Finite Group Databasebeta

← #13138829

Subgroups — #13138829

Order 1728.

400 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 73 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#11721129C211yesmaximal, characteristic
2#11721106C211yesmaximal, characteristic
3#11720550C211yesmaximal, characteristic, complemented
4#63647⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590686—31nomaximal
9#10590674—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88450—21no—
19#88449—21no—
20#88448—21no—
21#63900—31no—
22D3×C4⋊Dic3—31no—
23#63871—31no—
24#63857—31no—
25#63647D311yescharacteristic
26#63643D311yescharacteristic, complemented
27C3×(C2×D3)⋊C8—31no—
28C3×#817—31no—
29C3×#817—31no—
30C3×#817—61no—
31C3×#816—31no—
32#63574—31no—
33#63573—31no—
34#63567—31no—
35#63565—31no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4×(C3⋊D3)⋊C3—21no—
44C4×(C3⋊D3)⋊C3—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47((C3⋊D3)⋊C3)⋊C4—42no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#4862—91no—
51#4805—91no—
52#4721—91no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634C2×D311yescharacteristic
55C2×C3×C4×D3—31no—
56C2×C3×Dic6—31no—
57C2×#624—61no—
58C2×D3×Dic3—61no—
59C2×C32×C8—21no—
60C2×C32×C8C2×D311yescharacteristic
61C2×C32×C8C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×C4⋊Dic3—61no—
71C2×C3×C3⋊C8—31no—
72#3724—61no—
73Dic3⋊Dic3—61no—
74Dic3⋊Dic3—61no—
75Dic3⋊Dic3—61no—
76#3722—61no—
77(C2×D3)⋊Dic3—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3SD1612yes—
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—41no—
85#1084—41no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89#946—91no—
90C2×C4⋊Dic3—91no—
91#924—91no—
92#897—91no—
93D3×C4⋊C4—91no—
94D3×C4⋊C4—91no—
95C3×#106—91no—
96C3×#103—91no—
97C3×C22⋊C8—91no—
98#835—91no—
99(C2×D3)⋊C8—31no—
100(C2×D3)⋊C8—91no—
101#817—31no—
102#817—31no—
103#817—61no—
104#817—91no—
105#816—31no—
106#816—91no—
107#816—91no—
108#810—91no—
109#808—91no—
110#808—91no—
111C22×C3⋊D3D1211yescharacteristic
112C22×C3×D3—31no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3D1211yescharacteristic
119C4×C3⋊D3—21no—
120C4×C3⋊D3—21no—
121#634—21no—
122#634—21no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—121no—
131C2×C3×Dic3—121no—
132C3×C4×D3—61no—
133C3×C4×D3—61no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136#624—122no—
137D3×Dic3—122no—
138C32×C8—22no—
139C32×C8—21no—
140C32×C8—21no—
141C3⋊(C3⋊C8)—21no—
142C3×C3⋊C8—61no—
143#483—271noSylow
144C2×H(3,𝔽3)#12311yescharacteristic
145C2×H(3,𝔽3)C2×SD1611yescharacteristic
146C2×H(3,𝔽3)#13711yescharacteristic
147(C3⋊D3)⋊C3—22no—
148C2×C3×Q8—91no—
149C22×C3×C4—91no—
150C22×Dic3—181no—
151C2×C4×D3—31no—
152C2×C4×D3—91no—
153C2×C4×D3—181no—
154C2×C4×D3—181no—
155C2×Dic6—31no—
156C2×Dic6—91no—
157C2×C3×C8—31no—
158C2×C3×C8—31no—
159C2×C3×C8—61no—
160C2×C3×C8—91no—
161C2×C3×C8D3211yescharacteristic
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165#217—181no—
166(C2×D3)⋊C4—181no—
167(C2×D3)⋊C4—181no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—31no—
173C4⋊Dic3—61no—
174C4⋊Dic3—61no—
175C4⋊Dic3—91no—
176C4⋊Dic3—181no—
177C4⋊Dic3—181no—
178Dic3⋊C4—181no—
179Dic3⋊C4—181no—
180Dic3⋊C4—181no—
181Dic3⋊C4—181no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—91no—
184C22×C32—21no—
185C22×C32C2×D1211yescharacteristic
186C22×C32D3×D411yescharacteristic
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3#20412yes—
190C2×C3×D3—32no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4#23511yescharacteristic
196C32×C4#23711yescharacteristic
197C32×C4#23711yescharacteristic
198C32×C4#23711yescharacteristic
199C3⋊Dic3—21no—
200C3⋊Dic3—41no—
201C3⋊Dic3—41no—
202C3×Dic3—61no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—241no—
210C3×Dic3—241no—
211C22⋊Q8—271no—
212C2×C4⋊C4—271no—
213#106—91no—
214#106—271no—
215#103—91no—
216#103—271no—
217C22⋊C8—91no—
218H(3,𝔽3)#48311yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—31no—
221C22×D3—91no—
222C3×Q8—181no—
223C3×Q8—181no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—61no—
227C2×C3×C4—91no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4C2×D3211yescharacteristic
235C2×Dic3—31no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—91no—
243C2×Dic3—121no—
244C2×Dic3—121no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C2×Dic3—362no—
248C4×D3—61no—
249C4×D3—61no—
250C4×D3—181no—
251C4×D3—181no—
252C4×D3—362no—
253C4×D3—362no—
254Dic6—61no—
255Dic6—61no—
256Dic6—181no—
257Dic6—181no—
258C3×C8—21no—
259C3×C8—61no—
260C3×C8—61no—
261C3×C8—62no—
262C3×C8—181no—
263C3⋊C8—61no—
264C3⋊C8—181no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32#88311yescharacteristic
269C2×C32#88611yescharacteristic
270C2×C32C2×#20411yescharacteristic
271C2×C32#90911yescharacteristic
272C2×C32D3×SD1611yescharacteristic
273C2×C32#91811yescharacteristic
274C3⋊D3—22no—
275C3×D3—62no—
276C2×Q8—91no—
277C22×C4—91no—
278C22×C4—541no—
279C2×C8—91no—
280C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
281C4⋊C4—91no—
282C4⋊C4—91no—
283C4⋊C4—91no—
284C4⋊C4—541no—
285C4⋊C4—541no—
286C22⋊C4—541no—
287C22×C3—31no—
288C22×C3—31no—
289C22×C3—61no—
290C22×C3—92no—
291C22×C3—181no—
292C22×C3—181no—
293C22×C3D3×D1211yescharacteristic
294C2×D3—32no—
295C2×D3—61no—
296C2×D3—61no—
297C2×D3—92no—
298C2×D3—181no—
299C2×D3—181no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—61no—
305C3×C4—61no—
306C3×C4—181no—
307C3×C4—361no—
308C3×C4—361no—
309C3×C4—361no—
310C3×C4—361no—
311C3×C4#379511yescharacteristic
312C3×C4#380511yescharacteristic
313Dic3—61no—
314Dic3—121no—
315Dic3—121no—
316Dic3—121no—
317Dic3—121no—
318Dic3—121no—
319Dic3—121no—
320Dic3—181no—
321Dic3—241no—
322Dic3—241no—
323Dic3—361no—
324Dic3—361no—
325C32—21no—
326C32#472111yescharacteristic, complemented
327C32#480511yescharacteristic, complemented
328C23—91no—
329Q8—181no—
330Q8—181no—
331C2×C4—91no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—1082no—
339C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
340C8—21no—
341C8—181no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—61no—
349C2×C3—61no—
350C2×C3—61no—
351C2×C3—182no—
352C2×C3D3×#20411yescharacteristic
353C2×C3#6380111yescharacteristic
354C2×C3#6389111yescharacteristic
355D3—62no—
356D3—182no—
357C22—92no—
358C22—181no—
359C22—181no—
360C22#8865411yescharacteristic
361C4—181no—
362C4—361no—
363C4—361no—
364C4—361no—
365C4—361no—
366C4#8864811yescharacteristic
367C4#8867211yescharacteristic
368C3—31no—
369C3—31no—
370C3—61no—
371C3#1059067511yescharacteristic
372C2—182no—
373C2#1172084611yescharacteristic
374C2#1172085311yescharacteristic
375C2#1172113111yescharacteristic
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 radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#483
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. #11720550C2semidirect
5. #63643⋊C3C2semidirect
6. #11720309C2semidirect
7. #11720307C2semidirect
26. #63643D3semidirect
218. H(3,𝔽3)#483semidirect
326. C32#4721semidirect
327. C32#4805semidirect

Maximal subgroups

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

subgroupindex
#117211292
#117211062
#117205502
#63647⋊C32
#63643⋊C32
#117203092
#117203072
#105906863
#105906743