dic3—32no—
344C2×Dic3—91no—
345C2×Dic3—92no—
346C2×Dic3—91no—
347C2×Dic3—92no—
348C2×Dic3—181no—
349C2×Dic3—182no—
350C2×Dic3C22×D912yes—
351C2×Dic3C22×D911yescharacteristic
352C2×Dic3C22×D911yescharacteristic
353D12—94no—
354C4×D3—62no—
355C4×D3—122no—
356C4×D3—184no—
357C4×D3—182no—
358C4×D3—182no—
359C4×D3—362no—
360C4×D3—362no—
361C4×D3—364no—
362Dic6—22no—
363Dic6—362no—
364Dic6—362no—
365C2×C32C2×#23511yescharacteristic
366C2×C32C2×D3×Q811yescharacteristic
367C2×C32#100911yescharacteristic
368C3⋊D3—62no—
369C3×D3—62no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9C2×#23511yescharacteristic
374C2×C9#100411yescharacteristic
375C2×C9C2×D3×Q811yescharacteristic
376D9—22no—
377D9—64no—
378D9—62no—
379C2×Q8—31no—
380C2×Q8—272no—
381C2×D4—271no—
382C22×C4—91no—
383C22×C4—92no—
384C22×C4—271no—
385C22×C4—272no—
386C4⋊C4—32no—
387C4⋊C4—92no—
388C4⋊C4—91no—
389C4⋊C4—271no—
390C4⋊C4—272no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C4⋊C4—272no—
394C4⋊C4—271no—
395C4⋊C4D3×D911yescharacteristic
396C22⋊C4—271no—
397C22⋊C4—272no—
398C22⋊C4—271no—
399C22⋊C4—272no—
400C42—32no—
401C42—31no—
402C22×C3—21no—
403C22×C3—92no—
404C22×C3—181no—
405C22×C3—181no—
406C22×C3C23×D911yescharacteristic
407C22×C3C22×D3211yescharacteristic
408C2×D3—32no—
409C2×D3—61no—
410C2×D3—61no—
411C2×D3—92no—
412C2×D3—92no—
413C2×D3—182no—
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—181no—
417C2×D3—181no—
418C2×D3—181no—
419C2×D3—181no—
420C3×C4—22no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—32no—
424C3×C4—42no—
425C3×C4—62no—
426C3×C4—61no—
427C3×C4—181no—
428C3×C4—182no—
429C3×C4#369712yes—
430C3×C4#379812yes—
431Dic3—22no—
432Dic3—21no—
433Dic3—61no—
434Dic3—62no—
435Dic3—182no—
436Dic3—181no—
437Dic3—182no—
438Dic3—181no—
439Dic3—182no—
440Dic3—362no—
441Dic3Q8×D912yes—
442C32#557911yescharacteristic
443C9—21no—
444C9#551111yescharacteristic, complemented
445C23—91no—
446C23—271no—
447Q8—62no—
448Q8—1082no—
449D4—274no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—32no—
453C2×C4—91no—
454C2×C4—92no—
455C2×C4—182no—
456C2×C4—271no—
457C2×C4—272no—
458C2×C4—362no—
459C2×C4—542no—
460C2×C4—1082no—
461C2×C4C2×D3×D912yes—
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—182no—
467C2×C3C2×#369711yescharacteristic
468C2×C3C2×Q8×D911yescharacteristic
469C2×C3#6371511yescharacteristic
470C2×C3C2×#379811yescharacteristic
471C2×C3#6431411yescharacteristic
472C2×C3D32×Q811yescharacteristic
473D3—62no—
474D3—184no—
475D3—182no—
476D3—182no—
477C22—92no—
478C22—181no—
479C22—181no—
480C22—272no—
481C22—541no—
482C22—541no—
483C22C22×D3×D911yescharacteristic
484C4—22no—
485C4—32no—
486C4—61no—
487C4—62no—
488C4—182no—
489C4—181no—
490C4—542no—
491C4—541no—
492C4#8864012yes—
493C3—21no—
494C3#1058968111yescharacteristic, complemented
495C3#1059453611yescharacteristic
496C2—182no—
497C2—542no—
498C2C2×#8864011yescharacteristic
499C2#1172250111yescharacteristic
500C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6192C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#554
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×C3⋊D9C2semidirect
2. C3×C4⋊C4×D9C2semidirect
3. C9×#887C2semidirect
6. #11720970C2semidirect
7. #11720944C2semidirect
8. #11720939C2semidirect
9. #11720929C2semidirect
10. #11720925C2semidirect
11. #11720923C2semidirect
55. C4⋊C4×D9D3semidirect
76. #6192Q8semidirect
165. #887D9semidirect
321. C3×C9#554semidirect
444. C9#5511semidirect
494. C3#10589681semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊D92
C3×C4⋊C4×D92
C9×#8872
C4×#61922
#117209702
#117209442
#117209392
#117209292
#117209252
#117209232
#105945363
#105896813