Finite Group Databasebeta

← #13165579

Subgroups — #13165579

Order 1728.

514 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 56 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×#88936C211yesmaximal, characteristic, complemented
2C2×C3×#3775C211yesmaximal, characteristic, complemented
3#11723029C211yesmaximal, characteristic
4Dic3×#627C211yesmaximal, characteristic, complemented
5#11722793C211yesmaximal, characteristic, complemented
6#11722773C211yesmaximal, characteristic, complemented
7Dic3⋊#622C211yesmaximal, characteristic, complemented
8#10593439—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×#6290—21no—
11C2×C3×#627C2211yescharacteristic
12C2×C3×#626—21no—
13C2×C3×#625C2211yescharacteristic
14C2×C3⋊#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16#88936—22no—
17C3×#3775—22no—
18#88815C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3C2211yescharacteristic
21C3×Dic32—21no—
22C2×#3775—31no—
23C2×#3775D311yescharacteristic, complemented
24#63979—61no—
25C4×#627—31no—
26#63914—31no—
27Dic3×Dic6—61no—
28C4⋊#622—31no—
29#63739—31no—
30#63739—31no—
31C22×C3×C3⋊D3—21no—
32C22×C32×D3—21no—
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C2×C32×Dic3—21no—
37C2×C32×Dic3D411yescharacteristic
38#6295—21no—
39#6295—21no—
40#6290—42no—
41C3×#627—21no—
42C3×#627D412yescomplemented
43C3×#626—42no—
44C3×#625—21no—
45C3×#625—21no—
46C3⋊#622—21no—
47C3×#622—21no—
48#5163—91nomaximal
49C2×#638—61no—
50C2×C3×#67—61no—
51C2×C3×#67—61no—
52C2×C3×D12—61no—
53C2×C3×Dic6—61no—
54C2×#627C2×D311yescharacteristic
55C2×#626—21no—
56C2×#626—61no—
57C2×#625—31no—
58C2×#625—61no—
59C2×#625—61no—
60C2×#625—121no—
61C2×#625C2×D311yescharacteristic
62C2×#622—31no—
63C2×#622C2×D311yescharacteristic
64#3775—22no—
65#3775—62no—
66C4⋊(C3⋊Dic3)—61no—
67C4×C3⋊Dic3—31no—
68C3×C4×Dic3—61no—
69C3×C4×Dic3—61no—
70#3724—31no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74Dic3⋊Dic3—121no—
75Dic32—21no—
76Dic32—61no—
77Dic32—61no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3—41no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C2×C32×D3—41no—
85C3⋊(C3⋊Dic3)—21no—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3C2×D411yescharacteristic
88C3×C3⋊Dic3#4111yescharacteristic
89C32×Dic3—22no—
90C32×Dic3SD1612yescomplemented
91C2×C3×SD16—91no—
92Q8×Dic3—91no—
93C2×#215—91no—
94#940—91no—
95#940—181no—
96C4×Dic6—181no—
97#832—91no—
98#821—91no—
99Dic3⋊C8—91no—
100C23×C32—61no—
101C22×C3⋊D3—21no—
102C22×C3×D3—21no—
103C22×C3×D3—21no—
104C22×C3×D3—41no—
105C22×C3×D3—61no—
106C22×C3×D3—61no—
107C22×C3×D3—121no—
108C2×C32×C4—31no—
109C2×C32×C4—61no—
110#638—122no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—121no—
116C2×C3⋊Dic3—121no—
117C2×C3⋊Dic3C22×D311yescharacteristic
118C3×#67—121no—
119C3×#67—121no—
120C3×#67—122no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—41no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—121no—
131C3×D12—122no—
132C3×Dic6—62no—
133C3×Dic6—62no—
134#627—21no—
135#627#6712yescomplemented
136#626—42no—
137#626—121no—
138#626—121no—
139#625—21no—
140#625—21no—
141#625—61no—
142#625—61no—
143#625—122no—
144#625—122no—
145#625—124no—
146#622—21no—
147#622—61no—
148#460—271noSylow
149C2×C33#12111yescharacteristic
150C2×C33C2×SD1611yescharacteristic
151C2×C33AGL(1,ℤ/8)11yescharacteristic
152C3×C3⋊D3—81no—
153C32×D3—81no—
154C2×C3×Q8—91no—
155C2×C3×D4—91no—
156C2×C3×D4—181no—
157C2×#67—61no—
158C2×#67—61no—
159C2×#67—181no—
160C2×#67—181no—
161C2×#67—181no—
162C2×#67—181no—
163C2×#67—361no—
164C2×D12—61no—
165C2×D12—91no—
166C2×Dic6—61no—
167C3×SD16—182no—
168C2×C3×C8—91no—
169C3×C42—181no—
170#215—182no—
171C4⋊Dic3—91no—
172C4⋊Dic3—181no—
173C4⋊Dic3—181no—
174C4⋊Dic3—361no—
175Dic3⋊C4—181no—
176Dic3⋊C4—181no—
177C4×Dic3—61no—
178C4×Dic3—61no—
179C4×Dic3—91no—
180C4×Dic3—181no—
181C4×Dic3—181no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—121no—
188C22×C32—121no—
189C22×C32—121no—
190C22×C32D3×D411yescharacteristic
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C2×C3×D3—41no—
197C2×C3×D3—42no—
198C2×C3×D3—41no—
199C2×C3×D3—41no—
200C2×C3×D3—41no—
201C2×C3×D3—81no—
202C2×C3×D3—81no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C2×C3×D3—122no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—241no—
211C2×C3×D3—241no—
212C32×C4—32no—
213C32×C4—62no—
214C3⋊Dic3—41no—
215C3⋊Dic3—62no—
216C3⋊Dic3—61no—
217C3⋊Dic3—121no—
218C3⋊Dic3—122no—
219C3⋊Dic3—241no—
220C3⋊Dic3#23711yescharacteristic
221C3⋊Dic3C2×#6711yescharacteristic
222C3×Dic3—22no—
223C3×Dic3—22no—
224C3×Dic3—22no—
225C3×Dic3—22no—
226C3×Dic3—42no—
227C3×Dic3—61no—
228C3×Dic3—61no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—121no—
232C3×Dic3—121no—
233C3×Dic3—241no—
234C2×SD16—91no—
235C2×SD16—271no—
236#127—271no—
237C4×Q8—271no—
238C4⋊C8—271no—
239#102—271no—
240#102—271no—
241C33#46011yescharacteristic, complemented, Sylow
242C23×C3—61no—
243C23×C3—61no—
244C23×C3—121no—
245C23×C3—181no—
246C22×D3—21no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—121no—
250C3×Q8—92no—
251C3×Q8—181no—
252C3×D4—181no—
253C3×D4—181no—
254C3×D4—362no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—91no—
260C2×C3×C4—121no—
261C2×C3×C4—181no—
262#67—121no—
263#67—121no—
264#67—122no—
265#67—361no—
266#67—361no—
267#67—362no—
268#67—362no—
269#67—362no—
270#67—364no—
271C2×Dic3—21no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—91no—
277C2×Dic3—121no—
278C2×Dic3—181no—
279C2×Dic3—181no—
280C2×Dic3—361no—
281C2×Dic3—361no—
282C2×Dic3GO+(4,𝔽2)11yescharacteristic
283D12—122no—
284D12—181no—
285D12—181no—
286Dic6—62no—
287Dic6—62no—
288C3×C8—181no—
289C3⋊C8—181no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32—41no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32—241no—
303C2×C32D3×SD1611yescharacteristic
304C2×C32#91411yescharacteristic
305C2×C32#93911yescharacteristic
306C3⋊D3—81no—
307C3×D3—81no—
308C3×D3—82no—
309C3×D3—81no—
310C3×D3—241no—
311C3×D3—242no—
312C3×D3—241no—
313C2×Q8—91no—
314C2×D4—91no—
315C2×D4—181no—
316C2×D4—271no—
317C2×D4—541no—
318SD16—182no—
319SD16—542no—
320C2×C8—91no—
321C2×C8—271no—
322C4⋊C4—271no—
323C4⋊C4—541no—
324C42—181no—
325C42—271no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3—121no—
333C22×C3—122no—
334C22×C3—121no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—241no—
338C22×C3—241no—
339C22×C3—361no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3C2×GO+(4,𝔽2)11yescharacteristic
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—121no—
349C2×D3—122no—
350C2×D3—121no—
351C2×D3—121no—
352C2×D3—121no—
353C2×D3—241no—
354C2×D3—241no—
355C3×C4—62no—
356C3×C4—62no—
357C3×C4—62no—
358C3×C4—62no—
359C3×C4—91no—
360C3×C4—91no—
361C3×C4—122no—
362C3×C4—361no—
363Dic3—22no—
364Dic3—61no—
365Dic3—61no—
366Dic3—61no—
367Dic3—61no—
368Dic3—121no—
369Dic3—121no—
370Dic3—182no—
371Dic3—181no—
372Dic3—241no—
373Dic3—361no—
374Dic3—362no—
375Dic3—721no—
376Dic3#377512yescomplemented
377C32—21no—
378C32—21no—
379C32—41no—
380C32—41no—
381C32#516311yescharacteristic, complemented
382C23—61no—
383C23—181no—
384Q8—92no—
385Q8—181no—
386D4—181no—
387D4—181no—
388D4—362no—
389D4—541no—
390D4—541no—
391D4—1082no—
392C2×C4—31no—
393C2×C4—61no—
394C2×C4—91no—
395C2×C4—181no—
396C2×C4—271no—
397C8—181no—
398C8—541no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—241no—
412C2×C3—242no—
413C2×C3—241no—
414C2×C3—721no—
415C2×C3#6422411yescharacteristic
416C2×C3#6423011yescharacteristic
417C2×C3C2×#377511yescharacteristic
418D3—81no—
419D3—241no—
420D3—242no—
421D3—241no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22—361no—
426C22—361no—
427C22—361no—
428C22D3×GO+(4,𝔽2)11yescharacteristic
429C4—32no—
430C4—62no—
431C4—91no—
432C4—91no—
433C4—361no—
434C4—541no—
435C3—21no—
436C3—21no—
437C3—41no—
438C3—41no—
439C3#1059343911yescharacteristic, complemented
440C2—241no—
441C2—721no—
442C2#1172411711yescharacteristic
443C2D3×#377511yescharacteristic
444C2#1172414111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radical#11723029C2
hypocenterC33#460
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#88936C2semidirect
2. C2×C3×#3775C2semidirect
4. Dic3×#627C2semidirect
5. #11722793C2semidirect
6. #11722773C2semidirect
7. Dic3⋊#622C2semidirect
23. C2×#3775D3semidirect
42. C3×#627D4semidirect
90. C32×Dic3SD16semidirect
135. #627#67semidirect
241. C33#460semidirect
376. Dic3#3775semidirect
381. C32#5163semidirect
439. C3#10593439semidirect

Maximal subgroups

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

subgroupindex
C2×#889362
C2×C3×#37752
#117230292
Dic3×#6272
#117227932
#117227732
Dic3⋊#6222
#105934393
#51639