Finite Group Databasebeta

← #13165801

Subgroups — #13165801

Order 1728.

634 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C22×D3×#157C211yesmaximal, characteristic
2C2×#88946C211yesmaximal, characteristic, complemented
3C2×#88945C211yesmaximal, characteristic, complemented
4C2×(C2×D3)⋊#157C211yesmaximal, characteristic
5C2×(C2×D3)⋊#157C211yesmaximal, characteristic
6C3⋊#63747C211yesmaximal, characteristic, complemented
7C3×#63747C211yesmaximal, characteristic, complemented
8#10593569—31nomaximal
9C22×D3×C3⋊D3C2211yescharacteristic
10C22×C3⋊#157C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3⋊#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C22×C3×#157C2211yescharacteristic
15C22×C3×#157C2211yescharacteristic
16C2×D3×#157—22no—
17C2×D3×#157—22no—
18C2×D3×#157—22no—
19#88946—22no—
20#88945—22no—
21(C2×D3)⋊#157—22no—
22(C2×D3)⋊#157C414yescomplemented
23C23×#157—31no—
24C2×C22⋊#157—31no—
25C2×C22⋊#157—31no—
26C2×#3777—31no—
27C2×#3777—31no—
28#63747—31no—
29#63747D311yescharacteristic, complemented
30C22×C3⋊(C3⋊D3)Q811yescharacteristic
31C22×C3×C3⋊D3C2311yescharacteristic
32C22×C32×D3Q811yescharacteristic
33C2×D3×C3⋊D3—21no—
34C2×D3×C3⋊D3—21no—
35C2×D3×C3⋊D3—21no—
36C2×D3×C3⋊D3—21no—
37C2×D3×C3⋊D3C2×C412yes—
38C2×D3×C3⋊D3D412yes—
39C2×C3⋊#157—21no—
40C2×C3⋊#157—21no—
41C2×C3⋊#157—21no—
42C2×C3⋊#157—21no—
43C2×C3⋊#157—21no—
44C2×C3⋊#157—21no—
45C2×C3⋊#157—21no—
46C2×C3⋊#157C2×C412yescomplemented
47C2×C3⋊#157D412yes—
48C2×C3×#157—21no—
49C2×C3×#157—21no—
50C2×C3×#157—21no—
51C2×C3×#157—21no—
52C2×C3×#157—21no—
53C2×C3×#157—21no—
54C2×C3×#157—21no—
55C2×C3×#157C2×C412yescomplemented
56C2×C3×#157D412yes—
57D3×#157—28no—
58#4668—91nomaximal
59C23×C3⋊D3—31no—
60C22×D32—121no—
61C22×#157—31no—
62C22×#157—31no—
63C22×#157—31no—
64C22×#157—62no—
65C22×#157—62no—
66C22×#157—62no—
67C22×#157C2×D311yescharacteristic
68C22×#157C2×D311yescharacteristic
69C22×#157C2×D311yescharacteristic
70C22⋊#157—34no—
71C22⋊#157—62no—
72#3777—62no—
73#3777—62no—
74C22×C33C2×Q811yescharacteristic
75C2×C3⋊(C3⋊D3)—21no—
76C2×C3⋊(C3⋊D3)—21no—
77C2×C3⋊(C3⋊D3)C4⋊C412yescomplemented
78C2×C3×C3⋊D3C22×C411yescharacteristic
79C2×C3×C3⋊D3C2×D411yescharacteristic
80C2×C3×C3⋊D3C2×D411yescharacteristic
81C2×C3×C3⋊D3C2×D411yescharacteristic
82C2×C3×C3⋊D3#4111yescharacteristic
83C2×C3×C3⋊D3#4111yescharacteristic
84C2×C32×D3—21no—
85C2×C32×D3—21no—
86C2×C32×D3C4⋊C412yescomplemented
87D3×C3⋊D3—22no—
88D3×C3⋊D3—22no—
89D3×C3⋊D3—22no—
90D3×C3⋊D3—22no—
91C3⋊#157—22no—
92C3⋊#157—22no—
93C3⋊#157—41no—
94C3×#157—22no—
95C3×#157—22no—
96C3×#157—41no—
97C22×C4×D3—91no—
98C2×(C2×D3)⋊C4—91no—
99C2×(C2×D3)⋊C4—91no—
100C2×C4⋊Dic3—91no—
101C2×Dic3⋊C4—91no—
102C3×#95—91no—
103C3⋊#95—91no—
104C23×C32—31no—
105C22×C3⋊D3—31no—
106C22×C3⋊D3—32no—
107C22×C3⋊D3—32no—
108C22×C3⋊D3—61no—
109C22×C3⋊D3—61no—
110C22×C3⋊D3—61no—
111C22×C3⋊D3—61no—
112C22×C3⋊D3—121no—
113C22×C3⋊D3—241no—
114C22×C3⋊D3C22×D311yescharacteristic
115C22×C3×D3—41no—
116C22×C3×D3—121no—
117C2×D32—122no—
118C2×D32—122no—
119C2×D32—122no—
120C2×D32—122no—
121C2×D32—122no—
122C2×D32—122no—
123C2×#157—21no—
124C2×#157—21no—
125C2×#157—21no—
126C2×#157—21no—
127C2×#157—21no—
128C2×#157—21no—
129C2×#157—21no—
130C2×#157—32no—
131C2×#157—32no—
132C2×#157—68no—
133C2×#157—61no—
134C2×#157—61no—
135C2×#157—61no—
136C2×#157—61no—
137C2×#157—61no—
138C2×#157—61no—
139C2×#157—61no—
140C2×#157C4×D312yescomplemented
141C2×#157D1212yes—
142#385—271noSylow
143C2×C33C2×C4⋊C411yescharacteristic
144C2×C33C22⋊Q811yescharacteristic
145C2×C33C22⋊Q811yescharacteristic
146C3⋊(C3⋊D3)—22no—
147C3×C3⋊D3C4×D411yescharacteristic
148C3×C3⋊D3C4×D411yescharacteristic
149C3×C3⋊D3#12011yescharacteristic
150C3×C3⋊D3#12311yescharacteristic
151C32×D3—22no—
152C23×D3—91no—
153C23×D3—361no—
154C22×C3×C4—91no—
155C22×C3×C4—91no—
156C22×C3×C4—91no—
157C22×Dic3—91no—
158C22×Dic3—91no—
159C22×Dic3—91no—
160C2×C4×D3—182no—
161C2×C4×D3—182no—
162C2×C4×D3—182no—
163(C2×D3)⋊C4—94no—
164(C2×D3)⋊C4—182no—
165C4⋊Dic3—182no—
166Dic3⋊C4—182no—
167C22×C32—32no—
168C22×C32—41no—
169C22×C32—61no—
170C22×C32—61no—
171C22×C32—81no—
172C22×C32D3×Q811yescharacteristic
173C2×C3⋊D3—32no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—62no—
177C2×C3⋊D3—62no—
178C2×C3⋊D3—62no—
179C2×C3⋊D3—62no—
180C2×C3⋊D3—122no—
181C2×C3⋊D3—122no—
182C2×C3⋊D3—122no—
183C2×C3⋊D3—242no—
184C2×C3⋊D3—242no—
185C2×C3⋊D3—242no—
186C2×C3⋊D3C2×C4×D311yescharacteristic
187C2×C3⋊D3C2×D1211yescharacteristic
188C2×C3⋊D3#23511yescharacteristic
189C2×C3⋊D3D3×D411yescharacteristic
190C2×C3⋊D3D3×D411yescharacteristic
191C2×C3⋊D3#23711yescharacteristic
192C2×C3×D3—42no—
193C2×C3×D3—42no—
194C2×C3×D3—42no—
195C2×C3×D3—121no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201D32—124no—
202D32—124no—
203D32—124no—
204D32—124no—
205#157—22no—
206#157—22no—
207#157—41no—
208#157—62no—
209#157—62no—
210#157—121no—
211C23×C4—271no—
212C2×C4⋊C4—271no—
213C2×C4⋊C4—271no—
214C2×C22⋊C4—271no—
215C2×C22⋊C4—271no—
216#95—91no—
217#95—271no—
218C33#38511yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C23×C3—121no—
221C22×D3—91no—
222C22×D3—92no—
223C22×D3—92no—
224C22×D3—121no—
225C22×D3—181no—
226C22×D3—181no—
227C22×D3—181no—
228C22×D3—181no—
229C22×D3—362no—
230C22×D3—362no—
231C22×D3—361no—
232C22×D3—362no—
233C22×D3—362no—
234C22×D3—362no—
235C22×D3—362no—
236C22×D3—721no—
237C22×D3PSU(3,𝔽2)11yescharacteristic
238C2×C3×C4—92no—
239C2×C3×C4—92no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247C2×Dic3—92no—
248C2×Dic3—92no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—181no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C2×Dic3—181no—
255C2×Dic3—181no—
256C4×D3—188no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—62no—
261C2×C32—81no—
262C2×C32—81no—
263C2×C32—81no—
264C2×C32D3×C4⋊C411yescharacteristic
265C2×C32#89611yescharacteristic
266C2×C32#89711yescharacteristic
267C3⋊D3—62no—
268C3⋊D3—124no—
269C3⋊D3—244no—
270C3⋊D3C4×D1211yescharacteristic
271C3⋊D3#67⋊C411yescharacteristic
272C3⋊D3#88211yescharacteristic
273C3⋊D3#88311yescharacteristic
274C3×D3—44no—
275C3×D3—121no—
276C3×D3—121no—
277C3×D3—121no—
278C3×D3—121no—
279C24—271no—
280C22×C4—91no—
281C22×C4—91no—
282C22×C4—91no—
283C22×C4—271no—
284C22×C4—271no—
285C22×C4—271no—
286C22×C4—542no—
287C22×C4—542no—
288C22×C4—542no—
289C4⋊C4—542no—
290C4⋊C4—542no—
291C22⋊C4—274no—
292C22⋊C4—542no—
293C22×C3—41no—
294C22×C3—81no—
295C22×C3—91no—
296C22×C3—91no—
297C22×C3—91no—
298C22×C3—91no—
299C22×C3—91no—
300C22×C3—91no—
301C22×C3—122no—
302C22×C3—122no—
303C22×C3—122no—
304C22×C3C2×PSU(3,𝔽2)11yescharacteristic
305C2×D3—21no—
306C2×D3—21no—
307C2×D3—92no—
308C2×D3—121no—
309C2×D3—121no—
310C2×D3—121no—
311C2×D3—121no—
312C2×D3—121no—
313C2×D3—121no—
314C2×D3—182no—
315C2×D3—182no—
316C2×D3—182no—
317C2×D3—182no—
318C2×D3—181no—
319C2×D3—181no—
320C2×D3—362no—
321C2×D3—364no—
322C2×D3—364no—
323C2×D3—364no—
324C2×D3—364no—
325C2×D3—362no—
326C2×D3—362no—
327C2×D3—722no—
328C2×D3—722no—
329C2×D3—722no—
330C2×D3#377712yescomplemented
331C3×C4—182no—
332C3×C4—182no—
333C3×C4—361no—
334Dic3—182no—
335Dic3—182no—
336Dic3—361no—
337C32—41no—
338C32—81no—
339C32#466811yescharacteristic, complemented
340C23—31no—
341C23—91no—
342C23—272no—
343C23—272no—
344C23—271no—
345C23—541no—
346C23—541no—
347C23—541no—
348C23—541no—
349C2×C4—92no—
350C2×C4—92no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—272no—
359C2×C4—272no—
360C2×C4—548no—
361C2×C4—541no—
362C2×C4—541no—
363C2×C4—541no—
364C2×C4—541no—
365C2×C4—541no—
366C2×C4—541no—
367C2×C4—541no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—81no—
372C2×C3—81no—
373C2×C3—81no—
374C2×C3—91no—
375C2×C3—91no—
376C2×C3—91no—
377C2×C3—91no—
378C2×C3—124no—
379C2×C3C2×#377711yescharacteristic
380C2×C3C22⋊PSU(3,𝔽2)11yescharacteristic
381C2×C3C22⋊PSU(3,𝔽2)11yescharacteristic
382D3—22no—
383D3—121no—
384D3—121no—
385D3—121no—
386D3—121no—
387D3—182no—
388D3—364no—
389D3—724no—
390C22—32no—
391C22—61no—
392C22—61no—
393C22—91no—
394C22—91no—
395C22—91no—
396C22—91no—
397C22—91no—
398C22—91no—
399C22—272no—
400C22—542no—
401C22—542no—
402C22—542no—
403C22—542no—
404C22—541no—
405C22—541no—
406C22D3×PSU(3,𝔽2)11yescharacteristic
407C4—182no—
408C4—182no—
409C4—361no—
410C4—542no—
411C4—542no—
412C4—1081no—
413C3—41no—
414C3—81no—
415C3#1059356911yescharacteristic, complemented
416C2—62no—
417C2—91no—
418C2—91no—
419C2—91no—
420C2—91no—
421C2—542no—
422C2D3×#377711yescharacteristic
423C2#1172415011yescharacteristic
424C2#1172415111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×PSU(3,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×Q8
prime radicalC22×D3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypocenterC33#385
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#88946C2semidirect
3. C2×#88945C2semidirect
6. C3⋊#63747C2semidirect
7. C3×#63747C2semidirect
22. (C2×D3)⋊#157C4semidirect
29. #63747D3semidirect
46. C2×C3⋊#157C2×C4semidirect
55. C2×C3×#157C2×C4semidirect
77. C2×C3⋊(C3⋊D3)C4⋊C4semidirect
86. C2×C32×D3C4⋊C4semidirect
140. C2×#157C4×D3semidirect
218. C33#385semidirect
330. C2×D3#3777semidirect
339. C32#4668semidirect
415. C3#10593569semidirect

Maximal subgroups

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

subgroupindex
C22×D3×#1572
C2×#889462
C2×#889452
C2×(C2×D3)⋊#1572
C3⋊#637472
C3×#637472
#105935693
#46689