Finite Group Databasebeta

← #13140892

Subgroups — #13140892

Order 1728.

462 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 77 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
1Q8×#1084C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic
3C2×#3756⋊C3C211yesmaximal, characteristic, complemented
4#11721113C211yesmaximal, characteristic
5#11720309C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic
7#11720304C211yesmaximal, characteristic, complemented
8#10591607—31nomaximal
9#10591604—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊#1084—21no—
14C4⋊#1084C2211yescharacteristic
15C4×#1084—21no—
16C4×#1084C2211yescharacteristic
17#3755⋊C3—22no—
18#3756⋊C3—22no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88450—21no—
23C2×#3756D311yescharacteristic, complemented
24C2×#3755D311yescharacteristic
25C3×Q8×Dic3—31no—
26C3×Q8×Dic3—31no—
27C3×Q8×Dic3—61no—
28C2×C3×#216—31no—
29C2×C3×#215—31no—
30#63864—31no—
31#63860—31no—
32Dic3⋊(C3⋊C8)—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63567—31no—
35#63565—31no—
36#63564—31no—
37#63562—31no—
38C22×(C3⋊D3)⋊C3—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)D412yes—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×#1084D411yescharacteristic
45C2×#1084D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5185—91no—
54#5166—91no—
55#5028—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×Dic6—31no—
63#3756—22no—
64#3755—22no—
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×#216—62no—
68C3×#215—62no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—61no—
72C3×C4⋊Dic3—61no—
73C3×C4⋊Dic3—61no—
74C3×C4⋊Dic3—62no—
75C3×C4×Dic3—31no—
76C3×C4×Dic3—31no—
77C3×C4×Dic3—61no—
78C3×C4×Dic3—61no—
79C3×C4×Dic3—61no—
80C3×C4×Dic3—62no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83#3722—61no—
84#3722—61no—
85(C2×D3)⋊Dic3—61no—
86(C2×D3)⋊Dic3—61no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×(C3⋊D3)⋊C3—41no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
91C4×H(3,𝔽3)—22no—
92C4×H(3,𝔽3)C2×D411yescharacteristic
93C4×H(3,𝔽3)#4111yescharacteristic
94#1084—21no—
95#1084—21no—
96#1084—41no—
97(C3⋊Dic3)⋊C3—41no—
98C2×C3×Q16—91no—
99C2×C3×SD16—91no—
100C3×C4×Q8—91no—
101#947—91no—
102Q8×Dic3—31no—
103Q8×Dic3—31no—
104Q8×Dic3—61no—
105C2×#216—31no—
106C2×#216—91no—
107C2×#215—31no—
108C2×#215—91no—
109#936—91no—
110#875—91no—
111#835—91no—
112#832—91no—
113#821—91no—
114#816—91no—
115Dic3⋊C8—91no—
116Dic3⋊C8—91no—
117#810—91no—
118#809—91no—
119C4⋊(C3⋊C8)—91no—
120C22×C3⋊D3—21no—
121C22×C3×D3—61no—
122C32×Q8—22no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—22no—
126C32×Q8#6712yes—
127C32×Q8#6712yes—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—21no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137#634—21no—
138#634—21no—
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—62no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)—21no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—61no—
157#464—271noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)#13511yescharacteristic
160C2×H(3,𝔽3)#13711yescharacteristic
161(C3⋊D3)⋊C3—81no—
162C2×C3×Q8—31no—
163C2×C3×Q8—31no—
164C2×C3×Q8—61no—
165C2×C3×Q8—91no—
166C2×C3×Q8D3211yescharacteristic
167C2×C3×D4—91no—
168C2×D12—31no—
169C2×D12—91no—
170C2×Dic6—31no—
171C2×Dic6—91no—
172C3×Q16—182no—
173C3×SD16—182no—
174C2×C3×C8—91no—
175C2×C3×C8—91no—
176C3×C4⋊C4—91no—
177C3×C4⋊C4—181no—
178C3×C42—91no—
179C3×C42—181no—
180#217—181no—
181#217—181no—
182#216—62no—
183#216—182no—
184#215—62no—
185#215—182no—
186(C2×D3)⋊C4—181no—
187(C2×D3)⋊C4—181no—
188(C2×D3)⋊C4—181no—
189(C2×D3)⋊C4—181no—
190C4⋊Dic3—31no—
191C4⋊Dic3—31no—
192C4⋊Dic3—61no—
193C4⋊Dic3—61no—
194C4⋊Dic3—61no—
195C4⋊Dic3—62no—
196C4×Dic3—31no—
197C4×Dic3—31no—
198C4×Dic3—61no—
199C4×Dic3—61no—
200C4×Dic3—61no—
201C4×Dic3—62no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—91no—
206C22×C32—21no—
207C22×C32D3×D411yescharacteristic
208C22×C32D3×D411yescharacteristic
209C2×C3⋊D3—41no—
210C2×C3⋊D3—41no—
211C2×C3⋊D3—41no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C32×C4—24no—
216C32×C4—22no—
217C32×C4—22no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4#23711yescharacteristic
221C32×C4#23711yescharacteristic
222C32×C4C2×#6711yescharacteristic
223C32×C4C2×#6711yescharacteristic
224C3⋊Dic3—41no—
225C3×Dic3—62no—
226C3×Dic3—61no—
227C3×Dic3—61no—
228C3×Dic3—61no—
229C3×Dic3—61no—
230C3×Dic3—121no—
231C3×Dic3—122no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234C3×Dic3—121no—
235C2×Q16—91no—
236C2×SD16—91no—
237#124—271no—
238C4×Q8—91no—
239C4⋊C8—271no—
240#103—271no—
241#102—271no—
242H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
243C23×C3—181no—
244C22×D3—61no—
245C22×D3—181no—
246C3×Q8—21no—
247C3×Q8—32no—
248C3×Q8—32no—
249C3×Q8—61no—
250C3×Q8—61no—
251C3×Q8—62no—
252C3×Q8—62no—
253C3×Q8—181no—
254C3×Q8—181no—
255C3×Q8#62512yes—
256C3×D4—181no—
257C3×D4—181no—
258C2×C3×C4—21no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—62no—
264C2×C3×C4—61no—
265C2×C3×C4—91no—
266C2×C3×C4—91no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4C2×D3211yescharacteristic
270C2×Dic3—31no—
271C2×Dic3—31no—
272C2×Dic3—31no—
273C2×Dic3—31no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—62no—
279C2×Dic3—61no—
280C2×Dic3—181no—
281D12—61no—
282D12—61no—
283D12—181no—
284D12—181no—
285Dic6—61no—
286Dic6—61no—
287Dic6—181no—
288Dic6—181no—
289C3×C8—181no—
290C3×C8—181no—
291C3⋊C8—61no—
292C3⋊C8—61no—
293C3⋊C8—181no—
294C3⋊C8—181no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32#91511yescharacteristic
299C2×C32#91611yescharacteristic
300C2×C32#91811yescharacteristic
301C2×C32#91911yescharacteristic
302C2×C32#93911yescharacteristic
303C2×C32#93911yescharacteristic
304C3⋊D3—81no—
305C3×D3—241no—
306C2×Q8—91no—
307C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C2×D4—91no—
309Q16—182no—
310SD16—182no—
311C2×C8—91no—
312C2×C8—91no—
313C4⋊C4—91no—
314C4⋊C4—181no—
315C22⋊C4—541no—
316C22⋊C4—541no—
317C42—91no—
318C42—181no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—61no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3—361no—
325C22×C3#381111yescharacteristic
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—361no—
330C2×D3—361no—
331C2×D3—361no—
332C3×C4—22no—
333C3×C4—31no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—62no—
338C3×C4—62no—
339C3×C4—64no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—181no—
343C3×C4—181no—
344C3×C4—361no—
345C3×C4—361no—
346C3×C4#380511yescharacteristic
347C3×C4C2×#62511yescharacteristic
348Dic3—61no—
349Dic3—61no—
350Dic3—61no—
351Dic3—61no—
352Dic3—62no—
353Dic3—121no—
354Dic3—121no—
355Dic3—121no—
356Dic3—122no—
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#516611yescharacteristic, complemented
361C32#518511yescharacteristic, complemented
362C23—181no—
363Q8—21no—
364Q8—181no—
365Q8—181no—
366Q8#619812yes—
367D4—181no—
368D4—181no—
369C2×C4—21no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—181no—
373C2×C4—181no—
374C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
375C8—181no—
376C8—181no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—721no—
387C2×C3#6395011yescharacteristic
388C2×C3#6395111yescharacteristic
389C2×C3#6398011yescharacteristic
390D3—241no—
391D3—721no—
392C22—361no—
393C22—361no—
394C22—361no—
395C22#8867711yescharacteristic
396C4—22no—
397C4—181no—
398C4—181no—
399C4—361no—
400C4—361no—
401C4#8867211yescharacteristic
402C4C2×#619811yescharacteristic
403C3—31no—
404C3—31no—
405C3—61no—
406C3#1059161211yescharacteristic
407C2—721no—
408C2#1172123011yescharacteristic
409C2#1172123111yescharacteristic
410C2#1172131811yescharacteristic
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⋊((C3⋊D3)⋊C3)C22
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
1. Q8×#1084C2semidirect
3. C2×#3756⋊C3C2semidirect
5. #11720309C2semidirect
7. #11720304C2semidirect
23. C2×#3756D3semidirect
242. H(3,𝔽3)#464semidirect
360. C32#5166semidirect
361. C32#5185semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
C2×#3755⋊C32
C2×#3756⋊C32
#117211132
#117203092
#117203082
#117203042
#105916073
#105916043