Finite Group Databasebeta

← #13161769

Subgroups — #13161769

Order 1728.

748 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 115 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×#89040C211yesmaximal, characteristic
2C3×M16×C3⋊D3C211yesmaximal, characteristic
3#11723656C211yesmaximal, characteristic
4C3×#63651C212yesmaximal, complemented
5#11722074C212yesmaximal, complemented
6#10591003—31nomaximal
7D3×#824—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×#634C2212yes—
10C2×#6296C412yes—
11C2×#6294C412yes—
12#89040—21no—
13#89040—21no—
14#89040—21no—
15#89040—21no—
16C33×M16C2211yescharacteristic
17C3×#3748C2211yescharacteristic
18C3×#3748C2212yes—
19C3×#3743—21no—
20C3×C8×C3⋊D3—21no—
21#88809—21no—
22#88808—21no—
23C2×#3797—31no—
24C2×D3×Dic6—32no—
25M16×C3⋊D3D311yescharacteristic
26C3×D3×M16—32no—
27C3×D3×M16—61no—
28#63818—31no—
29D3×#208—32no—
30#63651D312yescomplemented
31C3×#824—32no—
32C3×#824—32no—
33C3×#824—62no—
34#63561—32no—
35#63560—32no—
36#63560—32no—
37C22×C3×C3⋊D3C2×C411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3C2×C414yes—
40C2×C3×C3⋊Dic3C2×C411yescharacteristic
41C3×C4×C3⋊D3D411yescharacteristic
42C3×C4×C3⋊D3D411yescharacteristic
43C3×C4×C3⋊D3D411yescharacteristic
44C3×C4×C3⋊D3D411yescharacteristic
45C3×#634—22no—
46C3×#634—22no—
47#6296—22no—
48#6296—22no—
49#6294—22no—
50#6294—22no—
51C33×C8—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×C3⋊(C3⋊C8)—22no—
54C2×#824—91no—
55D3×#101—92no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×#634C2×D312yes—
58C2×C3×C4×D3—32no—
59C2×C3×C4×D3—61no—
60C2×C3×Dic6—32no—
61C2×C3×Dic6—32no—
62C2×C3×Dic6—62no—
63C2×#627—34no—
64C2×#627—32no—
65C2×#624—32no—
66C2×D3×Dic3—34no—
67#3797—61no—
68#3797—61no—
69#3797—61no—
70#3797—61no—
71D3×Dic6—62no—
72D3×Dic6—62no—
73D3×Dic6—62no—
74D3×Dic6—62no—
75C32×M16—21no—
76C32×M16—22no—
77C32×M16—41no—
78C32×M16C2×D312yes—
79C32×M16C2×D311yescharacteristic
80#3748C2×D312yes—
81#3748C2×D311yescharacteristic
82#3743—21no—
83C8×C3⋊D3—21no—
84C3×#208—32no—
85C3×#208—32no—
86C3×#208—32no—
87C3×#208—61no—
88C3×#208—62no—
89C3×ΓL(1,𝔽25)—62no—
90C3×ΓL(1,𝔽25)—62no—
91C3×D3×C8—62no—
92C3×D3×C8—62no—
93#3712—61no—
94#3711—62no—
95#3710—61no—
96D3×C3⋊C8—62no—
97C22×C33C22×C411yescharacteristic
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3C22⋊C411yescharacteristic
100C2×C3×C3⋊D3C22⋊C411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C3×C3⋊Dic3—24no—
104C3×C3⋊Dic3C22⋊C411yescharacteristic
105C3×C3⋊Dic3C22⋊C411yescharacteristic
106C2×D3×Q8—92no—
107C22×Dic6—91no—
108C2×C3×M16—91no—
109C2×#208—91no—
110D3×M16—32no—
111D3×M16—61no—
112D3×M16—92no—
113C3×#101—92no—
114#836—92no—
115#824—32no—
116#824—32no—
117#824—62no—
118#824—92no—
119#824—92no—
120C22×C3⋊D3C4×D311yescharacteristic
121C22×C3×D3—32no—
122C22×C3×D3—61no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4—41no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D312yes—
128C2×C3⋊Dic3C4×D314yes—
129C2×C3⋊Dic3C4×D311yescharacteristic
130C4×C3⋊D3D1211yescharacteristic
131C4×C3⋊D3D1211yescharacteristic
132C4×C3⋊D3#6711yescharacteristic
133C4×C3⋊D3#6711yescharacteristic
134#634—22no—
135#634—22no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—64no—
141C3×C4×D3—32no—
142C3×C4×D3—32no—
143C3×C4×D3—32no—
144C3×C4×D3—32no—
145C3×C4×D3—61no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C3×C4×D3—61no—
149C3×Dic6—62no—
150C3×Dic6—62no—
151C3×Dic6—62no—
152C3×Dic6—62no—
153C3×Dic6—122no—
154C3×Dic6—122no—
155#627—64no—
156#627—64no—
157#627—62no—
158#627—62no—
159#624—62no—
160#624—62no—
161D3×Dic3—64no—
162D3×Dic3—64no—
163C32×C8—21no—
164C32×C8—22no—
165C32×C8—22no—
166C32×C8—42no—
167C32×C8—42no—
168C3⋊(C3⋊C8)—21no—
169C3⋊(C3⋊C8)—22no—
170C3×C3⋊C8—62no—
171C3×C3⋊C8—62no—
172C3×C3⋊C8—62no—
173C3×C3⋊C8—62no—
174C3×C3⋊C8—122no—
175C2×#101—271noSylow
176C2×C33—21no—
177C2×C33C2×C22⋊C411yescharacteristic
178C3×C3⋊D3—21no—
179C3×C3⋊D3#10112yescomplemented
180C2×C3×Q8—92no—
181C22×C3×C4—91no—
182C22×Dic3—92no—
183D3×Q8—182no—
184D3×Q8—182no—
185D3×Q8—182no—
186D3×Q8—182no—
187C2×C4×D3—32no—
188C2×C4×D3—61no—
189C2×C4×D3—94no—
190C2×Dic6—32no—
191C2×Dic6—32no—
192C2×Dic6—62no—
193C2×Dic6—92no—
194C2×Dic6—94no—
195C2×Dic6—181no—
196C2×Dic6—181no—
197C2×Dic6—181no—
198C2×Dic6—181no—
199C3×M16—21no—
200C3×M16—22no—
201C3×M16—41no—
202C3×M16—91no—
203C3×M16—92no—
204C3×M16—181no—
205C3×M16D3211yescharacteristic
206C3×M16D3212yes—
207C2×C3×C8—181no—
208#208—32no—
209#208—32no—
210#208—32no—
211#208—61no—
212#208—62no—
213#208—181no—
214C2×C3⋊C8—181no—
215ΓL(1,𝔽25)—62no—
216ΓL(1,𝔽25)—62no—
217ΓL(1,𝔽25)—182no—
218D3×C8—62no—
219D3×C8—62no—
220D3×C8—182no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32—41no—
224C22×C32C2×C4×D312yes—
225C22×C32C2×C4×D311yescharacteristic
226C2×C3⋊D3—22no—
227C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
228C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
229C2×C3×D3—32no—
230C2×C3×D3—32no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—64no—
234C2×C3×D3—122no—
235C32×C4—22no—
236C32×C4—21no—
237C32×C4—22no—
238C32×C4—21no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4C2×D1211yescharacteristic
242C32×C4D3×D412yes—
243C32×C4D3×D412yes—
244C32×C4C2×#6711yescharacteristic
245C3⋊Dic3—24no—
246C3⋊Dic3(C2×D3)⋊C411yescharacteristic
247C3⋊Dic3(C2×D3)⋊C411yescharacteristic
248C3×Dic3—32no—
249C3×Dic3—32no—
250C3×Dic3—61no—
251C3×Dic3—61no—
252C3×Dic3—64no—
253C3×Dic3—64no—
254C3×Dic3—124no—
255C22×Q8—271no—
256C2×M16—91no—
257C2×M16—271no—
258#101—92no—
259#101—272no—
260C33C2×#10111yescharacteristic, complemented, Sylow
261C23×C3—91no—
262C22×D3—32no—
263C22×D3—61no—
264C3×Q8—182no—
265C3×Q8—182no—
266C2×C3×C4—22no—
267C2×C3×C4—21no—
268C2×C3×C4—41no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—91no—
274C2×C3×C4—94no—
275C2×C3×C4C2×D3212yes—
276C2×C3×C4C2×D3211yescharacteristic
277C2×Dic3—32no—
278C2×Dic3—34no—
279C2×Dic3—34no—
280C2×Dic3—61no—
281C2×Dic3—64no—
282C2×Dic3—182no—
283C2×Dic3—182no—
284C4×D3—32no—
285C4×D3—32no—
286C4×D3—32no—
287C4×D3—32no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—184no—
293C4×D3—184no—
294Dic6—62no—
295Dic6—62no—
296Dic6—62no—
297Dic6—62no—
298Dic6—122no—
299Dic6—122no—
300Dic6—184no—
301Dic6—184no—
302Dic6—182no—
303Dic6—182no—
304C3×C8—22no—
305C3×C8—22no—
306C3×C8—21no—
307C3×C8—42no—
308C3×C8—42no—
309C3×C8—182no—
310C3×C8—181no—
311C3⋊C8—62no—
312C3⋊C8—62no—
313C3⋊C8—62no—
314C3⋊C8—62no—
315C3⋊C8—122no—
316C2×C32—21no—
317C2×C32—22no—
318C2×C32—22no—
319C2×C32—21no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—42no—
323C2×C32—81no—
324C2×C32D3×C22⋊C412yes—
325C2×C32C2×(C2×D3)⋊C411yescharacteristic
326C3⋊D3—21no—
327C3⋊D3#82412yescomplemented
328C3×D3—34no—
329C3×D3—62no—
330C3×D3—62no—
331C3×D3—121no—
332C2×Q8—92no—
333C2×Q8—272no—
334C2×Q8—541no—
335C2×Q8—541no—
336C2×Q8—541no—
337C2×Q8—541no—
338C22×C4—91no—
339C22×C4—272no—
340M16—92no—
341M16—91no—
342M16—181no—
343M16—541no—
344M16#111311yescharacteristic
345C2×C8—181no—
346C2×C8—541no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3—41no—
350C22×C3—91no—
351C22×C3—91no—
352C22×C3—182no—
353C22×C3C4×D3212yes—
354C22×C3C4×D3211yescharacteristic
355C2×D3—32no—
356C2×D3—32no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—64no—
360C2×D3—122no—
361C3×C4—22no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—22no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—91no—
368C3×C4—91no—
369C3×C4—184no—
370C3×C4D3×D1212yes—
371C3×C4#380211yescharacteristic
372C3×C4D3×#6712yes—
373C3×C4#381111yescharacteristic
374Dic3—32no—
375Dic3—32no—
376Dic3—61no—
377Dic3—61no—
378Dic3—64no—
379Dic3—64no—
380Dic3—124no—
381C32—22no—
382C32—21no—
383C32—41no—
384C32D3×#10112yescomplemented
385C32C2×#82411yescharacteristic, complemented
386C23—91no—
387Q8—182no—
388Q8—182no—
389Q8—542no—
390Q8—542no—
391C2×C4—91no—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—94no—
397C2×C4—542no—
398C2×C4—542no—
399C2×C4C2×#111311yescharacteristic
400C8—21no—
401C8—182no—
402C8—181no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—41no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—81no—
411C2×C3—92no—
412C2×C3—181no—
413C2×C3D3×(C2×D3)⋊C412yes—
414C2×C3#6392111yescharacteristic
415D3—34no—
416D3—62no—
417D3—62no—
418D3—121no—
419C22—91no—
420C22—91no—
421C22—182no—
422C22C4×#111311yescharacteristic
423C4—91no—
424C4—91no—
425C4—184no—
426C4C4⋊#111311yescharacteristic
427C4#8904911yescharacteristic
428C3—21no—
429C3—22no—
430C3—41no—
431C3D3×#82412yescomplemented
432C3#1059100311yescharacteristic, complemented
433C2—21no—
434C2—92no—
435C2—181no—
436C2#1172370011yescharacteristic
subgroupG/H
centerC2#11723700
commutatorC22×C33C22×C4
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33C2×C22⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterM16#1113
hypocenterC33C2×#101
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
4. C3×#63651C2semidirect
5. #11722074C2semidirect
30. #63651D3semidirect
179. C3×C3⋊D3#101semidirect
260. C33C2×#101semidirect
327. C3⋊D3#824semidirect
384. C32D3×#101semidirect
385. C32C2×#824semidirect
431. C3D3×#824semidirect
432. C3#10591003semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C3×M16×C3⋊D32
#117236562
C3×#636512
#117220742
#105910033
D3×#8243