Finite Group Databasebeta

← #13178698

Subgroups — #13178698

Order 1728.

1070 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C3×#64145C211yesmaximal, characteristic
2C3×#64144C212yesmaximal
3#11723721C214yesmaximal, complemented
4#11723716C214yesmaximal, complemented
5#11723679C212yesmaximal, complemented
6#11723666C212yesmaximal, complemented
7#10595276—32nomaximal
8#10595275—31nomaximal
9C2×C33×D4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C2×C3×#634C2211yescharacteristic
12C2×#6296C2212yes—
13C3×#3757C2212yes—
14C3×#3757C2214yes—
15C3×#3757C2214yes—
16C3×#3750C2214yes—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18#88815C2214yes—
19#88815C2212yes—
20#88815C2214yes—
21#88815C2212yes—
22#88813C2212yes—
23#64145D311yescharacteristic
24#64144D312yes—
25C3×#936—32no—
26C3×#936—61no—
27C3×#935—32no—
28C3×#935—32no—
29C3×#935—62no—
30#63982—34no—
31#63964—34no—
32#63964—34no—
33#63956—34no—
34#63956—34no—
35#63956—34no—
36#63870—32no—
37#63867—34no—
38#63840—32no—
39#63839—34no—
40C23×C33C2312yes—
41C33×D4—41no—
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3—42no—
44C2×C3×C3⋊Dic3C2312yes—
45C2×C3×C3⋊Dic3C2314yes—
46C2×C3×C3⋊Dic3C2314yes—
47C2×C3×C3⋊Dic3C2312yes—
48C3×#634—41no—
49#6296—42no—
50#5672—91no—
51#5651—92no—
52C2×C32×D4—22no—
53C2×C32×D4—21no—
54C2×C32×D4—41no—
55C2×C32×D4C2×D311yescharacteristic
56C2×C32×D4C2×D312yes—
57C22×C3⋊Dic3C2×D312yes—
58C2×#634C2×D311yescharacteristic
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—62no—
62C2×C3×Dic6—32no—
63C2×C3×Dic6—61no—
64C2×#627—34no—
65C2×#627—32no—
66#3757C2×D312yes—
67#3757C2×D314yes—
68#3757C2×D314yes—
69#3750C2×D314yes—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C3×#217—32no—
72C3×#217—32no—
73C3×#217—34no—
74C3×#217—34no—
75C3×#217—34no—
76C3×#217—34no—
77C3×#217—62no—
78C3×#217—64no—
79C3×#217—64no—
80C3×Dic3⋊C4—34no—
81C3×Dic3⋊C4—34no—
82C3×Dic3⋊C4—64no—
83C3×C4×Dic3—32no—
84C3×C4×Dic3—61no—
85#3724—34no—
86#3724—34no—
87#3724—32no—
88#3724—32no—
89Dic3⋊Dic3—34no—
90Dic3⋊Dic3—34no—
91Dic3⋊Dic3—34no—
92Dic3⋊Dic3—34no—
93Dic3⋊Dic3—34no—
94Dic3⋊Dic3—34no—
95Dic32—34no—
96Dic32—32no—
97C22×C33—22no—
98C22×C33—24no—
99C22×C33C2411yescharacteristic
100C33×C4—21no—
101C3×C3⋊Dic3—22no—
102C3×C3⋊Dic3—24no—
103C3×C3⋊Dic3—22no—
104C3×C3⋊Dic3—24no—
105C3×#124—91no—
106C3×#123—92no—
107#944—92no—
108#936—32no—
109#936—61no—
110#935—32no—
111#935—32no—
112#935—62no—
113#924—94no—
114#889—94no—
115#888—94no—
116#879—94no—
117#879—94no—
118#879—94no—
119#878—94no—
120#878—94no—
121#870—92no—
122C23×C32—22no—
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—42no—
126C23×C32C22×D312yes—
127C23×C32C22×D312yes—
128C23×C32C22×D312yes—
129C32×D4—41no—
130C32×D4—42no—
131C32×D4—42no—
132C32×D4—44no—
133C32×D4—44no—
134C2×C32×C4—21no—
135C2×C32×C4—22no—
136C2×C32×C4—41no—
137C2×C32×C4C22×D312yes—
138C2×C32×C4C22×D311yescharacteristic
139C2×C3⋊Dic3—42no—
140C2×C3⋊Dic3C22×D314yes—
141C2×C3⋊Dic3C22×D312yes—
142C2×C3⋊Dic3C22×D312yes—
143C2×C3⋊Dic3C22×D314yes—
144#634—41no—
145C2×C3×Dic3—34no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—34no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—64no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—122no—
156C2×C3×Dic3—122no—
157C2×C3×Dic3—124no—
158C3×Dic6—122no—
159C3×Dic6—122no—
160#627—124no—
161#627—122no—
162#563—271noSylow
163C2×C33—42no—
164C2×C33H(5,𝔽2)11yescharacteristic
165C2×C332−1+412yes—
166C2×C3×Q8—91no—
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—41no—
170C2×C3×D4D3211yescharacteristic
171C2×C3×D4D3212yes—
172C22×C3×C4—92no—
173C22×Dic3—32no—
174C22×Dic3—32no—
175C22×Dic3—62no—
176C2×Dic6—32no—
177C2×Dic6—61no—
178C2×Dic6—92no—
179C2×Dic6—94no—
180C3×C4⋊C4—94no—
181C3×C22⋊C4—94no—
182C3×C22⋊C4—94no—
183C3×C22⋊C4—92no—
184C3×C42—91no—
185#217—34no—
186#217—34no—
187#217—32no—
188#217—32no—
189#217—34no—
190#217—34no—
191#217—64no—
192#217—62no—
193#217—64no—
194C4⋊Dic3—92no—
195C4⋊Dic3—92no—
196C4⋊Dic3—94no—
197C4⋊Dic3—94no—
198Dic3⋊C4—34no—
199Dic3⋊C4—34no—
200Dic3⋊C4—64no—
201Dic3⋊C4—94no—
202Dic3⋊C4—94no—
203Dic3⋊C4—94no—
204Dic3⋊C4—94no—
205Dic3⋊C4—94no—
206Dic3⋊C4—94no—
207C4×Dic3—32no—
208C4×Dic3—61no—
209C4×Dic3—94no—
210C4×Dic3—92no—
211C22×C32—22no—
212C22×C32—24no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—24no—
216C22×C32—24no—
217C22×C32—24no—
218C22×C32—21no—
219C22×C32—22no—
220C22×C32—41no—
221C22×C32—42no—
222C22×C32—42no—
223C22×C32—44no—
224C22×C32—44no—
225C22×C32—44no—
226C22×C32—44no—
227C22×C32—84no—
228C22×C32C23×D312yes—
229C22×C32C23×D311yescharacteristic
230C32×C4—21no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4—42no—
234C32×C4—42no—
235C3⋊Dic3—22no—
236C3⋊Dic3—24no—
237C3⋊Dic3—24no—
238C3⋊Dic3—22no—
239C3×Dic3—64no—
240C3×Dic3—64no—
241C3×Dic3—64no—
242C3×Dic3—64no—
243C3×Dic3—64no—
244C3×Dic3—64no—
245C3×Dic3—122no—
246C3×Dic3—122no—
247C3×Dic3—124no—
248C3×Dic3—124no—
249C4⋊Q8—272no—
250#126—274no—
251#125—272no—
252#124—91no—
253#123—92no—
254C22⋊Q8—274no—
255C33#56311yescharacteristic, complemented, Sylow
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—42no—
260C23×C3C2×D3212yes—
261C23×C3C2×D3212yes—
262C23×C3C2×D3212yes—
263C3×Q8—361no—
264C3×D4—42no—
265C3×D4—41no—
266C3×D4—42no—
267C3×D4—44no—
268C3×D4—44no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—41no—
272C2×C3×C4—94no—
273C2×C3×C4—92no—
274C2×C3×C4—92no—
275C2×C3×C4—94no—
276C2×C3×C4—362no—
277C2×C3×C4C2×D3212yes—
278C2×C3×C4C2×D3211yescharacteristic
279C2×Dic3—34no—
280C2×Dic3—34no—
281C2×Dic3—34no—
282C2×Dic3—34no—
283C2×Dic3—34no—
284C2×Dic3—34no—
285C2×Dic3—62no—
286C2×Dic3—64no—
287C2×Dic3—64no—
288C2×Dic3—62no—
289C2×Dic3—122no—
290C2×Dic3—122no—
291C2×Dic3—124no—
292Dic6—122no—
293Dic6—122no—
294Dic6—364no—
295Dic6—362no—
296C2×C32—22no—
297C2×C32—24no—
298C2×C32—21no—
299C2×C32—22no—
300C2×C32—41no—
301C2×C32—42no—
302C2×C32—42no—
303C2×C32—44no—
304C2×C32—42no—
305C2×C32—42no—
306C2×C32—82no—
307C2×C32—82no—
308C2×C32—84no—
309C2×C32#100412yes—
310C2×C32#100411yescharacteristic
311C2×C32#101014yes—
312C2×C32#101012yes—
313C2×Q8—91no—
314C2×Q8—272no—
315C2×D4#111311yescharacteristic
316C22×C4—92no—
317C4⋊C4—94no—
318C4⋊C4—272no—
319C4⋊C4—274no—
320C4⋊C4—274no—
321C4⋊C4—272no—
322C22⋊C4—94no—
323C22⋊C4—94no—
324C22⋊C4—92no—
325C42—91no—
326C42—272no—
327C22×C3—24no—
328C22×C3—24no—
329C22×C3—24no—
330C22×C3—24no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—21no—
335C22×C3—22no—
336C22×C3—44no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—44no—
340C22×C3—44no—
341C22×C3—44no—
342C22×C3—41no—
343C22×C3—84no—
344C22×C3C22×D3211yescharacteristic
345C22×C3C22×D3212yes—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—21no—
349C3×C4—42no—
350C3×C4—42no—
351C3×C4—182no—
352C3×C4—184no—
353C3×C4—184no—
354C3×C4—182no—
355Dic3—64no—
356Dic3—64no—
357Dic3—64no—
358Dic3—64no—
359Dic3—64no—
360Dic3—64no—
361Dic3—122no—
362Dic3—124no—
363Dic3—122no—
364Dic3—124no—
365C32—21no—
366C32—22no—
367C32—41no—
368C32#565112yescomplemented
369C32#567211yescharacteristic, complemented
370C23C2×#111312yes—
371Q8—361no—
372Q8—1082no—
373D4—41no—
374C2×C4—92no—
375C2×C4—94no—
376C2×C4—92no—
377C2×C4—94no—
378C2×C4—362no—
379C2×C4C2×#111311yescharacteristic
380C2×C3—24no—
381C2×C3—22no—
382C2×C3—21no—
383C2×C3—22no—
384C2×C3—44no—
385C2×C3—42no—
386C2×C3—42no—
387C2×C3—42no—
388C2×C3—42no—
389C2×C3—41no—
390C2×C3—82no—
391C2×C3—82no—
392C2×C3—84no—
393C2×C3#6430912yes—
394C2×C3#6430914yes—
395C2×C3#6433011yescharacteristic
396C2×C3#6433012yes—
397C22—24no—
398C22—22no—
399C22C22×#111311yescharacteristic
400C4—21no—
401C4—182no—
402C4—184no—
403C4—184no—
404C4—182no—
405C3—22no—
406C3—21no—
407C3—41no—
408C3#1059527511yescharacteristic, complemented
409C3#1059527612yescomplemented
410C2—42no—
411C2#1172462012yes—
412C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#563
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11723721C2semidirect
4. #11723716C2semidirect
5. #11723679C2semidirect
6. #11723666C2semidirect
255. C33#563semidirect
368. C32#5651semidirect
369. C32#5672semidirect
408. C3#10595275semidirect
409. C3#10595276semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C3×#641442
#117237212
#117237162
#117236792
#117236662
#105952763
#105952753