Finite Group Databasebeta

← #13137969

Subgroups — #13137969

Order 1728.

459 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721765C211yesmaximal, characteristic
2#11721094C211yesmaximal, characteristic
3C2×#88439C211yesmaximal, characteristic, complemented
4C2×#88435C211yesmaximal, characteristic
5#63637⋊C3C211yesmaximal, characteristic, complemented
6#63635⋊C3C211yesmaximal, characteristic, complemented
7#11720326C211yesmaximal, characteristic, complemented
8#10590587—31nomaximal
9#10590520—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88542—21no—
15C4⋊#1084—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88448—21no—
21#88439—22no—
22#88435—22no—
23C3×#897—31no—
24C3×#897—31no—
25C3×#897—61no—
26#63857—31no—
27#63856—31no—
28C2×#3719—31no—
29C2×#3718—31no—
30C2×#3711—31no—
31C2×#3711—31no—
32#63637D311yescharacteristic, complemented
33#63635D311yescharacteristic, complemented
34C3×#810—31no—
35C3×#808—31no—
36#63566—31no—
37#63565—31no—
38C22×#272D411yescharacteristic
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C2×#1084D411yescharacteristic
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⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3D412yes—
53#5052—91no—
54#4867—91no—
55#4816—91no—
56C2×#634C2×D311yescharacteristic
57C2×C3×C4×D3—31no—
58C2×C3×C4×D3—31no—
59C2×C3×C4×D3—61no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C4⋊C4C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×(C2×D3)⋊C4—61no—
71C3×(C2×D3)⋊C4—61no—
72C3×(C2×D3)⋊C4—62no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—61no—
75C3×C4⋊Dic3—61no—
76C3×C4⋊Dic3—62no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79#3724—61no—
80Dic3⋊Dic3—61no—
81(C2×D3)⋊Dic3—61no—
82(C2×D3)⋊Dic3—61no—
83#3719—62no—
84#3718—62no—
85#3711—62no—
86#3711—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—21no—
89C2×#272—21no—
90C2×#272—21no—
91C4×H(3,𝔽3)—41no—
92C4×H(3,𝔽3)C2×D411yescharacteristic
93C4×H(3,𝔽3)#4111yescharacteristic
94#1084—21no—
95#1084—41no—
96(C3⋊Dic3)⋊C3—41no—
97(C3⋊Dic3)⋊C3—41no—
98C3×C22⋊Q8—91no—
99#946—91no—
100C2×#216—91no—
101C2×#216—91no—
102#924—91no—
103C2×#208—91no—
104C2×Dic12—91no—
105C2×ΓL(1,𝔽25)—91no—
106C2×ΓL(1,𝔽25)—91no—
107#897—31no—
108#897—31no—
109#897—61no—
110#897—91no—
111C3×#106—91no—
112C3×#103—91no—
113#835—91no—
114#816—91no—
115#810—31no—
116#810—91no—
117#810—91no—
118#808—31no—
119#808—91no—
120C22×C3×D3—31no—
121C22×C3×D3—31no—
122C22×C3×D3—61no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131#634—21no—
132#634—21no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×C4×D3—121no—
146C3×C4×D3—121no—
147C3×Dic6—61no—
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)D1212yes—
157C3×C3⋊C8—32no—
158C3×C3⋊C8—61no—
159#470—271noSylow
160C2×H(3,𝔽3)#12111yescharacteristic
161C2×H(3,𝔽3)#13711yescharacteristic
162C2×H(3,𝔽3)#13711yescharacteristic
163#272—41no—
164C2×C3×Q8—91no—
165C2×C3×Q8—91no—
166C22×C3×C4—91no—
167C2×C4×D3—31no—
168C2×C4×D3—31no—
169C2×C4×D3—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—31no—
178C3×C4⋊C4—31no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4—91no—
181C3×C4⋊C4—181no—
182C3×C4⋊C4D3211yescharacteristic
183C3×C22⋊C4—181no—
184#217—181no—
185#216—182no—
186#216—182no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—61no—
189(C2×D3)⋊C4—62no—
190(C2×D3)⋊C4—181no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—31no—
193C4⋊Dic3—61no—
194C4⋊Dic3—61no—
195C4⋊Dic3—62no—
196C4⋊Dic3—91no—
197C4⋊Dic3—181no—
198Dic3⋊C4—181no—
199Dic3⋊C4—181no—
200#208—182no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—91no—
205Dic12—182no—
206ΓL(1,𝔽25)—182no—
207ΓL(1,𝔽25)—182no—
208C22×C32—21no—
209C22×C32D3×D411yescharacteristic
210C22×C32D3×D411yescharacteristic
211C2×C3×D3—62no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—42no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4C2×D1211yescharacteristic
226C32×C4#23511yescharacteristic
227C32×C4D3×D411yescharacteristic
228C32×C4#23911yescharacteristic
229C3⋊Dic3—41no—
230C3⋊Dic3—41no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—121no—
234C3×Dic3—122no—
235C3×Dic3—121no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C2×Q16—271no—
240C2×M16—271no—
241C22⋊Q8—91no—
242C22⋊Q8—271no—
243#106—91no—
244#103—91no—
245#103—271no—
246H(3,𝔽3)#47011yescharacteristic, complemented, Sylow
247C23×C3—91no—
248C22×D3—31no—
249C22×D3—31no—
250C22×D3—61no—
251C3×Q8—181no—
252C3×Q8—181no—
253C3×Q8—181no—
254C3×Q8—181no—
255C2×C3×C4—21no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—62no—
261C2×C3×C4—61no—
262C2×C3×C4—91no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4C2×D3211yescharacteristic
269C2×Dic3—31no—
270C2×Dic3—31no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—62no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—181no—
278C2×Dic3—181no—
279C4×D3—61no—
280C4×D3—61no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—121no—
284C4×D3—121no—
285Dic6—61no—
286Dic6—61no—
287Dic6—61no—
288Dic6—61no—
289Dic6—61no—
290Dic6—61no—
291Dic6—121no—
292Dic6—121no—
293Dic6—181no—
294Dic6—181no—
295C3×C8—92no—
296C3×C8—181no—
297C3⋊C8—32no—
298C3⋊C8—61no—
299C3⋊C8—92no—
300C3⋊C8—181no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32#88411yescharacteristic
305C2×C32#89511yescharacteristic
306C2×C32#91511yescharacteristic
307C2×C32#91511yescharacteristic
308C2×C32#91511yescharacteristic
309C2×C32#91811yescharacteristic
310C3×D3—122no—
311C3×D3—121no—
312C3×D3—121no—
313C2×Q8—91no—
314C2×Q8—91no—
315C22×C4—91no—
316Q16—542no—
317M16—542no—
318C2×C8—91no—
319C2×C8—91no—
320C4⋊C4—91no—
321C4⋊C4—181no—
322C4⋊C4—541no—
323C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
324C22⋊C4—181no—
325C22⋊C4—541no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—181no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3#381111yescharacteristic
333C2×D3—61no—
334C2×D3—61no—
335C2×D3—61no—
336C2×D3—61no—
337C2×D3—61no—
338C2×D3—61no—
339C2×D3—62no—
340C2×D3—121no—
341C2×D3—121no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—41no—
347C3×C4—61no—
348C3×C4—61no—
349C3×C4—121no—
350C3×C4—121no—
351C3×C4—122no—
352C3×C4—181no—
353C3×C4—361no—
354C3×C4—361no—
355C3×C4—361no—
356C3×C4#379911yescharacteristic
357C3×C4D3×D1211yescharacteristic
358Dic3—61no—
359Dic3—61no—
360Dic3—121no—
361Dic3—121no—
362Dic3—121no—
363Dic3—122no—
364Dic3—121no—
365Dic3—121no—
366Dic3—361no—
367Dic3—361no—
368C32—21no—
369C32#481611yescharacteristic, complemented
370C32#486711yescharacteristic, complemented
371C23—91no—
372Q8—181no—
373Q8—181no—
374Q8—181no—
375Q8—181no—
376C2×C4—21no—
377C2×C4—91no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
384C8—92no—
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—361no—
396C2×C3#6391011yescharacteristic
397C2×C3#6393111yescharacteristic
398C2×C3#6395111yescharacteristic
399D3—121no—
400D3—121no—
401D3—122no—
402C22—181no—
403C22—181no—
404C22—181no—
405C22#8867711yescharacteristic
406C4—41no—
407C4—181no—
408C4—361no—
409C4—361no—
410C4—361no—
411C4#8865211yescharacteristic
412C4#8865411yescharacteristic
413C3—31no—
414C3—31no—
415C3—61no—
416C3#1059052411yescharacteristic
417C2—361no—
418C2#1172115711yescharacteristic
419C2#1172118911yescharacteristic
420C2#1172123111yescharacteristic
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 radicalC22×#272D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#470
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
3. C2×#88439C2semidirect
5. #63637⋊C3C2semidirect
6. #63635⋊C3C2semidirect
7. #11720326C2semidirect
32. #63637D3semidirect
33. #63635D3semidirect
246. H(3,𝔽3)#470semidirect
369. C32#4816semidirect
370. C32#4867semidirect

Maximal subgroups

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

subgroupindex
#117217652
#117210942
C2×#884392
C2×#884352
#63637⋊C32
#63635⋊C32
#117203262
#105905873
#105905203