Finite Group Databasebeta

← #10592865⋊C3 #13143369

Subgroups — #10592865⋊C3 #13143369

Order 1728.

460 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 96 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×SD16×H(3,𝔽3)C211yesmaximal, characteristic
2Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#3756⋊C3C211yesmaximal, characteristic, complemented
4#64145⋊C3C211yesmaximal, characteristic
5#63659⋊C3C211yesmaximal, characteristic
6#63643⋊C3C211yesmaximal, characteristic, complemented
7#63641⋊C3C211yesmaximal, characteristic, complemented
8#10592865C311yesmaximal, characteristic, complemented
9C3×#5166—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13SD16×H(3,𝔽3)—22no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#3757⋊C3—21no—
16#3757⋊C3—21no—
17#3756⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C4×(C3⋊Dic3)⋊C3—21no—
21C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C2×C32×SD16—21no—
24C2×C32×SD16D311yescharacteristic
25C2×C32×SD16C2×C311yescharacteristic
26Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
27C2×#3756C2×C311yescharacteristic, complemented
28#64145C2×C311yescharacteristic
29C3×Q8×Dic3—31no—
30C2×C3×#216—31no—
31C3×#936—31no—
32#63659C2×C311yescharacteristic
33#63643C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35C3×#832—31no—
36C3×#816—31no—
37C3×Dic3⋊C8—31no—
38C23×H(3,𝔽3)—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)D412yes—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49#634⋊C3—21no—
50#634⋊C3—21no—
51C8×H(3,𝔽3)—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53C3×#464—91no—
54#5166—31no—
55#5166—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C22×C311yescharacteristic
59C2×C32×D4—21no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C22×C311yescharacteristic
62C2×#634C22×C311yescharacteristic
63C2×C3×Dic6—31no—
64C32×SD16—24no—
65C32×SD16—22no—
66C32×SD16—22no—
67C2×C32×C8—21no—
68C2×C32×C8C2×D311yescharacteristic
69C2×C32×C8C22×C311yescharacteristic
70#3757—21no—
71#3757—21no—
72#3756—22no—
73C4⋊(C3⋊Dic3)—21no—
74C4⋊(C3⋊Dic3)C22×C311yescharacteristic
75C4×C3⋊Dic3—21no—
76C4×C3⋊Dic3C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78C3×#217—61no—
79C3×#217—61no—
80C3×#216—62no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—61no—
83C3×C4×Dic3—31no—
84C3×C4×Dic3—61no—
85C2×C3×C3⋊C8—31no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C4×H(3,𝔽3)—22no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)#4111yescharacteristic
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—41no—
96(C3⋊Dic3)⋊C3—41no—
97C2×C3×Q16—91no—
98C2×C3×SD16—31no—
99C2×C3×SD16—31no—
100C2×C3×SD16—61no—
101C2×C3×SD16C3×D311yescharacteristic
102C3×#124—91no—
103C3×C4×Q8—91no—
104Q8×Dic3—31no—
105Q8×Dic3—91no—
106C2×#216—31no—
107C2×#216—91no—
108#936—31no—
109#936—91no—
110C3×C4⋊C8—91no—
111C3×#103—91no—
112C3×#102—91no—
113#832—31no—
114#832—91no—
115#816—31no—
116#816—91no—
117Dic3⋊C8—31no—
118Dic3⋊C8—91no—
119C23×C32—21no—
120C23×C32—21no—
121C23×C32—41no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—22no—
125C32×Q8—41no—
126C32×Q8#6712yes—
127C32×Q8C3×D412yes—
128C32×D4—22no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—41no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—41no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C23×C311yescharacteristic
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3C3×D411yescharacteristic
143C2×C3⋊Dic3C3×D411yescharacteristic
144#634—21no—
145#634—21no—
146C2×C3×Dic3—31no—
147C2×C3×Dic3—31no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C32×C8—22no—
153C32×C8—21no—
154C32×C8—21no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—61no—
157#464—91noSylow
158C2×H(3,𝔽3)—81no—
159C2×H(3,𝔽3)#12111yescharacteristic
160C2×H(3,𝔽3)#13511yescharacteristic
161C2×H(3,𝔽3)#13711yescharacteristic
162C2×C3×Q8—31no—
163C2×C3×Q8—31no—
164C2×C3×Q8—61no—
165C2×C3×Q8—91no—
166C2×C3×Q8C2×C3×D311yescharacteristic
167C2×C3×D4—31no—
168C2×C3×D4—31no—
169C2×C3×D4—61no—
170C2×C3×D4C2×C3×D311yescharacteristic
171C2×Dic6—31no—
172C2×Dic6—91no—
173C3×Q16—182no—
174C3×SD16—22no—
175C3×SD16—62no—
176C3×SD16—62no—
177C3×SD16—64no—
178C2×C3×C8—31no—
179C2×C3×C8—31no—
180C2×C3×C8—61no—
181C2×C3×C8—91no—
182C2×C3×C8C2×C3×D311yescharacteristic
183C3×C4⋊C4—91no—
184C3×C4⋊C4—181no—
185C3×C22⋊C4—181no—
186C3×C22⋊C4—181no—
187C3×C42—91no—
188C3×C42—181no—
189#217—61no—
190#217—61no—
191#217—181no—
192#217—181no—
193#216—62no—
194#216—182no—
195C4⋊Dic3—31no—
196C4⋊Dic3—61no—
197C4⋊Dic3—91no—
198C4⋊Dic3—181no—
199C4×Dic3—31no—
200C4×Dic3—61no—
201C4×Dic3—91no—
202C4×Dic3—181no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205C22×C32—21no—
206C22×C32—42no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—81no—
214C22×C32—81no—
215C22×C32D3×D411yescharacteristic
216C22×C32C2×C3×D411yescharacteristic
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—42no—
222C32×C4#23711yescharacteristic
223C32×C4C2×#6711yescharacteristic
224C32×C4C2×C3×D411yescharacteristic
225C32×C4C3×#4111yescharacteristic
226C3⋊Dic3—21no—
227C3⋊Dic3—21no—
228C3⋊Dic3—41no—
229C3⋊Dic3—41no—
230C3×Dic3—61no—
231C3×Dic3—61no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234C2×Q16—91no—
235C2×SD16(C3⋊D3)⋊C311yescharacteristic
236#124—91no—
237C4×Q8—91no—
238C4⋊C8—91no—
239#103—91no—
240#102—91no—
241H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
242C23×C3—21no—
243C23×C3—61no—
244C23×C3—61no—
245C23×C3—121no—
246C3×Q8—21no—
247C3×Q8—32no—
248C3×Q8—32no—
249C3×Q8—61no—
250C3×Q8—61no—
251C3×Q8—62no—
252C3×Q8—121no—
253C3×Q8—181no—
254C3×Q8—181no—
255C3×Q8C3×#6712yes—
256C3×D4—21no—
257C3×D4—21no—
258C3×D4—61no—
259C3×D4—61no—
260C3×D4—61no—
261C3×D4—61no—
262C3×D4—62no—
263C3×D4—121no—
264C2×C3×C4—21no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—121no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4C22×C3×D311yescharacteristic
276C2×Dic3—31no—
277C2×Dic3—31no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—91no—
281C2×Dic3—91no—
282C2×Dic3—181no—
283C2×Dic3—181no—
284Dic6—61no—
285Dic6—61no—
286Dic6—181no—
287Dic6—181no—
288C3×C8—21no—
289C3×C8—61no—
290C3×C8—61no—
291C3×C8—62no—
292C3×C8—181no—
293C3⋊C8—61no—
294C3⋊C8—181no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—82no—
299C2×C32—81no—
300C2×C32—81no—
301C2×C32#91511yescharacteristic
302C2×C32#91611yescharacteristic
303C2×C32#93911yescharacteristic
304C2×C32C3×#12111yescharacteristic
305C2×C32C3×#13511yescharacteristic
306C2×C32C3×#13711yescharacteristic
307C2×Q8—91no—
308C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
309C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
310Q16—182no—
311SD16—22no—
312C2×C8—91no—
313C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
314C4⋊C4—91no—
315C4⋊C4—181no—
316C22⋊C4—181no—
317C22⋊C4—181no—
318C42—91no—
319C42—181no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—61no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3—122no—
333C22×C3—241no—
334C22×C3—241no—
335C22×C3C3×D3×D411yescharacteristic
336C3×C4—22no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—62no—
342C3×C4—62no—
343C3×C4—61no—
344C3×C4—61no—
345C3×C4—122no—
346C3×C4—181no—
347C3×C4—181no—
348C3×C4—361no—
349C3×C4—361no—
350C3×C4C3×#23711yescharacteristic
351C3×C4C2×C3×#6711yescharacteristic
352Dic3—61no—
353Dic3—61no—
354Dic3—121no—
355Dic3—121no—
356Dic3—181no—
357Dic3—181no—
358Dic3—361no—
359Dic3—361no—
360C32—21no—
361C32#516611yescharacteristic, complemented
362C32C3×#46411yescharacteristic, complemented
363C23—21no—
364Q8—21no—
365Q8—181no—
366Q8—181no—
367Q8#638⋊C312yes—
368D4—21no—
369D4—21no—
370C2×C4—21no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
376C8—21no—
377C8—181no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—61no—
387C2×C3—81no—
388C2×C3—241no—
389C2×C3—241no—
390C2×C3—242no—
391C2×C3C3×#91511yescharacteristic
392C2×C3C3×#91611yescharacteristic
393C2×C3C3×#93911yescharacteristic
394C22—41no—
395C22—41no—
396C22—41no—
397C22D4×(C3⋊D3)⋊C311yescharacteristic
398C4—22no—
399C4—181no—
400C4—181no—
401C4—361no—
402C4—361no—
403C4#3830⋊C311yescharacteristic
404C4C2×#638⋊C311yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3C3×#516611yescharacteristic
409C2—81no—
410C2#64124⋊C311yescharacteristic
411C2#64125⋊C311yescharacteristic
412C2#64148⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
2. Q8×(C3⋊Dic3)⋊C3C2semidirect
3. C2×#3756⋊C3C2semidirect
6. #63643⋊C3C2semidirect
7. #63641⋊C3C2semidirect
8. #10592865C3semidirect
26. Q8×C3⋊Dic3C2×C3semidirect
27. C2×#3756C2×C3semidirect
33. #63643C2×C3semidirect
34. #63641C2×C3semidirect
241. H(3,𝔽3)#464semidirect
361. C32#5166semidirect
362. C32C3×#464semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×H(3,𝔽3)2
Q8×(C3⋊Dic3)⋊C32
C2×#3756⋊C32
#64145⋊C32
#63659⋊C32
#63643⋊C32
#63641⋊C32
#105928653
C3×#51663