Finite Group Databasebeta

← #13165470

Subgroups — #13165470

Order 1728.

660 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 68 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×#88964C211yesmaximal, characteristic
2C2×#88925C211yesmaximal, characteristic, complemented
3D3×#3724C211yesmaximal, characteristic, complemented
4#11722793C211yesmaximal, characteristic, complemented
5#11722775C211yesmaximal, characteristic, complemented
6#11722772C211yesmaximal, characteristic, complemented
7C3×#63740C211yesmaximal, characteristic, complemented
8#10593496—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×Dic3×C3⋊D3—21no—
11C2×D3×C3⋊Dic3C2211yescharacteristic
12C2×C3×#627C2211yescharacteristic
13C2×C3×#626—21no—
14C2×C3×D3×Dic3—21no—
15C2×C3⋊#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17#88964—22no—
18#88964C414yescomplemented
19#88925—22no—
20#88815C2211yescharacteristic
21#88806—21no—
22C3×#3724C2211yescharacteristic
23C2×#3805—31no—
24C2×#3796—61no—
25C2×#3792—31no—
26C2×#3724—31no—
27D3×Dic3⋊C4—61no—
28#63740—31no—
29#63740D311yescharacteristic, complemented
30#63739—31no—
31#63739—31no—
32C22×C3×C3⋊D3—21no—
33C22×C32×D3D411yescharacteristic
34C2×C3⋊(C3⋊Dic3)D411yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C2×C32×Dic3—21no—
38C2×C32×Dic3—21no—
39#6295—21no—
40#6295C2×C412yescomplemented
41Dic3×C3⋊D3—24no—
42D3×C3⋊Dic3C2×C412yes—
43D3×C3⋊Dic3D412yes—
44C3×#627—21no—
45C3×#627C2×C412yescomplemented
46C3×#626—24no—
47C3×D3×Dic3—42no—
48C3⋊#622—21no—
49C3×#622—21no—
50#4803—91nomaximal
51C22×C3⋊Dic3—31no—
52C2×C4×C3⋊D3—61no—
53C2×C3×#67—61no—
54C22×C3×Dic3—61no—
55C2×C3×D12—61no—
56C2×C3×C4×D3—61no—
57C2×C3×Dic6—61no—
58C2×#627C2×D311yescharacteristic
59C2×#626—21no—
60C2×#626—61no—
61C2×#625—31no—
62#3805—34no—
63#3805—62no—
64C2×D3×Dic3—21no—
65C2×D3×Dic3—61no—
66C2×D3×Dic3—61no—
67C2×D3×Dic3—61no—
68C2×D3×Dic3—61no—
69C2×D3×Dic3—121no—
70#3796—64no—
71#3796—64no—
72#3792—62no—
73C2×#622—31no—
74C2×#622C2×D311yescharacteristic
75#3750—61no—
76C3×Dic3⋊C4—61no—
77#3724—31no—
78#3724—61no—
79#3724—61no—
80#3724—121no—
81#3724C2×D311yescharacteristic
82Dic3⋊Dic3—61no—
83C22×C33C2×D411yescharacteristic
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3—41no—
86C2×C3×C3⋊D3—41no—
87C2×C32×D3—21no—
88C2×C32×D3—21no—
89C2×C32×D3C22⋊C412yes—
90C3⋊(C3⋊Dic3)C22⋊C412yes—
91C3×C3⋊Dic3—41no—
92C3×C3⋊Dic3C22×C411yescharacteristic
93C3×C3⋊Dic3C2×D411yescharacteristic
94C32×Dic3—22no—
95C32×Dic3—41no—
96C2×#239—91no—
97C2×#237—181no—
98C2×Dic3⋊C4—181no—
99C2×ΓL(1,𝔽25)—91no—
100D3×C4⋊C4—91no—
101C3×#103—91no—
102#835—91no—
103#821—91no—
104#809—91no—
105C23×C32—31no—
106C22×C3⋊D3—21no—
107C22×C3×D3—21no—
108C22×C3×D3—21no—
109C22×C3×D3—61no—
110C22×C3×D3—61no—
111C22×C3×D3—121no—
112C2×C32×C4—61no—
113C2×C32×C4—61no—
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—32no—
117C2×C3⋊Dic3—32no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—121no—
121C2×C3⋊Dic3—121no—
122C2×C3⋊Dic3C22×D311yescharacteristic
123C4×C3⋊D3—64no—
124C3×#67—64no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—41no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—121no—
136C2×C3×Dic3—122no—
137C3×D12—64no—
138C3×C4×D3—122no—
139C3×Dic6—62no—
140C3×Dic6—62no—
141#627—21no—
142#627C4×D312yescomplemented
143#626—24no—
144#626—62no—
145#626—62no—
146#625—32no—
147#625—61no—
148D3×Dic3—42no—
149D3×Dic3—62no—
150D3×Dic3—62no—
151D3×Dic3—62no—
152D3×Dic3—62no—
153D3×Dic3—64no—
154D3×Dic3—64no—
155D3×Dic3—124no—
156#622—21no—
157#622—61no—
158#417—271noSylow
159C2×C33C2×C22⋊C411yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C2×C33#13711yescharacteristic
162C3×C3⋊D3—42no—
163C32×D3—22no—
164C2×C3×Q8—91no—
165C2×C3×D4—181no—
166C22×C3×C4—181no—
167C2×#67—61no—
168C2×#67—181no—
169C22×Dic3—61no—
170C22×Dic3—181no—
171C22×Dic3—181no—
172C22×Dic3—181no—
173#239—94no—
174#239—182no—
175#237—184no—
176#237—184no—
177C2×D12—61no—
178C2×D12—91no—
179C2×C4×D3—61no—
180C2×C4×D3—91no—
181C2×C4×D3—181no—
182C2×C4×D3—181no—
183C2×C4×D3—361no—
184C2×Dic6—61no—
185C2×C3×C8—91no—
186C3×C4⋊C4—91no—
187C4⋊Dic3—91no—
188Dic3⋊C4—61no—
189Dic3⋊C4—181no—
190Dic3⋊C4—181no—
191Dic3⋊C4—181no—
192Dic3⋊C4—181no—
193Dic3⋊C4—361no—
194C2×C3⋊C8—91no—
195ΓL(1,𝔽25)—182no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—32no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—61no—
202C22×C32—61no—
203C22×C32D3×D411yescharacteristic
204C2×C3⋊D3—22no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3×D3—22no—
208C2×C3×D3—22no—
209C2×C3×D3—22no—
210C2×C3×D3—22no—
211C2×C3×D3—41no—
212C2×C3×D3—41no—
213C2×C3×D3—62no—
214C2×C3×D3—62no—
215C2×C3×D3—122no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C2×C3×D3—122no—
219C2×C3×D3—122no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C32×C4—62no—
223C32×C4—121no—
224C3⋊Dic3—32no—
225C3⋊Dic3—41no—
226C3⋊Dic3—62no—
227C3⋊Dic3—62no—
228C3⋊Dic3—122no—
229C3⋊Dic3—122no—
230C3⋊Dic3C2×C4×D311yescharacteristic
231C3⋊Dic3D3×D411yescharacteristic
232C3×Dic3—22no—
233C3×Dic3—22no—
234C3×Dic3—42no—
235C3×Dic3—41no—
236C3×Dic3—41no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C3×Dic3—61no—
241C3×Dic3—81no—
242C3×Dic3—121no—
243C3×Dic3—121no—
244C3×Dic3—241no—
245C2×#41—271no—
246C2×M16—271no—
247C2×C4⋊C4—271no—
248#103—91no—
249#103—271no—
250#102—271no—
251#102—271no—
252C33#41711yescharacteristic, complemented, Sylow
253C23×C3—61no—
254C23×C3—61no—
255C23×C3—181no—
256C22×D3—61no—
257C22×D3—61no—
258C22×D3—121no—
259C22×D3GO+(4,𝔽2)11yescharacteristic
260C3×Q8—92no—
261C3×Q8—181no—
262C3×D4—184no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—91no—
268C2×C3×C4—121no—
269C2×C3×C4—121no—
270C2×C3×C4—181no—
271C2×C3×C4—362no—
272#67—64no—
273#67—182no—
274#67—182no—
275C2×Dic3—21no—
276C2×Dic3—21no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—91no—
282C2×Dic3—121no—
283C2×Dic3—122no—
284C2×Dic3—182no—
285C2×Dic3—182no—
286C2×Dic3—182no—
287C2×Dic3—182no—
288C2×Dic3—181no—
289C2×Dic3—181no—
290C2×Dic3—184no—
291C2×Dic3—361no—
292C2×Dic3—361no—
293D12—64no—
294D12—92no—
295D12—181no—
296C4×D3—92no—
297C4×D3—92no—
298C4×D3—122no—
299C4×D3—184no—
300C4×D3—184no—
301C4×D3—364no—
302Dic6—62no—
303Dic6—62no—
304C3×C8—181no—
305C3⋊C8—181no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—62no—
319C2×C32D3×C22⋊C411yescharacteristic
320C2×C32#91411yescharacteristic
321C2×C32#91811yescharacteristic
322C3⋊D3—42no—
323C3×D3—24no—
324C3×D3—42no—
325C3×D3—122no—
326C3×D3—124no—
327C3×D3—122no—
328#41—274no—
329#41—542no—
330C2×Q8—91no—
331C2×D4—181no—
332C2×D4—271no—
333C22×C4—181no—
334C22×C4—271no—
335C22×C4—541no—
336M16—542no—
337C2×C8—91no—
338C2×C8—271no—
339C4⋊C4—91no—
340C4⋊C4—271no—
341C4⋊C4—541no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—62no—
347C22×C3—62no—
348C22×C3—62no—
349C22×C3—62no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3—182no—
353C22×C3—361no—
354C22×C3—361no—
355C22×C3C2×GO+(4,𝔽2)11yescharacteristic
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—62no—
359C2×D3—62no—
360C2×D3—122no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—122no—
364C2×D3—122no—
365C2×D3—121no—
366C2×D3—121no—
367C2×D3D32⋊C412yes—
368C3×C4—62no—
369C3×C4—62no—
370C3×C4—91no—
371C3×C4—91no—
372C3×C4—122no—
373C3×C4—121no—
374C3×C4—121no—
375C3×C4—241no—
376C3×C4—361no—
377Dic3—22no—
378Dic3—41no—
379Dic3—61no—
380Dic3—61no—
381Dic3—61no—
382Dic3—61no—
383Dic3—92no—
384Dic3—121no—
385Dic3—121no—
386Dic3—182no—
387Dic3—182no—
388Dic3—241no—
389Dic3—362no—
390Dic3—362no—
391C32—21no—
392C32—21no—
393C32—41no—
394C32—41no—
395C32#480311yescharacteristic, complemented
396C23—31no—
397C23—181no—
398Q8—92no—
399Q8—181no—
400D4—184no—
401D4—272no—
402D4—541no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—91no—
406C2×C4—181no—
407C2×C4—272no—
408C2×C4—272no—
409C2×C4—271no—
410C2×C4—362no—
411C2×C4—544no—
412C8—181no—
413C8—541no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—41no—
426C2×C3—64no—
427C2×C3—122no—
428C2×C3—362no—
429C2×C3C2×D32⋊C411yescharacteristic
430C2×C3#6423611yescharacteristic
431C2×C3#6423911yescharacteristic
432D3—22no—
433D3—122no—
434D3—124no—
435D3—122no—
436C22—32no—
437C22—61no—
438C22—61no—
439C22—182no—
440C22—361no—
441C22—361no—
442C22D3×GO+(4,𝔽2)11yescharacteristic
443C4—62no—
444C4—91no—
445C4—91no—
446C4—121no—
447C4—272no—
448C4—361no—
449C3—21no—
450C3—21no—
451C3—41no—
452C3—41no—
453C3#1059349611yescharacteristic, complemented
454C2—62no—
455C2—362no—
456C2D3×D32⋊C411yescharacteristic
457C2#1172413911yescharacteristic
458C2#1172414311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC2×#88964C2
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33#417
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#88925C2semidirect
3. D3×#3724C2semidirect
4. #11722793C2semidirect
5. #11722775C2semidirect
6. #11722772C2semidirect
7. C3×#63740C2semidirect
18. #88964C4semidirect
29. #63740D3semidirect
40. #6295C2×C4semidirect
45. C3×#627C2×C4semidirect
142. #627C4×D3semidirect
252. C33#417semidirect
395. C32#4803semidirect
453. C3#10593496semidirect

Maximal subgroups

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

subgroupindex
C2×#889642
C2×#889252
D3×#37242
#117227932
#117227752
#117227722
C3×#637402
#105934963
#48039