Finite Group Databasebeta

← #13159979

Subgroups — #13159979

Order 1728.

992 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 144 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×#88850C211yesmaximal, characteristic, complemented
2#11723532C211yesmaximal, characteristic
3#11723456C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic, complemented
6C32×#816C211yesmaximal, characteristic, complemented
7#11722056C211yesmaximal, characteristic, complemented
8#10592511—31nomaximal
9#10590842—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6292—21no—
14C2×C33×C8C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16#88850—22no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88807—21no—
20#88804—21no—
21#88803—21no—
22#88793—22no—
23#64155—31no—
24C2×#3755—31no—
25C2×#3743—31no—
26C2×#3743—34no—
27C2×#3743—64no—
28#64104—31no—
29#63916—34no—
30#63871—34no—
31C2×#3715—34no—
32C32×#103—31no—
33#63648D311yescharacteristic, complemented
34#63634—31no—
35C3×#821—34no—
36C3×#816D314yescomplemented
37#63576—34no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C22×C3×C3⋊D3—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C4×C3⋊(C3⋊D3)—21no—
43C4×C3⋊(C3⋊D3)—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C2×C32×Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51#6292—42no—
52C33×C8—21no—
53C3⋊(C3⋊(C3⋊C8))D412yescomplemented
54#4812—94no—
55#4780—91no—
56C2×C32×Q8—31no—
57C2×C4⋊(C3⋊D3)—61no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×C4×C3⋊D3—31no—
60C2×C4×C3⋊D3—34no—
61C2×C4×C3⋊D3—64no—
62C2×C3×D12—34no—
63C2×C3×Dic6C2×D314yes—
64C2×#626—64no—
65C2×C32×C8—24no—
66C2×C32×C8C2×D314yes—
67C2×C32×C8C2×D311yescharacteristic
68C32×C4⋊C4—31no—
69#3755—62no—
70#3752—61no—
71#3752—61no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73#3750—61no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77#3743—62no—
78#3743—68no—
79#3743—616no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3C2×D314yes—
82Dic3⋊Dic3—64no—
83#3722—64no—
84(C2×D3)⋊Dic3—64no—
85#3715—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—21no—
88C2×C3⋊(C3⋊D3)—21no—
89C2×C3⋊(C3⋊D3)—21no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—21no—
96C3×C3⋊Dic3—41no—
97C32×Dic3—41no—
98C32×Dic3—41no—
99#946—94no—
100C2×#215—94no—
101#938—91no—
102C2×#214—91no—
103C2×ΓL(1,𝔽25)—94no—
104C2×ΓL(1,𝔽25)—91no—
105C2×ΓL(1,𝔽25)—184no—
106#897—91no—
107#895—94no—
108C3×#103—34no—
109C3×#102—91no—
110#821—34no—
111#816C3⋊D311yescharacteristic, complemented
112#807—94no—
113#807—91no—
114C22×C3⋊D3—21no—
115C22×C3⋊D3—31no—
116C22×C3⋊D3—34no—
117C22×C3⋊D3—64no—
118C22×C3×D3—64no—
119C32×Q8—61no—
120C32×Q8—61no—
121C2×C32×C4—24no—
122C2×C32×C4—61no—
123C2×C32×C4—61no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—31no—
128C2×C3⋊Dic3—34no—
129C2×C3⋊Dic3—64no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—122no—
133C4×C3⋊D3—68no—
134C4×C3⋊D3—68no—
135C4×C3⋊D3—64no—
136C4×C3⋊D3—61no—
137C4×C3⋊D3—64no—
138C4×C3⋊D3—61no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—64no—
142C3×D12—64no—
143C3×D12—64no—
144C3×Dic6—24no—
145C3×Dic6—24no—
146#626—128no—
147C32×C8—28no—
148C32×C8—24no—
149C32×C8—21no—
150C3⋊(C3⋊C8)—32no—
151C3⋊(C3⋊C8)—38no—
152C3⋊(C3⋊C8)—68no—
153#468—271noSylow
154C2×C33#12111yescharacteristic
155C2×C33AGL(1,ℤ/8)11yescharacteristic
156C2×C33#13711yescharacteristic
157C3⋊(C3⋊D3)—41no—
158C3×C3⋊D3—81no—
159C2×C3×Q8—34no—
160C2×C3×D4—91no—
161C2×#67—181no—
162C2×D12—34no—
163C2×D12—184no—
164C2×C4×D3—94no—
165C2×C4×D3—91no—
166C2×C4×D3—184no—
167C2×Dic6C2×C3⋊D311yescharacteristic
168C2×C3×C8—24no—
169C2×C3×C8D3214yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C3×C4⋊C4—34no—
172C3×C4⋊C4—91no—
173#217—181no—
174#215—188no—
175#214—182no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—184no—
178(C2×D3)⋊C4—184no—
179C4⋊Dic3—34no—
180C4⋊Dic3—181no—
181C4⋊Dic3C2×C3⋊D311yescharacteristic
182Dic3⋊C4—184no—
183C2×C3⋊C8—94no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—184no—
186ΓL(1,𝔽25)—188no—
187ΓL(1,𝔽25)—182no—
188ΓL(1,𝔽25)—1816no—
189C22×C32—24no—
190C22×C32D3×D414yes—
191C22×C32D3×D411yescharacteristic
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—68no—
196C2×C3⋊D3—64no—
197C2×C3⋊D3—61no—
198C2×C3⋊D3—61no—
199C2×C3⋊D3—61no—
200C2×C3⋊D3—64no—
201C2×C3⋊D3—64no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—124no—
204C2×C3×D3—124no—
205C2×C3×D3—124no—
206C2×C3×D3—124no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—121no—
210C32×C4—121no—
211C32×C4#23514yes—
212C32×C4#23511yescharacteristic
213C32×C4D3×D414yes—
214C32×C4D3×D411yescharacteristic
215C3⋊Dic3—41no—
216C3⋊Dic3—68no—
217C3⋊Dic3—64no—
218C3⋊Dic3—61no—
219C3×Dic3—44no—
220C3×Dic3—44no—
221C3×Dic3—124no—
222C2×SD16—271no—
223C2×M16—271no—
224C22⋊Q8—271no—
225#121—271no—
226#107—271no—
227#103—31no—
228#102—91no—
229C33#46811yescharacteristic, complemented, Sylow
230C23×C3—181no—
231C22×D3—64no—
232C22×D3—94no—
233C22×D3—91no—
234C22×D3—184no—
235C3×Q8—64no—
236C3×Q8—64no—
237C3×D4—181no—
238C3×D4—181no—
239C2×C3×C4—24no—
240C2×C3×C4—64no—
241C2×C3×C4—64no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245#67—362no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—64no—
249C2×Dic3—94no—
250C2×Dic3—91no—
251C2×Dic3—184no—
252D12—64no—
253D12—64no—
254D12—368no—
255C4×D3—188no—
256C4×D3—188no—
257C4×D3—184no—
258C4×D3—184no—
259C4×D3—181no—
260C4×D3—181no—
261Dic6—21no—
262Dic6—21no—
263C3×C8—28no—
264C3×C8—24no—
265C3×C8—21no—
266C3⋊C8—98no—
267C3⋊C8—92no—
268C3⋊C8—188no—
269C2×C32—24no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32#88414yes—
273C2×C32#88411yescharacteristic
274C2×C32#91111yescharacteristic
275C2×C32#91414yes—
276C2×C32#91511yescharacteristic
277C2×C32#91814yes—
278C3⋊D3—81no—
279C3⋊D3—128no—
280C3⋊D3—121no—
281C3⋊D3—124no—
282C3×D3—244no—
283C2×Q8—31no—
284C2×D4—91no—
285C2×D4—541no—
286C22×C4—271no—
287SD16—542no—
288M16—542no—
289C2×C8—271no—
290C2×C8D3×C3⋊D311yescharacteristic
291C4⋊C4—31no—
292C4⋊C4—91no—
293C4⋊C4—541no—
294C22⋊C4—541no—
295C22⋊C4—541no—
296C22×C3—24no—
297C22×C3—361no—
298C22×C3—361no—
299C22×C3—361no—
300C22×C3#380214yes—
301C22×C3D4×C3⋊D311yescharacteristic
302C2×D3—124no—
303C2×D3—124no—
304C2×D3—124no—
305C2×D3—188no—
306C2×D3—184no—
307C2×D3—184no—
308C2×D3—181no—
309C2×D3—181no—
310C2×D3—181no—
311C2×D3—184no—
312C2×D3—364no—
313C2×D3—364no—
314C3×C4—24no—
315C3×C4—24no—
316C3×C4—124no—
317C3×C4—124no—
318C3×C4—361no—
319C3×C4#379814yes—
320C3×C4#381114yes—
321C3×C4#382811yescharacteristic
322C3×C4D4×C3⋊D311yescharacteristic
323Dic3—41no—
324Dic3—41no—
325Dic3—124no—
326Dic3—188no—
327Dic3—181no—
328Dic3—184no—
329C32—24no—
330C32#478011yescharacteristic, complemented
331C32#481214yescomplemented
332C23—181no—
333C23—271no—
334Q8—61no—
335Q8—61no—
336D4—181no—
337D4—181no—
338D4—1082no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—181no—
342C2×C4—271no—
343C2×C4—541no—
344C2×C4—541no—
345C2×C4C2×D3×C3⋊D311yescharacteristic
346C8—21no—
347C8—272no—
348C2×C3—24no—
349C2×C3—24no—
350C2×C3—24no—
351C2×C3—721no—
352C2×C3#6379814yes—
353C2×C3#6381114yes—
354C2×C3#6391214yes—
355C2×C3#6409311yescharacteristic
356C2×C3#6412311yescharacteristic
357C2×C3#6412711yescharacteristic
358D3—244no—
359D3—368no—
360D3—364no—
361D3—361no—
362C22—361no—
363C22—361no—
364C22—361no—
365C22—541no—
366C22—541no—
367C22—541no—
368C22#8902611yescharacteristic
369C4—121no—
370C4—121no—
371C4—361no—
372C4—541no—
373C4#8902011yescharacteristic
374C4#8903911yescharacteristic
375C3—24no—
376C3#1059084214yescomplemented
377C3#1059251111yescharacteristic, complemented
378C2—721no—
379C2—1081no—
380C2#1172332311yescharacteristic
381C2#1172334611yescharacteristic
382C2#1172352711yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723532C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88850C2semidirect
3. #11723456C2semidirect
4. C2×#88793C2semidirect
5. C3×#63648C2semidirect
6. C32×#816C2semidirect
7. #11722056C2semidirect
33. #63648D3semidirect
36. C3×#816D3semidirect
53. C3⋊(C3⋊(C3⋊C8))D4semidirect
111. #816C3⋊D3semidirect
229. C33#468semidirect
330. C32#4780semidirect
331. C32#4812semidirect
376. C3#10590842semidirect
377. C3#10592511semidirect

Maximal subgroups

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

subgroupindex
C2×#888502
#117235322
#117234562
C2×#887932
C3×#636482
C32×#8162
#117220562
#105925113
#105908423