Finite Group Databasebeta

← #13140983

Subgroups — #13140983

Order 1728.

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

normal subgroups only

This group has 578 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×#88550C211yesmaximal, characteristic, complemented
3#11721107C211yesmaximal, characteristic
4C2×#88439C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic
7#11720313C211yesmaximal, characteristic, complemented
8#10591603—31nomaximal
9#10591594—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12Q8×(C3⋊D3)⋊C3—22no—
13Q8×(C3⋊D3)⋊C3—22no—
14C2×C4×(C3⋊D3)⋊C3—21no—
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3—21no—
17C2×#634⋊C3C2211yescharacteristic
18#88550—22no—
19C2×#6180C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88450—21no—
23#88448—21no—
24#88439—22no—
25C2×Q8×C3⋊D3D311yescharacteristic
26C2×C3×D3×Q8—31no—
27C2×C3×#216—31no—
28C2×C3×#216—31no—
29C2×C3×#216—61no—
30#63871—31no—
31#63856—31no—
32C2×#3719—31no—
33C2×#3718—31no—
34#63662D311yescharacteristic, complemented
35C3×#835—31no—
36#63567—31no—
37#63566—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—
46#6229—21no—
47#6229—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4×(C3⋊D3)⋊C3—24no—
52C4×(C3⋊D3)⋊C3—21no—
53C4×(C3⋊D3)⋊C3—21no—
54#634⋊C3—24no—
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
71C3×D3×Q8—62no—
72C3×D3×Q8—62no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—61no—
75C2×C3×Dic6—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6—61no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×#216—64no—
83C3×#216—62no—
84C3×#216—62no—
85C3×C4⋊Dic3—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—61no—
90#3724—61no—
91Dic3⋊Dic3—61no—
92#3722—61no—
93(C2×D3)⋊Dic3—61no—
94#3719—62no—
95#3718—62no—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3—21no—
99C2×(C3⋊D3)⋊C3Q1612yescomplemented
100C4×H(3,𝔽3)—22no—
101C4×H(3,𝔽3)C2×D411yescharacteristic
102C4×H(3,𝔽3)C2×D411yescharacteristic
103#1084—41no—
104(C3⋊Dic3)⋊C3—21no—
105(C3⋊Dic3)⋊C3—22no—
106(C3⋊Dic3)⋊C3—41no—
107C22×C3×Q8—91no—
108C2×D3×Q8—31no—
109C2×D3×Q8—91no—
110C2×C3×Q16—91no—
111#946—91no—
112C2×#216—31no—
113C2×#216—31no—
114C2×#216—61no—
115C2×#216—91no—
116C2×#216—91no—
117#924—91no—
118C2×Dic12—91no—
119#897—91no—
120C3×#103—91no—
121#835—31no—
122#835—91no—
123#835—91no—
124#830—91no—
125(C2×D3)⋊C8—91no—
126(C2×D3)⋊C8—91no—
127#816—91no—
128#810—91no—
129C22×C3⋊D3#6711yescharacteristic
130C22×C3×D3—31no—
131C32×Q8—21no—
132C32×Q8—21no—
133C32×Q8—22no—
134C32×Q8—41no—
135C32×Q8D1212yes—
136C32×Q8#6712yes—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—41no—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3#6711yescharacteristic
146C4×C3⋊D3—24no—
147C4×C3⋊D3—21no—
148C4×C3⋊D3—21no—
149#634—24no—
150#634—21no—
151#634#6712yes—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—121no—
158C3×C4×D3—64no—
159C3×C4×D3—61no—
160C3×C4×D3—61no—
161C3×Dic6—32no—
162C3×Dic6—64no—
163C3×Dic6—61no—
164C3×Dic6—62no—
165C3×Dic6—61no—
166C3×Dic6—61no—
167C3×Dic6—61no—
168C3×Dic6—61no—
169C3×Dic6—121no—
170C3⋊(C3⋊C8)—21no—
171C3×C3⋊C8—62no—
172C3×C3⋊C8—61no—
173C3×C3⋊C8—61no—
174C3×C3⋊C8—61no—
175#451—271noSylow
176C2×H(3,𝔽3)#12011yescharacteristic
177C2×H(3,𝔽3)C2×Q1611yescharacteristic
178C2×H(3,𝔽3)#13711yescharacteristic
179(C3⋊D3)⋊C3—22no—
180C2×C3×Q8—31no—
181C2×C3×Q8—31no—
182C2×C3×Q8—61no—
183C2×C3×Q8—91no—
184C2×C3×Q8—91no—
185C2×C3×Q8—181no—
186C2×C3×Q8—182no—
187C2×C3×Q8—182no—
188C2×C3×Q8D3211yescharacteristic
189C22×C3×C4—91no—
190C22×C3×C4—181no—
191D3×Q8—62no—
192D3×Q8—62no—
193D3×Q8—182no—
194D3×Q8—182no—
195C2×C4×D3—31no—
196C2×C4×D3—61no—
197C2×C4×D3—91no—
198C2×C4×D3—181no—
199C2×Dic6—31no—
200C2×Dic6—31no—
201C2×Dic6—31no—
202C2×Dic6—61no—
203C2×Dic6—61no—
204C2×Dic6—91no—
205C2×Dic6—181no—
206C3×Q16—182no—
207C2×C3×C8—91no—
208C2×C3×C8—91no—
209C3×C4⋊C4—91no—
210#217—181no—
211#216—62no—
212#216—62no—
213#216—64no—
214#216—182no—
215#216—182no—
216(C2×D3)⋊C4—181no—
217(C2×D3)⋊C4—181no—
218C4⋊Dic3—31no—
219C4⋊Dic3—91no—
220C4⋊Dic3—181no—
221Dic3⋊C4—181no—
222Dic3⋊C4—181no—
223C2×C3⋊C8—31no—
224C2×C3⋊C8—31no—
225C2×C3⋊C8—31no—
226C2×C3⋊C8—61no—
227C2×C3⋊C8—91no—
228Dic12—182no—
229C22×C32—21no—
230C22×C32D3×D411yescharacteristic
231C22×C32C2×#6711yescharacteristic
232C2×C3⋊D3—21no—
233C2×C3⋊D3—21no—
234C2×C3⋊D3#21612yescomplemented
235C2×C3×D3—32no—
236C2×C3×D3—61no—
237C2×C3×D3—61no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—42no—
243C32×C4C2×D1211yescharacteristic
244C32×C4D3×D411yescharacteristic
245C32×C4C2×#6711yescharacteristic
246C32×C4C2×#6711yescharacteristic
247C3⋊Dic3—21no—
248C3⋊Dic3—22no—
249C3⋊Dic3—41no—
250C3×Dic3—62no—
251C3×Dic3—61no—
252C3×Dic3—122no—
253C3×Dic3—121no—
254C3×Dic3—121no—
255C3×Dic3—121no—
256C22×Q8—91no—
257C2×Q16—91no—
258C2×Q16—271no—
259C22⋊Q8—271no—
260#103—91no—
261#103—271no—
262C22⋊C8—271no—
263H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
264C23×C3—91no—
265C22×D3—31no—
266C22×D3—91no—
267C3×Q8—21no—
268C3×Q8—32no—
269C3×Q8—32no—
270C3×Q8—61no—
271C3×Q8—61no—
272C3×Q8—62no—
273C3×Q8—92no—
274C3×Q8—121no—
275C3×Q8—181no—
276C3×Q8—184no—
277C3×Q8—181no—
278C3×Q8—181no—
279C3×Q8#62612yes—
280C2×C3×C4—21no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—91no—
287C2×C3×C4—121no—
288C2×C3×C4—181no—
289C2×C3×C4—181no—
290C2×C3×C4—184no—
291C2×C3×C4—181no—
292C2×C3×C4—181no—
293C2×C3×C4—181no—
294C2×C3×C4C2×D3211yescharacteristic
295C2×Dic3—31no—
296C2×Dic3—61no—
297C2×Dic3—61no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300C2×Dic3—91no—
301C2×Dic3—121no—
302C2×Dic3—181no—
303C2×Dic3—181no—
304C4×D3—64no—
305C4×D3—61no—
306C4×D3—61no—
307C4×D3—181no—
308C4×D3—181no—
309C4×D3—184no—
310Dic6—32no—
311Dic6—64no—
312Dic6—61no—
313Dic6—61no—
314Dic6—61no—
315Dic6—61no—
316Dic6—61no—
317Dic6—62no—
318Dic6—92no—
319Dic6—121no—
320Dic6—181no—
321Dic6—184no—
322C3×C8—181no—
323C3×C8—181no—
324C3⋊C8—61no—
325C3⋊C8—61no—
326C3⋊C8—62no—
327C3⋊C8—61no—
328C3⋊C8—181no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32#88211yescharacteristic
333C2×C32#91511yescharacteristic
334C2×C32D3×Q1611yescharacteristic
335C2×C32#94211yescharacteristic
336C2×C32C2×#21611yescharacteristic
337C2×C32#95311yescharacteristic
338C3⋊D3—22no—
339C3×D3—62no—
340C2×Q8—91no—
341C2×Q8—91no—
342C2×Q8—181no—
343C2×Q8—182no—
344C2×Q8—182no—
345C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
346C22×C4—91no—
347C22×C4—181no—
348Q16—182no—
349Q16—542no—
350C2×C8—91no—
351C2×C8—91no—
352C4⋊C4—91no—
353C4⋊C4—541no—
354C22⋊C4—541no—
355C22×C3—31no—
356C22×C3—31no—
357C22×C3—61no—
358C22×C3—92no—
359C22×C3—181no—
360C22×C3—181no—
361C22×C3D3×#6711yescharacteristic
362C2×D3—32no—
363C2×D3—61no—
364C2×D3—61no—
365C2×D3—92no—
366C2×D3—181no—
367C2×D3—181no—
368C3×C4—22no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—31no—
372C3×C4—31no—
373C3×C4—62no—
374C3×C4—62no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—122no—
378C3×C4—181no—
379C3×C4—182no—
380C3×C4—361no—
381C3×C4—361no—
382C3×C4C2×#62611yescharacteristic
383C3×C4D3×#6711yescharacteristic
384Dic3—62no—
385Dic3—61no—
386Dic3—121no—
387Dic3—121no—
388Dic3—122no—
389Dic3—121no—
390Dic3—182no—
391Dic3—181no—
392Dic3—361no—
393C32—21no—
394C32#480911yescharacteristic, complemented
395C32#522811yescharacteristic, complemented
396C23—91no—
397Q8—21no—
398Q8—92no—
399Q8—184no—
400Q8—181no—
401Q8—181no—
402Q8—181no—
403Q8#620012yes—
404C2×C4—21no—
405C2×C4—91no—
406C2×C4—184no—
407C2×C4—181no—
408C2×C4—181no—
409C2×C4—181no—
410C2×C4—181no—
411C2×C4—181no—
412C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C8—181no—
414C8—181no—
415C2×C3—31no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—61no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—182no—
425C2×C3D3×#21611yescharacteristic
426C2×C3#6394611yescharacteristic
427C2×C3#6397711yescharacteristic
428D3—62no—
429D3—182no—
430C22—92no—
431C22—181no—
432C22—181no—
433C22#8867611yescharacteristic
434C4—22no—
435C4—181no—
436C4—182no—
437C4—361no—
438C4—361no—
439C4C2×#620011yescharacteristic
440C4#8867611yescharacteristic
441C3—31no—
442C3—31no—
443C3—61no—
444C3#1059159511yescharacteristic
445C2—182no—
446C2#1172123611yescharacteristic
447C2#1172123811yescharacteristic
448C2#1172132111yescharacteristic
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×#88550C2semidirect
4. C2×#88439C2semidirect
5. #63662⋊C3C2semidirect
7. #11720313C2semidirect
34. #63662D3semidirect
99. C2×(C3⋊D3)⋊C3Q16semidirect
234. C2×C3⋊D3#216semidirect
263. H(3,𝔽3)#451semidirect
394. C32#4809semidirect
395. 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×#885502
#117211072
C2×#884392
#63662⋊C32
#117203172
#117203132
#105916033
#105915943