Finite Group Databasebeta

← #13150923

Subgroups — #13150923

Order 1728.

778 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C3⋊Dic9C211yesmaximal, characteristic
2D4×C9×Dic3C211yesmaximal, characteristic
3C3×D4×Dic9C211yesmaximal, characteristic
4#11721267C212yesmaximal, complemented
5C2×Dic3⋊Dic9C212yesmaximal, complemented
6#11721253C212yesmaximal, complemented
7#11721249C212yesmaximal, complemented
8#11720970C211yesmaximal, characteristic, complemented
9C4×#6189C211yesmaximal, characteristic, complemented
10#11720937C211yesmaximal, characteristic, complemented
11#11720935C211yesmaximal, characteristic, complemented
12#10595272—31nomaximal
13#10589645—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C22×C9×Dic3C2212yes—
17C22×C3×Dic9C2212yes—
18C2×#6189C2211yescharacteristic
19C2×#6189C2212yes—
20#88551C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4×C3⋊Dic9C2211yescharacteristic
23C9×#217C2212yes—
24C3×#3676C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C4×C9×Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29Dic3⋊Dic9C2211yescharacteristic
30Dic3⋊Dic9C2212yes—
31Dic3⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2211yescharacteristic
34#88442C2212yes—
35Dic9⋊Dic3C2211yescharacteristic
36Dic9⋊Dic3C2212yes—
37D4×C3⋊Dic3—31no—
38C3×D4×Dic3—31no—
39C3×D4×Dic3D311yescharacteristic
40#63982—32no—
41C2×Dic3⋊Dic3—32no—
42#63964—32no—
43#63964—32no—
44#63918—31no—
45C4×#627—31no—
46#63872—31no—
47#63869—31no—
48C4×D4×C9—31no—
49D4×Dic9—31no—
50D4×Dic9—61no—
51D4×Dic9D311yescharacteristic
52C2×C4⋊Dic9—32no—
53#63486—32no—
54#63452—31no—
55#63452—31no—
56#63440—32no—
57#63440—32no—
58#63431—31no—
59C4×Dic18—31no—
60C23×C3×C9C2312yes—
61C3×D4×C9Q814yescomplemented
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9—42no—
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C9×Dic3—24no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9—42no—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75#6189—22no—
76#6189—42no—
77#5543—91no—
78#5539—91no—
79C2×C32×D4C2×D311yescharacteristic
80C22×C3⋊Dic3—32no—
81C22×C3×Dic3—32no—
82C22×C3×Dic3C2×D312yes—
83C2×#627—31no—
84C2×#627—32no—
85#3757—32no—
86C4⋊(C3⋊Dic3)—31no—
87C4×C3⋊Dic3—31no—
88C3×#217—32no—
89C3×#217C2×D312yes—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4×Dic3—31no—
93C3×C4×Dic3C2×D311yescharacteristic
94#3724—32no—
95#3724—31no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—31no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100Dic3⋊Dic3—31no—
101Dic3⋊Dic3—32no—
102C2×D4×C9—21no—
103C2×D4×C9C2×D311yescharacteristic
104C22×C4×C9—32no—
105C22×Dic9—32no—
106C22×Dic9—62no—
107C22×Dic9C2×D312yes—
108C2×Dic18—31no—
109C2×Dic18—32no—
110C9×C4⋊C4—31no—
111C9×C22⋊C4—32no—
112C42×C9—31no—
113#3676—32no—
114#3676—62no—
115#3676C2×D312yes—
116C4⋊Dic9—31no—
117C4⋊Dic9—31no—
118C4⋊Dic9—32no—
119C4⋊Dic9—32no—
120C4⋊Dic9—32no—
121C4⋊Dic9—61no—
122C4⋊Dic9C2×D311yescharacteristic
123Dic9⋊C4—31no—
124Dic9⋊C4—32no—
125Dic9⋊C4—31no—
126Dic9⋊C4—32no—
127C4×Dic9—31no—
128C4×Dic9—61no—
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9—22no—
131C22×C3×C9—22no—
132C22×C3×C9C2×Q814yes—
133C22×C3×C9C2411yescharacteristic
134C3×C4×C9C2×Q812yes—
135C3⋊Dic9—21no—
136C3⋊Dic9—21no—
137C3⋊Dic9—22no—
138C9×Dic3—22no—
139C9×Dic3—21no—
140C9×Dic3#4112yes—
141C3×Dic9—21no—
142C3×Dic9—22no—
143C3×Dic9—21no—
144C3×C4×D4—31no—
145C3×C4×D4—91no—
146D4×Dic3—31no—
147D4×Dic3—91no—
148D4×Dic3—91no—
149D4×Dic3—181no—
150D4×Dic3D911yescharacteristic
151C2×C4⋊Dic3—92no—
152#924—92no—
153#924—92no—
154C2×Dic3⋊C4—92no—
155#890—91no—
156#890—91no—
157#890—91no—
158#888—91no—
159#878—92no—
160#878—92no—
161#878—92no—
162#878—92no—
163#870—91no—
164#869—91no—
165C4×Dic6—91no—
166C4×Dic6—91no—
167C23×C32C22×D312yes—
168C32×D4Dic614yes—
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—122no—
174C2×C3×Dic3—24no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—122no—
179C2×C3×Dic3C22×D312yes—
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D311yescharacteristic
182#627—62no—
183#627—122no—
184C23×C9—22no—
185C23×C9C22×D312yes—
186D4×C9—24no—
187D4×C9Dic614yescomplemented
188C2×C4×C9—21no—
189C2×C4×C9—31no—
190C2×C4×C9—32no—
191C2×C4×C9—31no—
192C2×C4×C9—64no—
193C2×C4×C9C22×D311yescharacteristic
194C2×Dic9—31no—
195C2×Dic9—31no—
196C2×Dic9—32no—
197C2×Dic9—42no—
198C2×Dic9—61no—
199C2×Dic9—61no—
200C2×Dic9—62no—
201C2×Dic9—122no—
202C2×Dic9—124no—
203C2×Dic9C22×D311yescharacteristic
204C2×Dic9C22×D311yescharacteristic
205C2×Dic9C22×D312yes—
206Dic18—62no—
207Dic18—122no—
208#554—271noSylow
209C2×C3×C9—24no—
210C2×C3×C9C22×Q811yescharacteristic
211C2×C3×C9C2×#4111yescharacteristic
212C2×C3×C9H(5,𝔽2)11yescharacteristic
213C2×C3×D4—21no—
214C2×C3×D4C2×D911yescharacteristic
215C2×C3×D4D3211yescharacteristic
216C22×C3×C4—32no—
217C22×C3×C4—92no—
218C22×Dic3—32no—
219C22×Dic3—92no—
220C22×Dic3—92no—
221C22×Dic3—182no—
222C22×Dic3C2×D912yes—
223C2×Dic6—91no—
224C2×Dic6—91no—
225C2×Dic6—92no—
226C2×Dic6—92no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—91no—
229C3×C22⋊C4—32no—
230C3×C22⋊C4—92no—
231C3×C42—31no—
232C3×C42—91no—
233#217—32no—
234#217—92no—
235#217—92no—
236#217—182no—
237#217C2×D912yes—
238C4⋊Dic3—31no—
239C4⋊Dic3—91no—
240C4⋊Dic3—92no—
241C4⋊Dic3—92no—
242C4⋊Dic3—92no—
243C4⋊Dic3—91no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—92no—
247C4⋊Dic3—181no—
248C4⋊Dic3C2×D911yescharacteristic
249Dic3⋊C4—91no—
250Dic3⋊C4—91no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—91no—
254Dic3⋊C4—91no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259C4×Dic3—31no—
260C4×Dic3—91no—
261C4×Dic3—91no—
262C4×Dic3—181no—
263C4×Dic3C2×D911yescharacteristic
264C22×C32—22no—
265C22×C32—22no—
266C22×C32C2×Dic614yes—
267C22×C32C23×D311yescharacteristic
268C32×C4C2×Dic612yes—
269C3⋊Dic3—61no—
270C3⋊Dic3—61no—
271C3⋊Dic3—62no—
272C3×Dic3—22no—
273C3×Dic3—21no—
274C3×Dic3—61no—
275C3×Dic3—62no—
276C3×Dic3—61no—
277C3×Dic3#23712yes—
278C22×C9—22no—
279C22×C9—22no—
280C22×C9—21no—
281C22×C9—24no—
282C22×C9—42no—
283C22×C9—42no—
284C22×C9C2×Dic614yes—
285C22×C9C23×D311yescharacteristic
286C4×C9—22no—
287C4×C9—32no—
288C4×C9—61no—
289C4×C9—62no—
290C4×C9C2×Dic612yes—
291Dic9—22no—
292Dic9—21no—
293Dic9—21no—
294Dic9—62no—
295Dic9—62no—
296Dic9—61no—
297Dic9—61no—
298Dic9—121no—
299Dic9—122no—
300C4⋊Q8—271no—
301#125—271no—
302#125—271no—
303C22⋊Q8—272no—
304C22⋊Q8—272no—
305C22⋊Q8—272no—
306C4×Q8—271no—
307C4×D4—31no—
308C4×D4—91no—
309C4×D4—271no—
310C2×C4⋊C4—272no—
311C3×C9#55411yescharacteristic, complemented, Sylow
312C23×C3—22no—
313C23×C3C22×D912yes—
314C23×C3C2×D3212yes—
315C3×D4—24no—
316C3×D4Dic1814yescomplemented
317C3×D4#62714yes—
318C2×C3×C4—21no—
319C2×C3×C4—31no—
320C2×C3×C4—32no—
321C2×C3×C4—31no—
322C2×C3×C4—64no—
323C2×C3×C4—91no—
324C2×C3×C4—91no—
325C2×C3×C4—92no—
326C2×C3×C4—362no—
327C2×C3×C4C22×D911yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×Dic3—24no—
330C2×Dic3—31no—
331C2×Dic3—31no—
332C2×Dic3—32no—
333C2×Dic3—91no—
334C2×Dic3—91no—
335C2×Dic3—92no—
336C2×Dic3—91no—
337C2×Dic3—91no—
338C2×Dic3—92no—
339C2×Dic3—122no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×Dic3—182no—
343C2×Dic3—362no—
344C2×Dic3—362no—
345C2×Dic3—364no—
346C2×Dic3C22×D911yescharacteristic
347C2×Dic3C22×D912yes—
348C2×Dic3C22×D911yescharacteristic
349Dic6—182no—
350Dic6—182no—
351Dic6—362no—
352Dic6—362no—
353C2×C32—24no—
354C2×C32C22×Dic611yescharacteristic
355C2×C32C2×#23711yescharacteristic
356C2×C32#100911yescharacteristic
357C2×C9—24no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—44no—
362C2×C9C22×Dic611yescharacteristic
363C2×C9#100411yescharacteristic
364C2×C9D3×#4111yescharacteristic
365C2×Q8—271no—
366C2×Q8—272no—
367C2×D4D3×D911yescharacteristic
368C22×C4—32no—
369C22×C4—92no—
370C22×C4—272no—
371C4⋊C4—31no—
372C4⋊C4—91no—
373C4⋊C4—271no—
374C4⋊C4—272no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—271no—
378C4⋊C4—271no—
379C4⋊C4—272no—
380C4⋊C4—272no—
381C4⋊C4—271no—
382C22⋊C4—32no—
383C22⋊C4—92no—
384C22⋊C4—272no—
385C42—31no—
386C42—91no—
387C42—271no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—21no—
393C22×C3—24no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3C2×Dic1814yes—
397C22×C3C23×D911yescharacteristic
398C22×C3C2×#62714yes—
399C22×C3C22×D3211yescharacteristic
400C3×C4—22no—
401C3×C4—32no—
402C3×C4—62no—
403C3×C4—61no—
404C3×C4—181no—
405C3×C4—182no—
406C3×C4—181no—
407C3×C4C2×Dic1812yes—
408C3×C4C2×#62712yes—
409Dic3—22no—
410Dic3—21no—
411Dic3—61no—
412Dic3—62no—
413Dic3—61no—
414Dic3—182no—
415Dic3—181no—
416Dic3—182no—
417Dic3—181no—
418Dic3—181no—
419Dic3—182no—
420Dic3—181no—
421Dic3—362no—
422Dic3—361no—
423Dic3#369912yes—
424C32#554311yescharacteristic
425C9—21no—
426C9#553911yescharacteristic, complemented
427C23C2×D3×D912yes—
428Q8—542no—
429Q8—1082no—
430D4#618914yescomplemented
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—32no—
434C2×C4—64no—
435C2×C4—91no—
436C2×C4—91no—
437C2×C4—92no—
438C2×C4—271no—
439C2×C4—271no—
440C2×C4—272no—
441C2×C4—362no—
442C2×C4—1082no—
443C2×C4C2×D3×D911yescharacteristic
444C2×C3—24no—
445C2×C3—24no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—44no—
450C2×C3C22×Dic1811yescharacteristic
451C2×C3C2×#369911yescharacteristic
452C2×C3#6371511yescharacteristic
453C2×C3D3×#23711yescharacteristic
454C2×C3#6431411yescharacteristic
455C2×C3C22×#62711yescharacteristic
456C22—22no—
457C22—22no—
458C22C2×#618914yes—
459C22C22×D3×D911yescharacteristic
460C4—32no—
461C4—62no—
462C4—61no—
463C4—181no—
464C4—181no—
465C4—182no—
466C4—541no—
467C4—541no—
468C4—542no—
469C4C2×#618912yes—
470C3—21no—
471C3#1058964511yescharacteristic, complemented
472C3#1059527211yescharacteristic
473C2—24no—
474C2D3×#369911yescharacteristic
475C2#1172250111yescharacteristic
476C2C22×#618911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC2×C32×D4C2×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#554
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #11721267C2semidirect
5. C2×Dic3⋊Dic9C2semidirect
6. #11721253C2semidirect
7. #11721249C2semidirect
8. #11720970C2semidirect
9. C4×#6189C2semidirect
10. #11720937C2semidirect
11. #11720935C2semidirect
61. C3×D4×C9Q8semidirect
187. D4×C9Dic6semidirect
311. C3×C9#554semidirect
316. C3×D4Dic18semidirect
426. C9#5539semidirect
430. D4#6189semidirect
471. C3#10589645semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic92
D4×C9×Dic32
C3×D4×Dic92
#117212672
C2×Dic3⋊Dic92
#117212532
#117212492
#117209702
C4×#61892
#117209372
#117209352
#105952723
#105896453