Finite Group Databasebeta

← #13161982

Subgroups — #13161982

Order 1728.

916 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89040C211yesmaximal, characteristic, complemented
2C3×#64147C211yesmaximal, characteristic
3C2×C3×#3754C212yesmaximal, complemented
4#11722076C212yesmaximal, complemented
5#11722072C211yesmaximal, characteristic, complemented
6#10591518—31nomaximal
7#10591517—32nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#634C2212yes—
12C2×#6296—21no—
13C2×#6294—21no—
14#89040—24no—
15C3×#3757—21no—
16C3×#3754—24no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2212yes—
19C2×#3797—31no—
20C2×D3×Dic6—32no—
21#64147D311yescharacteristic
22C2×#3754D312yescomplemented
23C3×#938—32no—
24C3×#938—61no—
25C2×C3×#214—32no—
26C2×C3×#214—32no—
27C2×C3×#214—62no—
28#63567—32no—
29#63566—32no—
30#63565—32no—
31#63557—31no—
32(C2×D3)⋊(C3⋊C8)—32no—
33C23×C33—21no—
34C22×C3×C3⋊D3D411yescharacteristic
35C33×D4—22no—
36C2×C33×C4C2311yescharacteristic
37C3×#638—42no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C3×C4×C3⋊D3—22no—
42C3×#634—22no—
43C3×#634D414yescomplemented
44#6296—24no—
45#6294—24no—
46C3×C3⋊(C3⋊C8)—22no—
47#5170—92no—
48#5040—91no—
49C2×C32×D4—22no—
50C2×C32×D4—21no—
51C2×C32×D4—41no—
52C2×C32×D4C2×D311yescharacteristic
53C2×C32×D4C2×D312yes—
54C2×#638—21no—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×#634C2×D312yes—
57C2×C3×#67—62no—
58C2×C3×#67—62no—
59C2×C3×C4×D3—32no—
60C2×C3×C4×D3—61no—
61C2×C3×Dic6—32no—
62C2×C3×Dic6—32no—
63C2×C3×Dic6—62no—
64C2×#627—62no—
65C2×#627—61no—
66C2×#624—61no—
67C2×D3×Dic3—62no—
68#3797—64no—
69D3×Dic6—64no—
70D3×Dic6—64no—
71#3757—21no—
72#3754—24no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D312yes—
75C3×#217—62no—
76C3×#217—62no—
77C3×#214—64no—
78C3×#214—64no—
79C3×#214—68no—
80C3×C4⋊Dic3—32no—
81C3×C4⋊Dic3—61no—
82C2×C3×C3⋊C8—32no—
83C2×C3×C3⋊C8—32no—
84C2×C3×C3⋊C8—62no—
85C22×C33—41no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3SD1612yescomplemented
92C33×C4C2×D412yes—
93C3×C3⋊Dic3—24no—
94C3×C3⋊Dic3—21no—
95C3×C3⋊Dic3—41no—
96C2×D3×Q8—92no—
97C22×Dic6—91no—
98C2×C3×SD16—92no—
99C3×#121—91no—
100#938—32no—
101#938—61no—
102C2×#214—32no—
103C2×#214—32no—
104C2×#214—62no—
105#835—92no—
106#830—91no—
107(C2×D3)⋊C8—92no—
108#816—92no—
109#810—92no—
110C23×C32—21no—
111C23×C32—22no—
112C23×C32—22no—
113C23×C32—42no—
114C23×C32—42no—
115C22×C3⋊D3#6711yescharacteristic
116C22×C3×D3—32no—
117C22×C3×D3—61no—
118C32×D4—22no—
119C32×D4—24no—
120C32×D4—22no—
121C32×D4—22no—
122C32×D4—42no—
123C32×D4—42no—
124C32×D4—44no—
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—41no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D312yes—
130#638—42no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3#6711yescharacteristic
134C4×C3⋊D3—22no—
135#634—22no—
136#634#6714yescomplemented
137C3×#67—124no—
138C3×#67—124no—
139C2×C3×Dic3—32no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—122no—
146C3×C4×D3—62no—
147C3×C4×D3—62no—
148C3×C4×D3—122no—
149C3×Dic6—34no—
150C3×Dic6—34no—
151C3×Dic6—64no—
152C3×Dic6—62no—
153C3×Dic6—62no—
154C3×Dic6—122no—
155#627—68no—
156#627—64no—
157#624—64no—
158D3×Dic3—68no—
159C3⋊(C3⋊C8)—22no—
160C3×C3⋊C8—64no—
161C3×C3⋊C8—62no—
162C3×C3⋊C8—62no—
163#448—271noSylow
164C2×C33—81no—
165C2×C33#12011yescharacteristic
166C2×C33C2×SD1611yescharacteristic
167C2×C33#13711yescharacteristic
168C3×C3⋊D3—22no—
169C2×C3×Q8—92no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—41no—
173C2×C3×D4—181no—
174C2×C3×D4D3211yescharacteristic
175C2×C3×D4D3212yes—
176C22×C3×C4—91no—
177C2×#67—62no—
178C2×#67—62no—
179C22×Dic3—181no—
180D3×Q8—184no—
181D3×Q8—184no—
182C2×C4×D3—32no—
183C2×C4×D3—61no—
184C2×C4×D3—182no—
185C2×Dic6—32no—
186C2×Dic6—32no—
187C2×Dic6—62no—
188C2×Dic6—184no—
189C2×Dic6—182no—
190C2×Dic6—181no—
191C3×SD16—184no—
192C2×C3×C8—92no—
193C3×C4⋊C4—91no—
194C3×C22⋊C4—181no—
195#217—62no—
196#217—62no—
197#214—64no—
198#214—64no—
199#214—68no—
200C4⋊Dic3—32no—
201C4⋊Dic3—61no—
202C2×C3⋊C8—32no—
203C2×C3⋊C8—32no—
204C2×C3⋊C8—62no—
205C22×C32—22no—
206C22×C32—21no—
207C22×C32—41no—
208C22×C32—44no—
209C22×C32—42no—
210C22×C32—44no—
211C22×C32—41no—
212C22×C32—42no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—42no—
216C22×C32—42no—
217C22×C32—42no—
218C22×C32—42no—
219C22×C32—82no—
220C22×C32—82no—
221C22×C32—82no—
222C22×C32—82no—
223C22×C32D3×D412yes—
224C22×C32C2×#6711yescharacteristic
225C2×C3⋊D3—21no—
226C2×C3⋊D3—21no—
227C2×C3⋊D3#21412yescomplemented
228C2×C3×D3—34no—
229C2×C3×D3—62no—
230C2×C3×D3—62no—
231C2×C3×D3—62no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—22no—
237C32×C4—42no—
238C32×C4D3×D412yes—
239C32×C4D3×D412yes—
240C32×C4C2×#6712yes—
241C3⋊Dic3—24no—
242C3⋊Dic3—21no—
243C3⋊Dic3—41no—
244C3×Dic3—64no—
245C3×Dic3—64no—
246C3×Dic3—62no—
247C3×Dic3—124no—
248C3×Dic3—121no—
249C3×Dic3—122no—
250C3×Dic3—122no—
251C22×Q8—271no—
252C2×SD16—92no—
253#121—91no—
254#103—272no—
255C22⋊C8—271no—
256C33#44811yescharacteristic, complemented, Sylow
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—21no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—91no—
263C22×D3—32no—
264C22×D3—61no—
265C3×Q8—94no—
266C3×Q8—182no—
267C3×D4—24no—
268C3×D4—22no—
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—42no—
272C3×D4—42no—
273C3×D4—44no—
274C3×D4—362no—
275C2×C3×C4—22no—
276C2×C3×C4—21no—
277C2×C3×C4—41no—
278C2×C3×C4—91no—
279C2×C3×C4—182no—
280C2×C3×C4—182no—
281C2×C3×C4—181no—
282C2×C3×C4C2×D3211yescharacteristic
283C2×C3×C4C2×D3212yes—
284#67—124no—
285#67—124no—
286C2×Dic3—32no—
287C2×Dic3—61no—
288C2×Dic3—62no—
289C2×Dic3—62no—
290C2×Dic3—62no—
291C2×Dic3—62no—
292C2×Dic3—122no—
293C2×Dic3—184no—
294C4×D3—62no—
295C4×D3—62no—
296C4×D3—122no—
297C4×D3—188no—
298Dic6—34no—
299Dic6—34no—
300Dic6—64no—
301Dic6—62no—
302Dic6—62no—
303Dic6—122no—
304Dic6—184no—
305Dic6—188no—
306C3×C8—182no—
307C3⋊C8—64no—
308C3⋊C8—62no—
309C3⋊C8—62no—
310C2×C32—22no—
311C2×C32—22no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—21no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—82no—
320C2×C32—84no—
321C2×C32—84no—
322C2×C32—82no—
323C2×C32—81no—
324C2×C32D3×SD1612yes—
325C2×C32#91512yes—
326C2×C32C2×#21411yescharacteristic
327C2×C32#93712yes—
328C2×C32#93711yescharacteristic
329C2×C32#95111yescharacteristic
330C3⋊D3—22no—
331C3×D3—64no—
332C3×D3—122no—
333C2×Q8—92no—
334C2×Q8—541no—
335C2×Q8—544no—
336C2×D4—181no—
337C2×D4#111311yescharacteristic
338C22×C4—91no—
339C22×C4—541no—
340SD16—184no—
341C2×C8—92no—
342C4⋊C4—91no—
343C22⋊C4—181no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—42no—
347C22×C3—44no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—44no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—42no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—92no—
363C22×C3—181no—
364C22×C3—181no—
365C22×C3D3×#6712yes—
366C22×C3#381111yescharacteristic
367C2×D3—34no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—121no—
372C2×D3—121no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—42no—
377C3×C4—181no—
378C3×C4—184no—
379C3×C4—361no—
380C3×C4D3×#6712yes—
381C3×C4D3×#6712yes—
382C3×C4#381112yes—
383Dic3—64no—
384Dic3—64no—
385Dic3—62no—
386Dic3—124no—
387Dic3—121no—
388Dic3—122no—
389Dic3—122no—
390C32—22no—
391C32—21no—
392C32—41no—
393C32#504011yescharacteristic, complemented
394C32#517012yescomplemented
395C23—21no—
396C23—91no—
397Q8—94no—
398Q8—182no—
399Q8—544no—
400D4—22no—
401D4—362no—
402C2×C4—91no—
403C2×C4—182no—
404C2×C4—182no—
405C2×C4—181no—
406C2×C4—544no—
407C2×C4C2×#111311yescharacteristic
408C8—182no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—21no—
412C2×C3—22no—
413C2×C3—22no—
414C2×C3—21no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—82no—
419C2×C3—84no—
420C2×C3—84no—
421C2×C3—82no—
422C2×C3—81no—
423C2×C3—182no—
424C2×C3D3×#21412yes—
425C2×C3#6392912yes—
426C2×C3#6393011yescharacteristic
427C2×C3#6393111yescharacteristic
428C2×C3#6398312yes—
429C2×C3#6398311yescharacteristic
430D3—64no—
431D3—122no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—92no—
436C22—181no—
437C22—181no—
438C22#8904911yescharacteristic
439C4—181no—
440C4—184no—
441C4—361no—
442C4#8904912yes—
443C3—21no—
444C3—22no—
445C3—41no—
446C3#1059151712yescomplemented
447C3#1059151811yescharacteristic, complemented
448C2—81no—
449C2—182no—
450C2#1172370411yescharacteristic
451C2#1172370511yescharacteristic
452C2#1172372911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64147C2
hypercenterC2×D4#1113
hypocenterC33#448
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89040C2semidirect
3. C2×C3×#3754C2semidirect
4. #11722076C2semidirect
5. #11722072C2semidirect
22. C2×#3754D3semidirect
43. C3×#634D4semidirect
91. C2×C3×C3⋊D3SD16semidirect
136. #634#67semidirect
227. C2×C3⋊D3#214semidirect
256. C33#448semidirect
393. C32#5040semidirect
394. C32#5170semidirect
446. C3#10591517semidirect
447. C3#10591518semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C3×#641472
C2×C3×#37542
#117220762
#117220722
#105915183
#105915173