Finite Group Databasebeta

← #13138793

Subgroups — #13138793

Order 1728.

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

normal subgroups only

This group has 416 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#11721136C211yesmaximal, characteristic
2#11721117C211yesmaximal, characteristic, complemented
3#11720547C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic
5#63645⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic
7#11720315C211yesmaximal, characteristic, complemented
8#10590764—31nomaximal
9#10590762—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×((C3⋊D3)⋊C3)⋊C4—21no—
12C2×#6196—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4×#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#88450—21no—
20#88449—21no—
21#63918—31no—
22#63918—31no—
23#63894—31no—
24Dic3×D12—31no—
25#63648D311yescharacteristic
26#63645D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#818—31no—
29C3×Dic3⋊C8—31no—
30C3×Dic3⋊C8—31no—
31C3×Dic3⋊C8—61no—
32#63575—31no—
33#63575—31no—
34#63564—31no—
35#63563—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084D411yescharacteristic
39C2×#1084D411yescharacteristic
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)Q812yescomplemented
45((C3⋊D3)⋊C3)⋊C4—42no—
46#6196—42no—
47C8×H(3,𝔽3)—21no—
48#6180—21no—
49#4890—91no—
50#4846—91no—
51#4759—91no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×C3×D12—31no—
54C2×#627—61no—
55C2×#627—61no—
56C2×#624—61no—
57C2×D3×Dic3—61no—
58C2×C32×C8—21no—
59C2×C32×C8C2×D311yescharacteristic
60C2×C32×C8C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)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—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—61no—
73Dic3⋊Dic3—61no—
74Dic3⋊Dic3—61no—
75#3722—61no—
76(C2×D3)⋊Dic3—61no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—22no—
79C2×(C3⋊D3)⋊C3—41no—
80C2×(C3⋊D3)⋊C3—41no—
81C4×H(3,𝔽3)C2×Q811yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—21no—
84#1084D812yes—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88D4×Dic3—91no—
89#893—91no—
90#888—91no—
91#888—91no—
92C4×D12—91no—
93#869—91no—
94C3×#107—91no—
95C3×C4⋊C8—91no—
96C3×#102—91no—
97#832—91no—
98#821—31no—
99#821—91no—
100#821—91no—
101#818—31no—
102#818—91no—
103#818—91no—
104Dic3⋊C8—31no—
105Dic3⋊C8—31no—
106Dic3⋊C8—61no—
107#809—91no—
108#807—91no—
109#807—91no—
110C22×C3⋊D3—21no—
111C22×C3×D3—61no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C4⋊(C3⋊D3)—21no—
119C4⋊(C3⋊D3)Dic612yescomplemented
120C2×C3×Dic3—31no—
121C2×C3×Dic3—31no—
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—61no—
129C3×D12—32no—
130C3×D12—61no—
131#627—122no—
132#627—122no—
133#624—122no—
134D3×Dic3—122no—
135C32×C8—21no—
136C32×C8—21no—
137C32×C8—41no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—121no—
141#474—271noSylow
142C2×H(3,𝔽3)C22⋊Q811yescharacteristic
143C2×H(3,𝔽3)C2×D811yescharacteristic
144C2×H(3,𝔽3)#13711yescharacteristic
145(C3⋊D3)⋊C3—42no—
146C2×C3×D4—91no—
147C22×Dic3—181no—
148C2×D12—31no—
149C2×D12—91no—
150C2×C4×D3—181no—
151C2×C4×D3—181no—
152C2×Dic6—181no—
153C2×Dic6—181no—
154C2×Dic6—181no—
155C2×C3×C8—31no—
156C2×C3×C8—31no—
157C2×C3×C8—61no—
158C2×C3×C8—91no—
159C2×C3×C8D3211yescharacteristic
160C3×C4⋊C4—91no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—91no—
163C3×C42—91no—
164#217—181no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—181no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—91no—
171C4⋊Dic3—91no—
172C4⋊Dic3—91no—
173C4⋊Dic3—181no—
174Dic3⋊C4—181no—
175Dic3⋊C4—181no—
176C4×Dic3—31no—
177C4×Dic3—31no—
178C4×Dic3—61no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—61no—
182C22×C32—21no—
183C22×C32D3×D411yescharacteristic
184C22×C32D3×D411yescharacteristic
185C2×C3⋊D3—22no—
186C2×C3⋊D3—41no—
187C2×C3⋊D3—41no—
188C2×C3×D3—62no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4C2×Dic611yescharacteristic
194C32×C4#23511yescharacteristic
195C32×C4#23711yescharacteristic
196C32×C4D3×Q811yescharacteristic
197C3⋊Dic3—41no—
198C3⋊Dic3—41no—
199C3⋊Dic3—41no—
200C3×Dic3—32no—
201C3×Dic3—32no—
202C3×Dic3—62no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C4⋊Q8—271no—
210C4×D4—271no—
211#107—91no—
212#107—271no—
213C4⋊C8—91no—
214#102—91no—
215#102—271no—
216H(3,𝔽3)#47411yescharacteristic, complemented, Sylow
217C23×C3—181no—
218C22×D3—61no—
219C22×D3—181no—
220C3×D4—92no—
221C3×D4—181no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—61no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4C2×D3211yescharacteristic
231C2×Dic3—31no—
232C2×Dic3—31no—
233C2×Dic3—31no—
234C2×Dic3—31no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—181no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243C2×Dic3—362no—
244D12—32no—
245D12—61no—
246D12—92no—
247D12—181no—
248C4×D3—362no—
249C4×D3—362no—
250Dic6—362no—
251Dic6—362no—
252Dic6—362no—
253C3×C8—21no—
254C3×C8—61no—
255C3×C8—61no—
256C3×C8—121no—
257C3×C8—181no—
258C3⋊C8—61no—
259C3⋊C8—61no—
260C3⋊C8—121no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32#87811yescharacteristic
265C2×C32#89611yescharacteristic
266C2×C32D3×D811yescharacteristic
267C2×C32D3×D811yescharacteristic
268C2×C32#91511yescharacteristic
269C2×C32#91811yescharacteristic
270C3⋊D3—42no—
271C3×D3—122no—
272C2×Q8—541no—
273C2×D4—91no—
274C22×C4—541no—
275C2×C8—91no—
276C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
277C4⋊C4—91no—
278C4⋊C4—91no—
279C4⋊C4—91no—
280C4⋊C4—541no—
281C22⋊C4—541no—
282C42—91no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—61no—
286C22×C3—182no—
287C22×C3—361no—
288C22×C3—361no—
289C22×C3#380211yescharacteristic
290C2×D3—62no—
291C2×D3—121no—
292C2×D3—121no—
293C2×D3—182no—
294C2×D3—361no—
295C2×D3—361no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4—92no—
303C3×C4—181no—
304C3×C4—361no—
305C3×C4—361no—
306C3×C4—361no—
307C3×C4D3×Dic611yescharacteristic
308C3×C4#380411yescharacteristic
309Dic3—32no—
310Dic3—32no—
311Dic3—62no—
312Dic3—61no—
313Dic3—61no—
314Dic3—121no—
315Dic3—121no—
316Dic3—121no—
317Dic3—121no—
318Dic3—361no—
319Dic3—361no—
320Dic3—361no—
321C32—21no—
322C32#475911yescharacteristic, complemented
323C32#489011yescharacteristic, complemented
324C23—181no—
325Q8—1082no—
326D4—92no—
327D4—181no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—1082no—
334C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C8—21no—
336C8—181no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—362no—
347C2×C3#6379711yescharacteristic
348C2×C3#6381111yescharacteristic
349C2×C3#6389911yescharacteristic
350D3—122no—
351D3—362no—
352C22—182no—
353C22—361no—
354C22—361no—
355C22#8865511yescharacteristic
356C4—92no—
357C4—181no—
358C4—361no—
359C4—361no—
360C4—361no—
361C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
362C4#8867111yescharacteristic
363C3—31no—
364C3—31no—
365C3—61no—
366C3#1059077811yescharacteristic
367C2—362no—
368C2#1172083811yescharacteristic
369C2#1172084511yescharacteristic
370C2#1172113411yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#474
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
2. #11721117C2semidirect
3. #11720547C2semidirect
5. #63645⋊C3C2semidirect
7. #11720315C2semidirect
26. #63645D3semidirect
44. C4⋊((C3⋊D3)⋊C3)Q8semidirect
119. C4⋊(C3⋊D3)Dic6semidirect
216. H(3,𝔽3)#474semidirect
322. C32#4759semidirect
323. C32#4890semidirect

Maximal subgroups

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

subgroupindex
#117211362
#117211172
#117205472
#63648⋊C32
#63645⋊C32
#117203182
#117203152
#105907643
#105907623