Finite Group Databasebeta

← #13137427

Subgroups — #13137427

Order 1728.

468 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721721C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×#88439C211yesmaximal, characteristic, complemented
4C2×#88437C211yesmaximal, characteristic
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic
8#10590378—31nomaximal
9#10590267—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14#88542—21no—
15#88542—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88439—22no—
22#88437—22no—
23C4×#634D311yescharacteristic, complemented
24C3×#875—31no—
25C3×#875—31no—
26C3×#875—61no—
27C3×C4×Dic6—31no—
28C2×#3719—31no—
29C2×#3718—31no—
30C2×#3716—31no—
31C2×#3715—31no—
32C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
33C3×C4⋊(C3⋊C8)—31no—
34#63566—31no—
35#63565—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C4⋊#272—21no—
44C4⋊#272—21no—
45#6229—21no—
46#6229—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3D412yes—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5054—91no—
54#5028—91no—
55#4845—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—
64C32×C42—21no—
65C32×C42C2×D311yescharacteristic
66C32×C42C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—21no—
69C4×C3⋊Dic3—21no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×(C2×D3)⋊C4—62no—
73C3×(C2×D3)⋊C4—62no—
74C3×(C2×D3)⋊C4—61no—
75C3×(C2×D3)⋊C4—61no—
76C3×(C2×D3)⋊C4—61no—
77C3×(C2×D3)⋊C4—61no—
78C3×C4⋊Dic3—31no—
79C3×Dic3⋊C4—61no—
80C3×C4×Dic3—61no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83#3719—62no—
84#3718—62no—
85#3716—62no—
86#3715—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—41no—
89C2×#272—41no—
90C2×#272—41no—
91C4×H(3,𝔽3)—21no—
92C4×H(3,𝔽3)—21no—
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)#4111yescharacteristic
95#1084—41no—
96(C3⋊Dic3)⋊C3—22no—
97(C3⋊Dic3)⋊C3—41no—
98C3×#124—91no—
99C3×C4×Q8—91no—
100C2×#216—91no—
101C2×#216—91no—
102C2×#215—91no—
103C2×#214—91no—
104C2×Dic12—91no—
105C2×#204—91no—
106#875—31no—
107#875—31no—
108#875—61no—
109C4×Dic6—31no—
110C4×Dic6—91no—
111C3×C4⋊C8—91no—
112#835—91no—
113#832—91no—
114#821—91no—
115#816—91no—
116#810—91no—
117#809—91no—
118C4⋊(C3⋊C8)—31no—
119C4⋊(C3⋊C8)—91no—
120C22×C3×D3—62no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4D1211yescharacteristic
127C2×C32×C4D1211yescharacteristic
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4#6711yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134#634—21no—
135#634D1212yes—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C3×D12—61no—
142C3×D12—61no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—121no—
146C3×D12—121no—
147C3×Dic6—32no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—121no—
154C3×Dic6—121no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)—21no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—61no—
159#464—271noSylow
160C2×H(3,𝔽3)#12111yescharacteristic
161C2×H(3,𝔽3)#13511yescharacteristic
162C2×H(3,𝔽3)#13711yescharacteristic
163#272—81no—
164C2×C3×Q8—91no—
165C2×C3×Q8—91no—
166C2×C3×D4—91no—
167C2×D12—31no—
168C2×D12—31no—
169C2×D12—61no—
170C2×Dic6—31no—
171C2×Dic6—31no—
172C2×Dic6—31no—
173C2×Dic6—61no—
174C2×Dic6—91no—
175C2×C3×C8—91no—
176C2×C3×C8—91no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—181no—
179C3×C22⋊C4—181no—
180C3×C22⋊C4—181no—
181C3×C42—31no—
182C3×C42—31no—
183C3×C42—61no—
184C3×C42—181no—
185C3×C42D3211yescharacteristic
186#216—182no—
187#216—182no—
188#215—182no—
189#214—182no—
190(C2×D3)⋊C4—61no—
191(C2×D3)⋊C4—61no—
192(C2×D3)⋊C4—61no—
193(C2×D3)⋊C4—61no—
194(C2×D3)⋊C4—62no—
195(C2×D3)⋊C4—62no—
196C4⋊Dic3—31no—
197C4⋊Dic3—91no—
198Dic3⋊C4—61no—
199Dic3⋊C4—181no—
200C4×Dic3—61no—
201C4×Dic3—181no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—91no—
206Dic12—182no—
207#204—182no—
208C22×C32—21no—
209C22×C32C2×D1211yescharacteristic
210C22×C32C2×#6711yescharacteristic
211C2×C3×D3—122no—
212C2×C3×D3—122no—
213C2×C3×D3—122no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4C2×D1211yescharacteristic
229C32×C4#23511yescharacteristic
230C32×C4D3×D411yescharacteristic
231C32×C4#23911yescharacteristic
232C3⋊Dic3—22no—
233C3⋊Dic3—41no—
234C3×Dic3—62no—
235C3×Dic3—122no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C2×Q16—271no—
240C2×SD16—271no—
241#124—91no—
242C4×Q8—91no—
243C4⋊C8—91no—
244#103—271no—
245#102—271no—
246H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
247C23×C3—181no—
248C22×D3—61no—
249C22×D3—61no—
250C22×D3—62no—
251C3×Q8—92no—
252C3×Q8—181no—
253C3×Q8—181no—
254C3×Q8—181no—
255C3×D4—181no—
256C3×D4—181no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4#62611yescharacteristic
270C2×C3×C4#62611yescharacteristic
271C2×C3×C4C2×D3211yescharacteristic
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—62no—
276C2×Dic3—61no—
277C2×Dic3—181no—
278C2×Dic3—181no—
279D12—61no—
280D12—61no—
281D12—61no—
282D12—61no—
283D12—121no—
284D12—121no—
285Dic6—32no—
286Dic6—61no—
287Dic6—61no—
288Dic6—61no—
289Dic6—61no—
290Dic6—61no—
291Dic6—92no—
292Dic6—121no—
293Dic6—121no—
294Dic6—181no—
295C3×C8—181no—
296C3×C8—181no—
297C3⋊C8—61no—
298C3⋊C8—61no—
299C3⋊C8—181no—
300C3⋊C8—181no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32#89511yescharacteristic
305C2×C32#90411yescharacteristic
306C2×C32#90911yescharacteristic
307C2×C32#93011yescharacteristic
308C2×C32#94211yescharacteristic
309C2×C32#95011yescharacteristic
310C3×D3—242no—
311C3×D3—241no—
312C3×D3—241no—
313C2×Q8—91no—
314C2×Q8—91no—
315C2×D4—91no—
316Q16—542no—
317SD16—542no—
318C2×C8—91no—
319C2×C8—91no—
320C4⋊C4—91no—
321C4⋊C4—181no—
322C22⋊C4—181no—
323C22⋊C4—181no—
324C42—181no—
325C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—361no—
330C22×C3—361no—
331C22×C3—361no—
332C22×C3C2×#62611yescharacteristic
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—122no—
340C2×D3—122no—
341C2×D3—122no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—61no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—121no—
355C3×C4—121no—
356C3×C4—182no—
357C3×C4—361no—
358C3×C4—361no—
359C3×C4#379911yescharacteristic
360C3×C4D3×D1211yescharacteristic
361Dic3—62no—
362Dic3—121no—
363Dic3—121no—
364Dic3—122no—
365Dic3—121no—
366Dic3—182no—
367Dic3—361no—
368C32—21no—
369C32#484511yescharacteristic, complemented
370C32#502811yescharacteristic, complemented
371C23—181no—
372Q8—92no—
373Q8—181no—
374Q8—181no—
375Q8—181no—
376D4—181no—
377D4—181no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4#620011yescharacteristic
382C2×C4#620011yescharacteristic
383C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
384C8—181no—
385C8—181no—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—61no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—721no—
396C2×C3#6382911yescharacteristic
397C2×C3#6383311yescharacteristic
398C2×C3#6390911yescharacteristic
399D3—241no—
400D3—241no—
401D3—242no—
402C22—361no—
403C22—361no—
404C22—361no—
405C22C2×#620011yescharacteristic
406C4—21no—
407C4—21no—
408C4—182no—
409C4—361no—
410C4—361no—
411C4#8865211yescharacteristic
412C4#8865411yescharacteristic
413C3—31no—
414C3—31no—
415C3—61no—
416C3#1059036811yescharacteristic
417C2—721no—
418C2#1172106911yescharacteristic
419C2#1172107311yescharacteristic
420C2#1172115611yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#464
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#634⋊C3C2semidirect
3. C2×#88439C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #11720326C2semidirect
23. C4×#634D3semidirect
32. C4⋊(C3⋊(C3⋊C8))D3semidirect
246. H(3,𝔽3)#464semidirect
369. C32#4845semidirect
370. C32#5028semidirect

Maximal subgroups

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

subgroupindex
#117217212
C4×#634⋊C32
C2×#884392
C2×#884372
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203262
#117203242
#105903783
#105902673