Finite Group Databasebeta

← #13140988

Subgroups — #13140988

Order 1728.

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

normal subgroups only

This group has 494 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×#88550C211yesmaximal, characteristic, complemented
2#64153⋊C3C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic
4#11721110C211yesmaximal, characteristic
5C2×#88439C211yesmaximal, characteristic, complemented
6C2×#88431C211yesmaximal, characteristic
7#11720312C211yesmaximal, characteristic, complemented
8#10591611—31nomaximal
9#10591608—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14#88550—22no—
15C2×#6180C2211yescharacteristic
16#3755⋊C3—22no—
17#3750⋊C3—22no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—22no—
21#88439—22no—
22#88431—22no—
23#64153D311yescharacteristic, complemented
24C2×#3755D311yescharacteristic
25C3×#944—31no—
26C2×C3×#216—31no—
27C2×C3×#216—31no—
28C2×C3×#216—61no—
29C2×C3×#215—31no—
30#63864—31no—
31#63863—31no—
32C2×#3719—31no—
33C2×#3718—31no—
34C2×#3717—31no—
35C2×#3716—31no—
36#63555—31no—
37#63554—31no—
38C22×(C3⋊D3)⋊C3—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44#6229—21no—
45#6229—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D412yes—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊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×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67#3755—22no—
68#3750—22no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#216—62no—
72C3×#216—62no—
73C3×#216—64no—
74C3×#215—62no—
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#3722—62no—
82(C2×D3)⋊Dic3—62no—
83#3719—62no—
84#3718—62no—
85#3717—62no—
86#3716—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×(C3⋊D3)⋊C3—41no—
89C2×(C3⋊D3)⋊C3—42no—
90C4×H(3,𝔽3)—41no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)C2×D411yescharacteristic
93#1084—41no—
94(C3⋊Dic3)⋊C3—22no—
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—31no—
104C2×#216—61no—
105C2×#216—91no—
106C2×#216—91no—
107C2×#215—31no—
108C2×#215—91no—
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—21no—
119C22×C3×D3—61no—
120C32×Q8—22no—
121C32×Q8—21no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—41no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3#6712yes—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136#634—21no—
137#634—21no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—121no—
143C3×D12—61no—
144C3×D12—61no—
145C3×Dic6—62no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—121no—
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(C3⋊D3)⋊C3—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—91no—
171C2×Dic6—31no—
172C2×Dic6—31no—
173C2×Dic6—31no—
174C2×Dic6—61no—
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—62no—
185#216—64no—
186#216—182no—
187#216—182no—
188#215—62no—
189#215—182no—
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—42no—
209C2×C3⋊D3—41no—
210C2×C3×D3—121no—
211C2×C3×D3—122no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—42no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4D3×D411yescharacteristic
218C32×C4D3×D411yescharacteristic
219C32×C4D3×D411yescharacteristic
220C32×C4C2×#6711yescharacteristic
221C3⋊Dic3—22no—
222C3⋊Dic3—41no—
223C3×Dic3—62no—
224C3×Dic3—122no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C3×Dic3—121no—
228C2×Q16—91no—
229C2×Q16—271no—
230C2×SD16—91no—
231C2×SD16—271no—
232C4⋊Q8—91no—
233#124—271no—
234#97—271no—
235H(3,𝔽3)#49711yescharacteristic, complemented, Sylow
236C23×C3—181no—
237C22×D3—61no—
238C22×D3—181no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—61no—
242C3×Q8—61no—
243C3×Q8—61no—
244C3×Q8—61no—
245C3×Q8—62no—
246C3×Q8—121no—
247C3×Q8—181no—
248C3×Q8—181no—
249C3×Q8—181no—
250C3×Q8—181no—
251C3×D4—181no—
252C3×D4—181no—
253C2×C3×C4—21no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—92no—
260C2×C3×C4—121no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4C2×D3211yescharacteristic
264C2×Dic3—32no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—92no—
269C2×Dic3—121no—
270C2×Dic3—181no—
271D12—61no—
272D12—61no—
273D12—181no—
274D12—181no—
275Dic6—61no—
276Dic6—61no—
277Dic6—61no—
278Dic6—61no—
279Dic6—62no—
280Dic6—61no—
281Dic6—61no—
282Dic6—121no—
283Dic6—181no—
284Dic6—181no—
285C3×C8—92no—
286C3×C8—92no—
287C3⋊C8—32no—
288C3⋊C8—32no—
289C3⋊C8—32no—
290C3⋊C8—62no—
291C3⋊C8—92no—
292C2×C32—22no—
293C2×C32—21no—
294C2×C32#91512yes—
295C2×C32#94011yescharacteristic
296C2×C32#94011yescharacteristic
297C2×C32#94212yes—
298C3⋊D3—81no—
299C3×D3—241no—
300C2×Q8—91no—
301C2×Q8—91no—
302C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
303C2×D4—91no—
304Q16—182no—
305Q16—542no—
306SD16—182no—
307SD16—542no—
308C2×C8—91no—
309C2×C8—91no—
310C4⋊C4—182no—
311C22⋊C4—542no—
312C42—91no—
313C22×C3—31no—
314C22×C3—31no—
315C22×C3—61no—
316C22×C3—361no—
317C22×C3—362no—
318C22×C3D3×#6711yescharacteristic
319C2×D3—122no—
320C2×D3—121no—
321C2×D3—361no—
322C2×D3—362no—
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—41no—
328C3×C4—61no—
329C3×C4—61no—
330C3×C4—121no—
331C3×C4—121no—
332C3×C4—122no—
333C3×C4—182no—
334C3×C4—361no—
335C3×C4—361no—
336C3×C4D3×#6711yescharacteristic
337C3×C4#381111yescharacteristic
338Dic3—62no—
339Dic3—121no—
340Dic3—121no—
341Dic3—122no—
342Dic3—121no—
343Dic3—182no—
344Dic3—361no—
345C32—21no—
346C32#508611yescharacteristic, complemented
347C32#516711yescharacteristic, complemented
348C23—181no—
349Q8—21no—
350Q8—21no—
351Q8—181no—
352Q8—181no—
353Q8—181no—
354Q8—181no—
355D4—181no—
356D4—181no—
357C2×C4—21no—
358C2×C4—92no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
362C8—92no—
363C8—92no—
364C2×C3—32no—
365C2×C3—32no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—62no—
369C2×C3—61no—
370C2×C3—721no—
371C2×C3#6394612yes—
372C2×C3#6397911yescharacteristic
373D3—241no—
374D3—721no—
375C22—362no—
376C22—361no—
377C22#8867611yescharacteristic
378C4—41no—
379C4—182no—
380C4—361no—
381C4—361no—
382C4#8867611yescharacteristic
383C4#8867711yescharacteristic
384C3—31no—
385C3—31no—
386C3—61no—
387C3#1059160511yescharacteristic
388C2—721no—
389C2#1172123812yes—
390C2#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⋊((C3⋊D3)⋊C3)C22
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
1. C2×#88550C2semidirect
2. #64153⋊C3C2semidirect
5. C2×#88439C2semidirect
7. #11720312C2semidirect
23. #64153D3semidirect
235. H(3,𝔽3)#497semidirect
346. C32#5086semidirect
347. C32#5167semidirect

Maximal subgroups

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

subgroupindex
C2×#885502
#64153⋊C32
C2×#3755⋊C32
#117211102
C2×#884392
C2×#884312
#117203122
#105916113
#105916083