Finite Group Databasebeta

← #13137879

Subgroups — #13137879

Order 1728.

451 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 69 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#11721761C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic, complemented
3#63635⋊C3C211yesmaximal, characteristic, complemented
4#63634⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic
6#11720324C211yesmaximal, characteristic
7#11720304C211yesmaximal, characteristic, complemented
8#10590545—31nomaximal
9#10590542—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13#88542—21no—
14C4×#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88448—21no—
21C3×#893—31no—
22C3×#893—31no—
23C3×#893—61no—
24#63869—31no—
25#63869—31no—
26#63635D311yescharacteristic, complemented
27#63634D311yescharacteristic, complemented
28C3×#808—31no—
29C3×#807—31no—
30Dic3⋊(C3⋊C8)—31no—
31Dic3⋊(C3⋊C8)—31no—
32#63563—31no—
33#63563—31no—
34#63562—31no—
35#63562—31no—
36C22×#272—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084D411yescharacteristic
40C2×#1084D411yescharacteristic
41C4⋊#272—21no—
42C4⋊#272Q812yes—
43C4×#272—42no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48#5009—91no—
49#4891—91no—
50#4871—91no—
51C2×C3×D12—31no—
52C2×C3×D12—31no—
53C2×C3×D12—61no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56C2×C3×C4×D3—62no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—61no—
66C3×(C2×D3)⋊C4—62no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4×Dic3—31no—
70C3×C4×Dic3—31no—
71C3×C4×Dic3—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74#3724—61no—
75#3724—61no—
76Dic3⋊Dic3—61no—
77Dic3⋊Dic3—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×#272—22no—
80C2×#272—41no—
81C2×#272—41no—
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)C2×Q811yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85#1084—21no—
86#1084—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89C3×C4×D4—91no—
90#893—31no—
91#893—31no—
92#893—61no—
93#890—91no—
94#890—91no—
95#870—91no—
96C3×#107—91no—
97C3×#106—91no—
98#832—91no—
99#832—91no—
100#821—91no—
101#821—91no—
102Dic3⋊C8—91no—
103Dic3⋊C8—91no—
104#809—91no—
105#809—91no—
106#808—31no—
107#808—91no—
108#807—31no—
109#807—91no—
110C4⋊(C3⋊C8)—91no—
111C22×C3×D3—62no—
112C22×C3×D3—61no—
113C22×C3×D3—61no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C3×D12—32no—
131C3×D12—32no—
132C3×D12—62no—
133C3×D12—62no—
134C3×D12—61no—
135C3×D12—61no—
136C3×C4×D3—124no—
137C3×C4×D3—122no—
138C3×C4×D3—122no—
139C3⋊(C3⋊C8)—21no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143#478—271noSylow
144C2×H(3,𝔽3)C22⋊Q811yescharacteristic
145C2×H(3,𝔽3)#13511yescharacteristic
146C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
147#272—42no—
148C2×C3×D4—91no—
149C22×C3×C4—181no—
150C2×D12—31no—
151C2×D12—31no—
152C2×D12—61no—
153C2×C4×D3—61no—
154C2×C4×D3—61no—
155C2×C4×D3—62no—
156C2×C3×C8—91no—
157C2×C3×C8—91no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—61no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4D3211yescharacteristic
164C3×C22⋊C4—181no—
165C3×C42—91no—
166(C2×D3)⋊C4—61no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—62no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—91no—
172C4⋊Dic3—91no—
173C4⋊Dic3—181no—
174C4⋊Dic3—181no—
175Dic3⋊C4—181no—
176Dic3⋊C4—181no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—61no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—91no—
186C22×C32—21no—
187C22×C32D3×D411yescharacteristic
188C22×C32D3×D411yescharacteristic
189C2×C3×D3—64no—
190C2×C3×D3—62no—
191C2×C3×D3—62no—
192C2×C3×D3—122no—
193C2×C3×D3—122no—
194C2×C3×D3—121no—
195C2×C3×D3—121no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C32×C4—21no—
199C32×C4—21no—
200C32×C4—42no—
201C32×C4—41no—
202C32×C4—41no—
203C32×C4#23511yescharacteristic
204C32×C4#23511yescharacteristic
205C32×C4D3×Q811yescharacteristic
206C32×C4D3×Q811yescharacteristic
207C3⋊Dic3—41no—
208C3⋊Dic3—41no—
209C3×Dic3—62no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217#125—271no—
218C4×D4—91no—
219#107—91no—
220#106—91no—
221C4⋊C8—271no—
222#102—271no—
223#102—271no—
224H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
225C23×C3—181no—
226C22×D3—61no—
227C22×D3—61no—
228C22×D3—62no—
229C3×D4—92no—
230C3×D4—181no—
231C2×C3×C4—21no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—62no—
237C2×C3×C4—61no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—362no—
243C2×C3×C4C2×D3211yescharacteristic
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254D12—32no—
255D12—32no—
256D12—62no—
257D12—61no—
258D12—61no—
259D12—62no—
260C4×D3—122no—
261C4×D3—122no—
262C4×D3—124no—
263C3×C8—181no—
264C3×C8—181no—
265C3⋊C8—61no—
266C3⋊C8—61no—
267C3⋊C8—181no—
268C3⋊C8—181no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32#89611yescharacteristic
273C2×C32#89611yescharacteristic
274C2×C32#91111yescharacteristic
275C2×C32#91411yescharacteristic
276C2×C32#91611yescharacteristic
277C2×C32#91911yescharacteristic
278C3×D3—124no—
279C3×D3—122no—
280C3×D3—122no—
281C2×D4—91no—
282C22×C4—181no—
283C2×C8—91no—
284C2×C8—91no—
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—541no—
288C4⋊C4—541no—
289C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C22⋊C4—181no—
291C42—91no—
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—61no—
295C22×C3—182no—
296C22×C3—361no—
297C22×C3—361no—
298C22×C3#381111yescharacteristic
299C2×D3—62no—
300C2×D3—62no—
301C2×D3—64no—
302C2×D3—121no—
303C2×D3—121no—
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—122no—
307C2×D3—122no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—41no—
313C3×C4—61no—
314C3×C4—61no—
315C3×C4—121no—
316C3×C4—121no—
317C3×C4—122no—
318C3×C4—181no—
319C3×C4—181no—
320C3×C4—361no—
321C3×C4—361no—
322C3×C4#379711yescharacteristic
323C3×C4#379811yescharacteristic
324Dic3—61no—
325Dic3—61no—
326Dic3—61no—
327Dic3—61no—
328Dic3—62no—
329Dic3—121no—
330Dic3—121no—
331Dic3—121no—
332Dic3—361no—
333Dic3—361no—
334C32—21no—
335C32#487111yescharacteristic, complemented
336C32#489111yescharacteristic, complemented
337C23—181no—
338D4—92no—
339D4—181no—
340C2×C4—21no—
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—362no—
346C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
347C8—181no—
348C8—181no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—61no—
356C2×C3—61no—
357C2×C3—61no—
358C2×C3—362no—
359C2×C3#6386511yescharacteristic
360C2×C3#6393711yescharacteristic
361C2×C3#6395011yescharacteristic
362D3—122no—
363D3—122no—
364D3—124no—
365C22—182no—
366C22—361no—
367C22—361no—
368C22#8867711yescharacteristic
369C4—41no—
370C4—181no—
371C4—181no—
372C4—361no—
373C4—361no—
374C4#8865011yescharacteristic
375C4#8865111yescharacteristic
376C3—31no—
377C3—31no—
378C3—61no—
379C3#1059043511yescharacteristic
380C2—362no—
381C2#1172109611yescharacteristic
382C2#1172119311yescharacteristic
383C2#1172123011yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊#272C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#478
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
2. #11721095C2semidirect
3. #63635⋊C3C2semidirect
4. #63634⋊C3C2semidirect
7. #11720304C2semidirect
26. #63635D3semidirect
27. #63634D3semidirect
224. H(3,𝔽3)#478semidirect
335. C32#4871semidirect
336. C32#4891semidirect

Maximal subgroups

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

subgroupindex
#117217612
#117210952
#63635⋊C32
#63634⋊C32
#117203242
#117203042
#105905453
#105905423