Finite Group Databasebeta

← #13138983

Subgroups — #13138983

Order 1728.

580 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88529C211yesmaximal, characteristic, complemented
2C2×#3744⋊C3C211yesmaximal, characteristic, complemented
3#11721158C211yesmaximal, characteristic, complemented
4#11721114C211yesmaximal, characteristic, complemented
5C2×#88437C211yesmaximal, characteristic, complemented
6C2×#88436C211yesmaximal, characteristic, complemented
7#63642⋊C3C211yesmaximal, characteristic, complemented
8#10590887—31nomaximal
9#10590883—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×#6200—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#88529—22no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#3744⋊C3—22no—
20#88451—22no—
21#88437—22no—
22#88436—22no—
23C2×#3744D311yescharacteristic, complemented
24C2×C3×D24—31no—
25C2×C3×D24—31no—
26C2×C3×D24—61no—
27C2×C3×#204—31no—
28#63914—31no—
29#63911—31no—
30#63863—31no—
31#63860—31no—
32C2×#3716—31no—
33C2×#3715—31no—
34C2×#3714—31no—
35C2×#3713—31no—
36#63642D311yescharacteristic, complemented
37C3×#815—31no—
38C22×#272—21no—
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C4⋊#272—21no—
43C4⋊#272—21no—
44C4⋊#272—21no—
45C4⋊#272—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D412yes—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52#6200—44no—
53C8×H(3,𝔽3)D412yescomplemented
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55#5176—91no—
56#5154—91no—
57#4712—91no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—61no—
66C2×C3×D12—61no—
67C2×C3×Dic6—31no—
68C2×#626—62no—
69C2×#625—62no—
70C2×C32×C8—21no—
71C2×C32×C8C2×D311yescharacteristic
72C2×C32×C8C2×D311yescharacteristic
73C4×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75#3744—22no—
76C3×C4×Dic3—31no—
77C2×C3×C3⋊C8—31no—
78C3×D24—64no—
79C3×D24—62no—
80C3×D24—62no—
81C3×#204—62no—
82(C2×D3)⋊Dic3—62no—
83(C2×D3)⋊Dic3—62no—
84#3716—62no—
85#3715—62no—
86#3714—62no—
87#3713—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×#272—41no—
90C2×#272—41no—
91C2×#272—42no—
92C2×#272—42no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—42no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)C2×D411yescharacteristic
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×SD16—91no—
100C2×C3×D8—91no—
101#947—91no—
102C2×#215—91no—
103#940—91no—
104#940—91no—
105#936—91no—
106C2×#214—91no—
107C2×#213—91no—
108C2×#213—91no—
109C2×D24—31no—
110C2×D24—31no—
111C2×D24—61no—
112C2×D24—91no—
113C2×#204—31no—
114C2×#204—91no—
115C2×#204—91no—
116#875—91no—
117#874—91no—
118C3×#97—91no—
119#815—31no—
120#815—91no—
121C22×C3⋊D3—21no—
122C22×C3×D3—61no—
123C22×C3×D3—61no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C22×C3×D3—121no—
128C22×C3×D3—121no—
129C2×C32×C4—21no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3D1212yes—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136#634—21no—
137#634—21no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—61no—
140C3×D12—62no—
141C3×D12—62no—
142C3×D12—61no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—121no—
153C3×D12—121no—
154C3×Dic6—61no—
155C3×Dic6—61no—
156#626—124no—
157#625—124no—
158C32×C8—22no—
159C32×C8D1212yescomplemented
160C32×C8#6712yescomplemented
161C3⋊(C3⋊C8)D1212yescomplemented
162C3×C3⋊C8—32no—
163#496—271noSylow
164C2×H(3,𝔽3)#12711yescharacteristic
165C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
166#272—81no—
167#272—81no—
168(C3⋊D3)⋊C3—81no—
169C2×C3×Q8—91no—
170C2×C3×D4—91no—
171C2×C3×D4—91no—
172C2×C3×D4—91no—
173C2×#67—182no—
174C2×#67—182no—
175C2×D12—31no—
176C2×D12—31no—
177C2×D12—31no—
178C2×D12—31no—
179C2×D12—31no—
180C2×D12—61no—
181C2×D12—61no—
182C2×D12—91no—
183C2×D12—182no—
184C2×Dic6—31no—
185C2×Dic6—91no—
186C3×SD16—182no—
187C3×D8—182no—
188C2×C3×C8—31no—
189C2×C3×C8—31no—
190C2×C3×C8—61no—
191C2×C3×C8—91no—
192C2×C3×C8D3211yescharacteristic
193C3×C42—91no—
194#217—182no—
195#215—182no—
196#214—182no—
197#213—182no—
198#213—182no—
199(C2×D3)⋊C4—182no—
200(C2×D3)⋊C4—182no—
201C4×Dic3—31no—
202C4×Dic3—91no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205D24—62no—
206D24—62no—
207D24—64no—
208D24—182no—
209#204—62no—
210#204—182no—
211#204—182no—
212C22×C32—21no—
213C22×C32C2×D1211yescharacteristic
214C22×C32D3×D411yescharacteristic
215C2×C3⋊D3—42no—
216C2×C3⋊D3—41no—
217C2×C3×D3—122no—
218C2×C3×D3—122no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C2×C3×D3—122no—
223C2×C3×D3—122no—
224C2×C3×D3—122no—
225C2×C3×D3—122no—
226C2×C3×D3—122no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—242no—
230C2×C3×D3—242no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4C2×D1211yescharacteristic
234C32×C4C2×D1211yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4C2×#6711yescharacteristic
237C3⋊Dic3—22no—
238C3⋊Dic3—41no—
239C3×Dic3—62no—
240C3×Dic3—121no—
241C2×SD16—91no—
242C2×SD16—271no—
243C2×D8—91no—
244C2×D8—271no—
245#127—271no—
246#124—271no—
247#97—91no—
248H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
249C23×C3—181no—
250C23×C3—181no—
251C23×C3—181no—
252C22×D3—61no—
253C22×D3—61no—
254C22×D3—61no—
255C22×D3—61no—
256C22×D3—61no—
257C22×D3—121no—
258C22×D3—121no—
259C22×D3—181no—
260C3×Q8—181no—
261C3×Q8—181no—
262C3×D4—181no—
263C3×D4—181no—
264C3×D4—181no—
265C3×D4—181no—
266C3×D4—181no—
267C3×D4—181no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—92no—
272C2×C3×C4—181no—
273C2×C3×C4C2×D3211yescharacteristic
274#67—364no—
275#67—364no—
276C2×Dic3—32no—
277C2×Dic3—61no—
278C2×Dic3—92no—
279C2×Dic3—181no—
280D12—61no—
281D12—61no—
282D12—61no—
283D12—61no—
284D12—61no—
285D12—61no—
286D12—61no—
287D12—61no—
288D12—62no—
289D12—62no—
290D12—61no—
291D12—61no—
292D12—121no—
293D12—121no—
294D12—181no—
295D12—181no—
296D12—364no—
297Dic6—61no—
298Dic6—61no—
299Dic6—181no—
300Dic6—181no—
301C3×C8—32no—
302C3×C8—32no—
303C3×C8—62no—
304C3×C8—92no—
305C3×C8#62612yes—
306C3⋊C8—32no—
307C3⋊C8—92no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32#87411yescharacteristic
311C2×C32#90812yes—
312C2×C32#91412yes—
313C2×C32#94011yescharacteristic
314C3⋊D3—81no—
315C3×D3—242no—
316C3×D3—242no—
317C3×D3—241no—
318C3×D3—241no—
319C3×D3—241no—
320C3×D3—241no—
321C3×D3—241no—
322C2×Q8—91no—
323C2×D4—91no—
324C2×D4—91no—
325C2×D4—91no—
326C2×D4—542no—
327SD16—182no—
328SD16—542no—
329D8—182no—
330D8—542no—
331C2×C8—91no—
332C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C22⋊C4—542no—
334C42—91no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—61no—
338C22×C3—362no—
339C22×C3—361no—
340C22×C3—362no—
341C22×C3—362no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3D3×D1211yescharacteristic
345C2×D3—121no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—121no—
349C2×D3—122no—
350C2×D3—122no—
351C2×D3—122no—
352C2×D3—122no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—122no—
356C2×D3—121no—
357C2×D3—242no—
358C2×D3—242no—
359C2×D3—361no—
360C2×D3—362no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—31no—
365C3×C4—61no—
366C3×C4—61no—
367C3×C4—182no—
368C3×C4—361no—
369C3×C4D3×D1211yescharacteristic
370C3×C4C2×#62611yescharacteristic
371Dic3—62no—
372Dic3—121no—
373Dic3—182no—
374Dic3—361no—
375C32—21no—
376C32#471211yescharacteristic, complemented
377C32#517611yescharacteristic, complemented
378C23—181no—
379C23—181no—
380C23—181no—
381Q8—181no—
382Q8—181no—
383D4—181no—
384D4—181no—
385D4—181no—
386D4—181no—
387D4—181no—
388D4—181no—
389D4—1084no—
390C2×C4—92no—
391C2×C4—181no—
392C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
393C8—92no—
394C8#620012yescomplemented
395C2×C3—32no—
396C2×C3—32no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—62no—
400C2×C3—61no—
401C2×C3—721no—
402C2×C3—721no—
403C2×C3—721no—
404C2×C3#6379412yes—
405C2×C3#6391111yescharacteristic
406D3—241no—
407D3—241no—
408D3—241no—
409D3—241no—
410D3—242no—
411D3—242no—
412D3—241no—
413D3—721no—
414C22—362no—
415C22—362no—
416C22—362no—
417C22—361no—
418C22—361no—
419C22—361no—
420C22#8865411yescharacteristic
421C4—182no—
422C4—361no—
423C4#8865411yescharacteristic
424C4C2×#620011yescharacteristic
425C3—31no—
426C3—31no—
427C3—61no—
428C3#1059088111yescharacteristic
429C2—721no—
430C2—721no—
431C2—721no—
432C2#1172084812yes—
433C2#1172115811yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#496
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88529C2semidirect
2. C2×#3744⋊C3C2semidirect
3. #11721158C2semidirect
4. #11721114C2semidirect
5. C2×#88437C2semidirect
6. C2×#88436C2semidirect
7. #63642⋊C3C2semidirect
23. C2×#3744D3semidirect
36. #63642D3semidirect
53. C8×H(3,𝔽3)D4semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
159. C32×C8D12semidirect
160. C32×C8#67semidirect
161. C3⋊(C3⋊C8)D12semidirect
248. H(3,𝔽3)#496semidirect
376. C32#4712semidirect
377. C32#5176semidirect
394. C8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
C2×#3744⋊C32
#117211582
#117211142
C2×#884372
C2×#884362
#63642⋊C32
#105908873
#105908833