Finite Group Databasebeta

← #13159944

Subgroups — #13159944

Order 1728.

1052 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1052 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, complemented
3#11723531C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722053C211yesmaximal, characteristic, complemented
8#10592481—31nomaximal
9#10590840—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6292—21no—
14C2×#6291—21no—
15C2×C33×C8C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17#88850—22no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20#88803—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88789—22no—
23#64147—31no—
24C2×#3753—31no—
25C2×#3743—31no—
26C2×#3743—34no—
27C2×#3743—64no—
28#64104—31no—
29#63916—34no—
30#63916—34no—
31C2×#3713—34no—
32C32×#102—31no—
33#63648D311yescharacteristic, complemented
34#63635—31no—
35C3×#821—34no—
36C3×#821D314yescomplemented
37#63574—34no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C22×C3×C3⋊D3—21no—
40C22×C32×D3—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4×C3⋊(C3⋊D3)—21no—
44C4×C3⋊(C3⋊D3)—21no—
45C2×C3×C3⋊Dic3—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C2×C32×Dic3—21no—
49C32×D12—21no—
50C32×D12—21no—
51#6292—42no—
52#6291—42no—
53C33×C8—21no—
54C3⋊(C3⋊(C3⋊C8))D412yescomplemented
55#4782—94no—
56#4782—91no—
57C2×C32×D4—31no—
58C2×#638—61no—
59C2×C4⋊(C3⋊D3)—61no—
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C4×C3⋊D3—31no—
62C2×C4×C3⋊D3—34no—
63C2×C4×C3⋊D3—64no—
64C2×C3×D12—34no—
65C2×C3×D12C2×D314yes—
66C2×#626—64no—
67C2×#626—64no—
68C2×C32×C8—24no—
69C2×C32×C8C2×D314yes—
70C2×C32×C8C2×D311yescharacteristic
71C32×C4⋊C4—31no—
72#3757—61no—
73#3753—62no—
74#3752—61no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—34no—
78C2×C3⋊(C3⋊C8)—64no—
79#3743—62no—
80#3743—68no—
81#3743—616no—
82C3×C4⋊Dic3—34no—
83C3×C4⋊Dic3C2×D314yes—
84(C2×D3)⋊Dic3—64no—
85(C2×D3)⋊Dic3—64no—
86#3713—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—21no—
89C2×C3⋊(C3⋊D3)—21no—
90C2×C3⋊(C3⋊D3)—21no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3⋊(C3⋊Dic3)—21no—
100C3×C3⋊Dic3—41no—
101C32×Dic3—41no—
102#938—94no—
103#938—91no—
104C2×#213—94no—
105C2×#213—91no—
106C2×ΓL(1,𝔽25)—94no—
107C2×ΓL(1,𝔽25)—91no—
108C2×ΓL(1,𝔽25)—184no—
109#895—94no—
110#895—91no—
111C3×#102—34no—
112C3×#102—91no—
113#821—34no—
114#821C3⋊D311yescharacteristic, complemented
115#808—94no—
116#808—91no—
117C23×C32—61no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3—31no—
120C22×C3⋊D3—34no—
121C22×C3⋊D3—64no—
122C22×C3×D3—24no—
123C22×C3×D3—64no—
124C32×D4—61no—
125C32×D4—61no—
126C2×C32×C4—24no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130#638—122no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—31no—
133C2×C3⋊Dic3—34no—
134C2×C3⋊Dic3—64no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—122no—
138C4×C3⋊D3—68no—
139C4×C3⋊D3—68no—
140C4×C3⋊D3—64no—
141C4×C3⋊D3—61no—
142C4×C3⋊D3—64no—
143C4×C3⋊D3—61no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—64no—
146C3×D12—24no—
147C3×D12—24no—
148C3×D12—64no—
149C3×D12—64no—
150#626—128no—
151#626—128no—
152C32×C8—28no—
153C32×C8—24no—
154C32×C8—21no—
155C3⋊(C3⋊C8)—32no—
156C3⋊(C3⋊C8)—38no—
157C3⋊(C3⋊C8)—68no—
158#469—271noSylow
159C2×C33#12111yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C2×C33AGL(1,ℤ/8)11yescharacteristic
162C3⋊(C3⋊D3)—41no—
163C3×C3⋊D3—81no—
164C32×D3—81no—
165C2×C3×D4—34no—
166C2×C3×D4—91no—
167C2×#67—181no—
168C2×#67—184no—
169C2×D12—34no—
170C2×D12—184no—
171C2×D12—181no—
172C2×D12C2×C3⋊D311yescharacteristic
173C2×C4×D3—94no—
174C2×C4×D3—91no—
175C2×C4×D3—184no—
176C2×C3×C8—24no—
177C2×C3×C8D3214yes—
178C2×C3×C8C2×C3⋊D311yescharacteristic
179C3×C4⋊C4—34no—
180C3×C4⋊C4—91no—
181#217—181no—
182#217—184no—
183#213—182no—
184#213—188no—
185(C2×D3)⋊C4—184no—
186(C2×D3)⋊C4—181no—
187C4⋊Dic3—34no—
188C4⋊Dic3C2×C3⋊D311yescharacteristic
189C2×C3⋊C8—94no—
190C2×C3⋊C8—91no—
191C2×C3⋊C8—184no—
192ΓL(1,𝔽25)—188no—
193ΓL(1,𝔽25)—182no—
194ΓL(1,𝔽25)—1816no—
195C22×C32—24no—
196C22×C32—121no—
197C22×C32—121no—
198C22×C32—121no—
199C22×C32D3×D414yes—
200C22×C32D3×D411yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—68no—
205C2×C3⋊D3—64no—
206C2×C3⋊D3—61no—
207C2×C3⋊D3—61no—
208C2×C3⋊D3—61no—
209C2×C3⋊D3—64no—
210C2×C3⋊D3—64no—
211C2×C3⋊D3—124no—
212C2×C3⋊D3—124no—
213C2×C3×D3—44no—
214C2×C3×D3—44no—
215C2×C3×D3—44no—
216C2×C3×D3—124no—
217C2×C3×D3—124no—
218C2×C3×D3—124no—
219C32×C4—24no—
220C32×C4—24no—
221C32×C4—121no—
222C32×C4#23514yes—
223C32×C4#23511yescharacteristic
224C32×C4D3×D414yes—
225C32×C4D3×D411yescharacteristic
226C3⋊Dic3—41no—
227C3⋊Dic3—68no—
228C3⋊Dic3—64no—
229C3⋊Dic3—61no—
230C3×Dic3—44no—
231C3×Dic3—124no—
232C2×D8—271no—
233C2×M16—271no—
234#121—271no—
235#121—271no—
236#106—271no—
237#102—31no—
238#102—91no—
239C33#46911yescharacteristic, complemented, Sylow
240C23×C3—64no—
241C23×C3—181no—
242C22×D3—21no—
243C22×D3—64no—
244C22×D3—94no—
245C22×D3—91no—
246C22×D3—184no—
247C3×D4—64no—
248C3×D4—64no—
249C3×D4—181no—
250C3×D4—181no—
251C2×C3×C4—24no—
252C2×C3×C4—64no—
253C2×C3×C4—181no—
254C2×C3×C4C2×D3214yes—
255C2×C3×C4C22×C3⋊D311yescharacteristic
256#67—362no—
257#67—368no—
258C2×Dic3—21no—
259C2×Dic3—64no—
260C2×Dic3—94no—
261C2×Dic3—91no—
262C2×Dic3—184no—
263D12—21no—
264D12—21no—
265D12—64no—
266D12—64no—
267D12—368no—
268D12—362no—
269C4×D3—188no—
270C4×D3—188no—
271C4×D3—184no—
272C4×D3—184no—
273C4×D3—181no—
274C4×D3—181no—
275C3×C8—28no—
276C3×C8—24no—
277C3×C8—21no—
278C3⋊C8—98no—
279C3⋊C8—92no—
280C3⋊C8—188no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—241no—
285C2×C32#88414yes—
286C2×C32#88411yescharacteristic
287C2×C32#91114yes—
288C2×C32#91111yescharacteristic
289C2×C32#91414yes—
290C2×C32#91411yescharacteristic
291C3⋊D3—81no—
292C3⋊D3—128no—
293C3⋊D3—124no—
294C3⋊D3—121no—
295C3×D3—84no—
296C3×D3—244no—
297C2×D4—31no—
298C2×D4—91no—
299C2×D4—541no—
300C2×D4—541no—
301C22×C4—271no—
302D8—542no—
303M16—542no—
304C2×C8—271no—
305C2×C8D3×C3⋊D311yescharacteristic
306C4⋊C4—31no—
307C4⋊C4—91no—
308C22⋊C4—541no—
309C22⋊C4—541no—
310C22×C3—24no—
311C22×C3—124no—
312C22×C3—124no—
313C22×C3—124no—
314C22×C3—361no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3#380214yes—
318C22×C3D4×C3⋊D311yescharacteristic
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—124no—
323C2×D3—124no—
324C2×D3—124no—
325C2×D3—188no—
326C2×D3—184no—
327C2×D3—184no—
328C2×D3—181no—
329C2×D3—181no—
330C2×D3—181no—
331C2×D3—184no—
332C2×D3—364no—
333C2×D3—364no—
334C3×C4—24no—
335C3×C4—24no—
336C3×C4—124no—
337C3×C4—361no—
338C3×C4#379814yes—
339C3×C4#381114yes—
340C3×C4#382811yescharacteristic
341C3×C4D4×C3⋊D311yescharacteristic
342Dic3—41no—
343Dic3—124no—
344Dic3—188no—
345Dic3—181no—
346Dic3—184no—
347C32—24no—
348C32#478214yescomplemented
349C32#478211yescharacteristic, complemented
350C23—61no—
351C23—181no—
352C23—271no—
353D4—61no—
354D4—61no—
355D4—181no—
356D4—181no—
357D4—1082no—
358D4—1082no—
359C2×C4—61no—
360C2×C4—181no—
361C2×C4—271no—
362C2×C4—541no—
363C2×C4—541no—
364C2×C4C2×D3×C3⋊D311yescharacteristic
365C8—21no—
366C8—272no—
367C2×C3—24no—
368C2×C3—24no—
369C2×C3—24no—
370C2×C3—244no—
371C2×C3—721no—
372C2×C3#6379814yes—
373C2×C3#6379814yes—
374C2×C3#6391214yes—
375C2×C3#6409311yescharacteristic
376C2×C3#6412011yescharacteristic
377C2×C3#6412311yescharacteristic
378D3—81no—
379D3—244no—
380D3—368no—
381D3—364no—
382D3—361no—
383C22—121no—
384C22—121no—
385C22—121no—
386C22—361no—
387C22—361no—
388C22—361no—
389C22—541no—
390C22—541no—
391C22—541no—
392C22#8902611yescharacteristic
393C4—121no—
394C4—361no—
395C4—541no—
396C4#8902011yescharacteristic
397C4#8903911yescharacteristic
398C3—24no—
399C3#1059084014yescomplemented
400C3#1059248111yescharacteristic, complemented
401C2—241no—
402C2—721no—
403C2—1081no—
404C2#1172332311yescharacteristic
405C2#1172332411yescharacteristic
406C2#1172352711yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#469
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
2. #11723532C2semidirect
3. #11723531C2semidirect
4. C2×#88789C2semidirect
5. C3×#63648C2semidirect
6. C32×#821C2semidirect
7. #11722053C2semidirect
33. #63648D3semidirect
36. C3×#821D3semidirect
54. C3⋊(C3⋊(C3⋊C8))D4semidirect
114. #821C3⋊D3semidirect
239. C33#469semidirect
348. C32#4782semidirect
349. C32#4782semidirect
399. C3#10590840semidirect
400. C3#10592481semidirect

Maximal subgroups

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

subgroupindex
C2×#888502
#117235322
#117235312
C2×#887892
C3×#636482
C32×#8212
#117220532
#105924813
#105908403