Finite Group Databasebeta

← #13167493

Subgroups — #13167493

Order 1728.

506 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 53 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×#88942C211yesmaximal, characteristic, complemented
2C2×#88941C211yesmaximal, characteristic
3C4×#6295C211yesmaximal, characteristic, complemented
4#11723672C211yesmaximal, characteristic, complemented
5C3×#63779C211yesmaximal, characteristic, complemented
6#11722794C211yesmaximal, characteristic, complemented
7#11722793C211yesmaximal, characteristic, complemented
8#10593422—31nomaximal
9C2×C3×C4×C3⋊D3—21no—
10C2×#6296C2211yescharacteristic
11C2×#6295C2211yescharacteristic
12C2×#6295C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15#88942—22no—
16#88941—22no—
17C3×#3752—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19#88815C2211yescharacteristic
20#88814—21no—
21#88814—21no—
22C2×#3775—31no—
23C2×#3774—31no—
24#63915—31no—
25C4×#626—61no—
26C4×#625—31no—
27#63852—61no—
28#63779D311yescharacteristic, complemented
29#63740—31no—
30#63739—31no—
31C22×C3×C3⋊D3—21no—
32C22×C3×C3⋊D3—21no—
33C2×C33×C4D411yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C3×C4×C3⋊D3—42no—
39#6296—21no—
40#6296—21no—
41#6295—21no—
42#6295—21no—
43#6295—21no—
44#6295D412yes—
45C3×#622—21no—
46C3×#622—21no—
47#4844—91nomaximal
48C2×C4×C3⋊D3—21no—
49C2×C3×C4×D3—61no—
50C2×C3×C4×D3—61no—
51C2×C3×C4×D3—121no—
52C2×#627—31no—
53C2×#627—61no—
54C2×#626—61no—
55C2×#626—61no—
56C2×#625—31no—
57C2×#625—31no—
58C2×#622C2×D311yescharacteristic
59C2×#622C2×D311yescharacteristic
60#3775—62no—
61#3774—62no—
62#3752—21no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—61no—
66C3×(C2×D3)⋊C4—121no—
67C3×C4×Dic3—61no—
68C3×C4×Dic3—61no—
69#3724—31no—
70Dic3⋊Dic3—61no—
71#3722—61no—
72#3722—61no—
73(C2×D3)⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—22no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4—21no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3—41no—
88C3×C3⋊Dic3C2×D411yescharacteristic
89C3×C3⋊Dic3#4111yescharacteristic
90C4×#67—181no—
91C2×D24—91no—
92C2×#204—91no—
93#885—181no—
94#875—91no—
95C4×D12—91no—
96C3×C4⋊C8—91no—
97#821—91no—
98#816—91no—
99C22×C3⋊D3—21no—
100C22×C3⋊D3—21no—
101C22×C3×D3—61no—
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—61no—
105C22×C3×D3—121no—
106C22×C3×D3—121no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4D1211yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3D1211yescharacteristic
115C2×C3⋊Dic3C22×D311yescharacteristic
116C4×C3⋊D3—42no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—121no—
126C2×C3×Dic3—121no—
127C3×C4×D3—122no—
128C3×C4×D3—122no—
129C3×C4×D3—124no—
130#627—61no—
131#627—61no—
132#627—121no—
133#627—121no—
134#626—62no—
135#626—62no—
136#626—121no—
137#626—121no—
138#625—32no—
139#625—61no—
140#625—61no—
141#625—61no—
142#622—21no—
143#622—21no—
144#463—271noSylow
145C2×C33#12111yescharacteristic
146C2×C33#13511yescharacteristic
147C2×C33AGL(1,ℤ/8)11yescharacteristic
148C3×C3⋊D3—42no—
149C3×C3⋊D3—81no—
150C22×C3×C4—181no—
151C2×#67—181no—
152C2×#67—181no—
153C2×D12—91no—
154C2×D12—91no—
155C2×C4×D3—61no—
156C2×C4×D3—61no—
157C2×C4×D3—121no—
158C2×Dic6—91no—
159C2×Dic6—181no—
160C2×C3×C8—91no—
161C2×C3×C8—91no—
162C3×C22⋊C4—181no—
163C3×C42—91no—
164#217—181no—
165#217—181no—
166(C2×D3)⋊C4—61no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—121no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—91no—
174Dic3⋊C4—181no—
175C4×Dic3—61no—
176C4×Dic3—61no—
177D24—182no—
178#204—182no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—41no—
182C22×C32—41no—
183C22×C32C2×D1211yescharacteristic
184C2×C3⋊D3—22no—
185C2×C3⋊D3—41no—
186C2×C3⋊D3—41no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3⋊D3—41no—
190C2×C3×D3—62no—
191C2×C3×D3—62no—
192C2×C3×D3—122no—
193C2×C3×D3—121no—
194C2×C3×D3—121no—
195C2×C3×D3—122no—
196C2×C3×D3—122no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C2×C3×D3—122no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C2×C3×D3—241no—
207C2×C3×D3—241no—
208C32×C4—22no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4—42no—
212C32×C4—81no—
213C3⋊Dic3—21no—
214C3⋊Dic3—41no—
215C3⋊Dic3—41no—
216C3⋊Dic3D3×D411yescharacteristic
217C3⋊Dic3#23911yescharacteristic
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—61no—
222C3×Dic3—121no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C3×Dic3—122no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C3×Dic3—241no—
230C2×SD16—271no—
231C2×D8—271no—
232#124—271no—
233C4×D4—271no—
234C4⋊C8—91no—
235#103—271no—
236#102—271no—
237C33#46311yescharacteristic, complemented, Sylow
238C23×C3—181no—
239C23×C3—181no—
240C22×D3—61no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—61no—
244C22×D3—121no—
245C22×D3—121no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—91no—
251C2×C3×C4—91no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4—362no—
255C2×C3×C4GO+(4,𝔽2)11yescharacteristic
256#67—182no—
257#67—182no—
258#67—361no—
259#67—361no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—121no—
269C2×Dic3—121no—
270D12—92no—
271D12—181no—
272D12—181no—
273D12—181no—
274C4×D3—122no—
275C4×D3—122no—
276C4×D3—124no—
277Dic6—181no—
278Dic6—181no—
279Dic6—361no—
280Dic6—361no—
281C3×C8—181no—
282C3×C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32#89511yescharacteristic
296C2×C32#90411yescharacteristic
297C2×C32#90811yescharacteristic
298C3⋊D3—42no—
299C3⋊D3—81no—
300C3×D3—122no—
301C3×D3—124no—
302C3×D3—122no—
303C3×D3—241no—
304C3×D3—242no—
305C3×D3—241no—
306C2×Q8—271no—
307C2×D4—271no—
308C2×D4—271no—
309C22×C4—181no—
310SD16—542no—
311D8—542no—
312C2×C8—91no—
313C2×C8—91no—
314C4⋊C4—271no—
315C22⋊C4—181no—
316C22⋊C4—541no—
317C22⋊C4—541no—
318C42—91no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—182no—
324C22×C3—361no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3—361no—
329C22×C3C2×GO+(4,𝔽2)11yescharacteristic
330C2×D3—62no—
331C2×D3—62no—
332C2×D3—122no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—122no—
336C2×D3—122no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—122no—
343C2×D3—121no—
344C2×D3—121no—
345C2×D3—121no—
346C2×D3—241no—
347C2×D3—241no—
348C3×C4—22no—
349C3×C4—21no—
350C3×C4—41no—
351C3×C4—42no—
352C3×C4—81no—
353C3×C4—91no—
354C3×C4—91no—
355C3×C4—181no—
356C3×C4—361no—
357C3×C4—361no—
358Dic3—61no—
359Dic3—61no—
360Dic3—61no—
361Dic3—61no—
362Dic3—121no—
363Dic3—121no—
364Dic3—121no—
365Dic3—121no—
366Dic3—122no—
367Dic3—121no—
368Dic3—121no—
369Dic3—241no—
370C32—21no—
371C32—21no—
372C32—41no—
373C32—41no—
374C32#484411yescharacteristic, complemented
375C23—181no—
376C23—181no—
377Q8—541no—
378Q8—541no—
379D4—272no—
380D4—541no—
381D4—541no—
382D4—541no—
383C2×C4—91no—
384C2×C4—91no—
385C2×C4—181no—
386C2×C4—181no—
387C2×C4—362no—
388C2×C4#157⋊D311yescharacteristic
389C8—181no—
390C8—181no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3—362no—
404C2×C3—721no—
405C2×C3#6422311yescharacteristic
406C2×C3#6423011yescharacteristic
407C2×C3#6423611yescharacteristic
408D3—122no—
409D3—124no—
410D3—122no—
411D3—241no—
412D3—242no—
413D3—241no—
414C22—182no—
415C22—361no—
416C22—361no—
417C22—361no—
418C22—361no—
419C22—361no—
420C22C2×#157⋊D311yescharacteristic
421C4—21no—
422C4—91no—
423C4—91no—
424C4—181no—
425C4—361no—
426C4—361no—
427C3—21no—
428C3—21no—
429C3—41no—
430C3—41no—
431C3#1059342211yescharacteristic, complemented
432C2—362no—
433C2—721no—
434C2#1172433711yescharacteristic
435C2#1172434311yescharacteristic
436C2#1172435011yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#88941C2
hypercenterC2×C4#157⋊D3
hypocenterC33#463
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88942C2semidirect
3. C4×#6295C2semidirect
4. #11723672C2semidirect
5. C3×#63779C2semidirect
6. #11722794C2semidirect
7. #11722793C2semidirect
28. #63779D3semidirect
237. C33#463semidirect
374. C32#4844semidirect
431. C3#10593422semidirect

Maximal subgroups

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

subgroupindex
C2×#889422
C2×#889412
C4×#62952
#117236722
C3×#637792
#117227942
#117227932
#105934223
#48449