Finite Group Databasebeta

← #13165679

Subgroups — #13165679

Order 1728.

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

normal subgroups only

This group has 474 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
1#11722953C211yesmaximal, characteristic
2D3×#3724C211yesmaximal, characteristic, complemented
3#11722797C211yesmaximal, characteristic, complemented
4#11722779C211yesmaximal, characteristic, complemented
5#11722773C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#622C211yesmaximal, characteristic, complemented
8#10593492—31nomaximal
9C2×D3×C3⋊Dic3C2211yescharacteristic
10C2×C3×#625—21no—
11C2×C3×#625C2211yescharacteristic
12C2×C3×D3×Dic3—21no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15#88815C2211yescharacteristic
16#88815C2211yescharacteristic
17#88806—21no—
18(C2×D3)⋊(C3⋊Dic3)—21no—
19C3×#3724C2211yescharacteristic
20#63980—31no—
21#63970—61no—
22C2×#3724—31no—
23D3×Dic3⋊C4—61no—
24C22⋊#622—31no—
25#63743—31no—
26#63743—31no—
27#63739—31no—
28#63739D311yescharacteristic, complemented
29C22×C32×D3—21no—
30C22×C32×D3D411yescharacteristic
31C2×C3⋊(C3⋊Dic3)D411yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C2×C32×Dic3—21no—
36D3×C3⋊Dic3—21no—
37D3×C3⋊Dic3—21no—
38C3×#625—21no—
39C3×#625—21no—
40C3×#625—42no—
41C3×D3×Dic3—42no—
42C3⋊#622—21no—
43C3×#622—21no—
44#4774—91nomaximal
45C22×C3⋊Dic3—31no—
46C2×C3×#67—61no—
47C2×C3×#67—61no—
48C2×C3×#67—61no—
49C22×C3×Dic3—61no—
50C2×C3×C4×D3—61no—
51C2×#625—21no—
52C2×#625C2×D311yescharacteristic
53C2×D3×Dic3—21no—
54C2×D3×Dic3—61no—
55C2×D3×Dic3—61no—
56C2×#622—31no—
57C2×#622C2×D311yescharacteristic
58#3757—61no—
59#3750—61no—
60C3×Dic3⋊C4—61no—
61#3724—31no—
62#3724—31no—
63#3724—61no—
64#3724—61no—
65#3724—121no—
66#3724C2×D311yescharacteristic
67Dic3⋊Dic3—61no—
68Dic3⋊Dic3—61no—
69(C2×D3)⋊Dic3—61no—
70(C2×D3)⋊Dic3—61no—
71(C2×D3)⋊Dic3—121no—
72C22×C33C2×D411yescharacteristic
73C2×C32×D3—21no—
74C2×C32×D3—21no—
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C2×C32×D3D812yes—
79C3⋊(C3⋊Dic3)—21no—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3—41no—
82C3×C3⋊Dic3#4111yescharacteristic
83C3×C3⋊Dic3#4111yescharacteristic
84C32×Dic3—41no—
85#939—181no—
86#938—91no—
87C2×Dic3⋊C4—181no—
88D3×C4⋊C4—91no—
89C3×#102—91no—
90#832—91no—
91(C2×D3)⋊C8—91no—
92#818—91no—
93#807—91no—
94C23×C32—31no—
95C23×C32—61no—
96C22×C3×D3—21no—
97C22×C3×D3—21no—
98C22×C3×D3—21no—
99C22×C3×D3—21no—
100C22×C3×D3—41no—
101C2×C32×C4—61no—
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—31no—
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—61no—
107C2×C3⋊Dic3—61no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—121no—
110C2×C3⋊Dic3—121no—
111C2×C3⋊Dic3C22×D311yescharacteristic
112C3×#67—121no—
113C3×#67—121no—
114C3×#67—122no—
115C3×#67—122no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—41no—
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—
127C2×C3×Dic3—122no—
128C3×C4×D3—122no—
129#625—21no—
130#625—21no—
131#625—42no—
132D3×Dic3—42no—
133D3×Dic3—62no—
134D3×Dic3—62no—
135D3×Dic3—121no—
136D3×Dic3—121no—
137#622—21no—
138#622—61no—
139#480—271noSylow
140C2×C33#12311yescharacteristic
141C2×C33C2×D811yescharacteristic
142C2×C33#13711yescharacteristic
143C32×D3—22no—
144C32×D3—81no—
145C2×C3×D4—91no—
146C2×C3×D4—181no—
147C22×C3×C4—181no—
148C2×#67—61no—
149C2×#67—61no—
150C2×#67—61no—
151C22×Dic3—61no—
152C22×Dic3—181no—
153C22×Dic3—181no—
154C2×C4×D3—61no—
155C2×C4×D3—91no—
156C2×C3×C8—91no—
157C3×C4⋊C4—91no—
158#217—181no—
159#217—181no—
160#217—361no—
161(C2×D3)⋊C4—181no—
162C4⋊Dic3—91no—
163C4⋊Dic3—91no—
164Dic3⋊C4—61no—
165Dic3⋊C4—181no—
166Dic3⋊C4—181no—
167Dic3⋊C4—181no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—361no—
171C2×C3⋊C8—91no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32—32no—
175C22×C32—41no—
176C22×C32—41no—
177C22×C32—61no—
178C22×C32—61no—
179C22×C32—121no—
180C22×C32—121no—
181C22×C32—121no—
182C22×C32D3×D411yescharacteristic
183C2×C3×D3—22no—
184C2×C3×D3—22no—
185C2×C3×D3—22no—
186C2×C3×D3—22no—
187C2×C3×D3—41no—
188C2×C3×D3—41no—
189C2×C3×D3—41no—
190C2×C3×D3—41no—
191C2×C3×D3—41no—
192C2×C3×D3—42no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C2×C3×D3—81no—
197C2×C3×D3—81no—
198C32×C4—121no—
199C3⋊Dic3—41no—
200C3⋊Dic3—41no—
201C3⋊Dic3—62no—
202C3⋊Dic3—61no—
203C3⋊Dic3—121no—
204C3⋊Dic3—122no—
205C3⋊Dic3—241no—
206C3⋊Dic3#23511yescharacteristic
207C3⋊Dic3#23711yescharacteristic
208C3×Dic3—41no—
209C3×Dic3—41no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C3×Dic3—81no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3—241no—
220C3×Dic3—241no—
221#121—271no—
222C2×C4⋊C4—271no—
223#107—271no—
224#107—271no—
225#102—91no—
226#102—271no—
227C22⋊C8—271no—
228C33#48011yescharacteristic, complemented, Sylow
229C23×C3—61no—
230C23×C3—61no—
231C23×C3—61no—
232C23×C3—61no—
233C23×C3—121no—
234C22×D3—21no—
235C22×D3GO+(4,𝔽2)11yescharacteristic
236C3×D4—181no—
237C3×D4—181no—
238C3×D4—362no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—91no—
242C2×C3×C4—121no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4—362no—
246#67—121no—
247#67—121no—
248#67—122no—
249#67—122no—
250C2×Dic3—21no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—91no—
258C2×Dic3—121no—
259C2×Dic3—121no—
260C2×Dic3—122no—
261C2×Dic3—181no—
262C2×Dic3—181no—
263C2×Dic3—182no—
264C2×Dic3—182no—
265C2×Dic3—361no—
266C2×Dic3—361no—
267C2×Dic3—361no—
268C2×Dic3—361no—
269C4×D3—122no—
270C4×D3—181no—
271C4×D3—181no—
272C3×C8—181no—
273C3⋊C8—181no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—62no—
287C2×C32—241no—
288C2×C32#88311yescharacteristic
289C2×C32D3×D811yescharacteristic
290C2×C32#91511yescharacteristic
291C3×D3—24no—
292C3×D3—42no—
293C3×D3—81no—
294C3×D3—82no—
295C3×D3—81no—
296C2×D4—91no—
297C2×D4—181no—
298C22×C4—181no—
299C22×C4—271no—
300C2×C8—91no—
301C2×C8—271no—
302C4⋊C4—91no—
303C4⋊C4—271no—
304C4⋊C4—271no—
305C4⋊C4—541no—
306C22⋊C4—541no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—62no—
312C22×C3—62no—
313C22×C3—62no—
314C22×C3—62no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3—121no—
318C22×C3—121no—
319C22×C3—121no—
320C22×C3—122no—
321C22×C3—121no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3—241no—
325C22×C3—241no—
326C22×C3C2×GO+(4,𝔽2)11yescharacteristic
327C2×D3—21no—
328C2×D3—21no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C2×D3#377412yes—
333C3×C4—91no—
334C3×C4—91no—
335C3×C4—121no—
336C3×C4—121no—
337C3×C4—241no—
338C3×C4—361no—
339C3×C4—361no—
340Dic3—41no—
341Dic3—61no—
342Dic3—61no—
343Dic3—61no—
344Dic3—61no—
345Dic3—121no—
346Dic3—121no—
347Dic3—121no—
348Dic3—121no—
349Dic3—182no—
350Dic3—181no—
351Dic3—241no—
352Dic3—241no—
353Dic3—361no—
354Dic3—362no—
355Dic3—721no—
356C32—21no—
357C32—21no—
358C32—41no—
359C32—41no—
360C32#477411yescharacteristic, complemented
361C23—31no—
362C23—61no—
363D4—181no—
364D4—181no—
365D4—362no—
366C2×C4—61no—
367C2×C4—91no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—271no—
371C2×C4—362no—
372C2×C4—541no—
373C2×C4—541no—
374C8—181no—
375C8—541no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—41no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3—41no—
388C2×C3—64no—
389C2×C3—122no—
390C2×C3—241no—
391C2×C3—242no—
392C2×C3—241no—
393C2×C3#6423311yescharacteristic
394C2×C3C2×#377411yescharacteristic
395C2×C3#6423911yescharacteristic
396D3—22no—
397D3—81no—
398C22—32no—
399C22—61no—
400C22—61no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22D3×GO+(4,𝔽2)11yescharacteristic
405C4—91no—
406C4—91no—
407C4—121no—
408C4—361no—
409C4—361no—
410C4—541no—
411C3—21no—
412C3—21no—
413C3—41no—
414C3—41no—
415C3#1059349211yescharacteristic, complemented
416C2—62no—
417C2—241no—
418C2#1172412111yescharacteristic
419C2D3×#377411yescharacteristic
420C2#1172414211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radical#11722953C2
hypocenterC33#480
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. D3×#3724C2semidirect
3. #11722797C2semidirect
4. #11722779C2semidirect
5. #11722773C2semidirect
6. C3×#63739C2semidirect
7. (C2×D3)⋊#622C2semidirect
28. #63739D3semidirect
228. C33#480semidirect
360. C32#4774semidirect
415. C3#10593492semidirect

Maximal subgroups

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

subgroupindex
#117229532
D3×#37242
#117227972
#117227792
#117227732
C3×#637392
(C2×D3)⋊#6222
#105934923
#47749