Finite Group Databasebeta

← #13138062

Subgroups — #13138062

Order 1728.

567 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 567 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×((C3⋊D3)⋊C3)⋊Q8C211yesmaximal, characteristic
2#64101⋊C3C211yesmaximal, characteristic
3C2×#88430C211yesmaximal, characteristic
4#11720532C211yesmaximal, characteristic, complemented
5#63637⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10590602—31nomaximal
9#10590600—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×((C3⋊D3)⋊C3)⋊C4—21no—
14C2×#6196—21no—
15((C3⋊D3)⋊C3)⋊Q8—22no—
16((C3⋊D3)⋊C3)⋊Q8C414yescomplemented
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19#3752⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C4×(C3⋊Dic3)⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88430—22no—
24C2×#3797—31no—
25C2×D3×Dic6—31no—
26#64101D311yescharacteristic
27C3×#892—31no—
28C2×#3712—31no—
29C2×#3711—31no—
30#63637D311yescharacteristic, complemented
31C3×#810—31no—
32C3×#810—31no—
33C3×#810—31no—
34C3×#810—61no—
35#63567—31no—
36#63566—31no—
37#63566—31no—
38#63565—31no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43#6229—21no—
44#6229C2×C412yes—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4×(C3⋊D3)⋊C3C2×C412yes—
49C4×(C3⋊D3)⋊C3D412yes—
50#634⋊C3—21no—
51#634⋊C3C2×C412yes—
52((C3⋊D3)⋊C3)⋊C4—24no—
53#6196—24no—
54#6180—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5004—91no—
57#4802—91no—
58#4802—91no—
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×C4×D3—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—61no—
66C2×#627—61no—
67C2×#627—61no—
68C2×#624—61no—
69C2×D3×Dic3—61no—
70#3797—34no—
71#3797—62no—
72D3×Dic6—34no—
73D3×Dic6—62no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4C2×D311yescharacteristic
76C32×C4⋊C4C2×D311yescharacteristic
77#3752—21no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C4×C3⋊Dic3—21no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×(C2×D3)⋊C4—61no—
82C3×C4⋊Dic3—31no—
83C3×C4×Dic3—61no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—61no—
88#3712—62no—
89#3711—62no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3C22⋊C412yes—
94C4×H(3,𝔽3)—41no—
95C4×H(3,𝔽3)C22×C411yescharacteristic
96C4×H(3,𝔽3)C2×D411yescharacteristic
97#1084—22no—
98(C3⋊Dic3)⋊C3—22no—
99(C3⋊Dic3)⋊C3—41no—
100(C3⋊Dic3)⋊C3C22⋊C412yes—
101C2×D3×Q8—91no—
102C2×D3×Q8—91no—
103C22×Dic6—91no—
104C3×#117—91no—
105C2×#208—91no—
106C2×ΓL(1,𝔽25)—91no—
107C2×ΓL(1,𝔽25)—91no—
108#892—31no—
109#892—91no—
110C3×#103—91no—
111C3×#103—91no—
112#835—91no—
113#835—91no—
114#816—91no—
115#816—91no—
116#810—31no—
117#810—31no—
118#810—31no—
119#810—61no—
120#810—91no—
121#810—91no—
122#810—91no—
123C22×C3⋊D3#6711yescharacteristic
124C22×C3×D3—31no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—41no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3#6711yescharacteristic
134C4×C3⋊D3C4×D312yes—
135C4×C3⋊D3D1212yes—
136#634—21no—
137#634C4×D312yes—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—121no—
144C3×C4×D3—32no—
145C3×C4×D3—32no—
146C3×Dic6—32no—
147C3×Dic6—32no—
148C3×Dic6—32no—
149C3×Dic6—62no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—121no—
154#627—64no—
155#627—64no—
156#624—64no—
157D3×Dic3—64no—
158C3⋊(C3⋊C8)—21no—
159C3×C3⋊C8—61no—
160C3×C3⋊C8—61no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—121no—
163#419—271noSylow
164C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
165C2×H(3,𝔽3)#13711yescharacteristic
166C2×H(3,𝔽3)#13711yescharacteristic
167(C3⋊D3)⋊C3—22no—
168C2×C3×Q8—91no—
169C2×C3×Q8—91no—
170C22×C3×C4—91no—
171C22×Dic3—181no—
172D3×Q8—94no—
173D3×Q8—94no—
174D3×Q8—182no—
175D3×Q8—182no—
176C2×C4×D3—31no—
177C2×C4×D3—91no—
178C2×C4×D3—181no—
179C2×C4×D3—181no—
180C2×Dic6—31no—
181C2×Dic6—31no—
182C2×Dic6—31no—
183C2×Dic6—61no—
184C2×Dic6—91no—
185C2×Dic6—94no—
186C2×Dic6—182no—
187C2×Dic6—181no—
188C2×Dic6—181no—
189C2×Dic6—181no—
190C2×C3×C8—91no—
191C2×C3×C8—91no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—31no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—91no—
196C3×C4⋊C4D3211yescharacteristic
197C3×C22⋊C4—181no—
198C3×C42—181no—
199(C2×D3)⋊C4—61no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—31no—
202C4⋊Dic3—91no—
203C4×Dic3—61no—
204C4×Dic3—181no—
205#208—182no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—61no—
210C2×C3⋊C8—91no—
211ΓL(1,𝔽25)—182no—
212ΓL(1,𝔽25)—182no—
213C22×C32—21no—
214C22×C32D3×D411yescharacteristic
215C22×C32C2×#6711yescharacteristic
216C2×C3⋊D3—21no—
217C2×C3⋊D3—21no—
218C2×C3⋊D3(C2×D3)⋊C412yes—
219C2×C3×D3—32no—
220C2×C3×D3—61no—
221C2×C3×D3—61no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—81no—
227C32×C4C2×C4×D311yescharacteristic
228C32×C4C2×C4×D311yescharacteristic
229C32×C4C2×D1211yescharacteristic
230C32×C4D3×D411yescharacteristic
231C3⋊Dic3—22no—
232C3⋊Dic3—41no—
233C3⋊Dic3(C2×D3)⋊C412yes—
234C3×Dic3—32no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C3×Dic3—122no—
239C3×Dic3—121no—
240C22×Q8—271no—
241C2×M16—271no—
242#117—91no—
243#103—91no—
244#103—91no—
245#103—271no—
246#103—271no—
247H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
248C23×C3—91no—
249C22×D3—31no—
250C22×D3—91no—
251C3×Q8—92no—
252C3×Q8—92no—
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—61no—
261C2×C3×C4—92no—
262C2×C3×C4—92no—
263C2×C3×C4—91no—
264C2×C3×C4—121no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4C2×D3211yescharacteristic
269C2×Dic3—31no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—91no—
275C2×Dic3—121no—
276C2×Dic3—181no—
277C2×Dic3—181no—
278C2×Dic3—184no—
279C4×D3—32no—
280C4×D3—32no—
281C4×D3—92no—
282C4×D3—92no—
283C4×D3—184no—
284C4×D3—184no—
285Dic6—32no—
286Dic6—32no—
287Dic6—32no—
288Dic6—62no—
289Dic6—61no—
290Dic6—61no—
291Dic6—61no—
292Dic6—92no—
293Dic6—121no—
294Dic6—181no—
295Dic6—184no—
296Dic6—184no—
297Dic6—184no—
298C3×C8—181no—
299C3×C8—181no—
300C3⋊C8—61no—
301C3⋊C8—61no—
302C3⋊C8—61no—
303C3⋊C8—121no—
304C3⋊C8—181no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32D3×C22⋊C411yescharacteristic
309C2×C32#91511yescharacteristic
310C2×C32#91811yescharacteristic
311C2×C32C2×(C2×D3)⋊C411yescharacteristic
312C2×C32#95111yescharacteristic
313C2×C32#95111yescharacteristic
314C3⋊D3—22no—
315C3×D3—62no—
316C2×Q8—91no—
317C2×Q8—91no—
318C2×Q8—274no—
319C2×Q8—541no—
320C2×Q8—542no—
321C22×C4—91no—
322C22×C4—541no—
323M16—542no—
324C2×C8—91no—
325C2×C8—91no—
326C4⋊C4—91no—
327C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
328C22⋊C4—181no—
329C42—181no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—61no—
333C22×C3—92no—
334C22×C3—181no—
335C22×C3—181no—
336C22×C3D3×#6711yescharacteristic
337C2×D3—32no—
338C2×D3—61no—
339C2×D3—61no—
340C2×D3—92no—
341C2×D3—181no—
342C2×D3—181no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—41no—
348C3×C4—61no—
349C3×C4—61no—
350C3×C4—92no—
351C3×C4—121no—
352C3×C4—121no—
353C3×C4—182no—
354C3×C4—182no—
355C3×C4—241no—
356C3×C4—361no—
357C3×C4C4×D3211yescharacteristic
358C3×C4D3×D1211yescharacteristic
359Dic3—32no—
360Dic3—62no—
361Dic3—62no—
362Dic3—62no—
363Dic3—92no—
364Dic3—122no—
365Dic3—121no—
366Dic3—182no—
367Dic3—361no—
368C32—21no—
369C32#480211yescharacteristic, complemented
370C32#500411yescharacteristic, complemented
371C23—91no—
372Q8—92no—
373Q8—92no—
374Q8—181no—
375Q8—181no—
376Q8—544no—
377C2×C4—21no—
378C2×C4—91no—
379C2×C4—92no—
380C2×C4—92no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4—181no—
384C2×C4—544no—
385C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
386C8—181no—
387C8—181no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—61no—
397C2×C3—182no—
398C2×C3D3×(C2×D3)⋊C411yescharacteristic
399C2×C3#6392911yescharacteristic
400C2×C3#6394311yescharacteristic
401D3—62no—
402D3—182no—
403C22—92no—
404C22—181no—
405C22—181no—
406C22#8867611yescharacteristic
407C4—41no—
408C4—92no—
409C4—182no—
410C4—182no—
411C4—361no—
412C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C4#8865411yescharacteristic
414C3—31no—
415C3—31no—
416C3—61no—
417C3#1059060011yescharacteristic
418C2—182no—
419C2#1172116011yescharacteristic
420C2#1172119911yescharacteristic
421C2#1172123711yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×(C3⋊D3)⋊C3D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#419
LCSG ▷ 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
4. #11720532C2semidirect
5. #63637⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720313C2semidirect
16. ((C3⋊D3)⋊C3)⋊Q8C4semidirect
30. #63637D3semidirect
247. H(3,𝔽3)#419semidirect
369. C32#4802semidirect
370. C32#5004semidirect

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊Q82
#64101⋊C32
C2×#884302
#117205322
#63637⋊C32
#117203262
#117203132
#105906023
#105906003