Finite Group Databasebeta

← #13155391

Subgroups — #13155391

Order 1728.

934 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 128 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×#89014C211yesmaximal, characteristic
2C2×#88809C211yesmaximal, characteristic, complemented
3C3×#63900C211yesmaximal, characteristic, complemented
4#11722041C212yesmaximal, complemented
5C3×#63562C212yesmaximal, complemented
6#10591418—31nomaximal
7#10590607—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×D12C2211yescharacteristic
11C2×#6290—21no—
12C2×D3×C3⋊Dic3—21no—
13#89014—22no—
14#89014C414yescomplemented
15C2×C3×#624—21no—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×C4⋊Dic3C2212yes—
18#88809—22no—
19C3×#3724—21no—
20C2×#3830—31no—
21C2×#3795—32no—
22C2×#3795—61no—
23C3×D3×C4⋊C4—32no—
24#63900D311yescharacteristic, complemented
25C2×#3712—31no—
26C2×#3711—32no—
27#63637—32no—
28C3×#832—32no—
29C3×#809—32no—
30#63567—32no—
31#63567—32no—
32#63567—62no—
33#63562D312yescomplemented
34C22×C3×C3⋊D3D411yescharacteristic
35C22×C32×D3—21no—
36C2×C33×C4C2311yescharacteristic
37C2×C3⋊(C3⋊Dic3)—21no—
38#6308—21no—
39#6308C2×C412yescomplemented
40C2×C3×C3⋊Dic3D411yescharacteristic
41C3×C4×C3⋊D3C2×C412yes—
42C3×C4×C3⋊D3D412yes—
43C2×C32×Dic3—22no—
44C32×D12—21no—
45C32×D12C2×C412yescomplemented
46#6290—24no—
47D3×C3⋊Dic3—24no—
48C3×#624—42no—
49C3×C3⋊(C3⋊C8)—22no—
50#4966—91no—
51#4771—92no—
52C2×C32×D4—31no—
53C2×#638—61no—
54C22×C3⋊Dic3—61no—
55#3830—34no—
56#3830—62no—
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×#634—32no—
59C2×#634—31no—
60C2×#634—61no—
61C2×#634—62no—
62C2×#634—121no—
63C2×C3×D12—21no—
64C2×C3×D12C2×D312yes—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—61no—
67C2×C3×C4×D3—62no—
68C2×#625—62no—
69C2×#625—62no—
70C2×#624—21no—
71C2×D3×Dic3—62no—
72C2×D3×Dic3—62no—
73#3795—38no—
74#3795—64no—
75#3795—64no—
76#3795—64no—
77C32×C4⋊C4—32no—
78C2×C3⋊(C3⋊C8)C2×D312yes—
79C3×C4⋊Dic3—22no—
80C3×C4⋊Dic3C2×D312yes—
81C3×C4⋊Dic3C2×D312yes—
82C3×Dic3⋊C4—62no—
83C2×C3×C3⋊C8—32no—
84C2×C3×C3⋊C8—32no—
85C2×C3×C3⋊C8—62no—
86#3724—21no—
87#3712—62no—
88#3711—64no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3C22⋊C412yes—
93C2×C32×D3—22no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C33×C4C22×C411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C3⋊(C3⋊Dic3)—22no—
99C3×C3⋊Dic3C22⋊C412yes—
100C32×Dic3—42no—
101C2×#237—92no—
102C2×#237—181no—
103C2×#235—91no—
104C2×C3×C4⋊C4—91no—
105C2×#208—91no—
106C2×ΓL(1,𝔽25)—92no—
107D3×C4⋊C4—32no—
108C3×#102—92no—
109#832—32no—
110#816—92no—
111#810—92no—
112#810—92no—
113#810—182no—
114#809—32no—
115C23×C32—61no—
116C22×C3⋊D3#6711yescharacteristic
117C22×C3×D3—22no—
118C22×C3×D3—22no—
119C22×C3×D3—32no—
120C22×C3×D3—61no—
121C32×D4—32no—
122C32×D4—61no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4—41no—
126C2×C32×C4—62no—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D312yes—
129#638—64no—
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—62no—
133C2×C3⋊Dic3—62no—
134C2×C3⋊Dic3—122no—
135C2×C3⋊Dic3—122no—
136C2×C3⋊Dic3#6711yescharacteristic
137C4×C3⋊D3C4×D312yes—
138C4×C3⋊D3D1212yes—
139#634—34no—
140#634—32no—
141#634—62no—
142#634—64no—
143#634—61no—
144#634—62no—
145#634—62no—
146#634—122no—
147#634—122no—
148#634—122no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—42no—
153C2×C3×Dic3—61no—
154C3×D12—22no—
155C3×D12—22no—
156C3×D12—22no—
157C3×D12C4×D314yescomplemented
158C3×C4×D3—34no—
159C3×C4×D3—34no—
160C3×C4×D3—62no—
161C3×C4×D3—62no—
162C3×C4×D3—124no—
163#625—68no—
164#625—68no—
165#624—42no—
166D3×Dic3—68no—
167D3×Dic3—68no—
168C3⋊(C3⋊C8)—22no—
169C3×C3⋊C8—62no—
170C3×C3⋊C8—62no—
171C3×C3⋊C8—122no—
172#417—271noSylow
173C2×C33C2×C22⋊C411yescharacteristic
174C2×C33AGL(1,ℤ/8)11yescharacteristic
175C2×C33#13711yescharacteristic
176C3×C3⋊D3—22no—
177C32×D3—42no—
178C2×C3×D4—32no—
179C2×C3×D4—61no—
180C22×C3×C4—91no—
181C22×C3×C4—181no—
182C2×#67—181no—
183C2×#67—182no—
184C2×#67—182no—
185C22×Dic3—182no—
186C22×Dic3—182no—
187#237—98no—
188#237—184no—
189#237—184no—
190#237—184no—
191#235—94no—
192#235—182no—
193C2×D12D3211yescharacteristic
194C2×C4×D3—32no—
195C2×C4×D3—61no—
196C2×C4×D3—62no—
197C2×C4×D3—181no—
198C2×Dic6—91no—
199C2×Dic6—92no—
200C2×Dic6—181no—
201C2×Dic6—182no—
202C2×Dic6—361no—
203C2×C3×C8—92no—
204C3×C4⋊C4—32no—
205C3×C4⋊C4—32no—
206C3×C4⋊C4—62no—
207C3×C4⋊C4—181no—
208C4⋊Dic3D3212yes—
209Dic3⋊C4—62no—
210#208—182no—
211C2×C3⋊C8—32no—
212C2×C3⋊C8—32no—
213C2×C3⋊C8—62no—
214ΓL(1,𝔽25)—184no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32—62no—
219C22×C32—121no—
220C22×C32—121no—
221C22×C32D3×D412yes—
222C22×C32C2×#6711yescharacteristic
223C2×C3⋊D3—21no—
224C2×C3⋊D3—21no—
225C2×C3⋊D3(C2×D3)⋊C412yes—
226C2×C3×D3—24no—
227C2×C3×D3—24no—
228C2×C3×D3—34no—
229C2×C3×D3—42no—
230C2×C3×D3—42no—
231C2×C3×D3—42no—
232C2×C3×D3—42no—
233C2×C3×D3—62no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—41no—
243C32×C4—41no—
244C32×C4—122no—
245C32×C4C2×C4×D311yescharacteristic
246C32×C4C2×C4×D312yes—
247C32×C4C2×D1211yescharacteristic
248C32×C4D3×D412yes—
249C3⋊Dic3—62no—
250C3⋊Dic3—64no—
251C3⋊Dic3—64no—
252C3⋊Dic3—124no—
253C3⋊Dic3—124no—
254C3⋊Dic3(C2×D3)⋊C412yes—
255C3×Dic3—34no—
256C3×Dic3—42no—
257C3×Dic3—42no—
258C3×Dic3—62no—
259C3×Dic3—82no—
260C2×#41—271no—
261C2×M16—271no—
262C2×C4⋊C4—91no—
263#103—272no—
264#102—92no—
265C33#41711yescharacteristic, complemented, Sylow
266C23×C3—62no—
267C23×C3—62no—
268C23×C3—91no—
269C22×D3—21no—
270C22×D3—32no—
271C22×D3—61no—
272C3×D4—34no—
273C3×D4—62no—
274C3×D4—62no—
275C3×D4—62no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—41no—
279C2×C3×C4—62no—
280C2×C3×C4—62no—
281C2×C3×C4—91no—
282C2×C3×C4—92no—
283C2×C3×C4—92no—
284C2×C3×C4—122no—
285C2×C3×C4—362no—
286C2×C3×C4C2×D3212yes—
287C2×C3×C4C2×D3211yescharacteristic
288#67—184no—
289#67—188no—
290#67—188no—
291C2×Dic3—22no—
292C2×Dic3—32no—
293C2×Dic3—61no—
294C2×Dic3—188no—
295C2×Dic3—188no—
296C2×Dic3—182no—
297C2×Dic3—182no—
298C2×Dic3—181no—
299C2×Dic3—362no—
300C2×Dic3—362no—
301D12—21no—
302D12#62412yescomplemented
303C4×D3—34no—
304C4×D3—34no—
305C4×D3—62no—
306C4×D3—62no—
307C4×D3—124no—
308C4×D3—184no—
309Dic6—94no—
310Dic6—92no—
311Dic6—182no—
312Dic6—184no—
313Dic6—182no—
314Dic6—182no—
315Dic6—181no—
316Dic6—362no—
317Dic6—362no—
318Dic6—362no—
319C3×C8—182no—
320C3⋊C8—62no—
321C3⋊C8—62no—
322C3⋊C8—122no—
323C2×C32—22no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—22no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32—122no—
333C2×C32D3×C22⋊C412yes—
334C2×C32#91112yes—
335C2×C32#91512yes—
336C2×C32C2×(C2×D3)⋊C411yescharacteristic
337C2×C32#93211yescharacteristic
338C2×C32#94211yescharacteristic
339C3⋊D3—22no—
340C3×D3—44no—
341C3×D3—44no—
342C3×D3—64no—
343C3×D3—122no—
344#41—274no—
345#41—542no—
346C2×Q8—271no—
347C2×D4—31no—
348C2×D4—541no—
349C22×C4—91no—
350C22×C4—181no—
351C22×C4—541no—
352M16—542no—
353C2×C8—92no—
354C4⋊C4—32no—
355C4⋊C4—181no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—41no—
359C22×C3—64no—
360C22×C3—64no—
361C22×C3—92no—
362C22×C3—122no—
363C22×C3—122no—
364C22×C3—122no—
365C22×C3—122no—
366C22×C3—181no—
367C22×C3—181no—
368C22×C3D3×#6712yes—
369C22×C3#381111yescharacteristic
370C2×D3—22no—
371C2×D3—34no—
372C2×D3—41no—
373C2×D3—41no—
374C2×D3—62no—
375C2×D3—62no—
376C2×D3—62no—
377C2×D3—121no—
378C2×D3—121no—
379C3×C4—21no—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—21no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—92no—
386C3×C4—122no—
387C3×C4—122no—
388C3×C4—242no—
389C3×C4C4×D3212yes—
390C3×C4D3×D1212yes—
391C3×C4C2×#62411yescharacteristic
392C3×C4#381111yescharacteristic
393Dic3—34no—
394Dic3—42no—
395Dic3—62no—
396Dic3—184no—
397Dic3—184no—
398Dic3—182no—
399Dic3—364no—
400Dic3—364no—
401C32—22no—
402C32—21no—
403C32—41no—
404C32#477112yescomplemented
405C32#496611yescharacteristic, complemented
406C23—61no—
407C23—91no—
408Q8—272no—
409Q8—541no—
410D4—32no—
411D4—61no—
412D4—544no—
413C2×C4—62no—
414C2×C4—91no—
415C2×C4—92no—
416C2×C4—92no—
417C2×C4—362no—
418C2×C4—544no—
419C2×C4—541no—
420C2×C4D3311yescharacteristic
421C8—182no—
422C2×C3—22no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—22no—
426C2×C3—21no—
427C2×C3—22no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—124no—
432C2×C3—124no—
433C2×C3—182no—
434C2×C3D3×(C2×D3)⋊C412yes—
435C2×C3#6392711yescharacteristic
436C2×C3#6393111yescharacteristic
437C2×C3#6393212yes—
438C2×C3#6394612yes—
439C2×C3#6397411yescharacteristic
440D3—42no—
441D3—64no—
442D3—122no—
443C22—62no—
444C22—92no—
445C22—121no—
446C22—121no—
447C22—181no—
448C22—181no—
449C22#8895211yescharacteristic
450C4—92no—
451C4—122no—
452C4—542no—
453C4D3×#62411yescharacteristic
454C4#8895511yescharacteristic
455C3—21no—
456C3—22no—
457C3—41no—
458C3#1059060712yescomplemented
459C3#1059141811yescharacteristic, complemented
460C2—122no—
461C2—182no—
462C2#1172287211yescharacteristic
463C2#1172288911yescharacteristic
464C2#1172290411yescharacteristic
subgroupG/H
centerC22#88952
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#89014C2
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88809C2semidirect
3. C3×#63900C2semidirect
4. #11722041C2semidirect
5. C3×#63562C2semidirect
14. #89014C4semidirect
24. #63900D3semidirect
33. #63562D3semidirect
39. #6308C2×C4semidirect
45. C32×D12C2×C4semidirect
157. C3×D12C4×D3semidirect
265. C33#417semidirect
302. D12#624semidirect
404. C32#4771semidirect
405. C32#4966semidirect
458. C3#10590607semidirect
459. C3#10591418semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
C2×#888092
C3×#639002
#117220412
C3×#635622
#105914183
#105906073