Finite Group Databasebeta

← #13167596

Subgroups — #13167596

Order 1728.

854 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C22×#6295C211yesmaximal, characteristic
2C2×#88942C212yesmaximal, complemented
3#11723721C211yesmaximal, characteristic, complemented
4C3×C22⋊#622C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593523—31nomaximal
7C23×C3×C3⋊D3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×#6296C2211yescharacteristic, complemented
10C2×#6295—21no—
11C2×#6295—24no—
12C2×#6295C2212yes—
13C2×C3×#622C2212yescomplemented
14#88942—24no—
15C3×#3757—21no—
16#88815—21no—
17#88815C2211yescharacteristic, complemented
18C22×#626—61no—
19C22×#625—31no—
20C2×#3775—32no—
21#63982—31no—
22#63964—61no—
23C22⋊#622D311yescharacteristic, complemented
24#63739—32no—
25C23×C33D411yescharacteristic
26C22×C3×C3⋊D3—24no—
27C22×C3×C3⋊D3—22no—
28C22×C3×C3⋊D3—42no—
29C22×C3×C3⋊D3—42no—
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35#6296—22no—
36#6295—24no—
37#6295—22no—
38#6295D414yes—
39C3×#622—22no—
40#4731—91nomaximal
41C23×C3⋊D3—21no—
42C23×C3×D3—61no—
43C23×C3×D3—61no—
44C23×C3×D3—121no—
45C22×C3⋊Dic3C2×D311yescharacteristic
46C22×C3×Dic3—61no—
47C22×C3×Dic3—61no—
48C2×#627—31no—
49C2×#627—61no—
50C2×#626—62no—
51C2×#626—68no—
52C2×#626—62no—
53C2×#625—32no—
54C2×#625—64no—
55C2×#625—61no—
56C2×#622C2×D312yescomplemented
57#3775—64no—
58#3757—21no—
59C3×#217—61no—
60C3×#217—61no—
61C3×#217—121no—
62#3724—31no—
63#3724—61no—
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67C22×C33—21no—
68C22×C33—21no—
69C22×C33SD1612yescomplemented
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3—24no—
72C2×C3×C3⋊D3—48no—
73C2×C3×C3⋊D3—42no—
74C2×C3×C3⋊D3—42no—
75C3×C3⋊Dic3—21no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3—41no—
78C3×C3⋊Dic3C2×D412yes—
79C22×#67—181no—
80C22×D12—91no—
81#924—91no—
82C2×#204—92no—
83#878—181no—
84C3×C22⋊C8—91no—
85#821—92no—
86C23×C32—21no—
87C23×C32—21no—
88C23×C32—41no—
89C23×C32—41no—
90C23×C32D1211yescharacteristic
91C22×C3⋊D3—24no—
92C22×C3⋊D3—22no—
93C22×C3⋊D3—42no—
94C22×C3⋊D3—42no—
95C22×C3×D3—64no—
96C22×C3×D3—64no—
97C22×C3×D3—62no—
98C22×C3×D3—62no—
99C22×C3×D3—122no—
100C22×C3×D3—124no—
101C22×C3×D3—122no—
102C22×C3×D3—122no—
103C22×C3×D3—122no—
104C22×C3×D3—124no—
105C22×C3×D3—124no—
106C22×C3×D3—122no—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3D1211yescharacteristic
111C2×C3⋊Dic3C22×D311yescharacteristic
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—64no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—122no—
122C2×C3×Dic3—121no—
123C2×C3×Dic3—121no—
124#627—62no—
125#627—122no—
126#626—64no—
127#626—64no—
128#626—68no—
129#625—34no—
130#625—64no—
131#625—62no—
132#622—22no—
133#450—271noSylow
134C2×C33—22no—
135C2×C33#12011yescharacteristic
136C2×C33C2×SD1611yescharacteristic
137C2×C33AGL(1,ℤ/8)11yescharacteristic
138C3×C3⋊D3—44no—
139C24×C3—181no—
140C23×D3—61no—
141C23×D3—61no—
142C23×D3—121no—
143C22×C3×C4—91no—
144C2×#67—182no—
145C2×#67—182no—
146C2×#67—188no—
147C22×Dic3—61no—
148C22×Dic3—61no—
149C2×D12—92no—
150C2×D12—184no—
151C2×D12—181no—
152C2×Dic6—91no—
153C2×Dic6—181no—
154C2×C3×C8—92no—
155C3×C22⋊C4—181no—
156#217—61no—
157#217—61no—
158#217—121no—
159C4⋊Dic3—91no—
160C4⋊Dic3—181no—
161Dic3⋊C4—181no—
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164#204—184no—
165C22×C32—22no—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—21no—
171C22×C32—22no—
172C22×C32—22no—
173C22×C32—42no—
174C22×C32—41no—
175C22×C32—41no—
176C22×C32—42no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—42no—
180C22×C32—42no—
181C22×C32—81no—
182C22×C32—81no—
183C22×C32#20412yescomplemented
184C22×C32C2×D1211yescharacteristic
185C2×C3⋊D3—24no—
186C2×C3⋊D3—48no—
187C2×C3⋊D3—42no—
188C2×C3⋊D3—42no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C2×C3×D3—124no—
192C2×C3×D3—128no—
193C2×C3×D3—128no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196C2×C3×D3—124no—
197C2×C3×D3—124no—
198C2×C3×D3—1216no—
199C2×C3×D3—122no—
200C2×C3×D3—122no—
201C3⋊Dic3—21no—
202C3⋊Dic3—41no—
203C3⋊Dic3—41no—
204C3⋊Dic3D3×D412yes—
205C3×Dic3—62no—
206C3×Dic3—62no—
207C3×Dic3—62no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C3×Dic3—241no—
214C3×Dic3—241no—
215C22×D4—271no—
216C2×SD16—272no—
217C22⋊Q8—271no—
218#102—272no—
219C22⋊C8—91no—
220C33#45011yescharacteristic, complemented, Sylow
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—184no—
226C23×C3—182no—
227C23×C3—362no—
228C23×C3—362no—
229C23×C3GO+(4,𝔽2)11yescharacteristic
230C22×D3—64no—
231C22×D3—64no—
232C22×D3—62no—
233C22×D3—62no—
234C22×D3—124no—
235C22×D3—122no—
236C22×D3—122no—
237C22×D3—122no—
238C22×D3—122no—
239C22×D3—124no—
240C22×D3—124no—
241C22×D3—122no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—182no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247#67—184no—
248#67—184no—
249#67—188no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—64no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—122no—
260C2×Dic3—121no—
261C2×Dic3—121no—
262D12—94no—
263D12—182no—
264D12—184no—
265Dic6—182no—
266Dic6—362no—
267C3×C8—182no—
268C2×C32—24no—
269C2×C32—22no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—42no—
283C2×C32—44no—
284C2×C32—82no—
285C2×C32#88211yescharacteristic
286C2×C32C2×#20411yescharacteristic
287C2×C32#90811yescharacteristic
288C3⋊D3—44no—
289C3×D3—124no—
290C3×D3—128no—
291C3×D3—124no—
292C24—181no—
293C2×Q8—271no—
294C2×D4—272no—
295C2×D4—541no—
296C2×D4—544no—
297C22×C4—91no—
298SD16—544no—
299C2×C8—92no—
300C4⋊C4—271no—
301C4⋊C4—541no—
302C22⋊C4—181no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—42no—
314C22×C3—42no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—42no—
318C22×C3—42no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—184no—
322C22×C3—368no—
323C22×C3—362no—
324C22×C3—362no—
325C22×C3#377512yescomplemented
326C22×C3C2×GO+(4,𝔽2)11yescharacteristic
327C2×D3—64no—
328C2×D3—64no—
329C2×D3—124no—
330C2×D3—128no—
331C2×D3—128no—
332C2×D3—122no—
333C2×D3—122no—
334C2×D3—1216no—
335C2×D3—124no—
336C2×D3—124no—
337C2×D3—122no—
338C2×D3—122no—
339C3×C4—92no—
340C3×C4—181no—
341C3×C4—361no—
342C3×C4—361no—
343Dic3—62no—
344Dic3—62no—
345Dic3—62no—
346Dic3—121no—
347Dic3—121no—
348Dic3—121no—
349Dic3—121no—
350Dic3—121no—
351Dic3—241no—
352Dic3—241no—
353C32—21no—
354C32—21no—
355C32—41no—
356C32—41no—
357C32#473111yescharacteristic, complemented
358C23—184no—
359C23—182no—
360C23—362no—
361C23—362no—
362C23#157⋊D311yescharacteristic
363Q8—542no—
364D4—274no—
365D4—542no—
366D4—544no—
367C2×C4—91no—
368C2×C4—91no—
369C2×C4—182no—
370C2×C4—181no—
371C2×C4—181no—
372C8—182no—
373C2×C3—24no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—42no—
382C2×C3—44no—
383C2×C3—41no—
384C2×C3—41no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3—41no—
388C2×C3—41no—
389C2×C3—82no—
390C2×C3—364no—
391C2×C3#6423611yescharacteristic
392C2×C3C2×#377511yescharacteristic
393C2×C3#6424111yescharacteristic
394D3—124no—
395D3—128no—
396D3—124no—
397C22—21no—
398C22—21no—
399C22—184no—
400C22—368no—
401C22—362no—
402C22—362no—
403C22#8894212yescomplemented
404C22C2×#157⋊D311yescharacteristic
405C4—92no—
406C4—181no—
407C4—361no—
408C4—361no—
409C3—21no—
410C3—21no—
411C3—41no—
412C3—41no—
413C3#1059352311yescharacteristic, complemented
414C2—22no—
415C2—364no—
416C2C2×#8894211yescharacteristic
417C2#1172435011yescharacteristic
418C2#1172435411yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC22×#6295C2
hypercenterC23#157⋊D3
hypocenterC33#450
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88942C2semidirect
3. #11723721C2semidirect
4. C3×C22⋊#622C2semidirect
5. #11722793C2semidirect
9. C2×#6296C22semidirect
13. C2×C3×#622C22semidirect
17. #88815C22semidirect
23. C22⋊#622D3semidirect
56. C2×#622C2×D3semidirect
69. C22×C33SD16semidirect
183. C22×C32#204semidirect
220. C33#450semidirect
325. C22×C3#3775semidirect
357. C32#4731semidirect
403. C22#88942semidirect
413. C3#10593523semidirect

Maximal subgroups

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

subgroupindex
C22×#62952
C2×#889422
#117237212
C3×C22⋊#6222
#117227932
#105935233
#47319