Finite Group Databasebeta

← #33003047

Subgroups — #33003047

Order 18688.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556536C211yesmaximal, characteristic
2C2×C4×Dic292C211yesmaximal, characteristic, complemented
3C2×#17988392C211yesmaximal, characteristic, complemented
4C2×#17988389C211yesmaximal, characteristic
5#22610797C211yesmaximal, characteristic
6#22608023C211yesmaximal, characteristic, complemented
7C73×#1374C211yesmaximal, characteristic, complemented
8C22×Dic292C2211yescharacteristic
9C22×D584C2211yescharacteristic
10C2×C4×Dic146—21no—
11C2×#14556536—22no—
12C2×#14556536—22no—
13C2×#14556536C414yescomplemented
14C4×Dic292—22no—
15C4×Dic292—22no—
16C2×#14556525C2211yescharacteristic
17C2×#14556525C2211yescharacteristic
18C2×#14556516—21no—
19C2×#14556513—21no—
20#17988489—21no—
21#17988392—22no—
22#17988389—22no—
23C22×C16×C73C2211yescharacteristic
24C22×C16×C73C2211yescharacteristic
25C2×C4×C8×C73C2211yescharacteristic
26C22×Dic146—21no—
27C22×D292—21no—
28C2×Dic292—21no—
29C2×Dic292—21no—
30C2×Dic292—21no—
31C2×Dic292C2×C412yes—
32C2×Dic292D412yes—
33C2×Dic292D412yes—
34C2×D584—21no—
35C2×D584—21no—
36C2×D584—21no—
37C2×D584—21no—
38C2×D584—21no—
39C2×D584C2×C412yes—
40C4×Dic146—24no—
41C4×Dic146—42no—
42C2×C4⋊Dic73—21no—
43C2×C4⋊Dic73—21no—
44C2×#11924007—41no—
45C2×#11924005—41no—
46C2×#11924005—41no—
47C2×C4×Dic73—41no—
48#14556536—28no—
49#14556525—21no—
50#14556525—21no—
51#14556525—21no—
52#14556525D412yes—
53#14556516—24no—
54#14556513—42no—
55C22×C8×C73D411yescharacteristic
56C22×C8×C73C2311yescharacteristic
57C2×C42×C73D411yescharacteristic
58C2×C16×C73—21no—
59C2×C16×C73—21no—
60C2×C16×C73—21no—
61C2×C16×C73—21no—
62C2×C16×C73—21no—
63C2×C16×C73C2×C412yes—
64C4×C8×C73—21no—
65C4×C8×C73—21no—
66C23×D73—41no—
67C2×Dic146—22no—
68C2×Dic146—22no—
69C2×Dic146—22no—
70C2×Dic146—41no—
71C2×Dic146—41no—
72C2×Dic146—41no—
73C2×D292—22no—
74C2×D292—41no—
75C2×D292—41no—
76C2×D292—41no—
77C2×D292—41no—
78C2×D292—41no—
79C22×Dic73—41no—
80C22×Dic73—41no—
81C22×Dic73—41no—
82Dic292—22no—
83Dic292—22no—
84Dic292—22no—
85Dic292C22⋊C414yescomplemented
86D584—22no—
87D584—22no—
88D584—41no—
89D584—41no—
90C4⋊Dic73—22no—
91C4⋊Dic73—41no—
92C4⋊Dic73—41no—
93C4⋊Dic73—41no—
94#11924007—44no—
95#11924005—82no—
96#11924005—82no—
97C4×Dic73—44no—
98C22×C4×C73D811yescharacteristic
99C22×C4×C73D811yescharacteristic
100C22×C4×C73C2×D411yescharacteristic
101C2×C8×C73—21no—
102C2×C8×C73—21no—
103C2×C8×C73—21no—
104C2×C8×C73C22×C411yescharacteristic
105C2×C8×C73C2×D411yescharacteristic
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73C2×D411yescharacteristic
108C2×C8×C73#4111yescharacteristic
109C2×C8×C73#4111yescharacteristic
110C42×C73—21no—
111C42×C73—21no—
112C16×C73—22no—
113C16×C73—41no—
114C23×C73C2×D811yescharacteristic
115C22×D73—42no—
116C22×D73—81no—
117C22×D73—81no—
118C22×D73—81no—
119C22×D73—81no—
120C22×D73—81no—
121C22×D73—81no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73SD3212yescomplemented
128C2×C4×C73C4×D411yescharacteristic
129C2×C4×C73#12011yescharacteristic
130C2×C4×C73#12111yescharacteristic
131C2×C4×C73#13511yescharacteristic
132C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
134C2×Dic73—42no—
135C2×Dic73—42no—
136C2×Dic73—42no—
137C2×Dic73—42no—
138C2×Dic73—81no—
139C2×Dic73—81no—
140C2×Dic73—81no—
141C2×Dic73—81no—
142C2×Dic73—81no—
143D292—42no—
144D292—42no—
145D292—81no—
146D292—81no—
147Dic146—24no—
148Dic146—42no—
149Dic146—42no—
150Dic146—42no—
151C8×C73—41no—
152C8×C73C2×C22⋊C411yescharacteristic
153C8×C73C4×D411yescharacteristic
154C8×C73#12111yescharacteristic
155C8×C73#12311yescharacteristic
156C22×C73C4×D811yescharacteristic
157C22×C73#44711yescharacteristic
158C22×C73#45911yescharacteristic
159C22×C73C2×SD3211yescharacteristic
160C22×C73#50811yescharacteristic
161C22×C73#50911yescharacteristic
162C22×C73#51011yescharacteristic
163C2×D73—82no—
164C2×D73—82no—
165C2×D73—82no—
166C2×D73—161no—
167C2×D73—161no—
168C2×D73—161no—
169C2×D73—161no—
170C4×C73—22no—
171C4×C73—41no—
172C4×C73#38611yescharacteristic
173C4×C73#44211yescharacteristic
174C4×C73#44911yescharacteristic
175C4×C73#46311yescharacteristic
176Dic73—44no—
177Dic73—82no—
178Dic73—161no—
179#13013—731nomaximal, Sylow
180C2×C73#187911yescharacteristic
181C2×C73C4×SD3211yescharacteristic
182C2×C73#217511yescharacteristic
183C2×C73#218511yescharacteristic
184C2×C73#218611yescharacteristic
185C2×C73#220711yescharacteristic
186C2×C73#221111yescharacteristic
187D73—162no—
188C22×SD32—731no—
189C2×C4×Q16—731no—
190C2×#358—731no—
191C2×#357—731no—
192#1970—731no—
193#1380—731no—
194#1374D7311yescharacteristic, complemented
195C73#1301311yescharacteristic, complemented, Sylow
196C22×Q16—731no—
197C22×D8—731no—
198C2×C4×Q8—1461no—
199C2×SD32—734no—
200C2×SD32—1462no—
201C2×SD32—1462no—
202C22×C16C2×D7311yescharacteristic
203C22×C16C2×D7311yescharacteristic
204C4×Q16—1462no—
205C4×Q16—1462no—
206C2×#107—731no—
207C2×#107—731no—
208C2×#103—1461no—
209C2×#102—1461no—
210C2×C4×C8C2×D7311yescharacteristic
211#390—1461no—
212#358—1462no—
213#357—1462no—
214C22×Q8—1461no—
215C22×D4—1461no—
216C2×Q16—732no—
217C2×Q16—732no—
218C2×Q16—732no—
219C2×Q16—1461no—
220C2×Q16—1461no—
221C2×Q16—1461no—
222C2×D8—732no—
223C2×D8—1461no—
224C2×D8—1461no—
225C2×D8—1461no—
226C2×D8—1461no—
227C2×D8—1461no—
228C22×C8D29211yescharacteristic
229C22×C8C22×D7311yescharacteristic
230C4×Q8—2922no—
231C4×Q8—1464no—
232C2×C4⋊C4—2921no—
233C2×C4⋊C4—1461no—
234C2×C4⋊C4—1461no—
235C2×C22⋊C4—2921no—
236C2×C22⋊C4—2921no—
237C2×C42—2921no—
238C2×C42D29211yescharacteristic
239SD32—1468no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16C4×D7312yes—
246#107—732no—
247#107—1461no—
248#107—1461no—
249#107—1461no—
250#103—1464no—
251#102—2922no—
252C4×C8—21no—
253C4×C8—21no—
254C24—2921no—
255C2×Q8—2921no—
256C2×Q8—2921no—
257C2×Q8—2921no—
258C2×Q8—1462no—
259C2×Q8—1462no—
260C2×Q8—1462no—
261C2×D4—2921no—
262C2×D4—2921no—
263C2×D4—2921no—
264C2×D4—2921no—
265C2×D4—2921no—
266C2×D4—1462no—
267C22×C4—2921no—
268C22×C4—2921no—
269C22×C4—2921no—
270C22×C4D58411yescharacteristic
271C22×C4D58411yescharacteristic
272C22×C4C2×D29211yescharacteristic
273Q16—734no—
274Q16—1462no—
275Q16—1462no—
276Q16—1462no—
277D8—2921no—
278D8—2921no—
279D8—1462no—
280D8—1462no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C2×C4×D7311yescharacteristic
285C2×C8D4×D7311yescharacteristic
286C2×C8D4×D7311yescharacteristic
287C2×C8D4×D7311yescharacteristic
288C2×C8#1192402811yescharacteristic
289C2×C8#1192402911yescharacteristic
290C4⋊C4—2924no—
291C4⋊C4—2921no—
292C4⋊C4—2921no—
293C4⋊C4—2921no—
294C4⋊C4—1462no—
295C22⋊C4—5842no—
296C22⋊C4—5842no—
297C42—21no—
298C42—21no—
299C42—2924no—
300C16—22no—
301C16—41no—
302C23—2922no—
303C23—5841no—
304C23—5841no—
305C23—5841no—
306C23—5841no—
307C23—5841no—
308C23—5841no—
309C23C2×D58411yescharacteristic
310Q8—2922no—
311Q8—2922no—
312Q8—2922no—
313Q8—1464no—
314D4—2922no—
315D4—2922no—
316D4—5841no—
317D4—5841no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—2922no—
324C2×C4—2922no—
325C2×C4—2922no—
326C2×C4—2922no—
327C2×C4—5841no—
328C2×C4—5841no—
329C2×C4—5841no—
330C2×C4—5841no—
331C2×C4—5841no—
332C2×C4#1455653612yescomplemented
333C2×C4C4×D29211yescharacteristic
334C2×C4#1455656111yescharacteristic
335C2×C4#1455656611yescharacteristic
336C2×C4#1455661311yescharacteristic
337C2×C4#1455661911yescharacteristic
338C2×C4#1455661911yescharacteristic
339C8—41no—
340C8C22⋊C4×D7311yescharacteristic
341C8#1455655311yescharacteristic
342C8#1455656411yescharacteristic
343C8#1455657611yescharacteristic
344C22—5842no—
345C22—5842no—
346C22—5842no—
347C22—11681no—
348C22—11681no—
349C22—11681no—
350C22—11681no—
351C22C4×D58411yescharacteristic
352C22#1798872311yescharacteristic
353C22#1798878311yescharacteristic
354C22C2×#1455653611yescharacteristic
355C22#1798901911yescharacteristic
356C22#1798902511yescharacteristic
357C22#1798903011yescharacteristic
358C4—22no—
359C4—41no—
360C4—2924no—
361C4—5842no—
362C4—11681no—
363C4#1798846511yescharacteristic
364C4#1798870011yescharacteristic
365C4#1798873411yescharacteristic
366C4#1798880311yescharacteristic
367C2—11682no—
368C2#2261027811yescharacteristic
369C2C4×#1455653611yescharacteristic
370C2#2261179711yescharacteristic
371C2#2261185011yescharacteristic
372C2#2261185611yescharacteristic
373C2#2261197211yescharacteristic
374C2#2261199411yescharacteristic
subgroupG/H
centerC23C2×D584
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#1374C2
socleC23×C73C2×D8
prime radicalC22×D584C22
cosocleC22×C8×C73C23
hypercenter#1374D73
hypocenterC73#13013
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1374

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic292C2semidirect
3. C2×#17988392C2semidirect
6. #22608023C2semidirect
7. C73×#1374C2semidirect
13. C2×#14556536C4semidirect
85. Dic292C22⋊C4semidirect
127. C2×C4×C73SD32semidirect
194. #1374D73semidirect
195. C73#13013semidirect
332. C2×C4#14556536semidirect

Maximal subgroups

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

subgroupindex
C22×#145565362
C2×C4×Dic2922
C2×#179883922
C2×#179883892
#226107972
#226080232
C73×#13742
#1301373