Finite Group Databasebeta

← #13165637

Subgroups — #13165637

Order 1728.

516 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 50 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×#88935C211yesmaximal, characteristic, complemented
2C2×#88925C211yesmaximal, characteristic, complemented
3#11722955C211yesmaximal, characteristic, complemented
4#11722953C211yesmaximal, characteristic, complemented
5#11722793C211yesmaximal, characteristic, complemented
6#11722778C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593522—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×D3×C3⋊Dic3C2211yescharacteristic
11C2×C3×#626—21no—
12C2×C3×#625—21no—
13C2×C3×#625C2211yescharacteristic
14C2×C3⋊#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16#88935—22no—
17#88925—22no—
18#88815C2211yescharacteristic
19#88803—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21C3×#3724C2211yescharacteristic
22C2×#3792—31no—
23C2×#3774—31no—
24#63980—31no—
25#63980—31no—
26#63970—61no—
27#63886—61no—
28#63742—31no—
29#63739—31no—
30#63739D311yescharacteristic, complemented
31C22×C3×C3⋊D3—21no—
32C22×C32×D3—21no—
33C22×C32×D3D411yescharacteristic
34C2×C3⋊(C3⋊Dic3)D411yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C2×C32×Dic3—21no—
38#6295—21no—
39#6295—21no—
40D3×C3⋊Dic3—21no—
41D3×C3⋊Dic3—21no—
42C3×#626—42no—
43C3×#625—21no—
44C3×#625—21no—
45C3×#625—42no—
46C3⋊#622D412yescomplemented
47C3×#622—21no—
48#4782—91nomaximal
49C22×C3⋊Dic3—31no—
50C2×C3×#67—61no—
51C2×C3×#67—61no—
52C2×C3×#67—61no—
53C2×C3×#67—61no—
54C2×C3×D12—61no—
55C2×#626—21no—
56C2×#626—61no—
57C2×#625—21no—
58C2×#625—31no—
59C2×#625C2×D311yescharacteristic
60C2×D3×Dic3—61no—
61C2×D3×Dic3—61no—
62#3792—62no—
63C2×#622—31no—
64C2×#622C2×D311yescharacteristic
65#3774—62no—
66#3757—61no—
67#3752—61no—
68C3×Dic3⋊C4—61no—
69#3724—31no—
70#3724C2×D311yescharacteristic
71Dic3⋊Dic3—61no—
72(C2×D3)⋊Dic3—61no—
73(C2×D3)⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—121no—
77(C2×D3)⋊Dic3—121no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3—41no—
82C2×C32×D3—21no—
83C2×C32×D3—21no—
84C2×C32×D3—21no—
85C2×C32×D3—41no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C3⋊(C3⋊Dic3)—21no—
89C3×C3⋊Dic3—41no—
90C3×C3⋊Dic3C2×D411yescharacteristic
91C3×C3⋊Dic3#4111yescharacteristic
92C32×Dic3—41no—
93#939—181no—
94#939—181no—
95#938—91no—
96C2×#213—91no—
97C2×ΓL(1,𝔽25)—91no—
98#895—91no—
99C3×#102—91no—
100#821—91no—
101#808—91no—
102C23×C32—31no—
103C23×C32—61no—
104C22×C3⋊D3—21no—
105C22×C3×D3—21no—
106C22×C3×D3—21no—
107C22×C3×D3—21no—
108C22×C3×D3—21no—
109C22×C3×D3—41no—
110C22×C3×D3—61no—
111C22×C3×D3—61no—
112C22×C3×D3—121no—
113C2×C32×C4—61no—
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—121no—
121C2×C3⋊Dic3—121no—
122C2×C3⋊Dic3C22×D311yescharacteristic
123C3×#67—121no—
124C3×#67—121no—
125C3×#67—122no—
126C3×#67—122no—
127C3×#67—122no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—121no—
136C3×D12—122no—
137#626—42no—
138#626—121no—
139#626—121no—
140#625—21no—
141#625—21no—
142#625—42no—
143#625—61no—
144#625—61no—
145D3×Dic3—121no—
146D3×Dic3—121no—
147D3×Dic3—121no—
148D3×Dic3—121no—
149#622—21no—
150#622—32no—
151#469—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C2×C33AGL(1,ℤ/8)11yescharacteristic
155C3×C3⋊D3—81no—
156C32×D3—41no—
157C32×D3—81no—
158C2×C3×D4—91no—
159C2×C3×D4—181no—
160C2×C3×D4—181no—
161C2×#67—61no—
162C2×#67—61no—
163C2×#67—61no—
164C2×#67—61no—
165C2×#67—181no—
166C22×Dic3—181no—
167C22×Dic3—181no—
168C2×D12—61no—
169C2×D12—91no—
170C2×C4×D3—91no—
171C2×C3×C8—91no—
172C3×C4⋊C4—91no—
173#217—181no—
174#217—181no—
175#217—181no—
176#217—361no—
177#213—182no—
178(C2×D3)⋊C4—181no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—361no—
182C4⋊Dic3—91no—
183Dic3⋊C4—61no—
184Dic3⋊C4—181no—
185C2×C3⋊C8—91no—
186ΓL(1,𝔽25)—182no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—61no—
192C22×C32—61no—
193C22×C32—61no—
194C22×C32—121no—
195C22×C32—121no—
196C22×C32—121no—
197C22×C32D3×D411yescharacteristic
198C2×C3⋊D3—41no—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C2×C3×D3—22no—
202C2×C3×D3—22no—
203C2×C3×D3—41no—
204C2×C3×D3—41no—
205C2×C3×D3—41no—
206C2×C3×D3—41no—
207C2×C3×D3—41no—
208C2×C3×D3—41no—
209C2×C3×D3—41no—
210C2×C3×D3—42no—
211C2×C3×D3—41no—
212C2×C3×D3—41no—
213C2×C3×D3—41no—
214C2×C3×D3—81no—
215C2×C3×D3—81no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—122no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—241no—
224C2×C3×D3—241no—
225C32×C4—121no—
226C3⋊Dic3—41no—
227C3⋊Dic3—61no—
228C3⋊Dic3—121no—
229C3⋊Dic3—121no—
230C3⋊Dic3—241no—
231C3⋊Dic3—241no—
232C3⋊Dic3#23511yescharacteristic
233C3⋊Dic3D3×D411yescharacteristic
234C3×Dic3—41no—
235C3×Dic3—41no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C3×Dic3—81no—
241C3×Dic3—121no—
242C3×Dic3—121no—
243C3×Dic3—241no—
244C2×D8—271no—
245C2×M16—271no—
246#121—271no—
247#121—271no—
248#106—271no—
249#102—91no—
250#102—271no—
251C33#46911yescharacteristic, complemented, Sylow
252C23×C3—61no—
253C23×C3—61no—
254C23×C3—61no—
255C23×C3—61no—
256C23×C3—121no—
257C23×C3—181no—
258C22×D3—21no—
259C22×D3—61no—
260C22×D3—61no—
261C22×D3—121no—
262C22×D3GO+(4,𝔽2)11yescharacteristic
263C3×D4—181no—
264C3×D4—181no—
265C3×D4—362no—
266C3×D4—362no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—91no—
270C2×C3×C4—121no—
271C2×C3×C4—181no—
272#67—121no—
273#67—121no—
274#67—122no—
275#67—122no—
276#67—122no—
277#67—361no—
278#67—361no—
279C2×Dic3—21no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—91no—
285C2×Dic3—121no—
286C2×Dic3—181no—
287C2×Dic3—181no—
288C2×Dic3—361no—
289C2×Dic3—361no—
290C2×Dic3—361no—
291C2×Dic3—361no—
292C2×Dic3—361no—
293C2×Dic3—361no—
294D12—122no—
295D12—181no—
296D12—181no—
297C4×D3—181no—
298C4×D3—181no—
299C3×C8—181no—
300C3⋊C8—92no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—121no—
314C2×C32—241no—
315C2×C32#88411yescharacteristic
316C2×C32#91111yescharacteristic
317C2×C32#91411yescharacteristic
318C3⋊D3—81no—
319C3×D3—42no—
320C3×D3—42no—
321C3×D3—81no—
322C3×D3—82no—
323C3×D3—81no—
324C3×D3—241no—
325C3×D3—242no—
326C3×D3—241no—
327C2×D4—91no—
328C2×D4—181no—
329C2×D4—181no—
330C2×D4—271no—
331C22×C4—271no—
332D8—542no—
333M16—542no—
334C2×C8—91no—
335C2×C8—271no—
336C4⋊C4—91no—
337C4⋊C4—271no—
338C22⋊C4—541no—
339C22⋊C4—541no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—62no—
345C22×C3—62no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—121no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3—121no—
353C22×C3—122no—
354C22×C3—121no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—241no—
358C22×C3—241no—
359C22×C3—361no—
360C22×C3—361no—
361C22×C3—361no—
362C22×C3C2×GO+(4,𝔽2)11yescharacteristic
363C2×D3—21no—
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—41no—
367C2×D3—41no—
368C2×D3—41no—
369C2×D3—121no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—122no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—121no—
376C2×D3—241no—
377C2×D3—241no—
378C3×C4—91no—
379C3×C4—91no—
380C3×C4—121no—
381C3×C4—121no—
382C3×C4—241no—
383C3×C4—361no—
384Dic3—41no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—61no—
389Dic3—121no—
390Dic3—121no—
391Dic3—181no—
392Dic3—241no—
393Dic3—361no—
394Dic3—361no—
395Dic3—721no—
396Dic3—721no—
397C32—21no—
398C32—21no—
399C32—41no—
400C32—41no—
401C32#478211yescharacteristic, complemented
402C23—31no—
403C23—61no—
404C23—181no—
405D4—181no—
406D4—181no—
407D4—362no—
408D4—362no—
409D4—541no—
410D4—541no—
411C2×C4—61no—
412C2×C4—91no—
413C2×C4—181no—
414C2×C4—271no—
415C2×C4—541no—
416C2×C4—541no—
417C8—181no—
418C8—272no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—122no—
432C2×C3—122no—
433C2×C3—241no—
434C2×C3—242no—
435C2×C3—241no—
436C2×C3—721no—
437C2×C3#6423611yescharacteristic
438C2×C3#6423611yescharacteristic
439C2×C3#6424211yescharacteristic
440D3—41no—
441D3—81no—
442D3—241no—
443D3—242no—
444D3—241no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—121no—
449C22—121no—
450C22—121no—
451C22—361no—
452C22—361no—
453C22—361no—
454C22D3×GO+(4,𝔽2)11yescharacteristic
455C4—91no—
456C4—91no—
457C4—121no—
458C4—361no—
459C4—541no—
460C3—21no—
461C3—21no—
462C3—41no—
463C3—41no—
464C3#1059352211yescharacteristic, complemented
465C2—121no—
466C2—241no—
467C2—721no—
468C2#1172411511yescharacteristic
469C2#1172413811yescharacteristic
470C2#1172413911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
hypocenterC33#469
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#88935C2semidirect
2. C2×#88925C2semidirect
3. #11722955C2semidirect
4. #11722953C2semidirect
5. #11722793C2semidirect
6. #11722778C2semidirect
7. C3×#63739C2semidirect
30. #63739D3semidirect
46. C3⋊#622D4semidirect
251. C33#469semidirect
401. C32#4782semidirect
464. C3#10593522semidirect

Maximal subgroups

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

subgroupindex
C2×#889352
C2×#889252
#117229552
#117229532
#117227932
#117227782
C3×#637392
#105935223
#47829