Finite Group Databasebeta

← #10592508⋊C3 #13141921

Subgroups — #10592508⋊C3 #13141921

Order 1728.

560 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 104 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×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic, complemented
3C2×#3746⋊C3C211yesmaximal, characteristic, complemented
4#64106⋊C3C211yesmaximal, characteristic
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592508C311yesmaximal, characteristic, complemented
9C3×#4809—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×(C3⋊D3)⋊C3—22no—
12Q8×(C3⋊D3)⋊C3—22no—
13C2×C4×(C3⋊D3)⋊C3—21no—
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#634⋊C3—21no—
16C2×#634⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#3756⋊C3—22no—
21#3752⋊C3—21no—
22C4⋊((C3⋊Dic3)⋊C3)—21no—
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
24#3746⋊C3—22no—
25C2×Q8×C3⋊D3C2×C311yescharacteristic
26C2×C3×D3×Q8—31no—
27C2×#3756C2×C311yescharacteristic, complemented
28C2×#3746C2×C311yescharacteristic, complemented
29#64106C2×C311yescharacteristic
30C2×C3×#216—31no—
31C2×C3×Dic12—31no—
32C3×#897—31no—
33C32×#103—21no—
34C32×#103D311yescharacteristic, complemented
35C32×#103C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic
37#63637C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—31no—
39C3×#810—31no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)D412yes—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4×(C3⋊D3)⋊C3—24no—
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3—21no—
52#634⋊C3—24no—
53#634⋊C3—21no—
54#634⋊C3—21no—
55#634⋊C3—21no—
56#634⋊C3D412yes—
57C8×H(3,𝔽3)—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59C3×#451—91no—
60#4809—31no—
61#4809—91no—
62C2×C32×Q8—21no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C22×C311yescharacteristic
65Q8×C3⋊D3—22no—
66Q8×C3⋊D3—22no—
67C2×C4×C3⋊D3—21no—
68C2×C4×C3⋊D3C22×C311yescharacteristic
69C2×#634—21no—
70C2×#634C22×C311yescharacteristic
71C2×#634C22×C311yescharacteristic
72C3×D3×Q8—62no—
73C3×D3×Q8—62no—
74C2×C3×C4×D3—31no—
75C2×C3×C4×D3—61no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C32×C8—21no—
80C2×C32×C8C2×D311yescharacteristic
81C2×C32×C8C22×C311yescharacteristic
82C32×C4⋊C4—21no—
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C22×C311yescharacteristic
85#3756—22no—
86#3752—21no—
87C4⋊(C3⋊Dic3)—21no—
88C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
89#3746—22no—
90C3×#216—62no—
91C3×(C2×D3)⋊C4—61no—
92C3×C4⋊Dic3—61no—
93C2×C3×C3⋊C8—31no—
94C3×Dic12—62no—
95C22×H(3,𝔽3)C2×D411yescharacteristic
96C2×(C3⋊D3)⋊C3—21no—
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3Q1612yescomplemented
99C4×H(3,𝔽3)—22no—
100C4×H(3,𝔽3)—41no—
101C4×H(3,𝔽3)C2×D411yescharacteristic
102C4×H(3,𝔽3)C2×D411yescharacteristic
103(C3⋊Dic3)⋊C3—21no—
104(C3⋊Dic3)⋊C3—22no—
105(C3⋊Dic3)⋊C3—41no—
106C22×C3×Q8—91no—
107C2×D3×Q8—31no—
108C2×D3×Q8—91no—
109C2×C3×Q16—91no—
110C2×C3×Q16—91no—
111C3×C22⋊Q8—91no—
112C2×#216—31no—
113C2×#216—91no—
114C2×Dic12—31no—
115C2×Dic12—91no—
116#897—31no—
117#897—91no—
118C3×#103—31no—
119C3×#103—31no—
120C3×#103—61no—
121C3×#103—91no—
122C3×#103C3×D311yescharacteristic
123C3×C22⋊C8—91no—
124(C2×D3)⋊C8—31no—
125(C2×D3)⋊C8—91no—
126#810—31no—
127#810—91no—
128C22×C3⋊D3C3×D411yescharacteristic
129C22×C3×D3—31no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—22no—
133C32×Q8—41no—
134C32×Q8D1212yes—
135C32×Q8C3×D412yes—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4—41no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C23×C311yescharacteristic
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3C3×D411yescharacteristic
148C4×C3⋊D3—24no—
149C4×C3⋊D3—21no—
150C4×C3⋊D3—21no—
151#634—24no—
152#634—21no—
153#634—21no—
154#634—21no—
155#634C3×D412yes—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C3×C4×D3—64no—
160C3×C4×D3—61no—
161C3×C4×D3—61no—
162C3×Dic6—32no—
163C3×Dic6—64no—
164C3×Dic6—61no—
165C3×Dic6—61no—
166C3×Dic6—61no—
167C32×C8—21no—
168C32×C8—21no—
169C32×C8—41no—
170C3⋊(C3⋊C8)—21no—
171C3×C3⋊C8—61no—
172#451—91noSylow
173C2×H(3,𝔽3)#12011yescharacteristic
174C2×H(3,𝔽3)C2×Q1611yescharacteristic
175C2×H(3,𝔽3)#13711yescharacteristic
176(C3⋊D3)⋊C3—22no—
177C2×C3×Q8—31no—
178C2×C3×Q8—31no—
179C2×C3×Q8—61no—
180C2×C3×Q8—91no—
181C2×C3×Q8—91no—
182C2×C3×Q8—181no—
183C2×C3×Q8—182no—
184C2×C3×Q8—182no—
185C2×C3×Q8C2×C3×D311yescharacteristic
186C22×C3×C4—91no—
187C22×C3×C4—181no—
188D3×Q8—62no—
189D3×Q8—62no—
190D3×Q8—182no—
191D3×Q8—182no—
192C2×C4×D3—31no—
193C2×C4×D3—61no—
194C2×C4×D3—91no—
195C2×C4×D3—181no—
196C2×Dic6—31no—
197C2×Dic6—31no—
198C2×Dic6—61no—
199C2×Dic6—91no—
200C2×Dic6—91no—
201C2×Dic6—181no—
202C3×Q16—182no—
203C3×Q16—182no—
204C2×C3×C8—31no—
205C2×C3×C8—31no—
206C2×C3×C8—61no—
207C2×C3×C8—91no—
208C2×C3×C8C2×C3×D311yescharacteristic
209C3×C4⋊C4—31no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—181no—
213C3×C4⋊C4C2×C3×D311yescharacteristic
214C3×C22⋊C4—181no—
215#216—62no—
216#216—182no—
217(C2×D3)⋊C4—61no—
218(C2×D3)⋊C4—181no—
219C4⋊Dic3—61no—
220C4⋊Dic3—181no—
221C2×C3⋊C8—31no—
222C2×C3⋊C8—91no—
223Dic12—62no—
224Dic12—182no—
225C22×C32—21no—
226C22×C32D3×D411yescharacteristic
227C22×C32C2×C3×D411yescharacteristic
228C2×C3⋊D3—21no—
229C2×C3⋊D3—21no—
230C2×C3⋊D3C3×Q1612yescomplemented
231C2×C3×D3—32no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—42no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4—81no—
242C32×C4C2×D1211yescharacteristic
243C32×C4D3×D411yescharacteristic
244C32×C4C2×C3×D411yescharacteristic
245C32×C4C2×C3×D411yescharacteristic
246C3⋊Dic3—21no—
247C3⋊Dic3—22no—
248C3⋊Dic3—41no—
249C3×Dic3—62no—
250C3×Dic3—61no—
251C3×Dic3—121no—
252C22×Q8—91no—
253C2×Q16—91no—
254C2×Q16—91no—
255C22⋊Q8—91no—
256#103—91no—
257#103(C3⋊D3)⋊C311yescharacteristic, complemented
258C22⋊C8—91no—
259H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
260C23×C3—91no—
261C22×D3—31no—
262C22×D3—91no—
263C3×Q8—21no—
264C3×Q8—32no—
265C3×Q8—32no—
266C3×Q8—61no—
267C3×Q8—61no—
268C3×Q8—62no—
269C3×Q8—92no—
270C3×Q8—121no—
271C3×Q8—181no—
272C3×Q8—181no—
273C3×Q8—181no—
274C3×Q8—184no—
275C3×Q8C3×D1212yes—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—91no—
286C2×C3×C4—121no—
287C2×C3×C4—121no—
288C2×C3×C4—181no—
289C2×C3×C4—181no—
290C2×C3×C4—184no—
291C2×C3×C4—181no—
292C2×C3×C4—181no—
293C2×C3×C4C22×C3×D311yescharacteristic
294C2×Dic3—31no—
295C2×Dic3—61no—
296C2×Dic3—61no—
297C2×Dic3—91no—
298C2×Dic3—181no—
299C2×Dic3—181no—
300C4×D3—64no—
301C4×D3—61no—
302C4×D3—61no—
303C4×D3—181no—
304C4×D3—181no—
305C4×D3—184no—
306Dic6—32no—
307Dic6—64no—
308Dic6—61no—
309Dic6—61no—
310Dic6—61no—
311Dic6—92no—
312Dic6—181no—
313Dic6—184no—
314Dic6—181no—
315Dic6—181no—
316C3×C8—21no—
317C3×C8—61no—
318C3×C8—61no—
319C3×C8—121no—
320C3×C8—181no—
321C3⋊C8—61no—
322C3⋊C8—181no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32#88211yescharacteristic
327C2×C32#91511yescharacteristic
328C2×C32D3×Q1611yescharacteristic
329C2×C32C3×#12011yescharacteristic
330C2×C32C2×C3×Q1611yescharacteristic
331C2×C32C3×#13711yescharacteristic
332C3⋊D3—22no—
333C3×D3—62no—
334C2×Q8—91no—
335C2×Q8—91no—
336C2×Q8—181no—
337C2×Q8—182no—
338C2×Q8—182no—
339C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
340C22×C4—91no—
341C22×C4—181no—
342Q16—182no—
343Q16—182no—
344C2×C8—91no—
345C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
346C4⋊C4—181no—
347C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
348C22⋊C4—181no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—61no—
352C22×C3—92no—
353C22×C3—181no—
354C22×C3—181no—
355C22×C3C3×D3×D411yescharacteristic
356C2×D3—32no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—92no—
360C2×D3—181no—
361C2×D3—181no—
362C3×C4—22no—
363C3×C4—31no—
364C3×C4—31no—
365C3×C4—31no—
366C3×C4—31no—
367C3×C4—41no—
368C3×C4—62no—
369C3×C4—62no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—122no—
373C3×C4—121no—
374C3×C4—121no—
375C3×C4—181no—
376C3×C4—182no—
377C3×C4—241no—
378C3×C4—361no—
379C3×C4C2×C3×D1211yescharacteristic
380C3×C4C3×D3×D411yescharacteristic
381Dic3—62no—
382Dic3—61no—
383Dic3—121no—
384Dic3—182no—
385Dic3—181no—
386Dic3—361no—
387C32—21no—
388C32#480911yescharacteristic, complemented
389C32C3×#45111yescharacteristic, complemented
390C23—91no—
391Q8—21no—
392Q8—92no—
393Q8—184no—
394Q8—181no—
395Q8—181no—
396Q8—181no—
397Q8C4⋊((C3⋊D3)⋊C3)12yes—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—91no—
401C2×C4—184no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4—181no—
405C2×C4—181no—
406C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
407C8—21no—
408C8—181no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—61no—
416C2×C3—61no—
417C2×C3—61no—
418C2×C3—182no—
419C2×C3C3×#88211yescharacteristic
420C2×C3C3×#91511yescharacteristic
421C2×C3C3×D3×Q1611yescharacteristic
422D3—62no—
423D3—182no—
424C22—92no—
425C22—181no—
426C22—181no—
427C22D4×(C3⋊D3)⋊C311yescharacteristic
428C4—22no—
429C4—41no—
430C4—181no—
431C4—182no—
432C4—361no—
433C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
434C4D4×(C3⋊D3)⋊C311yescharacteristic
435C3—31no—
436C3—31no—
437C3—61no—
438C3C3×#480911yescharacteristic
439C2—182no—
440C2#64091⋊C311yescharacteristic
441C2#64124⋊C311yescharacteristic
442C2Q16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×#3756⋊C3C2semidirect
3. C2×#3746⋊C3C2semidirect
5. H(3,𝔽3)×#103C2semidirect
7. #63637⋊C3C2semidirect
8. #10592508C3semidirect
27. C2×#3756C2×C3semidirect
28. C2×#3746C2×C3semidirect
34. C32×#103D3semidirect
35. C32×#103C2×C3semidirect
37. #63637C2×C3semidirect
98. C2×(C3⋊D3)⋊C3Q16semidirect
230. C2×C3⋊D3C3×Q16semidirect
257. #103(C3⋊D3)⋊C3semidirect
259. H(3,𝔽3)#451semidirect
388. C32#4809semidirect
389. C32C3×#451semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
C2×#3756⋊C32
C2×#3746⋊C32
#64106⋊C32
H(3,𝔽3)×#1032
#63647⋊C32
#63637⋊C32
#105925083
C3×#48093