Finite Group Databasebeta

← #10592507⋊C3 #13141930

Subgroups — #10592507⋊C3 #13141930

Order 1728.

574 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 100 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×#88722C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic, complemented
3C2×#3744⋊C3C211yesmaximal, characteristic, complemented
4#64106⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8#10592507C311yesmaximal, characteristic, complemented
9C3×#4808—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88722—22no—
12#88722—22no—
13C2×C4⋊((C3⋊D3)⋊C3)—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#3756⋊C3—22no—
21#3752⋊C3—21no—
22C4⋊((C3⋊Dic3)⋊C3)—21no—
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
24#3744⋊C3—22no—
25C2×#3832C2×C311yescharacteristic
26C2×C3×#239—31no—
27C2×#3756C2×C311yescharacteristic, complemented
28C2×#3744C2×C311yescharacteristic, complemented
29#64106C2×C311yescharacteristic, complemented
30C2×C3×#216—31no—
31C2×C3×#204—31no—
32C3×#897—31no—
33C32×#103—21no—
34C32×#103D311yescharacteristic, complemented
35C32×#103C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63636C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—31no—
39C3×#809—31no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3D411yescharacteristic
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)D412yes—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—24no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)D412yes—
52C4×(C3⋊D3)⋊C3—24no—
53C4×(C3⋊D3)⋊C3—21no—
54C4×(C3⋊D3)⋊C3—21no—
55#634⋊C3—21no—
56#634⋊C3—21no—
57C8×H(3,𝔽3)—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59C3×#452—91no—
60#4808—31no—
61#4808—91no—
62C2×C32×Q8—21no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C22×C311yescharacteristic
65#3832—22no—
66#3832—22no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3C22×C311yescharacteristic
71C2×#634C22×C311yescharacteristic
72C3×#239—62no—
73C3×#239—62no—
74C2×C3×D12—31no—
75C2×C3×D12—61no—
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—61no—
78C2×C3×Dic6—31no—
79C2×C32×C8—21no—
80C2×C32×C8C2×D311yescharacteristic
81C2×C32×C8C22×C311yescharacteristic
82C32×C4⋊C4—21no—
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C22×C311yescharacteristic
85#3756—22no—
86#3752—21no—
87C4⋊(C3⋊Dic3)—21no—
88C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
89#3744—22no—
90C3×#216—62no—
91C3×(C2×D3)⋊C4—61no—
92C3×C4⋊Dic3—61no—
93C2×C3×C3⋊C8—31no—
94C3×#204—62no—
95C22×H(3,𝔽3)C2×D411yescharacteristic
96C2×(C3⋊D3)⋊C3—21no—
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—21no—
99C2×(C3⋊D3)⋊C3—21no—
100C2×(C3⋊D3)⋊C3—41no—
101C2×(C3⋊D3)⋊C3—41no—
102C4×H(3,𝔽3)—22no—
103C4×H(3,𝔽3)—41no—
104C4×H(3,𝔽3)C2×D411yescharacteristic
105C4×H(3,𝔽3)C2×D411yescharacteristic
106(C3⋊Dic3)⋊C3—21no—
107(C3⋊Dic3)⋊C3—41no—
108C2×C3×#41—91no—
109C2×#239—31no—
110C2×#239—91no—
111C2×C3×Q16—91no—
112C2×C3×SD16—91no—
113C3×C22⋊Q8—91no—
114C2×#216—31no—
115C2×#216—91no—
116C2×#204—31no—
117C2×#204—91no—
118#897—31no—
119#897—91no—
120C3×#103—31no—
121C3×#103—31no—
122C3×#103—61no—
123C3×#103C3×D311yescharacteristic
124C3×#102—91no—
125C3×C22⋊C8—91no—
126(C2×D3)⋊C8—31no—
127(C2×D3)⋊C8—91no—
128#809—31no—
129#809—91no—
130C22×C3⋊D3—21no—
131C22×C3⋊D3C3×D411yescharacteristic
132C22×C3×D3—31no—
133C22×C3×D3—61no—
134C32×Q8—21no—
135C32×Q8—21no—
136C32×Q8—22no—
137C32×Q8—41no—
138C32×Q8D1212yes—
139C32×Q8C3×D412yes—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—41no—
146C2×C32×C4—41no—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C23×C311yescharacteristic
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3C3×D411yescharacteristic
151C4⋊(C3⋊D3)—24no—
152C4⋊(C3⋊D3)—21no—
153C4⋊(C3⋊D3)C3×D412yes—
154C4×C3⋊D3—24no—
155C4×C3⋊D3—21no—
156C4×C3⋊D3—21no—
157#634—21no—
158#634—21no—
159C2×C3×Dic3—31no—
160C2×C3×Dic3—61no—
161C3×D12—32no—
162C3×D12—64no—
163C3×D12—61no—
164C3×C4×D3—64no—
165C3×C4×D3—61no—
166C3×C4×D3—61no—
167C3×Dic6—61no—
168C3×Dic6—61no—
169C32×C8—21no—
170C32×C8—21no—
171C32×C8—41no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—61no—
174#452—91noSylow
175C2×H(3,𝔽3)#12011yescharacteristic
176C2×H(3,𝔽3)#13511yescharacteristic
177C2×H(3,𝔽3)#13711yescharacteristic
178(C3⋊D3)⋊C3—41no—
179(C3⋊D3)⋊C3—42no—
180C3×#41—182no—
181C3×#41—182no—
182C2×C3×Q8—31no—
183C2×C3×Q8—31no—
184C2×C3×Q8—61no—
185C2×C3×Q8—91no—
186C2×C3×Q8C2×C3×D311yescharacteristic
187C2×C3×D4—91no—
188C2×C3×D4—181no—
189C22×C3×C4—91no—
190C22×C3×C4—181no—
191#239—62no—
192#239—62no—
193#239—182no—
194#239—182no—
195C2×D12—31no—
196C2×D12—61no—
197C2×D12—91no—
198C2×D12—181no—
199C2×C4×D3—31no—
200C2×C4×D3—61no—
201C2×C4×D3—91no—
202C2×C4×D3—181no—
203C2×Dic6—31no—
204C2×Dic6—91no—
205C3×Q16—182no—
206C3×SD16—182no—
207C2×C3×C8—31no—
208C2×C3×C8—31no—
209C2×C3×C8—61no—
210C2×C3×C8—91no—
211C2×C3×C8C2×C3×D311yescharacteristic
212C3×C4⋊C4—31no—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—61no—
215C3×C4⋊C4—181no—
216C3×C4⋊C4C2×C3×D311yescharacteristic
217C3×C22⋊C4—181no—
218#216—62no—
219#216—182no—
220(C2×D3)⋊C4—61no—
221(C2×D3)⋊C4—181no—
222C4⋊Dic3—61no—
223C4⋊Dic3—181no—
224C2×C3⋊C8—31no—
225C2×C3⋊C8—91no—
226#204—62no—
227#204—182no—
228C22×C32—21no—
229C22×C32D3×D411yescharacteristic
230C22×C32C2×C3×D411yescharacteristic
231C2×C3⋊D3—21no—
232C2×C3⋊D3—22no—
233C2×C3⋊D3—21no—
234C2×C3⋊D3—21no—
235C2×C3⋊D3—41no—
236C2×C3⋊D3—41no—
237C2×C3×D3—62no—
238C2×C3×D3—61no—
239C2×C3×D3—61no—
240C2×C3×D3—61no—
241C2×C3×D3—121no—
242C2×C3×D3—121no—
243C32×C4—22no—
244C32×C4—22no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—42no—
248C32×C4—41no—
249C32×C4—41no—
250C32×C4—81no—
251C32×C4C2×D1211yescharacteristic
252C32×C4D3×D411yescharacteristic
253C32×C4C2×C3×D411yescharacteristic
254C32×C4C2×C3×D411yescharacteristic
255C3⋊Dic3—21no—
256C3⋊Dic3—41no—
257C3×Dic3—61no—
258C3×Dic3—121no—
259C2×#41—91no—
260C2×Q16—91no—
261C2×SD16—91no—
262C22⋊Q8—91no—
263#103(C3⋊D3)⋊C311yescharacteristic, complemented
264#102—91no—
265C22⋊C8—91no—
266H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
267C23×C3—91no—
268C23×C3—181no—
269C22×D3—31no—
270C22×D3—61no—
271C22×D3—91no—
272C22×D3—181no—
273C3×Q8—21no—
274C3×Q8—32no—
275C3×Q8—32no—
276C3×Q8—61no—
277C3×Q8—61no—
278C3×Q8—62no—
279C3×Q8—121no—
280C3×Q8—181no—
281C3×Q8—181no—
282C3×Q8C3×D1212yes—
283C3×D4—92no—
284C3×D4—181no—
285C3×D4—184no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—91no—
296C2×C3×C4—121no—
297C2×C3×C4—121no—
298C2×C3×C4—181no—
299C2×C3×C4—181no—
300C2×C3×C4—184no—
301C2×C3×C4—181no—
302C2×C3×C4C22×C3×D311yescharacteristic
303C2×Dic3—31no—
304C2×Dic3—61no—
305C2×Dic3—91no—
306C2×Dic3—181no—
307D12—32no—
308D12—64no—
309D12—61no—
310D12—92no—
311D12—181no—
312D12—184no—
313C4×D3—64no—
314C4×D3—61no—
315C4×D3—61no—
316C4×D3—181no—
317C4×D3—181no—
318C4×D3—184no—
319Dic6—61no—
320Dic6—61no—
321Dic6—181no—
322Dic6—181no—
323C3×C8—21no—
324C3×C8—61no—
325C3×C8—61no—
326C3×C8—121no—
327C3×C8—181no—
328C3⋊C8—61no—
329C3⋊C8—181no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32#88211yescharacteristic
334C2×C32#91611yescharacteristic
335C2×C32#91811yescharacteristic
336C2×C32C3×#12011yescharacteristic
337C2×C32C3×#13511yescharacteristic
338C2×C32C3×#13711yescharacteristic
339C3⋊D3—41no—
340C3⋊D3—42no—
341C3×D3—121no—
342C3×D3—122no—
343#41—182no—
344#41—182no—
345C2×Q8—91no—
346C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
347C2×D4—91no—
348C2×D4—181no—
349C22×C4—91no—
350C22×C4—181no—
351Q16—182no—
352SD16—182no—
353C2×C8—91no—
354C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
355C4⋊C4—181no—
356C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
357C22⋊C4—181no—
358C22×C3—31no—
359C22×C3—31no—
360C22×C3—61no—
361C22×C3—182no—
362C22×C3—181no—
363C22×C3—181no—
364C22×C3—181no—
365C22×C3—361no—
366C22×C3—361no—
367C22×C3C3×D3×D411yescharacteristic
368C2×D3—62no—
369C2×D3—61no—
370C2×D3—61no—
371C2×D3—61no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—182no—
375C2×D3—181no—
376C2×D3—181no—
377C2×D3—181no—
378C2×D3—361no—
379C2×D3—361no—
380C3×C4—22no—
381C3×C4—31no—
382C3×C4—31no—
383C3×C4—31no—
384C3×C4—31no—
385C3×C4—41no—
386C3×C4—62no—
387C3×C4—62no—
388C3×C4—61no—
389C3×C4—61no—
390C3×C4—122no—
391C3×C4—121no—
392C3×C4—121no—
393C3×C4—181no—
394C3×C4—241no—
395C3×C4—361no—
396C3×C4C2×C3×D1211yescharacteristic
397C3×C4C3×D3×D411yescharacteristic
398Dic3—61no—
399Dic3—121no—
400Dic3—181no—
401Dic3—361no—
402C32—21no—
403C32#480811yescharacteristic, complemented
404C32C3×#45211yescharacteristic, complemented
405C23—91no—
406C23—181no—
407Q8—21no—
408Q8—181no—
409Q8—181no—
410Q8C4⋊((C3⋊D3)⋊C3)12yes—
411D4—92no—
412D4—184no—
413D4—181no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—91no—
417C2×C4—184no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4—181no—
421C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
422C8—21no—
423C8—181no—
424C2×C3—31no—
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—61no—
431C2×C3—61no—
432C2×C3—61no—
433C2×C3—361no—
434C2×C3—362no—
435C2×C3C3×#88211yescharacteristic
436C2×C3C3×#91611yescharacteristic
437C2×C3C3×#91811yescharacteristic
438D3—121no—
439D3—122no—
440D3—361no—
441D3—362no—
442C22—181no—
443C22—181no—
444C22—182no—
445C22—181no—
446C22—361no—
447C22—361no—
448C22D4×(C3⋊D3)⋊C311yescharacteristic
449C4—22no—
450C4—41no—
451C4—181no—
452C4—361no—
453C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
454C4D4×(C3⋊D3)⋊C311yescharacteristic
455C3—31no—
456C3—31no—
457C3—61no—
458C3C3×#480811yescharacteristic
459C2—361no—
460C2—362no—
461C2#64091⋊C311yescharacteristic
462C2#64125⋊C311yescharacteristic
463C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC2×#88722C2
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×#3756⋊C3C2semidirect
3. C2×#3744⋊C3C2semidirect
4. #64106⋊C3C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #63647⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. #10592507C3semidirect
27. C2×#3756C2×C3semidirect
28. C2×#3744C2×C3semidirect
29. #64106C2×C3semidirect
34. C32×#103D3semidirect
35. C32×#103C2×C3semidirect
36. #63647C2×C3semidirect
37. #63636C2×C3semidirect
263. #103(C3⋊D3)⋊C3semidirect
266. H(3,𝔽3)#452semidirect
403. C32#4808semidirect
404. C32C3×#452semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C2×#3756⋊C32
C2×#3744⋊C32
#64106⋊C32
H(3,𝔽3)×#1032
#63647⋊C32
#63636⋊C32
#105925073
C3×#48083