Finite Group Databasebeta

← #13139201

Subgroups — #13139201

Order 1728.

549 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 89 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
2M16×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3#11721053C211yesmaximal, characteristic
4#11720561C211yesmaximal, characteristic, complemented
5#63651⋊C3C211yesmaximal, characteristic, complemented
6#11720327C211yesmaximal, characteristic, complemented
7#11720321C211yesmaximal, characteristic, complemented
8#10591003—31nomaximal
9D3×#824—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×((C3⋊D3)⋊C3)⋊C4C412yes—
14C2×#6196C412yes—
15((C3⋊D3)⋊C3)⋊Q8—21no—
16((C3⋊D3)⋊C3)⋊Q8—21no—
17((C3⋊D3)⋊C3)⋊Q8—21no—
18((C3⋊D3)⋊C3)⋊Q8—21no—
19M16×H(3,𝔽3)C2211yescharacteristic
20#88538C2211yescharacteristic
21#3748⋊C3C2211yescharacteristic
22#3748⋊C3C2211yescharacteristic
23#3743⋊C3—21no—
24C8×(C3⋊D3)⋊C3—21no—
25#88430—21no—
26((C3⋊D3)⋊C3)⋊C8—21no—
27C2×#3797—31no—
28C2×D3×Dic6—31no—
29M16×C3⋊D3D311yescharacteristic
30C3×D3×M16—31no—
31#63818—31no—
32D3×#208—31no—
33#63651D311yescharacteristic, complemented
34C3×#824—31no—
35C3×#824—31no—
36C3×#824—31no—
37C3×#824—61no—
38#63561—31no—
39#63560—31no—
40#63560—31no—
41#63560—31no—
42C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084C2×C412yes—
45#6229—21no—
46#6229—21no—
47C2×(C3⋊Dic3)⋊C3C2×C412yes—
48C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
49C4×(C3⋊D3)⋊C3D411yescharacteristic
50C4×(C3⋊D3)⋊C3D411yescharacteristic
51C4×(C3⋊D3)⋊C3D411yescharacteristic
52C4×(C3⋊D3)⋊C3D411yescharacteristic
53#634⋊C3—21no—
54#634⋊C3—21no—
55((C3⋊D3)⋊C3)⋊C4—22no—
56((C3⋊D3)⋊C3)⋊C4—22no—
57#6196—22no—
58#6196—22no—
59C8×H(3,𝔽3)—21no—
60#6180—21no—
61(C3⋊(C3⋊C8))⋊C3—21no—
62(C3⋊(C3⋊C8))⋊C3—21no—
63C2×#824—91no—
64D3×#101—91no—
65D3×#101—91no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×#634C2×D311yescharacteristic
68C2×C3×C4×D3—31no—
69C2×C3×Dic6—31no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—61no—
73C2×#627—32no—
74C2×#627—32no—
75C2×#624—32no—
76C2×D3×Dic3—32no—
77#3797—61no—
78#3797—61no—
79#3797—61no—
80#3797—61no—
81D3×Dic6—61no—
82D3×Dic6—61no—
83D3×Dic6—61no—
84D3×Dic6—61no—
85C32×M16—21no—
86C32×M16C2×D311yescharacteristic
87C32×M16C2×D311yescharacteristic
88#3748C2×D311yescharacteristic
89#3748C2×D311yescharacteristic
90#3743—21no—
91C8×C3⋊D3—21no—
92C3×#208—31no—
93C3×#208—31no—
94C3×#208—31no—
95C3×#208—31no—
96C3×#208—61no—
97C3×ΓL(1,𝔽25)—61no—
98C3×D3×C8—61no—
99#3712—61no—
100#3711—61no—
101#3710—61no—
102D3×C3⋊C8—61no—
103C22×H(3,𝔽3)C22×C411yescharacteristic
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
106C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
107C4×H(3,𝔽3)C2×D411yescharacteristic
108C4×H(3,𝔽3)C2×D411yescharacteristic
109#1084—22no—
110(C3⋊Dic3)⋊C3—22no—
111(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
112(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
113C2×D3×Q8—91no—
114C2×D3×Q8—91no—
115C22×Dic6—91no—
116C2×C3×M16—91no—
117C2×#208—91no—
118D3×M16—31no—
119D3×M16—91no—
120D3×M16—91no—
121D3×M16—91no—
122C3×#101—91no—
123C3×#101—91no—
124#836—91no—
125#836—91no—
126#824—31no—
127#824—31no—
128#824—31no—
129#824—61no—
130#824—91no—
131#824—91no—
132#824—91no—
133#824—91no—
134#824—91no—
135C22×C3⋊D3C4×D311yescharacteristic
136C22×C3×D3—31no—
137C2×C32×C4—21no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3C4×D312yes—
141C2×C3⋊Dic3C4×D311yescharacteristic
142C4×C3⋊D3D1211yescharacteristic
143C4×C3⋊D3D1211yescharacteristic
144C4×C3⋊D3#6711yescharacteristic
145C4×C3⋊D3#6711yescharacteristic
146#634—21no—
147#634—21no—
148C2×C3×Dic3—31no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—62no—
153C3×C4×D3—31no—
154C3×C4×D3—31no—
155C3×C4×D3—31no—
156C3×C4×D3—31no—
157C3×Dic6—61no—
158C3×Dic6—61no—
159C3×Dic6—61no—
160C3×Dic6—61no—
161C3×Dic6—61no—
162C3×Dic6—61no—
163C3×Dic6—121no—
164C3×Dic6—121no—
165#627—62no—
166#627—62no—
167#627—62no—
168#627—62no—
169#624—62no—
170#624—62no—
171D3×Dic3—62no—
172D3×Dic3—62no—
173C32×C8—21no—
174C32×C8—21no—
175C32×C8—41no—
176C3⋊(C3⋊C8)—21no—
177C3⋊(C3⋊C8)—21no—
178C3×C3⋊C8—61no—
179C3×C3⋊C8—61no—
180C3×C3⋊C8—61no—
181C3×C3⋊C8—61no—
182C3×C3⋊C8—121no—
183C2×#101—271noSylow
184C2×H(3,𝔽3)—21no—
185C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
186(C3⋊D3)⋊C3—21no—
187(C3⋊D3)⋊C3#10112yescomplemented
188C2×C3×Q8—91no—
189C2×C3×Q8—91no—
190C22×C3×C4—91no—
191C22×Dic3—92no—
192D3×Q8—181no—
193D3×Q8—181no—
194D3×Q8—181no—
195D3×Q8—181no—
196D3×Q8—181no—
197D3×Q8—181no—
198D3×Q8—181no—
199D3×Q8—181no—
200C2×C4×D3—31no—
201C2×C4×D3—91no—
202C2×C4×D3—92no—
203C2×C4×D3—92no—
204C2×Dic6—31no—
205C2×Dic6—31no—
206C2×Dic6—31no—
207C2×Dic6—61no—
208C2×Dic6—91no—
209C2×Dic6—92no—
210C2×Dic6—92no—
211C2×Dic6—92no—
212C2×Dic6—181no—
213C2×Dic6—181no—
214C2×Dic6—181no—
215C2×Dic6—181no—
216C3×M16—31no—
217C3×M16—31no—
218C3×M16—61no—
219C3×M16—91no—
220C3×M16—91no—
221C3×M16—91no—
222C3×M16—181no—
223C3×M16D3211yescharacteristic
224C2×C3×C8—181no—
225#208—31no—
226#208—31no—
227#208—31no—
228#208—31no—
229#208—61no—
230#208—91no—
231#208—91no—
232#208—181no—
233C2×C3⋊C8—181no—
234ΓL(1,𝔽25)—61no—
235ΓL(1,𝔽25)—181no—
236ΓL(1,𝔽25)—181no—
237ΓL(1,𝔽25)—181no—
238D3×C8—61no—
239D3×C8—181no—
240D3×C8—181no—
241D3×C8—181no—
242C22×C32—21no—
243C22×C32C2×C4×D311yescharacteristic
244C22×C32C2×C4×D311yescharacteristic
245C2×C3⋊D3—22no—
246C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
247C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
248C2×C3×D3—31no—
249C2×C3×D3—31no—
250C2×C3×D3—62no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4C2×D1211yescharacteristic
254C32×C4D3×D411yescharacteristic
255C32×C4D3×D411yescharacteristic
256C32×C4C2×#6711yescharacteristic
257C3⋊Dic3—22no—
258C3⋊Dic3(C2×D3)⋊C411yescharacteristic
259C3⋊Dic3(C2×D3)⋊C411yescharacteristic
260C3×Dic3—31no—
261C3×Dic3—31no—
262C3×Dic3—62no—
263C3×Dic3—62no—
264C3×Dic3—62no—
265C3×Dic3—122no—
266C22×Q8—271no—
267C2×M16—91no—
268C2×M16—271no—
269#101—91no—
270#101—91no—
271#101—271no—
272#101—271no—
273H(3,𝔽3)C2×#10111yescharacteristic, complemented, Sylow
274C23×C3—91no—
275C22×D3—31no—
276C22×D3—91no—
277C3×Q8—181no—
278C3×Q8—181no—
279C3×Q8—181no—
280C3×Q8—181no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—61no—
284C2×C3×C4—91no—
285C2×C3×C4—91no—
286C2×C3×C4—91no—
287C2×C3×C4—91no—
288C2×C3×C4—91no—
289C2×C3×C4—92no—
290C2×C3×C4—92no—
291C2×C3×C4C2×D3211yescharacteristic
292C2×Dic3—31no—
293C2×Dic3—32no—
294C2×Dic3—32no—
295C2×Dic3—32no—
296C2×Dic3—62no—
297C2×Dic3—91no—
298C2×Dic3—92no—
299C2×Dic3—182no—
300C2×Dic3—182no—
301C4×D3—31no—
302C4×D3—31no—
303C4×D3—31no—
304C4×D3—31no—
305C4×D3—91no—
306C4×D3—91no—
307C4×D3—91no—
308C4×D3—91no—
309C4×D3—182no—
310C4×D3—182no—
311C4×D3—182no—
312C4×D3—182no—
313Dic6—61no—
314Dic6—61no—
315Dic6—61no—
316Dic6—61no—
317Dic6—61no—
318Dic6—61no—
319Dic6—121no—
320Dic6—121no—
321Dic6—181no—
322Dic6—181no—
323Dic6—182no—
324Dic6—182no—
325Dic6—182no—
326Dic6—182no—
327Dic6—182no—
328Dic6—182no—
329C3×C8—21no—
330C3×C8—61no—
331C3×C8—61no—
332C3×C8—121no—
333C3×C8—181no—
334C3×C8—181no—
335C3×C8—181no—
336C3⋊C8—61no—
337C3⋊C8—61no—
338C3⋊C8—61no—
339C3⋊C8—61no—
340C3⋊C8—121no—
341C3⋊C8—181no—
342C3⋊C8—181no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—41no—
347C2×C32D3×C22⋊C411yescharacteristic
348C2×C32C2×(C2×D3)⋊C411yescharacteristic
349C3⋊D3—21no—
350C3⋊D3#82412yescomplemented
351C3×D3—32no—
352C3×D3—61no—
353C2×Q8—91no—
354C2×Q8—91no—
355C2×Q8—272no—
356C2×Q8—541no—
357C2×Q8—541no—
358C2×Q8—541no—
359C2×Q8—541no—
360C22×C4—91no—
361C22×C4—272no—
362M16—91no—
363M16—91no—
364M16—91no—
365M16—181no—
366M16—541no—
367M16((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C2×C8—181no—
369C2×C8—541no—
370C22×C3—31no—
371C22×C3—31no—
372C22×C3—61no—
373C22×C3—91no—
374C22×C3—91no—
375C22×C3—182no—
376C22×C3C4×D3211yescharacteristic
377C2×D3—31no—
378C2×D3—31no—
379C2×D3—62no—
380C2×D3—91no—
381C2×D3—91no—
382C2×D3—182no—
383C3×C4—31no—
384C3×C4—31no—
385C3×C4—31no—
386C3×C4—31no—
387C3×C4—61no—
388C3×C4—61no—
389C3×C4—91no—
390C3×C4—91no—
391C3×C4—182no—
392C3×C4—182no—
393C3×C4D3×D1211yescharacteristic
394C3×C4D3×#6711yescharacteristic
395Dic3—31no—
396Dic3—31no—
397Dic3—62no—
398Dic3—62no—
399Dic3—62no—
400Dic3—91no—
401Dic3—91no—
402Dic3—122no—
403Dic3—182no—
404C32—21no—
405C32D3×#10111yescharacteristic, complemented
406C32C2×#82411yescharacteristic, complemented
407C23—91no—
408Q8—181no—
409Q8—181no—
410Q8—181no—
411Q8—181no—
412Q8—542no—
413Q8—542no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—91no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—92no—
420C2×C4—92no—
421C2×C4—542no—
422C2×C4—542no—
423C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
424C8—21no—
425C8—181no—
426C8—181no—
427C8—181no—
428C2×C3—21no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—61no—
432C2×C3—61no—
433C2×C3—61no—
434C2×C3—92no—
435C2×C3—121no—
436C2×C3—181no—
437C2×C3D3×(C2×D3)⋊C411yescharacteristic
438D3—32no—
439D3—61no—
440D3—92no—
441D3—181no—
442C22—91no—
443C22—91no—
444C22—182no—
445C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
446C4—91no—
447C4—91no—
448C4—182no—
449C4—182no—
450C4#8865411yescharacteristic
451C4#8867611yescharacteristic
452C3—31no—
453C3—31no—
454C3—61no—
455C3D3×#82411yescharacteristic
456C2—21no—
457C2—92no—
458C2—181no—
459C2#1172116011yescharacteristic
subgroupG/H
centerC2#11721160
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingM16×H(3,𝔽3)C22
socleC2×C3D3×(C2×D3)⋊C4
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterM16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#101
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
4. #11720561C2semidirect
5. #63651⋊C3C2semidirect
6. #11720327C2semidirect
7. #11720321C2semidirect
33. #63651D3semidirect
187. (C3⋊D3)⋊C3#101semidirect
273. H(3,𝔽3)C2×#101semidirect
350. C3⋊D3#824semidirect
405. C32D3×#101semidirect
406. C32C2×#824semidirect

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
M16×(C3⋊D3)⋊C32
#117210532
#117205612
#63651⋊C32
#117203272
#117203212
#105910033
D3×#8243