Finite Group Databasebeta

← #13139010

Subgroups — #13139010

Order 1728.

460 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 71 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
2#11721114C211yesmaximal, characteristic
3#11721083C211yesmaximal, characteristic, complemented
4C2×#88439C211yesmaximal, characteristic, complemented
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic
8#10590859—31nomaximal
9#10590796—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14#88528—22no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88451—21no—
19#88451—21no—
20#88449—21no—
21#88447—21no—
22#88439—22no—
23C2×C3×#204—31no—
24C2×C3×#204—31no—
25C2×C3×#204—61no—
26#63863—31no—
27#63860—31no—
28#63843—31no—
29Dic3×Dic6—31no—
30C2×#3719—31no—
31C2×#3718—31no—
32#63643D311yescharacteristic, complemented
33#63641D311yescharacteristic, complemented
34C3×#816—31no—
35C3×Dic3⋊C8—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C4⋊#272—21no—
42C4⋊#272—21no—
43#6229—21no—
44#6229D412yes—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49#634⋊C3—21no—
50#634⋊C3—21no—
51C8×H(3,𝔽3)—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5166—91no—
54#4845—91no—
55#4798—91no—
56C2×#634C2×D311yescharacteristic
57C2×C3×D12—31no—
58C2×C3×D12—31no—
59C2×C3×D12—61no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64C2×C32×C8—21no—
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C4⋊Dic3—31no—
71C3×C4×Dic3—31no—
72C2×C3×C3⋊C8—31no—
73C3×#204—62no—
74C3×#204—62no—
75C3×#204—64no—
76Dic3⋊Dic3—61no—
77Dic3⋊Dic3—61no—
78(C2×D3)⋊Dic3—61no—
79(C2×D3)⋊Dic3—61no—
80(C2×D3)⋊Dic3—61no—
81(C2×D3)⋊Dic3—61no—
82Dic32—61no—
83Dic32—61no—
84#3719—62no—
85#3718—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×#272—41no—
88C2×#272—41no—
89C2×#272—41no—
90C4×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92#1084—22no—
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—41no—
96(C3⋊Dic3)⋊C3—41no—
97C2×C3×SD16—91no—
98#947—91no—
99Q8×Dic3—91no—
100C2×#216—91no—
101C2×#216—91no—
102#936—91no—
103C2×Dic12—91no—
104C2×#204—31no—
105C2×#204—31no—
106C2×#204—61no—
107#887—91no—
108#875—91no—
109C4×Dic6—91no—
110C3×C4⋊C8—91no—
111C3×#103—91no—
112#832—91no—
113#821—91no—
114#816—31no—
115#816—91no—
116Dic3⋊C8—31no—
117Dic3⋊C8—91no—
118#809—91no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C22×C3×D3—121no—
122C2×C32×C4—21no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3D1211yescharacteristic
128C2×C3⋊Dic3D1211yescharacteristic
129#634—21no—
130#634—21no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—61no—
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—121no—
144C3×Dic6—32no—
145C3×Dic6—32no—
146C3×Dic6—62no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—121no—
152C32×C8—22no—
153C32×C8—21no—
154C32×C8—21no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—61no—
157#464—271noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)#13511yescharacteristic
160C2×H(3,𝔽3)#13711yescharacteristic
161#272—81no—
162C2×C3×Q8—91no—
163C2×C3×Q8—91no—
164C2×C3×D4—91no—
165C2×D12—31no—
166C2×D12—31no—
167C2×D12—61no—
168C2×Dic6—31no—
169C2×Dic6—31no—
170C2×Dic6—31no—
171C2×Dic6—61no—
172C2×Dic6—91no—
173C3×SD16—182no—
174C2×C3×C8—31no—
175C2×C3×C8—31no—
176C2×C3×C8—61no—
177C2×C3×C8—91no—
178C2×C3×C8D3211yescharacteristic
179C3×C4⋊C4—91no—
180C3×C42—91no—
181#217—181no—
182#217—181no—
183#216—182no—
184#216—182no—
185(C2×D3)⋊C4—181no—
186(C2×D3)⋊C4—181no—
187(C2×D3)⋊C4—181no—
188(C2×D3)⋊C4—181no—
189C4⋊Dic3—31no—
190C4⋊Dic3—91no—
191C4⋊Dic3—181no—
192Dic3⋊C4—181no—
193Dic3⋊C4—181no—
194C4×Dic3—31no—
195C4×Dic3—91no—
196C4×Dic3—181no—
197C4×Dic3—181no—
198C4×Dic3—181no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—91no—
201Dic12—182no—
202#204—62no—
203#204—62no—
204#204—64no—
205C22×C32—21no—
206C22×C32C2×D1211yescharacteristic
207C22×C32D3×D411yescharacteristic
208C2×C3×D3—122no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C2×C3×D3—241no—
216C2×C3×D3—241no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4#23511yescharacteristic
220C32×C4D3×D411yescharacteristic
221C32×C4D3×D411yescharacteristic
222C32×C4#23911yescharacteristic
223C3⋊Dic3—21no—
224C3⋊Dic3—21no—
225C3⋊Dic3—41no—
226C3⋊Dic3—41no—
227C3×Dic3—62no—
228C3×Dic3—62no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—122no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234C2×Q16—271no—
235C2×SD16—91no—
236#124—271no—
237C4×Q8—271no—
238C4⋊C8—91no—
239#103—91no—
240#102—271no—
241H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
242C23×C3—181no—
243C22×D3—61no—
244C22×D3—61no—
245C22×D3—121no—
246C3×Q8—92no—
247C3×Q8—181no—
248C3×Q8—181no—
249C3×Q8—181no—
250C3×D4—181no—
251C3×D4—181no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—61no—
255C2×C3×C4—91no—
256C2×C3×C4—91no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4C2×D3211yescharacteristic
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—91no—
268C2×Dic3—91no—
269C2×Dic3—121no—
270C2×Dic3—181no—
271C2×Dic3—181no—
272D12—61no—
273D12—61no—
274D12—61no—
275D12—61no—
276D12—62no—
277D12—121no—
278Dic6—32no—
279Dic6—32no—
280Dic6—62no—
281Dic6—61no—
282Dic6—61no—
283Dic6—61no—
284Dic6—61no—
285Dic6—121no—
286Dic6—181no—
287Dic6—181no—
288C3×C8—21no—
289C3×C8—61no—
290C3×C8—61no—
291C3×C8—62no—
292C3×C8—181no—
293C3⋊C8—61no—
294C3⋊C8—181no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32#88411yescharacteristic
299C2×C32#89511yescharacteristic
300C2×C32#90411yescharacteristic
301C2×C32#90911yescharacteristic
302C2×C32#91611yescharacteristic
303C2×C32#91811yescharacteristic
304C3×D3—242no—
305C3×D3—241no—
306C3×D3—241no—
307C2×Q8—91no—
308C2×Q8—91no—
309C2×D4—91no—
310Q16—542no—
311SD16—182no—
312C2×C8—91no—
313C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
314C4⋊C4—91no—
315C4⋊C4—541no—
316C22⋊C4—541no—
317C22⋊C4—541no—
318C42—91no—
319C42—541no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—61no—
323C22×C3—361no—
324C22×C3—361no—
325C22×C3—361no—
326C22×C3D3×D1211yescharacteristic
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—122no—
334C2×D3—241no—
335C2×D3—241no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—181no—
343C3×C4—181no—
344C3×C4—182no—
345C3×C4—361no—
346C3×C4—361no—
347C3×C4#379911yescharacteristic
348C3×C4#381111yescharacteristic
349Dic3—62no—
350Dic3—62no—
351Dic3—61no—
352Dic3—61no—
353Dic3—122no—
354Dic3—121no—
355Dic3—121no—
356Dic3—181no—
357Dic3—181no—
358Dic3—361no—
359Dic3—361no—
360C32—21no—
361C32#479811yescharacteristic, complemented
362C32#484511yescharacteristic, complemented
363C23—181no—
364Q8—92no—
365Q8—181no—
366Q8—181no—
367Q8—181no—
368D4—181no—
369D4—181no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—181no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
376C8—21no—
377C8—181no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—61no—
387C2×C3—721no—
388C2×C3#6380111yescharacteristic
389C2×C3#6380611yescharacteristic
390C2×C3#6391011yescharacteristic
391D3—241no—
392D3—241no—
393D3—242no—
394C22—361no—
395C22—361no—
396C22—361no—
397C22#8865411yescharacteristic
398C4—181no—
399C4—181no—
400C4—182no—
401C4—361no—
402C4—361no—
403C4#8865211yescharacteristic
404C4#8867711yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059081311yescharacteristic
409C2—721no—
410C2#1172085311yescharacteristic
411C2#1172085411yescharacteristic
412C2#1172115711yescharacteristic
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×C4⋊#272C22
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#464
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
3. #11721083C2semidirect
4. C2×#88439C2semidirect
5. #63643⋊C3C2semidirect
6. #63641⋊C3C2semidirect
32. #63643D3semidirect
33. #63641D3semidirect
241. H(3,𝔽3)#464semidirect
361. C32#4798semidirect
362. C32#4845semidirect

Maximal subgroups

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

subgroupindex
C2×#885282
#117211142
#117210832
C2×#884392
#63643⋊C32
#63641⋊C32
#117203242
#105908593
#105907963