Finite Group Databasebeta

← #13139013

Subgroups — #13139013

Order 1728.

523 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88528C211yesmaximal, characteristic, complemented
2C2×#3744⋊C3C211yesmaximal, characteristic, complemented
3#11721114C211yesmaximal, characteristic, complemented
4#11721110C211yesmaximal, characteristic, complemented
5C2×#88439C211yesmaximal, characteristic, complemented
6C2×#88436C211yesmaximal, characteristic
7C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
8#10590867—31nomaximal
9#10590862—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#88528—22no—
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#3744⋊C3—22no—
19#88451—22no—
20#88450—22no—
21#88439—22no—
22#88436—22no—
23C2×#3744D311yescharacteristic, complemented
24C2×C3×#204—31no—
25C2×C3×#204—31no—
26C2×C3×#204—31no—
27C2×C3×#204—61no—
28#63864—31no—
29#63863—31no—
30#63863—31no—
31#63860—31no—
32C2×#3719—31no—
33C2×#3718—31no—
34C2×#3714—31no—
35C2×#3713—31no—
36C8×C3⋊Dic3D311yescharacteristic, complemented
37C3×C8×Dic3—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C4⋊#272—21no—
43C4⋊#272—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—
52C8×H(3,𝔽3)D412yes—
53(C3⋊(C3⋊C8))⋊C3D412yes—
54#5168—91no—
55#5168—91no—
56#4696—91no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×#634C2×D311yescharacteristic
59C2×C3×D12—31no—
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—61no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67C2×C32×C8—21no—
68C2×C32×C8C2×D311yescharacteristic
69C2×C32×C8C2×D311yescharacteristic
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72#3744—22no—
73C3×C4×Dic3—31no—
74C2×C3×C3⋊C8—31no—
75C3×#204—64no—
76C3×#204—62no—
77C3×#204—62no—
78C3×#204—62no—
79#3722—62no—
80(C2×D3)⋊Dic3—62no—
81(C2×D3)⋊Dic3—62no—
82(C2×D3)⋊Dic3—62no—
83#3719—62no—
84#3718—62no—
85#3714—62no—
86#3713—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—41no—
89C2×#272—42no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—42no—
92C4×H(3,𝔽3)C2×D411yescharacteristic
93C4×H(3,𝔽3)C2×D411yescharacteristic
94#1084—41no—
95(C3⋊Dic3)⋊C3—22no—
96(C3⋊Dic3)⋊C3—41no—
97C2×C3×SD16—91no—
98C2×C3×SD16—91no—
99#947—91no—
100#947—91no—
101C2×#216—91no—
102C2×#216—91no—
103#936—91no—
104#936—91no—
105C2×#213—91no—
106C2×#213—91no—
107C2×Dic12—91no—
108C2×D24—91no—
109C2×#204—31no—
110C2×#204—31no—
111C2×#204—31no—
112C2×#204—61no—
113C2×#204—91no—
114#875—91no—
115#875—91no—
116C3×C4×C8—91no—
117C8×Dic3—31no—
118C8×Dic3—91no—
119C22×C3⋊D3—21no—
120C22×C3×D3—61no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
123C22×C3×D3—121no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3D1212yes—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—21no—
131#634—21no—
132#634—21no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—121no—
138C3×D12—62no—
139C3×D12—61no—
140C3×D12—61no—
141C3×D12—61no—
142C3×D12—61no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—121no—
146C3×Dic6—62no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—121no—
154C32×C8—22no—
155C32×C8D1212yes—
156C32×C8#6712yes—
157C3⋊(C3⋊C8)D1212yes—
158C3×C3⋊C8—32no—
159#495—271noSylow
160C2×H(3,𝔽3)#12711yescharacteristic
161C2×H(3,𝔽3)#13512yes—
162#272—81no—
163(C3⋊D3)⋊C3—81no—
164C2×C3×Q8—91no—
165C2×C3×Q8—91no—
166C2×C3×D4—91no—
167C2×C3×D4—91no—
168C2×D12—31no—
169C2×D12—31no—
170C2×D12—31no—
171C2×D12—61no—
172C2×D12—91no—
173C2×Dic6—31no—
174C2×Dic6—31no—
175C2×Dic6—31no—
176C2×Dic6—61no—
177C2×Dic6—91no—
178C3×SD16—182no—
179C3×SD16—182no—
180C2×C3×C8—31no—
181C2×C3×C8—31no—
182C2×C3×C8—61no—
183C2×C3×C8—91no—
184C2×C3×C8D3211yescharacteristic
185C3×C42—91no—
186#217—182no—
187#217—182no—
188#216—182no—
189#216—182no—
190#213—182no—
191#213—182no—
192(C2×D3)⋊C4—182no—
193(C2×D3)⋊C4—182no—
194(C2×D3)⋊C4—182no—
195(C2×D3)⋊C4—182no—
196C4×Dic3—31no—
197C4×Dic3—91no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—91no—
200Dic12—182no—
201D24—182no—
202#204—62no—
203#204—62no—
204#204—64no—
205#204—62no—
206#204—182no—
207C22×C32—21no—
208C22×C32C2×D1211yescharacteristic
209C22×C32D3×D411yescharacteristic
210C2×C3⋊D3—42no—
211C2×C3⋊D3—41no—
212C2×C3×D3—122no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C2×C3×D3—122no—
216C2×C3×D3—122no—
217C2×C3×D3—122no—
218C2×C3×D3—121no—
219C2×C3×D3—242no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4C2×D1211yescharacteristic
223C32×C4C2×D1211yescharacteristic
224C32×C4D3×D411yescharacteristic
225C32×C4C2×#6711yescharacteristic
226C3⋊Dic3—22no—
227C3⋊Dic3—41no—
228C3×Dic3—62no—
229C3×Dic3—122no—
230C3×Dic3—121no—
231C3×Dic3—121no—
232C3×Dic3—121no—
233C2×Q16—271no—
234C2×SD16—91no—
235C2×SD16—91no—
236C2×D8—271no—
237#124—271no—
238#124—271no—
239C4×C8—91no—
240H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
241C23×C3—181no—
242C23×C3—181no—
243C22×D3—61no—
244C22×D3—61no—
245C22×D3—61no—
246C22×D3—121no—
247C22×D3—181no—
248C3×Q8—181no—
249C3×Q8—181no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×D4—181no—
253C3×D4—181no—
254C3×D4—181no—
255C3×D4—181no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—92no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4C2×D3211yescharacteristic
263C2×Dic3—32no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—92no—
268C2×Dic3—121no—
269C2×Dic3—181no—
270D12—61no—
271D12—61no—
272D12—61no—
273D12—61no—
274D12—62no—
275D12—61no—
276D12—61no—
277D12—121no—
278D12—181no—
279D12—181no—
280Dic6—61no—
281Dic6—61no—
282Dic6—61no—
283Dic6—61no—
284Dic6—62no—
285Dic6—61no—
286Dic6—61no—
287Dic6—121no—
288Dic6—181no—
289Dic6—181no—
290C3×C8—32no—
291C3×C8—32no—
292C3×C8—62no—
293C3×C8—92no—
294C3×C8#62612yes—
295C3⋊C8—32no—
296C3⋊C8—92no—
297C2×C32—22no—
298C2×C32—21no—
299C2×C32#87411yescharacteristic
300C2×C32#90412yes—
301C2×C32#91612yes—
302C2×C32#94011yescharacteristic
303C3⋊D3—81no—
304C3×D3—242no—
305C3×D3—241no—
306C3×D3—241no—
307C3×D3—241no—
308C2×Q8—91no—
309C2×Q8—91no—
310C2×D4—91no—
311C2×D4—91no—
312Q16—542no—
313SD16—182no—
314SD16—182no—
315D8—542no—
316C2×C8—91no—
317C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
318C22⋊C4—542no—
319C22⋊C4—542no—
320C42—91no—
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—61no—
324C22×C3—361no—
325C22×C3—362no—
326C22×C3—362no—
327C22×C3—361no—
328C22×C3D3×D1211yescharacteristic
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—122no—
332C2×D3—122no—
333C2×D3—122no—
334C2×D3—122no—
335C2×D3—121no—
336C2×D3—242no—
337C2×D3—361no—
338C2×D3—362no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—61no—
344C3×C4—61no—
345C3×C4—182no—
346C3×C4—361no—
347C3×C4—361no—
348C3×C4D3×D1211yescharacteristic
349C3×C4C2×#62611yescharacteristic
350Dic3—62no—
351Dic3—121no—
352Dic3—121no—
353Dic3—122no—
354Dic3—121no—
355Dic3—182no—
356Dic3—361no—
357C32—21no—
358C32#469611yescharacteristic, complemented
359C32#516811yescharacteristic, complemented
360C23—181no—
361C23—181no—
362Q8—181no—
363Q8—181no—
364Q8—181no—
365Q8—181no—
366D4—181no—
367D4—181no—
368D4—181no—
369D4—181no—
370C2×C4—92no—
371C2×C4—181no—
372C2×C4—181no—
373C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
374C8—92no—
375C8#620012yes—
376C2×C3—32no—
377C2×C3—32no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—62no—
381C2×C3—61no—
382C2×C3—721no—
383C2×C3—721no—
384C2×C3#6380612yes—
385C2×C3#6391111yescharacteristic
386D3—241no—
387D3—241no—
388D3—242no—
389D3—241no—
390D3—721no—
391C22—362no—
392C22—362no—
393C22—361no—
394C22—361no—
395C22#8865411yescharacteristic
396C4—182no—
397C4—361no—
398C4—361no—
399C4#8865411yescharacteristic
400C4C2×#620011yescharacteristic
401C3—31no—
402C3—31no—
403C3—61no—
404C3#1059086211yescharacteristic
405C2—721no—
406C2—721no—
407C2#1172085412yes—
408C2#1172115811yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#88436C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#495
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88528C2semidirect
2. C2×#3744⋊C3C2semidirect
3. #11721114C2semidirect
4. #11721110C2semidirect
5. C2×#88439C2semidirect
7. C8×(C3⋊Dic3)⋊C3C2semidirect
23. C2×#3744D3semidirect
36. C8×C3⋊Dic3D3semidirect
240. H(3,𝔽3)#495semidirect
358. C32#4696semidirect
359. C32#5168semidirect

Maximal subgroups

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

subgroupindex
C2×#885282
C2×#3744⋊C32
#117211142
#117211102
C2×#884392
C2×#884362
C8×(C3⋊Dic3)⋊C32
#105908673
#105908623