Finite Group Databasebeta

← #13140920

Subgroups — #13140920

Order 1728.

560 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 87 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×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic, complemented
3#11721106C211yesmaximal, characteristic
4C2×#88433C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic
7#11720309C211yesmaximal, characteristic, complemented
8#10591603—31nomaximal
9#10591595—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×(C3⋊D3)⋊C3—22no—
12Q8×(C3⋊D3)⋊C3—22no—
13C2×C4×(C3⋊D3)⋊C3—21no—
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#634⋊C3—21no—
16C2×#634⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C4⋊#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20#3756⋊C3—22no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88450—21no—
23#88449—21no—
24#88433—22no—
25C2×Q8×C3⋊D3D311yescharacteristic
26C2×C3×D3×Q8—31no—
27C2×#3756D311yescharacteristic, complemented
28C2×C3×#216—31no—
29#63871—31no—
30#63857—31no—
31C2×#3719—31no—
32C2×#3719—31no—
33C3×#835—31no—
34C3×#835—31no—
35C3×#835—61no—
36#63567—31no—
37#63565—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)D412yes—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×#1084—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4×(C3⋊D3)⋊C3—24no—
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3—21no—
52#634⋊C3—24no—
53#634⋊C3—21no—
54#634⋊C3—21no—
55#634⋊C3—21no—
56#634⋊C3D412yes—
57#6180—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59#5228—91no—
60#5186—91no—
61#4809—91no—
62C2×C32×Q8—21no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C2×D311yescharacteristic
65Q8×C3⋊D3—22no—
66Q8×C3⋊D3—22no—
67C2×C4×C3⋊D3—21no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×#634—21no—
70C2×#634C2×D311yescharacteristic
71C2×#634C2×D311yescharacteristic
72C3×D3×Q8—62no—
73C3×D3×Q8—62no—
74C2×C3×C4×D3—31no—
75C2×C3×C4×D3—61no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79#3756—22no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×#216—62no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—61no—
89Dic3⋊Dic3—61no—
90Dic3⋊Dic3—61no—
91#3722—61no—
92(C2×D3)⋊Dic3—61no—
93#3719—62no—
94#3719—62no—
95C22×H(3,𝔽3)C2×D411yescharacteristic
96C2×(C3⋊D3)⋊C3—21no—
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3Q1612yescomplemented
99C4×H(3,𝔽3)—22no—
100C4×H(3,𝔽3)C2×D411yescharacteristic
101C4×H(3,𝔽3)C2×D411yescharacteristic
102#1084—41no—
103(C3⋊Dic3)⋊C3—21no—
104(C3⋊Dic3)⋊C3—22no—
105(C3⋊Dic3)⋊C3—41no—
106C22×C3×Q8—91no—
107C2×D3×Q8—31no—
108C2×D3×Q8—91no—
109C2×C3×Q16—91no—
110#946—91no—
111C2×#216—31no—
112C2×#216—91no—
113C2×#216—91no—
114C2×#216—91no—
115#924—91no—
116C2×Dic12—91no—
117#897—91no—
118C3×#103—91no—
119#835—31no—
120#835—31no—
121#835—61no—
122#835—91no—
123#830—91no—
124(C2×D3)⋊C8—91no—
125(C2×D3)⋊C8—91no—
126#816—91no—
127#810—91no—
128C22×C3⋊D3#6711yescharacteristic
129C22×C3×D3—31no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—22no—
133C32×Q8—41no—
134C32×Q8#6712yes—
135C32×Q8#6712yes—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3#6711yescharacteristic
145C4×C3⋊D3—24no—
146C4×C3⋊D3—21no—
147C4×C3⋊D3—21no—
148#634—24no—
149#634—21no—
150#634—21no—
151#634—21no—
152#634#6712yes—
153C2×C3×Dic3—31no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—121no—
159C3×C4×D3—64no—
160C3×C4×D3—61no—
161C3×C4×D3—61no—
162C3×Dic6—32no—
163C3×Dic6—64no—
164C3×Dic6—61no—
165C3×Dic6—61no—
166C3×Dic6—61no—
167C3⋊(C3⋊C8)—21no—
168C3×C3⋊C8—61no—
169C3×C3⋊C8—61no—
170C3×C3⋊C8—61no—
171C3×C3⋊C8—121no—
172#451—271noSylow
173C2×H(3,𝔽3)#12011yescharacteristic
174C2×H(3,𝔽3)C2×Q1611yescharacteristic
175C2×H(3,𝔽3)#13711yescharacteristic
176(C3⋊D3)⋊C3—22no—
177C2×C3×Q8—31no—
178C2×C3×Q8—31no—
179C2×C3×Q8—61no—
180C2×C3×Q8—91no—
181C2×C3×Q8—91no—
182C2×C3×Q8—181no—
183C2×C3×Q8—182no—
184C2×C3×Q8—182no—
185C2×C3×Q8D3211yescharacteristic
186C22×C3×C4—91no—
187C22×C3×C4—181no—
188D3×Q8—62no—
189D3×Q8—62no—
190D3×Q8—182no—
191D3×Q8—182no—
192C2×C4×D3—31no—
193C2×C4×D3—61no—
194C2×C4×D3—91no—
195C2×C4×D3—181no—
196C2×Dic6—31no—
197C2×Dic6—31no—
198C2×Dic6—61no—
199C2×Dic6—91no—
200C2×Dic6—91no—
201C2×Dic6—181no—
202C3×Q16—182no—
203C2×C3×C8—91no—
204C2×C3×C8—91no—
205C3×C4⋊C4—91no—
206#217—181no—
207#216—62no—
208#216—182no—
209#216—182no—
210#216—182no—
211(C2×D3)⋊C4—181no—
212(C2×D3)⋊C4—181no—
213C4⋊Dic3—31no—
214C4⋊Dic3—31no—
215C4⋊Dic3—61no—
216C4⋊Dic3—181no—
217Dic3⋊C4—181no—
218Dic3⋊C4—181no—
219C2×C3⋊C8—31no—
220C2×C3⋊C8—31no—
221C2×C3⋊C8—31no—
222C2×C3⋊C8—61no—
223C2×C3⋊C8—91no—
224Dic12—182no—
225C22×C32—21no—
226C22×C32D3×D411yescharacteristic
227C22×C32C2×#6711yescharacteristic
228C2×C3⋊D3—21no—
229C2×C3⋊D3—21no—
230C2×C3⋊D3#21612yescomplemented
231C2×C3×D3—32no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—42no—
239C32×C4D3×D411yescharacteristic
240C32×C4C2×#6711yescharacteristic
241C32×C4C2×#6711yescharacteristic
242C32×C4C2×#6711yescharacteristic
243C3⋊Dic3—21no—
244C3⋊Dic3—22no—
245C3⋊Dic3—41no—
246C3×Dic3—62no—
247C3×Dic3—61no—
248C3×Dic3—121no—
249C3×Dic3—121no—
250C3×Dic3—121no—
251C3×Dic3—241no—
252C22×Q8—91no—
253C2×Q16—91no—
254C2×Q16—271no—
255C22⋊Q8—271no—
256#103—91no—
257#103—271no—
258C22⋊C8—271no—
259H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
260C23×C3—91no—
261C22×D3—31no—
262C22×D3—91no—
263C3×Q8—21no—
264C3×Q8—32no—
265C3×Q8—32no—
266C3×Q8—61no—
267C3×Q8—61no—
268C3×Q8—62no—
269C3×Q8—92no—
270C3×Q8—121no—
271C3×Q8—181no—
272C3×Q8—181no—
273C3×Q8—181no—
274C3×Q8—184no—
275C3×Q8#62512yes—
276C2×C3×C4—21no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—91no—
283C2×C3×C4—121no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4—184no—
287C2×C3×C4—181no—
288C2×C3×C4—181no—
289C2×C3×C4—181no—
290C2×C3×C4C2×D3211yescharacteristic
291C2×Dic3—31no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—61no—
295C2×Dic3—61no—
296C2×Dic3—91no—
297C2×Dic3—121no—
298C2×Dic3—181no—
299C2×Dic3—181no—
300C4×D3—64no—
301C4×D3—61no—
302C4×D3—61no—
303C4×D3—181no—
304C4×D3—181no—
305C4×D3—184no—
306Dic6—32no—
307Dic6—64no—
308Dic6—61no—
309Dic6—61no—
310Dic6—61no—
311Dic6—92no—
312Dic6—181no—
313Dic6—184no—
314Dic6—181no—
315Dic6—181no—
316C3×C8—181no—
317C3×C8—181no—
318C3⋊C8—61no—
319C3⋊C8—61no—
320C3⋊C8—61no—
321C3⋊C8—121no—
322C3⋊C8—181no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32D3×Q1611yescharacteristic
327C2×C32#91811yescharacteristic
328C2×C32#93711yescharacteristic
329C2×C32#94211yescharacteristic
330C2×C32C2×#21611yescharacteristic
331C2×C32#95311yescharacteristic
332C3⋊D3—22no—
333C3×D3—62no—
334C2×Q8—91no—
335C2×Q8—91no—
336C2×Q8—181no—
337C2×Q8—182no—
338C2×Q8—182no—
339C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
340C22×C4—91no—
341C22×C4—181no—
342Q16—182no—
343Q16—542no—
344C2×C8—91no—
345C2×C8—91no—
346C4⋊C4—91no—
347C4⋊C4—541no—
348C22⋊C4—541no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—61no—
352C22×C3—92no—
353C22×C3—181no—
354C22×C3—181no—
355C22×C3D3×#6711yescharacteristic
356C2×D3—32no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—92no—
360C2×D3—181no—
361C2×D3—181no—
362C3×C4—22no—
363C3×C4—31no—
364C3×C4—31no—
365C3×C4—31no—
366C3×C4—31no—
367C3×C4—62no—
368C3×C4—62no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—122no—
372C3×C4—181no—
373C3×C4—182no—
374C3×C4—361no—
375C3×C4—361no—
376C3×C4C2×#62511yescharacteristic
377C3×C4D3×#6711yescharacteristic
378Dic3—62no—
379Dic3—61no—
380Dic3—121no—
381Dic3—121no—
382Dic3—121no—
383Dic3—182no—
384Dic3—181no—
385Dic3—241no—
386Dic3—361no—
387C32—21no—
388C32#518611yescharacteristic, complemented
389C32#522811yescharacteristic, complemented
390C23—91no—
391Q8—21no—
392Q8—92no—
393Q8—184no—
394Q8—181no—
395Q8—181no—
396Q8—181no—
397Q8#619812yes—
398C2×C4—21no—
399C2×C4—91no—
400C2×C4—184no—
401C2×C4—181no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4—181no—
405C2×C4—181no—
406C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
407C8—181no—
408C8—181no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—61no—
416C2×C3—61no—
417C2×C3—61no—
418C2×C3—182no—
419C2×C3D3×#21611yescharacteristic
420C2×C3#6394811yescharacteristic
421C2×C3#6397611yescharacteristic
422D3—62no—
423D3—182no—
424C22—92no—
425C22—181no—
426C22—181no—
427C22#8867611yescharacteristic
428C4—22no—
429C4—181no—
430C4—182no—
431C4—361no—
432C4—361no—
433C4C2×#619811yescharacteristic
434C4#8867611yescharacteristic
435C3—31no—
436C3—31no—
437C3—61no—
438C3#1059159411yescharacteristic
439C2—182no—
440C2#1172123411yescharacteristic
441C2#1172123611yescharacteristic
442C2#1172131611yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#451
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#3756⋊C3C2semidirect
4. C2×#88433C2semidirect
5. #11720583C2semidirect
7. #11720309C2semidirect
27. C2×#3756D3semidirect
98. C2×(C3⋊D3)⋊C3Q16semidirect
230. C2×C3⋊D3#216semidirect
259. H(3,𝔽3)#451semidirect
388. C32#5186semidirect
389. C32#5228semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
C2×#3756⋊C32
#117211062
C2×#884332
#117205832
#117203172
#117203092
#105916033
#105915953