Finite Group Databasebeta

← #13165627

Subgroups — #13165627

Order 1728.

468 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 52 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×C3×#3775C211yesmaximal, characteristic, complemented
2C2×#88925C211yesmaximal, characteristic, complemented
3#11722976C211yesmaximal, characteristic, complemented
4#11722953C211yesmaximal, characteristic
5#11722797C211yesmaximal, characteristic, complemented
6#11722775C211yesmaximal, characteristic, complemented
7#11722773C211yesmaximal, characteristic, complemented
8#10593495—31nomaximal
9C2×D3×C3⋊Dic3C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3×#625—21no—
12C2×C3×#625C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C3×#3775—22no—
16#88925—22no—
17#88815C2211yescharacteristic
18#88815C2211yescharacteristic
19#88807—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C2×#3792—31no—
23C2×#3775D311yescharacteristic, complemented
24#63982—31no—
25#63980—31no—
26#63970—61no—
27#63887—61no—
28#63743—31no—
29#63740—31no—
30#63739—31no—
31C22×C32×D3—21no—
32C22×C32×D3D411yescharacteristic
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C2×C32×Dic3—21no—
38D3×C3⋊Dic3—21no—
39D3×C3⋊Dic3—21no—
40C3×#627—21no—
41C3×#627—21no—
42C3×#625—21no—
43C3×#625—21no—
44C3×#625—42no—
45C3⋊#622—21no—
46C3×#622D412yescomplemented
47#5174—91nomaximal
48C22×C3⋊Dic3—31no—
49C2×C3×#67—61no—
50C2×C3×#67—61no—
51C2×C3×#67—61no—
52C2×C3×Dic6—61no—
53C2×#627C2×D311yescharacteristic
54C2×#625—21no—
55C2×#625C2×D311yescharacteristic
56C2×D3×Dic3—61no—
57C2×D3×Dic3—61no—
58#3792—62no—
59C2×#622—31no—
60C2×#622C2×D311yescharacteristic
61#3775—22no—
62#3757—61no—
63#3750—61no—
64C3×#217—61no—
65C3×(C2×D3)⋊C4—61no—
66#3724—31no—
67#3724—31no—
68Dic3⋊Dic3—61no—
69Dic3⋊Dic3—61no—
70Dic3⋊Dic3—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—121no—
73(C2×D3)⋊Dic3—21no—
74(C2×D3)⋊Dic3—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—121no—
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C2×C32×D3—21no—
81C2×C32×D3—41no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C3⋊(C3⋊Dic3)—21no—
85C3×C3⋊Dic3—41no—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3C2×D411yescharacteristic
88C3×C3⋊Dic3#4111yescharacteristic
89C32×Dic3—41no—
90C2×C3×SD16—91no—
91#946—91no—
92#939—181no—
93#938—91no—
94C2×ΓL(1,𝔽25)—91no—
95#878—181no—
96#835—91no—
97#832—91no—
98#818—91no—
99C23×C32—31no—
100C23×C32—61no—
101C22×C3×D3—21no—
102C22×C3×D3—21no—
103C22×C3×D3—21no—
104C22×C3×D3—21no—
105C22×C3×D3—41no—
106C2×C32×C4—61no—
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—121no—
115C2×C3⋊Dic3—121no—
116C2×C3⋊Dic3C22×D311yescharacteristic
117C3×#67—121no—
118C3×#67—121no—
119C3×#67—122no—
120C3×#67—122no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—41no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—121no—
131C2×C3×Dic3—121no—
132C3×Dic6—121no—
133C3×Dic6—121no—
134#627—21no—
135#627—21no—
136#625—21no—
137#625—21no—
138#625—42no—
139D3×Dic3—121no—
140D3×Dic3—121no—
141D3×Dic3—121no—
142D3×Dic3—121no—
143#622—61no—
144#622#6712yescomplemented
145#468—271noSylow
146C2×C33#12111yescharacteristic
147C2×C33AGL(1,ℤ/8)11yescharacteristic
148C2×C33#13711yescharacteristic
149C32×D3—41no—
150C32×D3—81no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C2×C3×D4—181no—
154C2×#67—61no—
155C2×#67—61no—
156C2×#67—61no—
157C22×Dic3—181no—
158C22×Dic3—181no—
159C2×C4×D3—91no—
160C2×Dic6—61no—
161C3×SD16—182no—
162C2×C3×C8—91no—
163C3×C22⋊C4—181no—
164#217—61no—
165#217—181no—
166#217—181no—
167#217—361no—
168(C2×D3)⋊C4—61no—
169(C2×D3)⋊C4—181no—
170C4⋊Dic3—91no—
171C4⋊Dic3—91no—
172C4⋊Dic3—181no—
173Dic3⋊C4—181no—
174Dic3⋊C4—181no—
175Dic3⋊C4—181no—
176Dic3⋊C4—181no—
177Dic3⋊C4—361no—
178C2×C3⋊C8—91no—
179ΓL(1,𝔽25)—182no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—61no—
185C22×C32—61no—
186C22×C32—61no—
187C22×C32—121no—
188C22×C32—121no—
189C22×C32—121no—
190C22×C32D3×D411yescharacteristic
191C2×C3×D3—22no—
192C2×C3×D3—22no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C2×C3×D3—41no—
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199C2×C3×D3—41no—
200C2×C3×D3—42no—
201C2×C3×D3—41no—
202C2×C3×D3—41no—
203C2×C3×D3—41no—
204C2×C3×D3—81no—
205C2×C3×D3—81no—
206C32×C4—121no—
207C3⋊Dic3—41no—
208C3⋊Dic3—41no—
209C3⋊Dic3—61no—
210C3⋊Dic3—121no—
211C3⋊Dic3—121no—
212C3⋊Dic3—241no—
213C3⋊Dic3—241no—
214C3⋊Dic3#23711yescharacteristic
215C3⋊Dic3C2×#6711yescharacteristic
216C3×Dic3—41no—
217C3×Dic3—41no—
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—61no—
222C3×Dic3—81no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C3×Dic3—122no—
226C3×Dic3—121no—
227C3×Dic3—121no—
228C3×Dic3—241no—
229C2×SD16—91no—
230C2×M16—271no—
231C22⋊Q8—271no—
232#121—271no—
233#107—271no—
234#103—271no—
235#102—271no—
236C33#46811yescharacteristic, complemented, Sylow
237C23×C3—61no—
238C23×C3—61no—
239C23×C3—61no—
240C23×C3—61no—
241C23×C3—121no—
242C22×D3—21no—
243C22×D3GO+(4,𝔽2)11yescharacteristic
244C3×Q8—181no—
245C3×Q8—181no—
246C3×D4—181no—
247C3×D4—181no—
248C3×D4—362no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—91no—
252C2×C3×C4—121no—
253C2×C3×C4—181no—
254C2×C3×C4—181no—
255#67—121no—
256#67—121no—
257#67—122no—
258#67—122no—
259C2×Dic3—21no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—91no—
267C2×Dic3—121no—
268C2×Dic3—121no—
269C2×Dic3—181no—
270C2×Dic3—181no—
271C2×Dic3—361no—
272C2×Dic3—361no—
273C2×Dic3—361no—
274C2×Dic3—361no—
275C2×Dic3—361no—
276C2×Dic3—361no—
277C4×D3—181no—
278C4×D3—181no—
279Dic6—121no—
280Dic6—121no—
281C3×C8—92no—
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—121no—
296C2×C32—241no—
297C2×C32#91411yescharacteristic
298C2×C32#91511yescharacteristic
299C2×C32#93811yescharacteristic
300C3×D3—42no—
301C3×D3—42no—
302C3×D3—81no—
303C3×D3—82no—
304C3×D3—81no—
305C2×Q8—91no—
306C2×D4—91no—
307C2×D4—181no—
308C22×C4—271no—
309SD16—182no—
310M16—542no—
311C2×C8—91no—
312C2×C8—271no—
313C4⋊C4—271no—
314C4⋊C4—271no—
315C4⋊C4—541no—
316C22⋊C4—181no—
317C22⋊C4—541no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—62no—
323C22×C3—62no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—121no—
331C22×C3—122no—
332C22×C3—121no—
333C22×C3—121no—
334C22×C3—121no—
335C22×C3—241no—
336C22×C3—241no—
337C22×C3C2×GO+(4,𝔽2)11yescharacteristic
338C2×D3—21no—
339C2×D3—21no—
340C2×D3—21no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—41no—
344C3×C4—91no—
345C3×C4—91no—
346C3×C4—121no—
347C3×C4—121no—
348C3×C4—241no—
349C3×C4—361no—
350C3×C4—361no—
351Dic3—41no—
352Dic3—61no—
353Dic3—61no—
354Dic3—61no—
355Dic3—61no—
356Dic3—121no—
357Dic3—121no—
358Dic3—122no—
359Dic3—121no—
360Dic3—121no—
361Dic3—181no—
362Dic3—241no—
363Dic3—361no—
364Dic3—361no—
365Dic3—721no—
366Dic3—721no—
367C32—21no—
368C32—21no—
369C32—41no—
370C32—41no—
371C32#517411yescharacteristic, complemented
372C23—31no—
373C23—61no—
374Q8—181no—
375Q8—181no—
376D4—181no—
377D4—181no—
378D4—362no—
379C2×C4—61no—
380C2×C4—91no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4—271no—
384C2×C4—541no—
385C2×C4—541no—
386C8—92no—
387C8—541no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—122no—
401C2×C3—122no—
402C2×C3—241no—
403C2×C3—242no—
404C2×C3—241no—
405C2×C3#6423611yescharacteristic
406C2×C3#6423911yescharacteristic
407C2×C3#6424211yescharacteristic
408D3—41no—
409D3—81no—
410C22—61no—
411C22—61no—
412C22—61no—
413C22—121no—
414C22—121no—
415C22—121no—
416C22D3×GO+(4,𝔽2)11yescharacteristic
417C4—91no—
418C4—91no—
419C4—121no—
420C4—361no—
421C4—361no—
422C4—541no—
423C3—21no—
424C3—21no—
425C3—41no—
426C3—41no—
427C3#1059349511yescharacteristic, complemented
428C2—121no—
429C2—241no—
430C2#1172411411yescharacteristic
431C2#1172413911yescharacteristic
432C2#1172414211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radical#11722953C2
hypocenterC33#468
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×C3×#3775C2semidirect
2. C2×#88925C2semidirect
3. #11722976C2semidirect
5. #11722797C2semidirect
6. #11722775C2semidirect
7. #11722773C2semidirect
23. C2×#3775D3semidirect
46. C3×#622D4semidirect
144. #622#67semidirect
236. C33#468semidirect
371. C32#5174semidirect
427. C3#10593495semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37752
C2×#889252
#117229762
#117229532
#117227972
#117227752
#117227732
#105934953
#51749