Finite Group Databasebeta

← #13138022

Subgroups — #13138022

Order 1728.

385 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 77 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
1#64097⋊C3C211yesmaximal, characteristic
2#11721090C211yesmaximal, characteristic
3#11720532C211yesmaximal, characteristic
4#63637⋊C3C211yesmaximal, characteristic
5#11720326C211yesmaximal, characteristic
6#11720313C211yesmaximal, characteristic
7#11720310C211yesmaximal, characteristic
8#10590567—31nomaximal
9#10590560—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×#634⋊C3—21no—
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88448—21no—
21#64097D311yescharacteristic
22C3×#888—31no—
23#63873—31no—
24#63866—31no—
25#63637D311yescharacteristic
26C3×#810—31no—
27C3×#810—31no—
28C3×#810—31no—
29C3×#810—61no—
30#63567—31no—
31#63566—31no—
32#63566—31no—
33#63565—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39#6229—21no—
40#6229—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#634⋊C3—21no—
46#634⋊C3—21no—
47#634⋊C3—42no—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5093—91no—
51#4792—91no—
52#4792—91no—
53C2×#634—21no—
54C2×#634C2×D311yescharacteristic
55C2×C3×Dic6—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—31no—
58C2×C3×Dic6—61no—
59C2×C3×Dic6—61no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—21no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×C4⋊Dic3—31no—
68C3×Dic3⋊C4—61no—
69C3×C4×Dic3—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—61no—
74#3724—61no—
75#3724—61no—
76Dic3⋊Dic3—61no—
77Dic3⋊Dic3—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C4×H(3,𝔽3)—41no—
80C4×H(3,𝔽3)#4111yescharacteristic
81C4×H(3,𝔽3)#4111yescharacteristic
82#1084—41no—
83(C3⋊Dic3)⋊C3—41no—
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3SD1612yes—
86(C3⋊Dic3)⋊C3Q1612yes—
87C3×C4⋊Q8—91no—
88#944—91no—
89#944—91no—
90#888—31no—
91#888—91no—
92#869—91no—
93C3×#103—91no—
94C3×#103—91no—
95#835—91no—
96#835—91no—
97#816—91no—
98#816—91no—
99C8×Dic3—91no—
100C8×Dic3—91no—
101#810—31no—
102#810—31no—
103#810—31no—
104#810—61no—
105#810—91no—
106#810—91no—
107#810—91no—
108C4×C3⋊C8—91no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—41no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3#6711yescharacteristic
118C2×C3⋊Dic3#6711yescharacteristic
119#634—21no—
120#634—21no—
121#634—42no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—121no—
129C3×Dic6—62no—
130C3×Dic6—61no—
131C3×Dic6—61no—
132C3×Dic6—61no—
133C3×Dic6—61no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136C3×Dic6—121no—
137C3×Dic6—122no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—121no—
143#487—271noSylow
144C2×H(3,𝔽3)#12411yescharacteristic
145C2×H(3,𝔽3)C2×SD1611yescharacteristic
146C2×H(3,𝔽3)C2×Q1611yescharacteristic
147C2×C3×Q8—91no—
148C2×C3×Q8—91no—
149C2×C3×Q8—181no—
150C2×Dic6—31no—
151C2×Dic6—31no—
152C2×Dic6—31no—
153C2×Dic6—61no—
154C2×Dic6—61no—
155C2×Dic6—91no—
156C2×Dic6—181no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—181no—
164C3×C4⋊C4D3211yescharacteristic
165C3×C42—91no—
166C4⋊Dic3—31no—
167C4⋊Dic3—91no—
168C4⋊Dic3—181no—
169C4⋊Dic3—181no—
170Dic3⋊C4—61no—
171Dic3⋊C4—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174Dic3⋊C4—181no—
175Dic3⋊C4—181no—
176C4×Dic3—31no—
177C4×Dic3—91no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—61no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32D3×D411yescharacteristic
185C22×C32C2×#6711yescharacteristic
186C32×C4—21no—
187C32×C4—21no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4—81no—
191C32×C4#23511yescharacteristic
192C32×C4#23711yescharacteristic
193C32×C4#23911yescharacteristic
194C32×C4#23911yescharacteristic
195C3⋊Dic3—41no—
196C3⋊Dic3—41no—
197C3⋊Dic3#21412yes—
198C3⋊Dic3#21612yes—
199C3×Dic3—32no—
200C3×Dic3—32no—
201C3×Dic3—122no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C4⋊Q8—91no—
207C4⋊Q8—271no—
208#103—91no—
209#103—91no—
210#103—271no—
211#103—271no—
212C4×C8—271no—
213H(3,𝔽3)#48711yescharacteristic, complemented, Sylow
214C3×Q8—181no—
215C3×Q8—181no—
216C3×Q8—181no—
217C3×Q8—181no—
218C3×Q8—362no—
219C2×C3×C4—21no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—121no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4C2×D3211yescharacteristic
232C2×Dic3—31no—
233C2×Dic3—31no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—91no—
239C2×Dic3—91no—
240C2×Dic3—121no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243Dic6—61no—
244Dic6—61no—
245Dic6—61no—
246Dic6—61no—
247Dic6—62no—
248Dic6—61no—
249Dic6—61no—
250Dic6—121no—
251Dic6—122no—
252Dic6—181no—
253Dic6—181no—
254Dic6—362no—
255C3×C8—181no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—61no—
259C3⋊C8—61no—
260C3⋊C8—121no—
261C3⋊C8—181no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32#88511yescharacteristic
266C2×C32D3×SD1611yescharacteristic
267C2×C32D3×Q1611yescharacteristic
268C2×C32C2×#21411yescharacteristic
269C2×C32C2×#21611yescharacteristic
270C2×C32#94711yescharacteristic
271C2×Q8—91no—
272C2×Q8—91no—
273C2×Q8—181no—
274C2×C8—91no—
275C2×C8—91no—
276C4⋊C4—91no—
277C4⋊C4—181no—
278C4⋊C4—541no—
279C4⋊C4—541no—
280C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
281C42—91no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—61no—
285C22×C3D3×#6711yescharacteristic
286C3×C4—31no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—41no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—92no—
294C3×C4—92no—
295C3×C4—121no—
296C3×C4—121no—
297C3×C4—241no—
298C3×C4—361no—
299C3×C4—361no—
300C3×C4—361no—
301C3×C4#379611yescharacteristic
302C3×C4#379911yescharacteristic
303Dic3—32no—
304Dic3—32no—
305Dic3—92no—
306Dic3—92no—
307Dic3—121no—
308Dic3—121no—
309Dic3—122no—
310Dic3—121no—
311Dic3—121no—
312Dic3—361no—
313Dic3—361no—
314C32—21no—
315C32#479211yescharacteristic, complemented
316C32#509311yescharacteristic, complemented
317Q8—181no—
318Q8—181no—
319Q8—181no—
320Q8—181no—
321Q8—362no—
322C2×C4—21no—
323C2×C4—91no—
324C2×C4—91no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C8—181no—
330C8—181no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—61no—
338C2×C3—61no—
339C2×C3—61no—
340C2×C3#6390711yescharacteristic
341C2×C3D3×#21411yescharacteristic
342C2×C3D3×#21611yescharacteristic
343C22#8867611yescharacteristic
344C4—41no—
345C4—92no—
346C4—92no—
347C4—361no—
348C4—361no—
349C4—361no—
350C4#8864911yescharacteristic
351C4#8865211yescharacteristic
352C3—31no—
353C3—31no—
354C3—61no—
355C3#1059056011yescharacteristic
356C2#1172115511yescharacteristic
357C2#1172119811yescharacteristic
358C2#1172123611yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#487
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
213. H(3,𝔽3)#487semidirect
315. C32#4792semidirect
316. C32#5093semidirect

Maximal subgroups

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

subgroupindex
#64097⋊C32
#117210902
#117205322
#63637⋊C32
#117203262
#117203132
#117203102
#105905673
#105905603