Finite Group Databasebeta

← #13140960

Subgroups — #13140960

Order 1728.

502 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 75 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×#88549C211yesmaximal, characteristic
2#64153⋊C3C211yesmaximal, characteristic, complemented
3C2×#3756⋊C3C211yesmaximal, characteristic, complemented
4#11721114C211yesmaximal, characteristic
5C2×#88437C211yesmaximal, characteristic
6C2×#88433C211yesmaximal, characteristic, complemented
7#11720312C211yesmaximal, characteristic, complemented
8#10591611—31nomaximal
9#10591605—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14#88549—22no—
15C2×#6180C2211yescharacteristic
16#3756⋊C3—22no—
17#3750⋊C3—22no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—22no—
21#88437—22no—
22#88433—22no—
23#64153D311yescharacteristic, complemented
24C2×#3756D311yescharacteristic, complemented
25C3×#944—31no—
26C2×C3×#216—31no—
27C2×C3×#215—31no—
28C2×C3×#215—31no—
29C2×C3×#215—61no—
30#63863—31no—
31#63860—31no—
32C2×#3719—31no—
33C2×#3719—31no—
34C2×#3716—31no—
35C2×#3715—31no—
36#63555—31no—
37#63554—31no—
38C22×#272—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C4⋊#272—21no—
44C4⋊#272—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D412yes—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52#6180D412yes—
53(C3⋊(C3⋊C8))⋊C3D412yes—
54#5190—91no—
55#5167—91no—
56#5086—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—61no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—31no—
67#3756—22no—
68#3750—22no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#216—62no—
72C3×#215—62no—
73C3×#215—62no—
74C3×#215—64no—
75C3×Dic3⋊C4—62no—
76C3×C4×Dic3—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—61no—
81(C2×D3)⋊Dic3—62no—
82(C2×D3)⋊Dic3—62no—
83#3719—62no—
84#3719—62no—
85#3716—62no—
86#3715—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—41no—
89C2×#272—42no—
90C4×H(3,𝔽3)—41no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)C2×D411yescharacteristic
93(C3⋊Dic3)⋊C3—22no—
94(C3⋊Dic3)⋊C3—41no—
95(C3⋊Dic3)⋊C3—41no—
96C2×C3×Q16—91no—
97C2×C3×SD16—91no—
98C3×C4⋊Q8—91no—
99#947—91no—
100#944—31no—
101#944—91no—
102C2×#216—31no—
103C2×#216—91no—
104C2×#216—91no—
105C2×#216—91no—
106C2×#215—31no—
107C2×#215—31no—
108C2×#215—61no—
109C2×#215—91no—
110#936—91no—
111C2×#214—91no—
112C2×Dic12—91no—
113C2×#204—91no—
114#875—91no—
115#815—91no—
116#815—91no—
117#803—91no—
118C22×C3×D3—61no—
119C22×C3×D3—61no—
120C22×C3×D3—121no—
121C32×Q8—22no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—41no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—41no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3#6712yes—
136#634—21no—
137#634—21no—
138#634—21no—
139#634—21no—
140C2×C3×Dic3—32no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C3×D12—62no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—121no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3⋊(C3⋊C8)#6712yes—
154C3×C3⋊C8—32no—
155C3×C3⋊C8—32no—
156C3×C3⋊C8—32no—
157C3×C3⋊C8—62no—
158#497—271noSylow
159C2×H(3,𝔽3)#12711yescharacteristic
160C2×H(3,𝔽3)#13712yes—
161#272—81no—
162C2×C3×Q8—31no—
163C2×C3×Q8—31no—
164C2×C3×Q8—61no—
165C2×C3×Q8—91no—
166C2×C3×Q8—91no—
167C2×C3×Q8D3211yescharacteristic
168C2×C3×D4—91no—
169C2×D12—31no—
170C2×D12—31no—
171C2×D12—61no—
172C2×Dic6—31no—
173C2×Dic6—31no—
174C2×Dic6—91no—
175C2×Dic6—91no—
176C3×Q16—182no—
177C3×SD16—182no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C3×C4⋊C4—182no—
181C3×C42—91no—
182#217—182no—
183#216—62no—
184#216—182no—
185#216—182no—
186#216—182no—
187#215—62no—
188#215—62no—
189#215—64no—
190#215—182no—
191#214—182no—
192(C2×D3)⋊C4—182no—
193(C2×D3)⋊C4—182no—
194Dic3⋊C4—62no—
195Dic3⋊C4—182no—
196C4×Dic3—31no—
197C4×Dic3—91no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—61no—
202C2×C3⋊C8—91no—
203Dic12—182no—
204#204—182no—
205C22×C32—21no—
206C22×C32D3×D411yescharacteristic
207C22×C32C2×#6711yescharacteristic
208C2×C3×D3—122no—
209C2×C3×D3—122no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C2×C3×D3—122no—
213C2×C3×D3—242no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—42no—
217C32×C4—41no—
218C32×C4—41no—
219C32×C4D3×D411yescharacteristic
220C32×C4D3×D411yescharacteristic
221C32×C4D3×D411yescharacteristic
222C32×C4C2×#6711yescharacteristic
223C3⋊Dic3—22no—
224C3⋊Dic3—41no—
225C3⋊Dic3—41no—
226C3×Dic3—62no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C2×Q16—91no—
230C2×Q16—271no—
231C2×SD16—91no—
232C2×SD16—271no—
233C4⋊Q8—91no—
234#124—271no—
235#97—271no—
236H(3,𝔽3)#49711yescharacteristic, complemented, Sylow
237C23×C3—181no—
238C22×D3—61no—
239C22×D3—61no—
240C22×D3—121no—
241C3×Q8—21no—
242C3×Q8—21no—
243C3×Q8—61no—
244C3×Q8—61no—
245C3×Q8—61no—
246C3×Q8—61no—
247C3×Q8—62no—
248C3×Q8—121no—
249C3×Q8—181no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×Q8—181no—
253C3×D4—181no—
254C3×D4—181no—
255C2×C3×C4—21no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—92no—
262C2×C3×C4—121no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4C2×D3211yescharacteristic
266C2×Dic3—32no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—92no—
270C2×Dic3—181no—
271C2×Dic3—181no—
272D12—61no—
273D12—61no—
274D12—61no—
275D12—61no—
276D12—62no—
277D12—121no—
278Dic6—61no—
279Dic6—61no—
280Dic6—61no—
281Dic6—61no—
282Dic6—181no—
283Dic6—181no—
284Dic6—181no—
285Dic6—181no—
286C3×C8—92no—
287C3×C8—92no—
288C3⋊C8—32no—
289C3⋊C8—32no—
290C3⋊C8—32no—
291C3⋊C8—62no—
292C3⋊C8—92no—
293C2×C32—22no—
294C2×C32—21no—
295C2×C32#91812yes—
296C2×C32#94011yescharacteristic
297C2×C32#94011yescharacteristic
298C2×C32#94212yes—
299C3×D3—242no—
300C3×D3—241no—
301C3×D3—241no—
302C2×Q8—91no—
303C2×Q8—91no—
304C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
305C2×D4—91no—
306Q16—182no—
307Q16—542no—
308SD16—182no—
309SD16—542no—
310C2×C8—91no—
311C2×C8—91no—
312C4⋊C4—182no—
313C22⋊C4—542no—
314C42—91no—
315C22×C3—31no—
316C22×C3—31no—
317C22×C3—61no—
318C22×C3—362no—
319C22×C3—361no—
320C22×C3D3×#6711yescharacteristic
321C2×D3—121no—
322C2×D3—121no—
323C2×D3—122no—
324C2×D3—122no—
325C2×D3—122no—
326C2×D3—242no—
327C3×C4—31no—
328C3×C4—31no—
329C3×C4—31no—
330C3×C4—31no—
331C3×C4—41no—
332C3×C4—61no—
333C3×C4—61no—
334C3×C4—121no—
335C3×C4—121no—
336C3×C4—122no—
337C3×C4—182no—
338C3×C4—361no—
339C3×C4—361no—
340C3×C4D3×#6711yescharacteristic
341C3×C4#381111yescharacteristic
342Dic3—62no—
343Dic3—121no—
344Dic3—121no—
345Dic3—182no—
346Dic3—361no—
347Dic3—361no—
348C32—21no—
349C32#508611yescharacteristic, complemented
350C32#519011yescharacteristic, complemented
351C23—181no—
352Q8—21no—
353Q8—21no—
354Q8—181no—
355Q8—181no—
356Q8—181no—
357Q8—181no—
358D4—181no—
359D4—181no—
360C2×C4—21no—
361C2×C4—92no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
365C8—92no—
366C8—92no—
367C2×C3—32no—
368C2×C3—32no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—62no—
372C2×C3—61no—
373C2×C3—721no—
374C2×C3#6394812yes—
375C2×C3#6397911yescharacteristic
376D3—241no—
377D3—241no—
378D3—242no—
379C22—362no—
380C22—361no—
381C22#8867611yescharacteristic
382C4—41no—
383C4—182no—
384C4—361no—
385C4—361no—
386C4#8867611yescharacteristic
387C4#8867711yescharacteristic
388C3—31no—
389C3—31no—
390C3—61no—
391C3#1059160811yescharacteristic
392C2—721no—
393C2#1172123412yes—
394C2#1172131711yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊#272C22
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#497
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #64153⋊C3C2semidirect
3. C2×#3756⋊C3C2semidirect
6. C2×#88433C2semidirect
7. #11720312C2semidirect
23. #64153D3semidirect
24. C2×#3756D3semidirect
236. H(3,𝔽3)#497semidirect
349. C32#5086semidirect
350. C32#5190semidirect

Maximal subgroups

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

subgroupindex
C2×#885492
#64153⋊C32
C2×#3756⋊C32
#117211142
C2×#884372
C2×#884332
#117203122
#105916113
#105916053