Finite Group Databasebeta

← #13140953

Subgroups — #13140953

Order 1728.

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

normal subgroups only

This group has 468 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#11721864C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic, complemented
3#11721084C211yesmaximal, characteristic, complemented
4C2×#88437C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic
7#11720304C211yesmaximal, characteristic, complemented
8#10591578—31nomaximal
9#10591577—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13#88542—21no—
14#88542—21no—
15C4×#1084C2211yescharacteristic
16#3756⋊C3—22no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#88447—21no—
22#88437—22no—
23C2×#3756D311yescharacteristic, complemented
24C3×#947—31no—
25C3×#947—31no—
26C3×#947—61no—
27C2×C3×#216—31no—
28#63844—31no—
29Dic3×Dic6—31no—
30C2×#3716—31no—
31C2×#3715—31no—
32#63662D311yescharacteristic, complemented
33C3×#835—31no—
34Dic3⋊(C3⋊C8)—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084D411yescharacteristic
44C2×#1084D411yescharacteristic
45C4⋊#272—21no—
46C4⋊#272—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#5185—91no—
54#5054—91no—
55#4798—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—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—61no—
72C3×(C2×D3)⋊C4—61no—
73C3×(C2×D3)⋊C4—62no—
74C3×(C2×D3)⋊C4—62no—
75C3×C4⋊Dic3—31no—
76C3×C4×Dic3—31no—
77C3×C4×Dic3—31no—
78C3×C4×Dic3—61no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81#3724—61no—
82Dic3⋊Dic3—61no—
83Dic32—61no—
84Dic32—61no—
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)—41no—
92C4×H(3,𝔽3)C2×D411yescharacteristic
93C4×H(3,𝔽3)#4111yescharacteristic
94#1084—21no—
95#1084—21no—
96(C3⋊Dic3)⋊C3—22no—
97(C3⋊Dic3)⋊C3—41no—
98C2×C3×Q16—91no—
99C3×#124—91no—
100#947—31no—
101#947—31no—
102#947—61no—
103Q8×Dic3—91no—
104C2×#216—31no—
105C2×#216—91no—
106C2×#215—91no—
107C2×#214—91no—
108C2×#204—91no—
109#887—91no—
110C4×Dic6—91no—
111C3×#103—91no—
112#835—31no—
113#835—91no—
114#832—91no—
115#821—91no—
116Dic3⋊C8—91no—
117Dic3⋊C8—91no—
118#809—91no—
119C4⋊(C3⋊C8)—91no—
120C22×C3×D3—62no—
121C22×C3×D3—61no—
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#6712yes—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—121no—
152C3×D12—121no—
153C3×Dic6—32no—
154C3×Dic6—61no—
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—31no—
165C2×C3×Q8—31no—
166C2×C3×Q8—61no—
167C2×C3×Q8—91no—
168C2×C3×Q8D3211yescharacteristic
169C2×C3×D4—91no—
170C2×D12—31no—
171C2×D12—31no—
172C2×D12—61no—
173C2×Dic6—31no—
174C2×Dic6—91no—
175C3×Q16—182no—
176C2×C3×C8—91no—
177C2×C3×C8—91no—
178C3×C4⋊C4—91no—
179C3×C22⋊C4—181no—
180C3×C22⋊C4—181no—
181C3×C42—91no—
182#216—62no—
183#216—182no—
184#215—182no—
185#214—182no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—61no—
189(C2×D3)⋊C4—61no—
190(C2×D3)⋊C4—62no—
191(C2×D3)⋊C4—62no—
192C4⋊Dic3—31no—
193C4⋊Dic3—91no—
194C4⋊Dic3—181no—
195Dic3⋊C4—181no—
196Dic3⋊C4—181no—
197C4×Dic3—31no—
198C4×Dic3—31no—
199C4×Dic3—61no—
200C4×Dic3—181no—
201C4×Dic3—181no—
202C4×Dic3—181no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—91no—
207#204—182no—
208C22×C32—21no—
209C22×C32D3×D411yescharacteristic
210C22×C32D3×D411yescharacteristic
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—42no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4#23511yescharacteristic
226C32×C4D3×D411yescharacteristic
227C32×C4#23711yescharacteristic
228C32×C4C2×#6711yescharacteristic
229C3⋊Dic3—22no—
230C3⋊Dic3—41no—
231C3×Dic3—62no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C2×Q16—91no—
240C2×SD16—271no—
241#124—91no—
242C4×Q8—271no—
243C4⋊C8—271no—
244#103—91no—
245#102—271no—
246H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
247C23×C3—181no—
248C22×D3—61no—
249C22×D3—61no—
250C22×D3—62no—
251C3×Q8—21no—
252C3×Q8—21no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—92no—
258C3×Q8—121no—
259C3×Q8—121no—
260C3×Q8—181no—
261C3×D4—181no—
262C3×D4—181no—
263C2×C3×C4—21no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—62no—
269C2×C3×C4—61no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4C2×D3211yescharacteristic
275C2×Dic3—31no—
276C2×Dic3—31no—
277C2×Dic3—31no—
278C2×Dic3—31no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—181no—
284C2×Dic3—181no—
285D12—61no—
286D12—61no—
287D12—61no—
288D12—61no—
289D12—121no—
290D12—121no—
291Dic6—32no—
292Dic6—61no—
293Dic6—92no—
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#88411yescharacteristic
305C2×C32#91611yescharacteristic
306C2×C32#91811yescharacteristic
307C2×C32#91811yescharacteristic
308C2×C32#91911yescharacteristic
309C2×C32#93911yescharacteristic
310C3×D3—242no—
311C3×D3—241no—
312C3×D3—241no—
313C2×Q8—91no—
314C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C2×D4—91no—
316Q16—182no—
317SD16—542no—
318C2×C8—91no—
319C2×C8—91no—
320C4⋊C4—91no—
321C4⋊C4—541no—
322C22⋊C4—181no—
323C22⋊C4—181no—
324C42—91no—
325C42—541no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—361no—
330C22×C3—361no—
331C22×C3—361no—
332C22×C3#381111yescharacteristic
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—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—182no—
353C3×C4—181no—
354C3×C4—181no—
355C3×C4—361no—
356C3×C4#380411yescharacteristic
357C3×C4D3×#6711yescharacteristic
358Dic3—61no—
359Dic3—61no—
360Dic3—61no—
361Dic3—61no—
362Dic3—62no—
363Dic3—121no—
364Dic3—121no—
365Dic3—121no—
366Dic3—182no—
367Dic3—361no—
368C32—21no—
369C32#479811yescharacteristic, complemented
370C32#518511yescharacteristic, complemented
371C23—181no—
372Q8—21no—
373Q8—21no—
374Q8—92no—
375Q8—181no—
376D4—181no—
377D4—181no—
378C2×C4—21no—
379C2×C4—91no—
380C2×C4—91no—
381C2×C4—181no—
382C2×C4—181no—
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#6394511yescharacteristic
397C2×C3#6395011yescharacteristic
398C2×C3#6397111yescharacteristic
399D3—241no—
400D3—241no—
401D3—242no—
402C22—361no—
403C22—361no—
404C22—361no—
405C22#8867711yescharacteristic
406C4—41no—
407C4—181no—
408C4—181no—
409C4—182no—
410C4—361no—
411C4#8867111yescharacteristic
412C4#8867611yescharacteristic
413C3—31no—
414C3—31no—
415C3—61no—
416C3#1059157911yescharacteristic
417C2—721no—
418C2#1172122711yescharacteristic
419C2#1172123011yescharacteristic
420C2#1172131211yescharacteristic
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 radicalC2×C4⋊#272C22
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#464
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
3. #11721084C2semidirect
5. #63662⋊C3C2semidirect
7. #11720304C2semidirect
23. C2×#3756D3semidirect
32. #63662D3semidirect
246. H(3,𝔽3)#464semidirect
369. C32#4798semidirect
370. C32#5185semidirect

Maximal subgroups

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

subgroupindex
#117218642
C2×#3756⋊C32
#117210842
C2×#884372
#63662⋊C32
#117203242
#117203042
#105915783
#105915773