Finite Group Databasebeta

← #10695987

Subgroups — #10695987

Order 768.

572 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 85 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×#2592C211yesmaximal, characteristic, complemented
2#82947C211yesmaximal, characteristic
3#82444C211yesmaximal, characteristic, complemented
4#71718C211yesmaximal, characteristic, complemented
5#71623C211yesmaximal, characteristic, complemented
6#69492C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#10006—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#397—21no—
13C3×#393—21no—
14C3×#391—21no—
15C3×#390—21no—
16C3×#388—22no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5227—21no—
19#5227—21no—
20#5217—21no—
21#5058—21no—
22#5057—22no—
23#5055—21no—
24#5027—21no—
25#5026—22no—
26#5021—21no—
27#4967—21no—
28C2×#805—22no—
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31C4×#208C2211yescharacteristic
32#4540—22no—
33#4535—22no—
34#3213—31no—
35#3112—31no—
36#2592D311yescharacteristic, complemented
37#1980—31no—
38#1527—31no—
39#1491—31no—
40#1285—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C2×#235—21no—
45C3×#124—21no—
46C3×#124—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C42—21no—
61C2×C3×C42D412yes—
62C2×C3×C42C2311yescharacteristic
63C2×#215—22no—
64C2×#215—22no—
65C2×#213—22no—
66#930—21no—
67#930—21no—
68C2×#208—21no—
69#875—21no—
70#874D411yescharacteristic
71C4×D12—21no—
72C4×D12—21no—
73#869D411yescharacteristic
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—21no—
77#835—22no—
78#835—22no—
79#832—22no—
80#830—21no—
81#830—21no—
82#809—22no—
83#807—22no—
84#805—44no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)D412yes—
87#803—21no—
88#803—21no—
89#803D412yes—
90C4×C3⋊C8—21no—
91#532—31no—
92C2×#124C2×D311yescharacteristic
93C2×C4×Q8C2×D311yescharacteristic
94#496—61no—
95#489—61no—
96#480—61no—
97#477—61no—
98#463—62no—
99#460—61no—
100#448—61no—
101#448—61no—
102#447—61no—
103#440—62no—
104#432—31no—
105#432—31no—
106C2×#104—62no—
107C4×M16—31no—
108#397—21no—
109#393—21no—
110#391—21no—
111#390—21no—
112#388—22no—
113C4×C4⋊C4C2×D311yescharacteristic
114#329—62no—
115Q8⋊C8—62no—
116C24×C3—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—42no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C2×#67—41no—
134#235—42no—
135#235—42no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×C4×D3—41no—
140C2×Dic6—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—42no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—44no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—82no—
152C3×C42—42no—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159#215—44no—
160#215—82no—
161#213—82no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166#208—22no—
167#208—41no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—41no—
171C2×#41—61no—
172C22×Q8#6711yescharacteristic
173C22×Q8#6711yescharacteristic
174C22×D4#6711yescharacteristic
175C2×SD16—62no—
176C2×SD16—62no—
177C2×D8—62no—
178C2×M16—61no—
179C4⋊Q8—31no—
180#127—31no—
181#124—21no—
182#124—21no—
183#124—21no—
184#124—21no—
185#124—61no—
186#121—61no—
187#121—61no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×D4—61no—
193C4×D4—61no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4#6711yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C42—21no—
202C2×C42#6712yes—
203C2×C42C22×D311yescharacteristic
204#107—62no—
205C4⋊C8—32no—
206C4⋊C8—61no—
207#104—124no—
208#103—62no—
209#103—62no—
210#102—62no—
211#102—62no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214#97—32no—
215#97—61no—
216#97—61no—
217C4×C8—61no—
218#95—22no—
219#95—22no—
220#95—21no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3#12011yescharacteristic
227C22×D3—41no—
228C3×Q8—22no—
229C3×Q8—22no—
230C3×Q8—41no—
231C3×Q8—41no—
232C3×Q8—42no—
233C3×Q8—81no—
234C3×D4—42no—
235C3×D4—81no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12711yescharacteristic
257#67—81no—
258#67—81no—
259C2×Dic3—41no—
260D12—42no—
261D12—42no—
262D12—42no—
263C4×D3—82no—
264Dic6—42no—
265C3⋊C8—42no—
266C3⋊C8—42no—
267C3⋊C8—42no—
268C24—21no—
269#41—122no—
270#41—122no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—61no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—121no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—121no—
290C22×C4C2×#6711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292C22×C4C2×#6711yescharacteristic
293SD16—124no—
294SD16—242no—
295D8—242no—
296M16—62no—
297M16—121no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—121no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C22⋊C4—44no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—82no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C42—42no—
317C42—41no—
318C42—41no—
319C42D3×D411yescharacteristic
320C42D3×D411yescharacteristic
321C42D3×D411yescharacteristic
322C42#23711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3#39211yescharacteristic
331C22×C3#45311yescharacteristic
332C22×C3#45411yescharacteristic
333C2×D3—81no—
334C2×D3—81no—
335C2×D3—81no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—42no—
342C3×C4—82no—
343C3×C4—81no—
344C3×C4—81no—
345C3×C4#45512yes—
346Dic3—81no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—121no—
353C23#95311yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358Q8—42no—
359Q8—81no—
360Q8—122no—
361D4—42no—
362D4—81no—
363D4—122no—
364D4—122no—
365D4—122no—
366D4—241no—
367D4—241no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—121no—
384C2×C4—242no—
385C2×C4#93711yescharacteristic
386C2×C4#93711yescharacteristic
387C2×C4#93811yescharacteristic
388C2×C4#93911yescharacteristic
389C2×C4#93911yescharacteristic
390C2×C4#94011yescharacteristic
391C8—122no—
392C8—122no—
393C8—122no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—161no—
397C2×C3#200411yescharacteristic
398C2×C3#200611yescharacteristic
399C2×C3#201011yescharacteristic
400D3—161no—
401C22—21no—
402C22—21no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22#522311yescharacteristic
412C22#524111yescharacteristic
413C22#524511yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—82no—
421C4—81no—
422C4—81no—
423C4—241no—
424C4#506112yes—
425C3#1000611yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—161no—
429C2—481no—
430C2#7171511yescharacteristic
431C2#7173111yescharacteristic
432C2#7187911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2592C2
socleC22×C3#392
prime radical#82947C2
hypercenter#2592D3
hypocenterC3#10006
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2592

Complements

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

NH ≅ G/Ntype
1. C3×#2592C2semidirect
3. #82444C2semidirect
4. #71718C2semidirect
5. #71623C2semidirect
6. #69492C2semidirect
7. #67621C2semidirect
36. #2592D3semidirect
425. C3#10006semidirect

Maximal subgroups

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

subgroupindex
C3×#25922
#829472
#824442
#717182
#716232
#694922
#676212
#100063