Finite Group Databasebeta

← #13140934

Subgroups — #13140934

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#11721863C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic, complemented
3#11721094C211yesmaximal, characteristic
4C2×#88435C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10591638—31nomaximal
9#10591598—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4×#272C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13#88542—21no—
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16#3756⋊C3—22no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—21no—
21#88448—21no—
22#88435—22no—
23C2×#3756D311yescharacteristic, complemented
24C3×#946—31no—
25C3×#946—31no—
26C3×#946—61no—
27C2×C3×#216—31no—
28#63857—31no—
29#63856—31no—
30C2×#3711—31no—
31C2×#3711—31no—
32#63662D311yescharacteristic, complemented
33C3×#835—31no—
34#63574—31no—
35#63573—31no—
36#63567—31no—
37#63565—31no—
38C22×#272D411yescharacteristic
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×#1084D411yescharacteristic
45C4×#272—21no—
46C4×#272—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#5189—91no—
54#5052—91no—
55#4816—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—31no—
64#3756—22no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×#216—62no—
69C3×(C2×D3)⋊C4—61no—
70C3×(C2×D3)⋊C4—61no—
71C3×(C2×D3)⋊C4—62no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—61no—
76C3×Dic3⋊C4—61no—
77C3×Dic3⋊C4—61no—
78C3×Dic3⋊C4—62no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81#3724—61no—
82Dic3⋊Dic3—61no—
83(C2×D3)⋊Dic3—61no—
84(C2×D3)⋊Dic3—61no—
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—
98C2×C3×Q16—91no—
99C3×C22⋊Q8—91no—
100#946—31no—
101#946—31no—
102#946—61no—
103#946—91no—
104C2×#216—31no—
105C2×#216—91no—
106#924—91no—
107C2×#208—91no—
108C2×ΓL(1,𝔽25)—91no—
109C2×ΓL(1,𝔽25)—91no—
110#897—91no—
111C3×#103—91no—
112#835—31no—
113#835—91no—
114#835—91no—
115#817—91no—
116#816—91no—
117#810—91no—
118#808—91no—
119#808—91no—
120C22×C3×D3—31no—
121C22×C3×D3—31no—
122C22×C3×D3—61no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—41no—
128C32×Q8—41no—
129C2×C32×C4—22no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137#634—21no—
138#634—21no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C3×C4×D3—61no—
148C3×C4×D3—61no—
149C3×C4×D3—61no—
150C3×C4×D3—61no—
151C3×C4×D3—121no—
152C3×C4×D3—121no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)#6712yes—
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—31no—
165C2×C3×Q8—31no—
166C2×C3×Q8—61no—
167C2×C3×Q8—91no—
168C2×C3×Q8D3211yescharacteristic
169C22×C3×C4—91no—
170C2×C4×D3—31no—
171C2×C4×D3—31no—
172C2×C4×D3—61no—
173C2×Dic6—31no—
174C2×Dic6—91no—
175C3×Q16—182no—
176C2×C3×C8—91no—
177C2×C3×C8—91no—
178C3×C4⋊C4—91no—
179C3×C4⋊C4—91no—
180C3×C4⋊C4—181no—
181C3×C22⋊C4—181no—
182#217—181no—
183#216—62no—
184#216—182no—
185(C2×D3)⋊C4—61no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—62no—
188(C2×D3)⋊C4—181no—
189(C2×D3)⋊C4—181no—
190C4⋊Dic3—31no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—61no—
194C4⋊Dic3—91no—
195C4⋊Dic3—181no—
196Dic3⋊C4—61no—
197Dic3⋊C4—61no—
198Dic3⋊C4—62no—
199Dic3⋊C4—181no—
200Dic3⋊C4—181no—
201#208—182no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—91no—
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×C4#23511yescharacteristic
226C32×C4D3×D411yescharacteristic
227C32×C4#23711yescharacteristic
228C32×C4C2×#6711yescharacteristic
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—91no—
240C2×M16—271no—
241C22⋊Q8—91no—
242C22⋊Q8—271no—
243#106—271no—
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—21no—
252C3×Q8—21no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—121no—
258C3×Q8—121no—
259C3×Q8—181no—
260C3×Q8—181no—
261C2×C3×C4—21no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—62no—
267C2×C3×C4—61no—
268C2×C3×C4—91no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4C2×D3211yescharacteristic
275C2×Dic3—31no—
276C2×Dic3—31no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—62no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—181no—
284C2×Dic3—181no—
285C4×D3—61no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—121no—
290C4×D3—121no—
291Dic6—61no—
292Dic6—61no—
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#91511yescharacteristic
306C2×C32#91811yescharacteristic
307C2×C32#91811yescharacteristic
308C2×C32#91811yescharacteristic
309C2×C32#93811yescharacteristic
310C3×D3—122no—
311C3×D3—121no—
312C3×D3—121no—
313C2×Q8—91no—
314C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C22×C4—91no—
316Q16—182no—
317M16—542no—
318C2×C8—91no—
319C2×C8—91no—
320C4⋊C4—91no—
321C4⋊C4—91no—
322C4⋊C4—181no—
323C4⋊C4—541no—
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#380411yescharacteristic
357C3×C4D3×#6711yescharacteristic
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#518911yescharacteristic, complemented
371C23—91no—
372Q8—21no—
373Q8—21no—
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#6394511yescharacteristic
397C2×C3#6395111yescharacteristic
398C2×C3#6395811yescharacteristic
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#8867111yescharacteristic
412C4#8867611yescharacteristic
413C3—31no—
414C3—31no—
415C3—61no—
416C3#1059160011yescharacteristic
417C2—361no—
418C2#1172122711yescharacteristic
419C2#1172123111yescharacteristic
420C2#1172130311yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
hypercenterC2×Q8((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 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#3756⋊C3C2semidirect
5. #63662⋊C3C2semidirect
6. #11720309C2semidirect
7. #11720307C2semidirect
23. C2×#3756D3semidirect
32. #63662D3semidirect
246. H(3,𝔽3)#470semidirect
369. C32#4816semidirect
370. C32#5189semidirect

Maximal subgroups

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

subgroupindex
#117218632
C2×#3756⋊C32
#117210942
C2×#884352
#63662⋊C32
#117203092
#117203072
#105916383
#105915983